Physics Reports vol.344

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COLOSSAL MAGNETORESISTANT MATERIALS: THE KEY ROLE OF PHASE SEPARATION

Elbio DAGOTTOa, Takashi HOTTAb, Adriana MOREOa National High Magnetic Field Laboratory and Department of Physics, Florida State University, Tallahassee, FL 32306, USA
Institute for Solid State Physics, University of Tokyo, 5-1-5 Kashiwa-no-ha, Kashiwa, Chiba 277-8581, Japan

AMSTERDAM } LONDON } NEW YORK } OXFORD } PARIS } SHANNON } TOKYO

Physics Reports 344 (2001) 1}153

Colossal magnetoresistant materials: the key role of phase separation Elbio Dagotto *, Takashi Hotta
, Adriana Moreo National High Magnetic Field Laboratory and Department of Physics, Florida State University, Tallahassee, FL 32306, USA
Institute for Solid State Physics, University of Tokyo, 5-1-5 Kashiwa-no-ha, Kashiwa, Chiba 277-8581, Japan Received October 2000; editor: D.L. Mills Contents 1. Introduction 2. Basic properties, phase diagrams, and CMR e!ect in manganites 2.1. Large-bandwidth manganites: the case of La Sr MnO \V V  2.2. Intermediate-bandwidth manganites: the case of La Ca MnO \V V  2.3. Low-bandwidth manganites: the case of Pr Ca MnO \V V  2.4. Other perovskite manganite compounds 2.5. Double-layer compound 2.6. Single-layer compound 2.7. Importance of tolerance factor 3. Theory of manganites 3.1. Early studies 3.2. More recent theories 3.3. Models and parameters 3.4. Main results: one orbital model 3.5. Main results: two orbital model 3.6. Pseudo-gap in mixed-phase states 3.7. Phase separation caused by the in#uence of disorder on "rst-order transitions 3.8. Resistivity of manganites in the mixed-phase regime

4 9 9 12 17 18 22 22 23 26 26 31 33 55 65 86 88

3.9. Related theoretical work on electronic phase separation applied to manganites 3.10. On-site Coulomb interactions and phase separation 3.11. Theories based on Anderson localization 4. Experimental evidence of inhomogeneities in manganites 4.1. La Ca MnO at density 0.04x(0.5 \V V  4.2. La Ca MnO at x&0.5 \V V  4.3. Electron-doped manganites 4.4. Large bandwidth manganites and inhomogeneities: the case of La Sr MnO \V V  4.5. Pr Ca MnO \V V  4.6. Mixed-phase tendencies in bilayered manganites 4.7. Mixed-phase tendencies in single-layered manganites 4.8. Possible mixed-phase tendencies in nonmanganite compounds 5. Discussion, open questions, and conclusions Acknowledgements References

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* Corresponding author. Fax: #18506445038. E-mail address: [email protected] (E. Dagotto). 0370-1573/01/$ - see front matter  2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 0 0 ) 0 0 1 2 1 - 6

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Abstract The study of the manganese oxides, widely known as manganites, that exhibit the `colossala magnetoresistance e!ect is among the main areas of research within the area of strongly correlated electrons. After considerable theoretical e!ort in recent years, mainly guided by computational and mean-"eld studies of realistic models, considerable progress has been achieved in understanding the curious properties of these compounds. These recent studies suggest that the ground states of manganite models tend to be intrinsically inhomogeneous due to the presence of strong tendencies toward phase separation, typically involving ferromagnetic metallic and antiferromagnetic charge and orbital ordered insulating domains. Calculations of the resistivity versus temperature using mixed states lead to a good agreement with experiments. The mixed-phase tendencies have two origins: (i) electronic phase separation between phases with di!erent densities that lead to nanometer scale coexisting clusters, and (ii) disorder-induced phase separation with percolative characteristics between equal-density phases, driven by disorder near "rst-order metal}insulator transitions. The coexisting clusters in the latter can be as large as a micrometer in size. It is argued that a large variety of experiments reviewed in detail here contain results compatible with the theoretical predictions. The main phenomenology of mixed-phase states appears to be independent of the "ne details of the model employed, since the microscopic origin of the competing phases does not in#uence the results at the phenomenological level. However, it is quite important to clarify the electronic properties of the various manganite phases based on microscopic Hamiltonians, including strong electron}phonon Jahn}Teller and/or Coulomb interactions. Thus, several issues are discussed here from the microscopic viewpoint as well, including the phase diagrams of manganite models, the stabilization of the charge/orbital/spin ordered half-doped correlated electronics (CE)-states, the importance of the naively small Heisenberg coupling among localized spins, the setup of accurate mean-"eld approximations, the existence of a new temperature scale ¹H where clusters start forming above the Curie temperature, the presence of stripes in the system, and many others. However, much work remains to be carried out, and a list of open questions is included here. It is also argued that the mixed-phase phenomenology of manganites may appear in a large variety of compounds as well, including ruthenates, diluted magnetic semiconductors, and others. It is concluded that manganites reveal such a wide variety of interesting physical phenomena that their detailed study is quite important for progress in the "eld of correlated electrons.  2001 Elsevier Science B.V. All rights reserved. PACS: 71.70.Ej; 71.15.-m; 71.38.#i; 71.45.Lr Keywords: Manganites; Colossal magnetoresistance; Computational physics; Inhomogeneities; Phase separation

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1. Introduction This is a review of theoretical and experimental work in the context of the manganese oxides widely known as manganites. These materials are currently being investigated by a sizable fraction of the condensed matter community, and their popularity is reaching levels comparable to that of the high-temperature superconducting cuprates. From this review hopefully the reader will be able to understand the reasons behind this wide interest in manganites, the problems that have been solved in this context, and those that remain to be investigated. The authors have made a considerable e!ort in trying to include in this review the majority of what they consider to be the most relevant literature on the subject. However, clearly it is not possible to cover all aspects of the problem in a single manuscript. Here the main focus has been directed into recent theoretical calculations that address the complex spin, charge, and/or orbital ordered phases of manganites, which have important and prominent intrinsic inhomogeneities, and also on the recent experimental results against which those calculations can be compared. Due to the complexity of the models needed to address manganites, it is natural that the most robust results have been obtained with computational tools, and those are the calculations that will be emphasized in the text. The continuous growth of available computer power has allowed simulations that were simply impossible not long ago. In addition, the physics of manganites appears dominated by intrinsic inhomogeneities and its description is quite di$cult in purely analytic frameworks that usually assume uniform states. However, several calculations, notably some mean-"eld approximations, have also reached a high-accuracy level and they are important in deciding which are the phases of relevance in manganites. These calculations are also discussed in detail below. Finally, it is reassuring for the success of manganite investigations that a variety of experiments, reviewed here, appear to be in qualitatively good agreement with the most recent theoretical calculations. Even quantitative agreement is slowly starting to emerge, although there are still many aspects of the problem that require further investigation. At a more general level, from this review it is expected that the readers will understand the richness of manganite physics and how it challenges aspects of our present understanding of condensed matter systems. The e!ort to fully unveil the behavior of electrons in manganites should continue at its current fast pace in the near future. The "eld of manganites started with the seminal paper of Jonker and Van Santen (1950) where the existence of ferromagnetism in mixed crystals of LaMnO }CaMnO , LaMnO }SrMnO , and     LaMnO }BaMnO was reported. The general chemical formula for the manganese oxides de  scribed in Jonker and van Santen's paper (1950), and many other compounds investigated later on, is T D MnO , with T a trivalent rare earth or Bi> cation, and D a divalent alkaline or Pb> \V V  cation. Oxygen is in a O\ state, and the relative fraction of Mn> and Mn> is regulated by `xa. The perovskite lattice structure of these materials is illustrated in Fig. 1.1a. Jonker and Van Santen (1950) adopted the terminology `manganitesa to refer to these mixed compounds, although it is not strictly correct, as they emphasized in a footnote, since the term manganite should in principle apply only to the 100% Mn> compound. More detailed information about La Ca MnO using neutron scattering techniques was \V V  obtained later by Wollan and Koehler (1955). In their study the antiferromagnetic (AF or AFM) and ferromagnetic (FM) phases were characterized, and the former was found to contain nontrivial arrangements of charge at particular hole densities. Wollan and Koehler (1955) noticed the mixture of C- and E-type magnetic unit cells in the structure at x"0.5, and labeled the insulating state at this density as a `CE-statea (the seven possible arrangements A, B, C, D, E, F, and G for the spin in

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the unit cell are shown in Fig. 1.1b, with the spins of relevance being those located in the manganese ions). Theoretical work at approximately the same time, to be reviewed below, explained the ferromagnetic phase as caused by an e!ect called `double exchangea (DE), and thus one of the most interesting properties of these materials appeared to have found a good rationalization in the early studies of these compounds. Perhaps as a consequence of the apparent initial theoretical success, studies of the manganites continued in subsequent years at a slow pace. The renewed surge of interest in manganites in the 1990s started with the experimental observation of large magnetoresistance (MR) e!ects in Nd Pb MnO by Kusters et al. (1989)      and in La Ba MnO by von Helmolt et al. (1993) (actually Searle and Wang (1969) were the   V "rst to report MR studies in manganites, which were carried out using La Pb MnO single \V V  crystals). Resistivity vs. temperature results for the (La,Ba) compound are shown in Fig. 1.2a, reproduced from von Helmolt et al. (1993). The MR e!ect was found to be as high as 60% at room temperature using thin "lms, and it was exciting to observe that this value was higher than found in arti"cial magnetic/nonmagnetic multilayers, allowing for potential applications in magnetic

Fig. 1.1. (a) Arrangement of ions in the perovskite structure of manganites (from Tokura, 1999). (b) Possible magnetic structures and their labels. The circles represent the position of Mn ions, and the sign that of their spin projections along the z-axis. The G-type is the familiar antiferromagnetic arrangement in the three directions, while B is the familiar ferromagnetic arrangement. Taken from Wollan and Koehler (1955).

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recording. However, as discussed extensively below, while a large body of subsequent experimental work has shown that the MR factor can actually be made very close to 100% (for a de"nition of the MR ratio see below), this occurs unfortunately at the cost of reducing the Curie temperature ¹ , ! which jeopardizes those possible technological applications. Consider for instance in Fig. 1.2b the results for Nd Pb MnO reproduced from Kusters et al. (1989). In this case the change in      resistivity is larger than for the (La,Ba) compound, however its Curie temperature is reduced to 184 K. Also complicating possible applications, it is known that giant MR multilayer structures

Fig. 1.2. (a) Temperature dependence of the resistivity of a La Ba MnO thin "lm at 0 and 5 T, taken from von   V Helmolt et al. (1993). The two panels are results as-deposited and after a subsequent annealing, respectively. For more details see von Helmolt et al. (1993). (b) Resistivity of Nd Pb MnO as a function of temperature and magnetic "elds      taken from Kusters et al. (1989). The inset is the magnetoresistance at the indicated temperatures.

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present their appreciable changes in resistivity with "elds as small as 0.01 T (Helman and Abeles, 1976; Fert and Campbell, 1976), while manganites typically need larger "elds of about 1 T or more for equivalent resistivity changes, which appear too large for potential use in magnetic recording. Although progress in the development of applications is frequently reported (for recent references see Chen et al., 2000; Venimadhav et al., 2000; Kida and Tonouchi, 2000b), in this review the manganites will be mainly considered as an interesting basic-physics problem, with emphasis focused on understanding the microscopic origin of the large MR e!ect which challenges our current knowledge of strongly correlated electron systems. The discussion of possible applications of manganites is left for future reviews. The big boost to the "eld of manganites that led to the present explosion of interest in the subject was produced by the discovery of the so-called `colossala magnetoresistance (CMR) e!ect. In studies of thin "lms of La Ca MnO , a MR e!ect three orders of magnitude larger than the typical     V `gianta MR of superlattice "lms was observed (the name colossal was coined mainly to distinguish the e!ect from this previously found giant MR e!ect). De"ning the MR ratio as
R/R(H)" (R(0)!R(H))/R(H), where R(0) and R(H) are the resistances without and with a magnetic "eld H, respectively, and expressing the result as a percentage (i.e., multiplying by an additional factor 100) it has been shown that MR ratios as large as 127,000% near 77 K can be obtained (Jin et al., 1994). This corresponds to more than a 1000-fold change in resistivity. Alternatively, expressed in terms of
R/R(0)"(R(0)!R(H))/R(0) the MR ratio in this case is as large as 99.92%. Xiong et al. (1995) reported thin-"lms studies of Nd Sr MnO and in this case
R/R(H) was as high as 10%, a truly     B colossal factor. Triggered by such huge numbers, the experimental and theoretical study of manganites reignited, and is presently carried out by dozens of groups around the world. Early work tended to focus on the x"0.3 doping due to its large ¹ . However, more recently the attention has shifted ! towards other densities such as x(0.2 or x'0.5, where the competition between the various states of manganites can be better analyzed. In fact, one of the main results of recent investigations is that in order to understand the CMR e!ect, knowledge of the ferromagnetic metallic phase is not su$cient. The competing phases must be understood as well. This issue will be discussed extensively below. Previous reviews on manganites are already available, but they have mainly focused on experiments. For instance, the reader can consult the reviews of Ramirez (1997), Coey et al. (1998), as well as the books recently edited by Rao and Raveau (1998), Kaplan and Mahanti (1999) and Tokura (1999). See also Loktev and Pogorelov (2000). The present review di!ers from previous ones in several aspects: (i) It addresses theoretical work in detail, especially regarding the stabilization in a variety of calculations of the many nontrivial charge/spin/orbital phases found in experiments; (ii) it highlights the importance of phase separation tendencies in models for manganites and the potential considerable in#uence of disorder on transitions that would be "rst order in the clean limit, leading to percolative processes; and (iii) it emphasizes the experimental results that have recently reported the presence of intrinsic inhomogeneities in manganites, results that appear in excellent agreement with the theoretical developments. If the review could be summarized in just a few words, the overall conclusion would be that theoretical and experimental work is rapidly converging to a uni"ed picture pointing toward a physics of manganites in the CMR regimes clearly dominated by inhomogeneities in the form of coexisting competing phases. This is an intrinsic feature of single crystals, unrelated to grain boundary e!ects of polycrystals, and its theoretical understanding and experimental control is a challenge that should be strongly pursued. In fact, in spite of the considerably progress in recent years reviewed here, it is

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Fig. 1.3. Field splitting of the "ve-fold degenerate atomic 3d levels into lower t and higher e levels. The particular   Jahn}Teller distortion sketched in the "gure further lifts each degeneracy as shown. Figure taken from Tokura (1999).

clear that the analysis of mixed-phase systems is at its early stages, and considerable more work should be devoted to the detailed study of manganese oxides and related compounds in such a regime. In this review, it is assumed that the reader is familiar with some basic phenomenology involving the d-orbitals of relevance in manganese oxides. In the cubic lattice environment, the "ve-fold degenerate 3d-orbitals of an isolated atom or ion are split into a manifold of three lower energy levels (d , d , and d ), usually referred to as `t a once mixing with the surrounding oxygens is VW WX XV  included, and two higher energy states (d   and d   ) called `e a. The valence of the Mn-ions X \P  V \W in this context is either four (Mn>) or three (Mn>), and their relative fraction is controlled through chemical doping. The large Hund coupling favors the population of the t levels with  three electrons forming a spin 3/2 state, and the e level either contains one electron or none.  A sketch with these results is shown in Fig. 1.3. Considerable more detail about all the theoretical and experimental issues reported in this review can be found in a book in preparation by one of the authors (E.D., Springer Verlag). The organization is the following. In Section 2, the most basic properties of manganites will be reviewed from the experimental viewpoint. Emphasis will be given to the phase diagrams and magnitude of the CMR e!ect in various manganites. For this section, the manganites will be divided into large, intermediate, and small bandwidth = compounds, a slightly unorthodox classi"cation since previous work simply labeled them as either large or small =. In Section 3, the theoretical aspects are presented, starting with the early developments in the subject. Models and approximations will be discussed in detail, and results will be described. Especially, the key importance of the recently found phase separation tendencies will be remarked. In Section 4, the experimental work that have reported evidence of intrinsic inhomogeneities in manganites compatible with the theoretical calculations will be reviewed. Finally, conclusions are presented in Section 5 including the problems still open.

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2. Basic properties, phase diagrams, and CMR e4ect in manganites 2.1. Large-bandwidth manganites: the case of ¸a Sr MnO \V V  In the renaissance of the study of manganites during the 1990s, a considerable emphasis has been given to the analysis of La Sr MnO , a material considered to be representative of the `largea \V V  bandwidth subset of manganese oxides. It is believed that in this compound the hopping amplitude for electrons in the e -band is larger than in other manganites, as a consequence of the sizes of the  ions involved in the chemical composition, as discussed below. Its Curie temperature ¹ as ! a function of hole doping is relatively high, increasing its chances for future practical applications. It has also been found that La Sr MnO presents a complex behavior in the vicinity of x"1/8 \V V  (see below), with a potential phenomenology as rich as found in the low-bandwidth manganites described later in this section. Resistivities vs. temperature for this compound at several densities are shown in Fig. 2.1.1a (taken from Urushibara et al., 1995). From these transport measurements, the phase diagram of this compound can be determined and it is shown in Fig. 2.1.1b (Y. Tokura and Y. Tomioka, prepared with data from Urushibara et al. (1995) and Fujishiro et al. (1998). See also Moritomo et al. (1998)). At hole concentrations such as x"0.4, the system is metallic (de"ned straightforwardly as regions where d /d¹'0) even above ¹ . At densities above x"0.5 an  ! interesting A-type antiferromagnetic metallic state is stabilized, with ferromagnetism in planes and antiferromagnetism between those planes. This phase was actually "rst observed in another compound Nd Sr MnO (Kawano et al., 1997) and is believed to have d   -type uniform orbital \V V  V \W order within the ferromagnetic planes (for a visual representation of this state see Fig. 4 of Kajimoto et al., 1999). Considering now lower hole densities, at concentrations slightly below x"0.30 the state above ¹ becomes an insulator, which is an unexpected property of a paramagnetic state that ! transforms into a metal upon reducing the temperature. This curious e!ect is present in all the intermediate and low-bandwidth manganites, and it is a key property of this family of compounds. At densities x:0.17, an insulating state is found even at low temperatures. As reviewed in more detail below, currently considerable work in the context of La Sr Mn O is being focused on the \V V  x&0.12 region, simply labeled `ferromagnetic insulatora in Fig. 2.1.1b. In this region, indications of charge ordering have been found even in this putative large-bandwidth material, establishing an interesting connection with intermediate- and low-bandwidth manganites where charge-ordering tendencies are very prominent. A revised phase diagram of La Sr MnO will be presented later \V V  when the experimental evidence of inhomogeneous states in this compound is discussed. Studies by Tokura et al. (1994) showed that the MR e!ect is maximized in the density region separating the insulating from metallic states at low temperature, namely x"0.175. The MR e!ect here is shown in Fig. 2.1.2a, taken from Tokura et al. (1994). The Curie temperature is still substantially large in this regime, which makes it attractive from the viewpoint of potential applications. A very important qualitative aspect of the results shown in Fig. 2.1.2a is that the MR e!ect is maximized at the smallest ¹ which leads to a metallic ferromagnetic state; this observa! tion let the authors to conclude that the large MR regime can only be clearly understood when the various e!ects which are in competition with DE are considered. This point will be a recurrent conclusion emerging from the theoretical calculations reviewed below, namely in order to achieve a large MR e!ect the insulating phase is at least as important as the metallic phase, and the region of the most interest should be the boundary between the metal and the insulator (Moreo et al.,

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Fig. 2.1.1. (a) Temperature dependence of resistivity for various single crystals of La Sr MnO . Arrows indicate the \V V  Curie temperature. The open triangles indicate anomalies due to structural transitions. For more details see Urushibara et al. (1995) from where this "gure is reproduced. (b) Phase diagram of La Sr MnO (courtesy of Y. Tokura and Y. \V V  Tomioka) prepared with data from Urushibara et al. (1995) and Fujishiro et al. (1998). The AFM phase at large x is an A-type AF metal with uniform orbital order. PM, PI, FM, FI, and CI denote paramagnetic metal, paramagnetic insulator, FM metal, FM insulator, and spin-canted insulator states, respectively. ¹ is the Curie temperature and ¹ is ! , the NeH el temperature. A more detailed version of this phase diagram is shown below in Fig. 4.4.1, with emphasis on the small hole-density region which presents tendencies to charge-ordering and mixed-phase states.

1999a; Moreo et al., 2000). These insulating properties occur at low temperature by changing the density, or, at "xed density, by increasing the temperature, at least in some density windows. It is at the metal}insulator boundary where the tendencies to form coexisting clusters and percolative transitions are the most important. This point of view is qualitatively di!erent from the approach followed in previous theories based on polaronic formation, Anderson localization, or on

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Fig. 2.1.2. (a) Temperature dependence of the resistivity in magnetic "elds corresponding to La Sr MnO at \V V  x"0.175 (from Tokura et al., 1994). (b) Temperature dependence of the total infrared-absorption spectral weight N (open circles) and Drude weight D (closed circles) for a single crystal of La Sr MnO (x"0.175). The solid line is  \V V  the square of the normalized ferromagnetic magnetization (M/M ), with M the saturated magnetization. Results   reproduced from Tokura (1999).

modi"cations of the double-exchange ideas, but it is crucial in the phase-separation-based approaches described here. Note, however, that the metallic phase of La Sr MnO at su$ciently \V V  large hole density seems quite properly described by double-exchange approaches, namely there is a simple relation between the resistivity and the magnetization in the metallic ferromagnetic phase (Tokura et al., 1994; Furukawa et al., 1998). Then, it is important to distinguish the theoretical understanding of the individual phases, far from others in parameter space, from the understanding of the competition among them. It is the latest issue that is the most important for the explanation of the MR e!ect according to recent calculations (Moreo et al., 1999a, 2000). It is also interesting to point out that the low-temperature ferromagnetic metallic state that appears prominently in Fig. 2.1.1b is actually `unconventionala in many respects. For instance, in Fig. 2.1.2b, taken from Tokura (1999), it is shown that the total low-energy spectral weight of the optical conductivity N is still changing even in the low-temperature region, where the spin is  already almost fully polarized. Clearly, there is another scattering mechanism in the system besides the spin. Actually, even within the FM-state the carrier motion is mostly incoherent since the

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Drude weight is only 1/5 of the total low-energy weight. The conventional Drude model is not applicable to the FM-state of manganites. Probably the orbital degrees of freedom are important to account for this e!ect. 2.2. Intermediate-bandwidth manganites: the case of ¸a

\V

Ca MnO V 

Currently, a large fraction of the work in manganites focuses on intermediate- to low-bandwidth materials since these are the ones that present the largest CMR e!ects, which are associated with the presence of charge ordering tendencies. Unfortunately, as discussed in the Introduction, this comes at a price: the Curie temperature decreases as the magnitude of the MR e!ects increases. In this section, the properties of La Ca MnO will be discussed in detail. This compound presents \V V  some characteristics of large bandwidth manganites, such as the presence of a robust ferromagnetic metallic phase. However, it also has features that indicate strong deviations from double-exchange behavior, including the existence of charge/orbital-ordered phases. For this reason, the authors consider that this compound should be labeled as of `intermediate bandwidtha, to distinguish it from the truly low-bandwidth compound Pr Ca MnO where a metallic ferromagnetic phase \V V  can only be stabilized by the application of magnetic "elds. La Ca MnO has been analyzed since the early days of manganite studies (Jonker and Van \V V  Santen, 1950; Wollan and Koehler, 1955; Matsumoto, 1970b), but it is only recently that it has been systematically scrutinized as a function of density and temperature. In particular, it has been

Fig. 2.2.1. The magnetization, resistivity, and magnetoresistance of La Ca MnO (x"0.25), as a function of \V V  temperature at various "elds. The inset is  at low temperatures. Reproduced from Schi!er et al. (1995).

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observed that La Ca MnO has a large MR e!ect. For example, Fig. 2.2.1 reproduces results \V V  from Schi!er et al. (1995) at x"0.25 showing the magnetization and resistivity as a function of temperature, and the existence of a robust MR, larger than 80%. The drop in  (¹) with  decreasing temperature and the peak in MR are correlated with the ferromagnetic transition in the magnetization. The insulating behavior above the Curie temperature is very prominent and the explanation of its origin is among the most important issues to be addressed in theories of manganites, as already discussed in the previous subsection. Below ¹ the presence of ferromag! netism was tentatively attributed to the double-exchange mechanism, but further work reviewed in Section 4 has actually revealed a far more complex structure with coexisting phases even in this metallic regime. In fact, hints of this behavior may already be present in Fig. 2.2.1 which already reveals a MR e!ect as large as 30% well below ¹ . In addition, it is also interesting to observe that ! hydrostatic pressure leads to large changes in resistivity comparable to those found using magnetic "elds [see for instance Fig. 2.2.2 where  (¹) is shown parametric with pressure at x"0.21, taken  from Neumeier et al. (1995). See also Hwang et al. (1995b)]. The qualitative features of Fig. 2.2.1 contribute to the understanding, at least in part, of the CMR e!ect found in thin "lms of this same compound (Jin et al., 1994). For references on thin "lm work in manganites, see Ramirez (1997, p. 8182). See also Kanki et al. (2000). In the work of Jin et al. (1994), ¹ was suppressed by substrate-induced strain, and, as a consequence, the  was much !  higher immediately above the transition than in crystals since the system was still in the insulating state, inducing an enormous change in resistivity. Thus, it appears that to understand the large MR values the insulating state is actually more important than the metallic state, and the relevance of the DE ideas is limited to the partial explanation of the low-temperature phase in a narrow density window, as explained in more detail later. Clearly, the DE framework is insu$cient for describing the physics of the manganites. The complete phase diagram of La Ca MnO , based on magnetization and resistivity \V V  measurements, is reproduced in Fig. 2.2.3, taken from Cheong and Hwang (1999). Note that the

Fig. 2.2.2. Resistivity vs. temperature at three hydrostatic pressures for La Ca MnO (x"0.21). In the inset the \V V  pressure dependence of ¹ and the activation energy e are plotted. For details see Neumeier et al. (1995), from where !  these results have been reproduced.

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FM phase actually occupies just a fraction of the whole diagram, illustrating once again that DE does not provide a full understanding of the manganites. For instance, equally prominent are the charge ordered (CO) states between x"0.50 and 0.87. The CO state at x"0.50 was already described by Wollan and Koehler (1955) as a CE-state, and the characteristics at other densities are discussed below. In the regime of CO-states, studies by Ramirez et al. (1996) of the sound velocity, speci"c heat, and electron di!raction were attributed to strong electron}phonon coupling, in agreement with the predictions of Millis et al. (1995). The `canteda state at x close to 1 could be a mixed-phase state with coexisting FM}AF characteristics based on recent theoretical calculations (see below Section 3), but the issue is still under discussion. The low hole-density regime is quite unusual and nontrivial, and it appears to involve a charge-ordered phase, and a curious ferromagnetic insulator. Actually at x"0.10, there is no large MR e!ect using "elds of 12 T, according to Fig. 6 of Ibarra and De Teresa (1998c). Fig. 25 of the same reference shows that at x"0.65, well inside the charge-ordered state, a 12 T "eld is also not su$cient to destabilize the insulating state into a metallic one. Thus, to search for a large MR e!ect, the density must be closer to that leading to the FM metallic regime, as emphasized before. In Fig. 2.2.3 note also the presence of well-de"ned features at commensurate carrier concentrations x"N/8 (N"1,3,4,5 and 7). The Curie temperature is maximized at x"3/8 according to Cheong and Hwang (1999), contrary to the x"0.30 believed by many to be the most optimal density for ferromagnetism. Cheong and Hwang (1999) also remarked that in the large-bandwidth compound La Sr MnO , ¹ is also maximized at the same x"3/8 concentration, implying \V V  ! that this phenomenon is universal. It is important to realize that within a simple one-orbital double-exchange model, as described later, the optimal density for ferromagnetism should be x"0.50. The fact that this is not observed is already indicative of the problems faced by a double-exchange description of manganites. Note also that Zhao et al. (1996, 1999) found a giant

Fig. 2.2.3. Phase diagram of La Ca MnO , constructed from measurements of macroscopic quantities such as the \V V  resistivity and magnetic susceptibility, reproduced from Cheong and Hwang (1999). FM: Ferromagnetic Metal, FI: Ferromagnetic Insulator, AF: Antiferromagnetism, CAF: Canted AF, and CO: Charge/Orbital Ordering. FI and/or CAF could be a spatially inhomogeneous states with FM and AF coexistence.

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oxygen isotope shift in ¹ of about 20 K at x"0.2, showing the relevance of electron}phonon ! couplings in manganites, a recurrent result of many papers in this context. The charge-ordering temperature ¹ peaks at x"5/8 (the same occurs in (Bi, Ca)-based !compounds), while at x"4/8"1/2 there is a sharp change from ferromagnetic to antiferromagnetic ground states. The whole phase diagram has a pronounced electron}hole asymmetry, showing again that simple double-exchange models with only one orbital are not realistic. At x"1/8 the low-density charge-ordered state appears to have the largest strength, while on the other side at x"7/8 charge ordering disappears into a mixed FM}AF state. Finally, at x"0 the ground state is an A-type antiferromagnet (see also Matsumoto, 1970a) with ferromagnetic spin correlations on a plane and antiferromagnetism between planes, while at x"1 it is a G-type antiferromagnet (AF in all directions), both of them insulating. The pattern of charge- and orbital order in the CO states of Fig. 2.2.3 is highly nontrivial and at several densities still under discussion (for early work in the context of orbital ordering see Kugel and Khomskii, 1974; Eremin and Kalinenkov, 1978, 1981). Some of the arrangements that have been identi"ed are those shown in Fig. 2.2.4, reproduced from Cheong and Hwang (1999). At x"0, the A-type spin state is orbitally ordered as it appears in Fig. 2.2.4a. At x"0.5 the famous CE-type arrangement (Fig. 2.2.4b) already found in early studies of manganites is certainly stabilized. This state has been recently observed experimentally using resonant X-ray scattering (Zimmermann et al., 1999, see also Zimmermann et al., 2000). At x"2/3, and also x"3/4, a novel `bi-stripea arrangement is found (Mori et al., 1998a). The x"0.65 state is very stable upon the application of a magnetic "eld (Fig. 25 of Ibarra and De Teresa, 1998c). The origin of the term bi-stripe is obvious from Fig. 2.2.4c. However, theoretical work (Hotta et al., 2000a and references therein) has shown that it is more appropriate to visualize this arrangement as formed by FM zigzag chains running in the direction perpendicular to those of the charge stripes of Fig. 2.2.4c. This issue will be discussed in more detail later in the review when the theoretical aspects are addressed. Based on electron microscopy techniques Cheong and Hwang (1999) believe that at the, e.g., x"5/8 concentration

Fig. 2.2.4. The charge and orbital ordering con"gurations forLa Ca MnO with x"0, 1/2, and 2/3. Open circles are Mn> \V V  and the lobes show the orbital ordering of the e -electrons of Mn>. Figure reproduced from Cheong and Hwang (1999). 

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a mixture of the x"1/2 and 2/3 con"gurations forms the ground state. The size of the coexisting clusters is approximately 100 As . Once again, it appears that phase separation tendencies are at work in manganese oxides. However, note that studies by Radaelli et al. (1999) on the x"2/3 compound arrived to the conclusion that a `Wigner crystala charge arrangement is stable at this density, with the charge ordered but spread as far from each other as possible. It appears that bi-stripes and Wigner crystal states must be very close in energy. While the results at x"0.0 and 0.5 have been reproduced in recent theoretical studies of manganite models, the more complex arrangements at other densities are still under analysis (Hotta et al., 2000b) and will be discussed in more detail below. Finally, there is an interesting observation that is related with some theoretical developments to be presented later in the review. In Fig. 2.2.5, the resistivity at 300 K and 100 K vs. hole density is shown, reproduced from Cheong and Hwang (1999). Note at 300 K the smooth behavior as x grows from 0, only interrupted close to x"1 when the G-type AF insulating state is reached. Then, at 300 K there is no precursor of the drastically di!erent physics found at, e.g., 100 K where for x(0.5 a FM-state is found while for x'0.5 the state is CO and AF. This lack of precursors is also in agreement with neutron scattering results that reported FM #uctuations above the CO and NeH el temperatures in the large x regime (Dai et al., 1996), similar to those observed at lower hole densities. These results are consistent with an abrupt "rst-order-like transition from the state with FM #uctuations to the

Fig. 2.2.5. Resistivity at 300 and 100 K vs. Ca concentration for La Ca MnO reproduced from Cheong and Hwang \V V  (1999).

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CO/AF-state, as observed in other experiments detailed in later sections (for very recent work see Ramos et al., 2000). These two states are so di!erent that a smooth transition between them is not possible. In addition, recent theoretical developments assign considerable importance to the in#uence of disorder on this type of "rst-order transitions to explain the large MR e!ect in manganites (Moreo et al., 2000), as shown elsewhere in this review. Then, the sudden character of the transition from ferromagnetism to charge-ordered antiferromagnetism appears to play a key role in the physics of manganites, and its importance is emphasized in many places in the text that follows. 2.3. Low-bandwidth manganites: the case of Pr Ca MnO \V V  As explained before, in perovskite manganites, such as La Ca MnO and the compound \V V  Pr Ca MnO described here, the bandwidth = is smaller than in other compounds that have \V V  a behavior more in line with the standard double-exchange ideas. In the low-bandwidth compounds, a charge-order state is stabilized in the vicinity of x"0.5, while manganites with a large = (La Sr MnO as example) present a metallic phase at this hole density. Let us focus in this \V V  subsection on Pr Ca MnO which presents a particularly stable CO-state in a broad density \V V  region between x"0.30 and 0.75, as Jirak et al. (1985) showed. Part of the phase diagram of this compound is in Fig. 2.3.1, reproduced from Tomioka et al. (1996) (see also Tomioka and Tokura, 1999). Note that a metallic ferromagnetic phase is not stabilized at zero magnetic "eld and ambient pressure in this low-bandwidth compound. Instead, a ferromagnetic insulating (FI) state exists in the range from x"0.15 to 0.30. This FI state has not been fully explored to the best of our knowledge, and it may itself present charge ordering as some recent theoretical studies have suggested (Hotta and Dagotto, 2000). For x50.30, an antiferromagnetic CO-state is stabilized. Neutron di!raction studies (Jirak et al., 1985) showed that at all densities between 0.30 and 0.75, the arrangement of charge/spin/orbital order of this state is similar to the CE-state (see Fig. 2.2.4b) already discussed in the context of x"0.5 (La, Ca)-based manganites. However, certainly the hole density is changing with x, and as a consequence the CE-state cannot be `perfecta at all densities but electrons have to be added or removed from the structure. Jirak et al. (1985) discussed a `pseudoa-CE-type structure for x"0.4 that has the proper density. Other authors simply refer to the xO0.5 CO-states as made out of the x"0.5 structure plus `defectsa. Hotta and Dagotto (2000) proposed an ordered state for x"3/8 based on mean-"eld and numerical approximations. Neutron di!raction studies have shown that the coupling along the c-axis changes from AF at x"0.5 to FM at x"0.3 (Yoshizawa et al., 1995) and a canted state has also been proposed to model this behavior. Certainly more work is needed to fully understand the distribution of charge in the ground-state away from x"0.5. The e!ect of magnetic "elds on the CO-state of Pr Ca MnO is remarkable. In Fig. 2.3.2 the \V V  resistivity vs. temperature is shown parametric with magnetic "elds of a few Teslas, which are small in typical electronic units. At low temperatures, changes in  by several orders of magnitude can  be observed. Note the stabilization of a metallic state upon the application of the "eld. This state is ferromagnetic according to magnetization measurements, and thus it is curious to observe that a state not present at zero "eld in the phase diagram, is nevertheless stabilized at "nite "elds, a puzzling result that is certainly di$cult to understand. The shapes of the curves in Fig. 2.3.2 resemble similar measurements carried out in other manganites which also present a large MR e!ect, and a possible origin based on percolation between the CO- and FM-phase will be discussed later in the review. First-order characteristics of the metal}insulator transitions in this context are

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Fig. 2.3.1. Phase diagram of Pr Ca MnO . PI and FI denote the paramagnetic insulating and ferromagnetic \V V  insulating states, respectively. For hole density between 0.3 and 0.5, the antiferromagnetic insulating (AFI) state exists in the charge/orbital-ordered insulating (COI) phase. The canted antiferromagnetic insulating (CAFI) state, which may be a mixed FM}AF state, also has been identi"ed between x"0.3 and 0.4. Reproduced from Tomioka and Tokura (1999). Fig. 2.3.2. Temperature dependence of the resistivity of Pr Ca MnO with x"0.3 at various magnetic "elds. The \V V  inset is the phase diagram in the temperature}magnetic "eld plane. The hatched region has hysteresis. Results reproduced from Tomioka and Tokura (1999).

very prominent, and they have been reviewed by Tomioka and Tokura (1999). It is interesting to observe that pressure leads to a colossal MR e!ect quite similar to that found upon the application of magnetic "elds (see for example Fig. 2.3.3, where results at x"0.30 from Moritomo et al. (1997) are reproduced). The abrupt metal}insulator transition at small magnetic "elds found in Pr Ca MnO at \V V  x"0.30 appears at other densities as well, as exempli"ed in Fig. 2.3.4, which shows the resistivity vs. temperature at x"0.35, 0.4 and 0.5, reproduced from Tomioka et al. (1996). Fig. 2.3.5 (from Tomioka and Tokura, 1999) shows that as x grows away from x"0.30, larger "elds are needed to destabilize the charge-ordered state at low temperatures (e.g., 27 T at x"0.50 compared with 4 T at x"0.30). It is also interesting to observe that the replacement of Ca by Sr at x"0.35 also leads to a metal-insulator transition, as shown in Fig. 2.3.6 taken from Tomioka et al. (1997). Clearly Pr Ca MnO presents a highly nontrivial behavior that challenges our theoretical \V V  understanding of the manganese oxide materials. The raw huge magnitude of the CMR e!ect in this compound highlights the relevance of the CO}FM competition. 2.4. Other perovskite manganite compounds Another interesting perovskite manganite compound is Nd Sr MnO , and its phase diagram \V V  is reproduced in Fig. 2.4.1 (from Kajimoto et al., 1999). This material could be labeled as

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Fig. 2.3.3. Temperature dependence of resistivity for Pr Ca MnO at x"0.3 under the various pressures indicated. \V V  Reproduced from Moritomo et al. (1997). Fig. 2.3.4. Temperature dependence of the resistivity corresponding to Pr Ca MnO at the hole concentrations and \V V  magnetic "elds indicated. Reproduced from Tomioka et al. (1996).

`intermediate bandwidtha due to the presence of a stable CO-phase at x"0.50, state which can be easily destroyed by a magnetic "eld in a "rst-order transition (Kuwahara et al., 1995). However, this phase appears only in a tiny range of densities and at low temperature. In fact, aside from this CO-phase, the rest of the phase diagram is very similar to the one of La Sr MnO . In particular, \V V  it is interesting to observe the presence of an A-type antiferromagnetic metallic structure which is believed to have ferromagnetic planes with uniform d   -type orbital order (Kawano et al., 1997), V \W making the system e!ectively anisotropic (Yoshizawa et al., 1998). A compound that behaves similarly to Nd Sr MnO is Pr Sr MnO , with the exception of x"0.5: In \V V  \V V  Pr Sr MnO (x"0.5) the CO-state is not stable. Actually, Tokura clari"ed to the authors \V V  (private communication) that the polycrystal results that appeared in Tomioka et al. (1995) showing a CO-phase in this compound were later proven incorrect after the preparation of single crystals. Nevertheless, such a result illustrates the fragile stability of the CO-phase in materials where the bandwidth is not su$ciently small. Results for this compound reporting mixed-phase tendencies were recently presented by Zvyagin et al. (2000). The corresponding phase diagram for a mixture (La Nd ) Sr MnO can be found in Akimoto et al. (1998) and it shows that the \X X \V V  CO-phase at x"0.5 of the pure (Nd,Sr) compound disappears for z smaller than &0.5. The phase

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Fig. 2.3.5. The charge/orbital-ordered state of Pr Ca MnO at several hole concentrations, plotted on the magnetic \V V  "eld-temperature plane. The hatched area indicates the hysteresis region. For more details see Tomioka and Tokura (1999).

Fig. 2.3.6. (a) Temperature dependence of the resistivity for Pr (Ca Sr ) MnO crystals with varying y. (b)   \W W    Resistivity vs. temperature of Pr (Ca Sr ) MnO (y"0.2) for several magnetic "elds. Reproduced from   \W W    Tomioka and Tokura (1999).

diagram of (La Nd ) Ca MnO investigated by Moritomo (1999b) also shows a competition \X X \V V  between FM and CO, with phase separation characteristics in between. Other manganites present CO-phases at x"0.5 as well, and the compound where this phase seems to be the strongest is Sm Ca MnO , as exempli"ed in Fig. 2.4.2, where the e!ect of      magnetic "elds on several low-bandwidth manganites is shown. An interesting way to visualize the relative tendencies of manganite compounds to form a CO-state at x"0.5 can be found in Fig. 2.4.3, taken from Tomioka and Tokura (1999). As discussed in more detail at the end of this

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Fig. 2.4.1. Phase diagram of Nd Sr MnO , reproduced from Kajimoto et al. (1999). The notation is standard. \V V 

Fig. 2.4.2. The charge-ordered phase of various compounds (RE) (AE) MnO plotted on the magnetic "eld   temperature plane. The hatched area indicates the hysteresis region. Reproduced from Tomioka and Tokura (1999).

section, the key ingredient determining the FM vs. CO character of a state at a "xed density is the size of the ions involved in the chemical composition. In Fig. 2.4.3 the radius of the trivalent and divalent ions, as well as their average radius at x"0.5, appear in the horizontal axes. The Curie and CO temperature are shown below in part (b). As an example, for the extreme case of (La,Sr) based manganites, a metallic ferromagnetic state is observed at x"0.5, while (Pr,Ca) compounds are charge-ordered. As a byproduct of Fig. 2.4.2, Fig. 2.4.3 it is quite interesting to note the similarities between the actual values of the critical temperatures ¹ and ¹ . Being two rather ! !di!erent states, there is no obvious reason why their critical temperatures are similar. A successful theory must certainly address this curious fact.

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Fig. 2.4.3. (a) Average ionic radius at x"0.5 corresponding to a mixture of a trivalent ion (upper abscissa) and a divalent ion (lower abscissa). (b) Critical Curie temperature ¹ and charge/orbital ordering transition ¹ for various trivalent! !divalent ion combinations. Reproduced from Tomioka and Tokura (1999).

2.5. Double-layer compound Not only three-dimensional perovskite-type structures are present in the family of manganite compounds, but layered ones as well. In fact, Moritomo et al. (1996) showed that it is possible to prepare double-layer compounds with a composition La Sr Mn O . Single-layer manga\V >V   nites can also be synthesized, as will be discussed in the next subsection. In fact, these are just special cases of the Ruddlesden}Popper series (T D ) Mn O , with ¹ a trivalent cation, \V V L> L L> D a divalent cation, and n"1 corresponding to the single layer, n"2 to the double layer, and n"R to the cubic perovskite structure (see Fig. 2.5.1 for the actual structure). The temperature dependence of the resistivity in representative multilayer structures is shown in Fig. 2.5.2 for the n"1, 2, and R (3D perovskite) compounds at a hole concentration of x"0.4. In the regime where the single layer is insulating and the n"R layer is metallic, the double layer has an intermediate behavior, with insulating properties above a critical temperature and metallic below. A large MR e!ect is observed in this double-layer system as shown in Fig. 2.5.3, larger than the one found for La Sr MnO . The full phase diagram of this compound will be discussed later in this \V V  review (Section 4) in connection with the presence of inhomogeneities and clustering tendencies. 2.6. Single-layer compound As mentioned in the previous subsection, the single-layer manganite has also been synthesized (see Rao et al., 1988). Its chemical formula is La Sr MnO . This compound does not have \V >V  a ferromagnetic phase in the range from x"0.0 to 0.7, which is curious. Note that other

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Fig. 2.5.1. Schematic crystal structure of four representatives of the Ruddlesden}Popper series of manganese oxides (taken from Tokura, 1999).

manganites have not presented a ferromagnetic metallic phase also, but they had at least a ferroinsulating regime (e.g., Pr Ca MnO ). A schematic phase diagram of the one-layer compound is \V V  given in Fig. 2.6.1, reproduced from Moritomo et al. (1995). For more recent results, see Larochelle et al. (2000). At all the densities shown, insulating behavior has been found. Note the prominent CO-phase near x"0.5, and especially the `spin-glassa phase in a wide range of densities between x"0.2 and x&0.5. The x"0.5 charge-ordered phase is of the CE-type (Sternlieb et al., 1996; Murakami et al., 1998a). The large x regime has phase separation according to Bao et al. (1996), as discussed in more detail below. The actual microscopic arrangement of charge and spin in the intermediate spin-glass regime has not been experimentally studied in detail, to the best of our knowledge, but it certainly deserves more attention since the two dimensionality of the system makes possible reliable theoretical studies and simulations. 2.7. Importance of tolerance factor It has been clearly shown experimentally that working at a "xed hole density the properties of manganites strongly depend on a geometrical quantity called the `tolerance factora, de"ned as
"d } /((2d } ). Here d } is the distance between the A site, where the trivalent or divalent + non-Mn ions are located, to the nearest oxygen. Remember that the A ion is at the center of a cube with Mn in the vertices and O in between the Mn's. d } is the Mn}O shortest distance. Since for + an undistorted cube with a straight Mn}O}Mn link, d } "(2 and d } "1 in units of the + Mn}O distance, then
"1 in this perfect system. However, sometimes the A ions are too small to

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Fig. 2.5.2. Temperature dependence of the resistivity in the n"1 (single layer), n"2 (double layer) and n"R (cubic) representatives of the Ruddlesden}Popper series of manganese oxides. The hole concentration is x"0.4. Results along the layers and perpendicular to them are shown for n"1 and 2. Reproduced from Moritomo et al. (1996). Fig. 2.5.3. Temperature dependence of the resistivity for single crystals of the n"2 compound at x"0.4 (from Moritomo et al., 1996), with an external "eld parallel to the layer.

Fig. 2.6.1. Phase diagram corresponding to the single layer compound La Sr MnO . AF, SG, and CO stand for \V >V  the antiferromagnetic, spin-glass, and charge-ordering phases, respectively. Solid lines are a guide to the eye. Reproduced from Moritomo et al. (1995).

"ll the space in the cube centers and for this reason the oxygens tend to move toward that center, reducing d } . In general d } also changes at the same time. For these reasons, the tolerance + factor becomes less than unity,
(1, as the A radius is reduced, and the Mn}O}Mn angle  becomes smaller than 1803. The hopping amplitude for carriers to move from Mn to Mn naturally decreases as  becomes smaller than 1803 (remember that for a 903 bond the hopping

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Fig. 2.7.1. (a) Phase diagram of temperature vs. tolerance factor for the system A A MnO , where A is a trivalent rare      earth ion and A is a divalent alkali earth ion. Open and closed symbols denote ¹ measured from the magnetization and ! resistivity, respectively. For more details see Cheong and Hwang (1999), from where this "gure is reproduced. A very similar "gure appeared in Hwang et al. (1995a). (b) Top panel: log (¹) in 0 and 5 T for a series of samples of La A Ca MnO , with A mainly Pr but also Y. Bottom panel: MR factor. For details see Hwang et al. (1995a).  \W W   

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involving a p-orbital at the oxygen simply cancels, as explained in more detail below). As a consequence, as the tolerance factor decreases, the tendencies to charge localization increase due to the reduction in the mobility of the carriers. Since in the general chemical composition for perovskite manganites A A MnO there are two possible ions at the `Aa site, then the tolerance \V V  factor for a given compound can be de"ned as a density-weighted average of the individual tolerance factors. In Fig. 2.4.3 the reader can "nd some of the ionic radius in As , for some of the most important elements in the manganite composition. Note that the distance Mn}Mn is actually reduced in the situation described so far (
(1), while the tolerance factor (monotonically related with the hopping amplitude) is also reduced, which is somewhat counterintuitive since it would be expected that having closer Mn-ions would increase the electron hopping between them. However, the hopping amplitude is not only proportional to 1/(d } )?, where '1 (see Harrison, 1989) but also to cos  due to the fact that it is the p-orbital of + oxygen that is involved in the process and if this orbital points toward one of the manganese ions, it cannot point toward the other one simultaneously for O1803. Hwang et al. (1995a) carried out a detailed study of the A A MnO compound for a variety      of A and A ions. Fig. 2.7.1a summarizes this e!ort, and it shows the presence of three dominant regimes: a paramagnetic insulator at high-temperature, a low-temperature ferromagnetic metal at large tolerance factor, and a low-temperature charge-ordered ferromagnetic insulator at small tolerance factor. This "gure clearly illustrates the drastic dependence with the tolerance factor of the properties of doped manganites. These same results will be discussed in more detail below in this review, when issues related with the presence of coexisting phases are addressed. In particular, experimental work have shown that the `FMIa regime may actually correspond to coexisting CO and FM large clusters. The CO-phase has both charge and orbital order. An example upon which Fig. 2.7.1a has been constructed is shown in Fig. 2.7.1b that mainly corresponds to results obtained for La Pr Ca MnO . The temperature dependence of  \V V     (¹) presents hysteresis e!ects, suggesting that the PMI}FMM transition has some xrst-order  characteristics, a feature that is of crucial importance in recent theoretical developments to be discussed later (Yunoki et al., 2000; Moreo et al., 2000). Note the huge MR ratios found in these compounds and the general trend that this ratio dramatically increases as ¹ is reduced, mainly as ! a consequence of the rapid increase of the resistivity of the PM insulating state as the temperature is reduced. Certainly, the state above ¹ is not a simple metal where ferromagnetic correlations ! slowly build up with decreasing temperature as in a second order transition, and as expected in the DE mechanism. A new theory is needed to explain these results.

3. Theory of manganites 3.1. Early studies 3.1.1. Double exchange Most of the early theoretical work on manganites focused on the qualitative aspects of the experimentally discovered relation between transport and magnetic properties, namely the increase in conductivity upon the polarization of the spins. Not much work was devoted to the magnitude of the magnetoresistance e!ect itself. The formation of coexisting clusters of competing phases was

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not included in the early considerations. The states of manganites were assumed to be uniform, and `double exchangea (DE) was proposed by Zener (1951b) as a way to allow for charge to move in manganites by the generation of a spin polarized state. The DE process has been historically explained in two somewhat di!erent ways. Originally, Zener (1951b) considered the explicit movement of electrons schematically written (Cieplak, 1978) as Mn>O Mn>P t ts Mn>O Mn> where 1, 2, and 3 label electrons that belong either to the oxygen ts t between manganese, or to the e -level of the Mn-ions. In this process there are two simultaneous  motions (thus the name double exchange) involving electron 2 moving from the oxygen to the right Mn-ion, and electron 1 from the left Mn-ion to the oxygen (see Fig. 3.1.1a). The second way to visualize DE processes was presented in detail by Anderson and Hasegawa (1955) and it involves a second-order process in which the two states described above go from one to the other using an intermediate state Mn>O Mn>. In this context the e!ective hopping for the t s t electron to move from one Mn-site to the next is proportional to the square of the hopping involving the p-oxygen and d-manganese orbitals (t ). In addition, if the localized spins are  considered classical and with an angle  between nearest-neighbor ones, the e!ective hopping becomes proportional to cos(/2), as shown by Anderson and Hasegawa (1955). If "0 the hopping is the largest, while if "
, corresponding to an antiferromagnetic background, then the hopping cancels. The quantum version of this process has been described by Kubo and Ohata (1972).

Fig. 3.1.1. (a) Sketch of the Double Exchange mechanism which involves two Mn ions and one O ion. (b) The mobility of e -electrons improves if the localized spins are polarized. (c) Spin-canted state which appears as the interpolation between  FM and AF states in some mean-"eld approximations. For more details see the text.

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Note that the oxygen linking the Mn-ions is crucial to understand the origin of the word `doublea in this process. Nevertheless, the majority of the theoretical work carried out in the context of manganites simply forgets the presence of the oxygen and uses a manganese-only Hamiltonian. It is interesting to observe that ferromagnetic states appear in this context even without the oxygen. It is clear that the electrons simply need a polarized background to improve their kinetic energy, in similar ways as the Nagaoka phase is generated in the one-band Hubbard model at large ;/t (for a high-¹ review, see Dagotto, 1994). This tendency to optimize the kinetic  energy is at work in a variety of models and the term double exchange appears unnecessary. However, in spite of this fact it has become customary to refer to virtually any ferromagnetic phase found in manganese models as `DE induceda or `DE generateda, forgetting the historical origin of the term. In this review a similar convention will be followed, namely the credit for the appearance of FM phases will be given to the DE mechanism, although a more general and simple kineticenergy optimization is certainly at work. 3.1.2. Ferromagnetism due to a large Hund coupling Regarding the stabilization of ferromagnetism, computer simulations (Yunoki et al., 1998a) and a variety of other approximations have clearly shown that models without the oxygen degrees of freedom (to be reviewed below) can also produce FM-phases, as long as the Hund coupling is large enough. In this situation, when the e electrons directly jump from Mn  to Mn their kinetic energy is minimized if all spins are aligned (see Fig. 3.1.1b). As explained in the previous subsection, this procedure to obtain ferromagnetism is usually also called double-exchange and even the models from where it emerges are called double-exchange models. However, there is little resemble of these models and physical process with the original DE ideas (Zener, 1951b) where two electrons were involved in the actual hopping. Actually, the FM phases recently generated in computer simulations and a variety of mean-"eld approximations resemble more closely the predictions of another work of Zener (1951a), where indeed a large Hund coupling is invoked as the main reason for ferromagnetism in some compounds. In addition, it has been questioned whether double-exchange or the large J mechanism & are su$cient to indeed produce the ferromagnetic phase of manganites. An alternative idea (Zhao, 2000) relies on the fact that holes are located mostly in the oxygens due to the charge-transfer character of manganites and these holes are linked antiferromagnetically with the spins in the adjacent Mn-ions due to the standard exchange coupling, leading to an e!ective Mn}Mn ferromagnetic interaction. In this context the movement of holes would be improved if all Mn spins are aligned leading to a FM-phase, although many-body calculations are needed to prove that this is indeed the case for realistic couplings. A comment about this idea: in the context of the cuprates a similar concept was discussed time ago (for references see Dagotto, 1994) and after considerable discussion it was concluded that this process has to be contrasted against the so-called Zhang}Rice singlet formation where the spin of the hole at the oxygen couples in a singlet state with the spin at the Cu. As explained in Riera et al. (1997) the analogous process in manganites would lead to the formation of e!ective S"3/2 `holea states, between the S"2 of manganese (3#) and the S"1/2 of the oxygen hole. Thus, the competition between these two tendencies should be addressed in detail, similarly as done for the cuprates to clarify the proposal of Zhao (2000).

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3.1.3. Spin-canted state At this point it is useful to discuss the well-known proposed `spin-canteda state for manganites. Work by de Gennes (1960) using mean-"eld approximations suggested that the interpolation between the antiferromagnetic state of the undoped limit and the ferromagnetic state at "nite hole density, where the DE mechanism works, occurs through a `canted statea, similar as the state produced by a magnetic "eld acting over an antiferromagnet (Fig. 3.1.1c). In this state the spins develop a moment in one direction, while being mostly antiparallel within the plane perpendicular to that moment. The coexistence of FM and AF features in several experiments carried out at low hole doping (some of them reviewed below) led to the widely spread belief until recently that this spin-canted state was indeed found in real materials. However, a plethora of recent theoretical work (also discussed below) has shown that the canted state is actually not realized in the model of manganites studied by de Gennes (i.e., the simple one-orbital model). Instead phase separation occurs between the AF- and FM-states, as extensively reviewed below. Nevertheless, a spin-canted state is certainly still a possibility in real low-doped manganites but its origin, if their presence is con"rmed, needs to be revised. It may occur that substantial Dzyaloshinskii}Moriya (DM) interactions appear in manganese oxides, but the authors are not aware of experimental papers con"rming or denying their relevance, although some estimations (Solovyev et al., 1996; LyandaGeller et al., 1999; Chun et al., 1999a) appear to indicate that the DM couplings are small. However, even if the DM coupling were large there are still subtle issues to be addressed. For instance, it is widely believed that DM interactions lead to canting. However, Co!ey et al. (1990) showed that in the simple case where the D factor for the DM interaction is a constant, as in most GH early work on the subject, the DM term leads to a spiral state rather than a truly canted state. In addition, the authors of this review are not aware of reliable calculations showing that a canted state can indeed be stabilized in a model without terms added that break explicitly the invariance under rotations of the system in such a way that a given direction, along which the moment develops, is made by hand di!erent from the others. For all these reasons and from the discussion below it may appear that the simplest way to explain the experimental data at low doping is to assume an AF}FM-phase coexistence instead of a canted state. However, the issue is still open and more experimental work should be devoted to its clari"cation. 3.1.4. Charge-ordered state at x"0.5 Early theoretical work on manganites carried out by Goodenough (1955) (see also Goodenough, 1963) explained many of the features observed in the neutron scattering La Ca MnO experi\V V  ments by Wollan and Koehler (1955), notably the appearance of the A-type AF phase at x"0 and the CE-type phase at x"0.5. The approach of Goodenough (1955) was based on the notion of `semicovalent exchangea and the main idea can be roughly explained as follows. Suppose one considers a Mn}O bond directed, say, along the x-axis, and let us assume that the Mn-cation has an occupied orbital pointing along y or z instead of x (in other words, there is an empty orbital along x). The oxygen, being in a (2!) state, will try to move towards this Mn site since it does not have a negative cloud of Mn electrons to "ght against. This process shortens the distance Mn}O and makes this bond quite stable. This is a semicovalent bond. Suppose now the occupied Mn-orbital has an electron with an up spin. Of the two relevant electrons of oxygen, the one with spin up will feel the exchange force toward the Mn electron, i.e., if both electrons involved have the same spin, the space part of their common wave function has nodes which reduce the

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electron}electron repulsion (as in the Hund's rules). Then, e!ectively the considered Mn}O bond becomes ferromagnetic between the Mn electron and one of the oxygen electrons. Consider now the left-side O}Mn portion of the Mn}O}Mn bond. In the example under consideration, the second electron of oxygen must be down and it spends most of the time away from the left Mn-ion (rather than close to it as the oxygen spin-up electron does). If the Mn-ion on the right of the link Mn}O}Mn also has an occupied orbital pointing perpendicular to the x-axis, then O}Mn and Mn}O behave similarly (individually FM) but with pairs of spins pointing in opposite directions. As a consequence an e!ective antiferromagnetic Mn}Mn interaction has been generated (see Fig. 3.1.2a), and both Mn}O and O}Mn are shorten in length. However, if the right Mn has an electron in an orbital pointing along x, namely along the relevant p-orbital of the oxygen, the Hund-rule-like argument does not apply anymore since now a simple direct exchange is more important, leading to an AF O}Mn bond. In this case, the overall Mn}Mn interaction is ferromagnetic, as sketched in Fig. 3.1.2b. Then, simple arguments lead to the notion that both AF and FM couplings among the Mn-ions can be e!ectively generated, depending on the orientation of the orbitals involved. Analyzing the various possibilities for the orbital directions and generalizing to the case where Mn> ions are also present, Goodenough (1955) arrived to the A- and CE-type phases of manganites very early in the theoretical study of these compounds (the shape of these states was shown in Fig. 2.2.4). In this line of reasoning, note that the Coulomb interactions are important to generate Hund-like rules and the oxygen is also important to produce the covalent bonds. The lattice distortions are also quite relevant in deciding which of the many possible states minimizes the energy. However, it is interesting to observe that in more recent theoretical work described below in this review, both the A- and CE-type phases can be generated without the explicit appearance of oxygens in the models and also without including long-range Coulombic terms. Summarizing, there appears to be three mechanisms to produce e!ective FM interactions: (i) double exchange, where electrons are mobile, which is valid for noncharge-ordered states and where the oxygen plays a key role, (ii) Goodenough's approach where covalent bonds are

Fig. 3.1.2. Generation of antiferromagnetic (a) or ferromagnetic (b) e!ective interactions between the spins of Mn ions mediated by oxygen, depending on the orientation of the Mn orbitals. For details see text.

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important (here the electrons do not have mobility in spite of the FM e!ective coupling), and it mainly applies to charge-ordered states, and (iii) the approach already described in this subsection based on purely Mn models (no oxygens) which leads to FM interactions mainly as a consequence of the large Hund coupling in the system. If phonons are introduced in the model it can be shown that the A- and CE-type states are generated, as reviewed later in this section. In the remaining theoretical part of the review most of the emphasis will be given to approach (iii) to induce FM bonds since a large number of experimental results can be reproduced by this procedure, but it is important to keep in mind the historical path followed in the understanding of manganites. Based on all this discussion, it is clear that reasonable proposals to understand the stabilization of AF- and FM-phase in manganites have been around since the early theoretical studies of manganese oxides. However, these approaches (double exchange, ferromagnetic covalent bonds, and large Hund coupling) are still not su$cient to handle the very complex phase diagram of manganites. For instance, there are compounds such as La Sr MnO that actually do not have \V V  the CE-phase at x"0.5, while others do. There are compounds that are never metallic, while others have a paramagnetic state with standard metallic characteristics. And even more important, in the early studies of manganites there was no proper rationalization for the large MR e!ect. It is only with the use of state-of-the-art many-body tools that the large magnetotransport e!ects are starting to be understood, owing to theoretical developments in recent years that can address the competition among the di!erent phases of manganites, their clustering and mixed-phase tendencies, and dynamical Jahn}Teller polaron formation. 3.2. More recent theories The prevailing ideas to explain the curious magnetotransport behavior of manganites changed in the mid-1990s from the simple double-exchange scenario to a more elaborated picture where a large Jahn}Teller (JT) e!ect, which occurs in the Mn> ions, produces a strong electron}phonon coupling that persists even at densities where a ferromagnetic ground state is observed. In fact, in the undoped limit x"0, and even at "nite but small x, it is well known that a robust static structural distortion is present in the manganites (see Goodenough, 1955; Elemans et al., 1971). In this context it is natural to imagine the existence of small lattice polarons in the paramagnetic phase above ¹ , and it was believed that these polarons lead to the insulating behavior of this ! regime. Actually, the term polaron (see Holstein, 1959) is somewhat ambiguous. In the context of manganites it is usually associated with a local distortion of the lattice around the charge, sometimes together with a magnetic cloud or region with ferromagnetic correlations (magneto polaron or lattice-magneto polaron). 3.2.1. Double-exchange is not enough The fact that double exchange cannot be enough to understand the physics of manganites is clear from several di!erent points of view. For instance, Millis et al. (1995) arrived at this conclusion by presenting estimations of the critical Curie temperature and of the resistivity using the DE framework. Regarding ferromagnetism, their calculations for a model having as an interaction only a large Hund coupling between e - and t -electron led to a ¹ prediction between 0.1 and 0.3 eV,   ! namely of the order of the bare hopping amplitude and considerably higher than the experimental results. Thus, it was argued that DE produces the wrong ¹ by a large factor. However, note that !

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computational work led to a much smaller estimation of the Curie temperature of the order of 0.1t for the double-exchange model (t is the e -electron hopping amplitude), and compatible with  experiments (Yunoki et al., 1998a; Calderon and Brey, 1998; Yi et al., 1999a; Motome and Furukawa, 1999; Motome and Furukawa, 2000b; some of which will be reviewed in more detail later. Results for S"1/2 localized spins can be found in RoK der et al., 1997). For this reason arguments based on the value of ¹ are not su$cient to exclude the double-exchange model. ! Regarding the resistance, using the memory function method (in principle valid at large frequency) to estimate the dc component, Millis et al. (1995) found a resistivity that grows with reducing temperature (insulating behavior) even below ¹ . For this reason Millis et al. (1995) concluded that ! the model based only on a large J is not adequate for the manganites, and instead the relevance of & the Jahn}Teller phonons was invoked. These results have to be contrasted with computer-based calculations of the resistivity for the one-orbital model at J "R by Calderon et al. (1999) that & reported instead a metallic behavior for the double-exchange model, actually both above and below ¹ . Paradoxically, this behavior also leads to the same conclusion, namely that double ! exchange is not su$cient to explain the manganite behavior of, e.g., La Ca MnO which has \V V  insulating characteristics above ¹ but it is metallic below. However, both lines of attack to the DE ! model may need further revision, since the computational work of Yunoki et al. (1998a) at a large but not in"nite Hund coupling has established that the simple one-orbital double-exchange model has regions with mixed-phase tendencies presenting an insulating resistivity (Moreo et al., 1999a) at and near n"1 (n is the e electron number per site), which becomes metallic as the electronic  density is further reduced. The existence of a metal}insulator transition in this model opens the possibility that the one-orbital system may still present physics qualitatively similar to that found experimentally, where such a transition is crucial in manganites. For this reason, using the one-orbital model as a toy model for manganites is still quite acceptable, as long as the region of study is close to the metal}insulator regime. In fact, recent work reporting percolative e!ects in this context use both the one- and two-orbital models with or without a strong JT coupling (Moreo et al., 2000). However, it is clear that the one-orbital model is incomplete for quantitative studies since it cannot describe, e.g., the key orbital ordering of manganites and the proper charge-order states at x near 0.5, which are so important for the truly CMR e!ect found in low-bandwidth manganites. Then, the authors of this review fully agree with the conclusions of Millis et al. (1995), although the arguments leading to such conclusion are di!erent. It is clear that not even a fully disordered set of classical spins can scatter electrons as much as needed to reproduce the experiments (again, unless large antiferromagnetic regions appear in a mixed-phase regime). 3.2.2. Jahn}Teller phonons and polarons Millis et al. (1996) (see also Millis et al., 1996a; Millis, 1998) argued that the physics of manganites is dominated by the interplay between a strong electron}phonon coupling and the large Hund coupling e!ect that optimizes the electronic kinetic energy by the generation of a FM-phase. The large value of the electron}phonon coupling is clear in the regime of manganites below x"0.20 where a static JT distortion plays a key role in the physics of the material. Millis et al. (1996b) argued that a dynamical JT e!ect may persist at higher hole densities, without leading to long-range order but producing important #uctuations that localize electrons by splitting the degenerate e levels at a given MnO octahedron. The calculations were carried out using the   in"nite-dimensional approximation that corresponds to a local mean-"eld technique where the

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polarons can have only a one site extension, and the classical limit for the phonons and spins was used. The latter approximation is not expected to be severe unless the temperatures are very low (for a discussion see Millis et al., 1996b). The Coulomb interactions were neglected, but further work reviewed below showed that JT and Coulombic interactions lead to very similar results (Hotta et al., 2000c), and, as a consequence, this approximation is not severe either. Orbital or charge ordering were not considered in the formalism of Millis et al. (1996). Following the work of Millis et al. (1995), phonons were also argued to be of much importance in manganites by RoK der et al. (1996), who found a tendency toward the formation of polarons in a single-orbital DE model with quantum phonons, treating the localized spins in the mean-"eld approximation and the polaron formation with the Lang}Firsov variational approximation. Coulomb interactions were later incorporated using the Gutzwiller approximation (Zang et al., 1996). Millis et al. (1996) argued that the ratio  "E /t dominates the physics of the problem.  (2  Here E is the static trapping energy at a given octahedron, and t is an e!ective hopping that is (2  temperature dependent following the standard DE discussion. In this context it was conjectured that when the temperature is larger than ¹ the e!ective coupling  could be above the critical !  value that leads to insulating behavior due to electron localization, while it becomes smaller than the critical value below ¹ , thus inducing metallic behavior. The calculations were carried out ! using classical phonons and t spins. The results of Millis et al. (1996) for ¹ and the resistivity at  ! a "xed density n"1 when plotted as a function of  had formal similarities with experimental  results (which are produced as a function of density). In particular, if  is tuned to be very close to  the metal}insulator transition, the resistivity naturally strongly depends on even small external magnetic "elds. However, in order to describe the percolative nature of the transition found experimentally and the notorious phase separation tendencies, calculations beyond mean-"eld approximations are needed, as reviewed later in this paper. The existence of a critical value of the electron}phonon coupling constant  of order unity at n"1 leading to a metal}insulator transition is natural and it was also obtained in Monte Carlo (MC) simulations by Yunoki et al. (1998b). However, computational studies of the conductivity led to either insulating or metallic behavior at all temperatures, for values of  above or below the critical temperature, respectively. A mixture of metal/insulator behavior in the resistivity at a "xed  was not observed at n"1. 3.3. Models and parameters In the previous subsections, the theoretical work on manganites has been reviewed mainly in a historical order. In this section, the "rst steps toward a description of the latest theoretical developments in this context are taken. First, it is important to clearly write down the model Hamiltonian for manganites. For complex material such as the Mn-oxides, unfortunately, the full Hamiltonian includes several competing tendencies and couplings. However, as shown below, the essential physics can be obtained using relatively simple models, deduced from the complicated full Hamiltonian. 3.3.1. Ewect of crystal xeld In order to construct the model Hamiltonian for manganites, let us start our discussion at the level of the atomic problem, in which just one electron occupies a certain orbital in the 3d shell of

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a manganese ion. Although for an isolated ion a "ve-fold degeneracy exists for the occupation of the 3d-orbitals, this degeneracy is partially lifted by the crystal "eld due to the six oxygen ions surrounding the manganese forming an octahedron. This is analyzed by the ligand "eld theory that shows that the "ve-fold degeneracy is lifted into doubly degenerate e -orbitals (d   and d   ) X \P  V \W and triply degenerate t -orbitals (d , d , and d ). The energy di!erence between those two levels  VW WX XV is usually expressed as 10 Dq, by following the traditional notation in the ligand "eld theory (see, for instance, Gerloch and Slade, 1973). Here note that the energy level for the t -orbitals is lower than that for e -orbitals. Qualitat  ively, this can be understood as follows: The energy di!erence originates in the Coulomb interaction between the 3d electrons and the oxygen ions surrounding manganese. While the wave functions of the e -orbitals is extended along the direction of the bond between manganese and  oxygen ions, those in the t -orbitals avoid this direction. Thus, an electron in t -orbitals is not   heavily in#uenced by the Coulomb repulsion due to the negatively charged oxygen ions, and the energy level for t -orbitals is lower than that for e -orbitals.   As for the value of 10 Dq, it is explicitly written as (see Gerloch and Slade, 1973) 5 Ze r , 10 Dq" 3 a a

(1)

where Z is the atomic number of the ligand ion, e is the electron charge, a is the distance between manganese and oxygen ions, r is the coordinate of the 3d-orbital, and 2 denotes the average value by using the radial wave function of the 3d-orbital. Estimations by Yoshida (1998, p. 29) suggest that 10 Dq is about 10,000}15,000 cm\ (remember that 1 eV"8063 cm\). 3.3.2. Coulomb interactions Consider now a Mn> ion, in which three electrons exist in the 3d shells. Although those electrons will occupy t -orbitals due to the crystalline "eld splitting, the con"guration is not  uniquely determined. To con"gure three electrons appropriately, it is necessary to take into account the e!ect of the Coulomb interactions. In the localized ion system, the Coulomb interaction term among d-electrons is generally given by   ,    ,    diR   diR   di   di   , (2) H!i "(1/2)          AN AN AN AN   A A A A N N N N where di is the annihilation operator for a d-electron with spin in the -orbital at site i, and the AN Coulomb matrix element is given by



 ,    ,   "        

dr drH  (r)H  (r)gr r    (r)   (r) . \Y A N AN AN AN

(3)

Here gr r is the screened Coulomb potential, and  (r) is the Wannier function for an electron \Y AN with spin in the -orbital at position r. By using the Coulomb matrix element, the so-called `Kanamori parametersa, ;, ;, J, and J, are de"ned as follows (see Kanamori, 1963; Dworin and Narath, 1970; Castellani et al., 1978). ; is the intraband Coulomb interaction, given by ;" ,  , 

(4)

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with O . ; is the interband Coulomb interaction, expressed by ;" ,   ,  

(5)

with O . J is the interband exchange interaction, written as J" ,    , 

(6)

with O . Finally, J is the pair-hopping amplitude between di!erent orbitals, given by J" ,   ,  

(7)

with O  and O . Note the relation J"J, which is simply due to the fact that each of the parameters above is given by an integral of the Coulomb interaction sandwiched with appropriate orbital wave functions. Analyzing the form of those integrals the equality between J and J can be deduced [see equation Eq. (2.6) of Castellani et al. (1978); See also the appendix of FreH sard and Kotliar (1997)]. Using the above parameters, it is convenient to rewrite the Coulomb interaction term in the following form: H!i "(;/2)  ni ni #(;/2)  ni ni AN ANY AN AYNY AN$NY NNYA$AY diR diR di di , (8) # (J/2)  diR diR di di #(J/2)  AN AYNY ANY AYN AN ANY AYNY AYN N$NYA$AY NNYA$AY where ni "diR di . Here it is important to clarify that the parameters ;, ;, and J in Eq. (8) are AN AN AN not independent (here J"J is used). The relation among them in the localized ion problem has been clari"ed by group theory arguments, showing that all the above Coulomb interactions can be expressed by the so-called `Racah parametersa A, B, and C (for more details, see Gri$th, 1961. See also Tang et al., 1998). Here only the main results are summarized in Table 1, following Tang et al. (1998). Note that the values of ; and J depend on the combination of orbitals, namely they take di!erent values depending on the orbitals used (Table 1), while ;"A#4B#3C is independent of the orbital choice. Thus, it is easily checked that the relation (9)

;";#2J holds in any combination of orbitals.

Table 1 Expressions for ; and J by using Racah parameters A, B, and C. Note that ;"A#4B#3C for each orbital. For more information, see Tang et al. (1998)



;

J

xy, yz, zx x!y, 3z!r xy xy yz, zx yz, zx

xy, yz, zx x!y, 3z!r x!y 3z!r x!y 3z!r

A!2B#C A!4B#C A#4B#C A!4B#C A!2B#C A#2B#C

3B#C 4B#C C 4B#C 3B#C B#C

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Although Eq. (9) has been clearly shown to be valid using the Racah parameters, the discussions in the current literature regarding this issue are somewhat confusing, probably since the arguments usually rely directly on Hamiltonian Eq. (8), rather than Eqs. (2) and (3). Thus, it is instructive to discuss the above-mentioned relation among the several couplings using arguments directly based on the model (8), without using the Racah parameters. First note that even using J"J, the electron}electron interaction is still not invariant under rotations in orbital space. This can be easily understood simply using two orbitals as an example, and two particles. In the absence of hopping terms, the problem involves just one site and it can be easily diagonalized, leading to four eigenenergies. The lowest one is ;!J, has degeneracy three, and it corresponds to a spin-triplet and orbital-singlet state. In order to verify that indeed this state is a singlet in orbital space, the operators ¹Wi "!(i/2) (diR di !diR di ) , ¹Vi "(1/2) (diR di #diR di ) , N
N
N N N
N
N N N N (10) ¹Xi "(1/2) (diR di !diR di ) N N
N
N N are needed. The next state is nondegenerate, it has energy ;#J and it is a spin singlet. Regarding the orbital component, it corresponds to the ¹Xi "0 part of an orbital triplet. This result already suggests us that orbital invariance is not respected in the system unless restrictions are imposed on the couplings, since a state of an orbital triplet is energetically separated from another state of the same triplet. The next two states have energies ;#J and ;!J, each is nondegenerate and spin singlet, and they are combinations of orbital triplets with ¹Xi "#1 and !1. Note that the state characterized by ;#J is invariant under rotations in orbital space (using a real rotation matrix parametrized by only one angle), while the other one with ;!J is not. Then, it is clear now how to proceed to restore rotational invariance. It should be demanded that ;!J";#J, namely, ;";#J#J (see, for instance, Kuei and Scalettar, 1997). In addition, following Castellani et al. (1978), it is known that J"J, as already discussed. Then, Eq. (9) is again obtained as a condition for the rotational invariance in orbital space. It should be noted that spin rotational invariance does not impose any constraints on the parameters. Also it should be noted that the orbital rotational invariance achieved here is not a full SU (2) one, but a subgroup, similarly as it occurs in the anisotropic Heisenberg model that has invariance under rotation in the xy plane only. For this reason the states are either singlets or doublets, but not triplets, in orbital space. For the case of three orbitals, a similar study can be carried out, although it is more tedious. For two particles, the energy levels now are at ;!J (degeneracy nine, spin triplet and orbital triplet), ;#J (degeneracy three, spin singlet, contains parts of an orbital quintuplet), ;!J (degeneracy two, spin singlet, contains portions of an orbital quintuplet), and ;#2J (nondegenerate, spin singlet and orbital singlet). In order to have the proper orbital multiplets that are characteristic of a rotational orbital invariant system, it is necessary to require that ;#J";!J. If the relation J"J is further used, then the condition again becomes ;";#2J. A better proof of this condition can be carried out by rewriting the H amiltonian in terms of spin and orbital rotational invariant operators such as Ni " ni , Si " Si ) Si , and Li " Li ) Li , where AN AN AAY A AY NNY N NY Si and Li are the spin and orbital operators, respectively. By this somewhat tedious procedure, a "nal expression is reached in which only one term is not in the form of explicitly invariant operators. To cancel that term, the condition mentioned above is needed.

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Now let us move to the discussion of the con"guration of three electrons for the Mn> ion. Since the largest energy scale among the several Coulombic interactions is ;, the orbitals are not doubly occupied by both up- and down-spin electrons. Thus, only one electron can exist in each orbital of the triply degenerate t sector. Furthermore, in order to take advantage of J, the spins of those  three electrons point along the same direction. This is the so-called `Hund's rulea. By adding one more electron to Mn> with three up-spin t -electrons, let us consider the  con"guration for the Mn> ion. Note here that there are two possibilities due to the balance between the crystalline-"eld splitting and the Hund coupling: One is the `high-spin statea in which an electron occupies the e -orbital with up spin if the Hund coupling is dominant. In this case, the energy level  appears at ;!J#10 Dq. Another is the `low-spin statea in which one of the t -orbitals is  occupied with a down-spin electron, when the crystalline-"eld splitting is much larger than the Hund coupling. In this case, the energy level occurs at ;#2J. Thus, the high-spin state appears if 10 Dq(5J holds. Since J is a few eV and 10 Dq is about 1 eV in the manganese oxide, the inequality 10 Dq(5J is considered to hold. Namely, in the Mn> ion, the high-spin state is realized. In order to simplify the model without loss of essential physics, it is reasonable to treat the three spin-polarized t -electrons as a localized `core-spina expressed by Si at site i, since the overlap  integral between t and oxygen p orbital is small compared to that between e and p orbitals.   Moreover, due to the large value of the total spin S"3/2, it is usually approximated by a classical spin (this approximation will be tested later using computational techniques). Thus, the e!ect of the strong Hund coupling between the e -electron spin and localized t -spins is considered by   introducing H "!J  si ) Sj , & & i

(11)

where si " dRi di , J ('0) is the Hund coupling between localized t -spin and mobile  A?@ A? ?@ A@ & e -electron, and "( , , ) are the Pauli matrices. The magnitude of J is of the order of J. Here  V W X & note that Si is normalized as Si "1. Thus, the direction of the classical t -spin at site i is de"ned as  Si "(sin i cos i , sin i sin i , cos i ), (12) by using the polar angle i and the azimuthal angle i . Unfortunately, the e!ect of the Coulomb interaction is not fully taken into account only by H since there remains the direct electrostatic repulsion between e -electrons, which will be &  referred to as the `Coulomb interactiona hereafter. Then, the following term should be added to the Hamiltonian H } " H!i #<  i j , (13)   i 6i j7 where i " ni . Note here that in this expression, the index for the orbital degree of freedom AN AN runs only in the e -sector. Note also that in order to consider the e!ect of the long-range Coulomb  repulsion between e -electrons, the term including < is added, where < is the nearest-neighbor  Coulomb interaction. 3.3.3. Electron}phonon coupling Another important ingredient in manganites is the lattice distortion coupled to the e -electrons.  In particular, the double degeneracy in the e -orbitals is lifted by the Jahn}Teller distortion of the 

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MnO octahedron (Jahn and Teller, 1937). The basic formalism for the study of electrons coupled to  Jahn}Teller modes has been set up by Kanamori (1960). He focused on cases where the electronic orbitals are degenerate in the undistorted crystal structure, as in the case of Mn in an octahedron of oxygens. As explained by Kanamori (1960), the Jahn}Teller e!ect (Jahn and Teller, 1937) in this context can be simply stated as follows: when a given electronic level of a cluster is degenerate in a structure of high symmetry, this structure is generally unstable, and the cluster will present a distortion toward a lower symmetry ionic arrangement. In the case of Mn>, which is doubly degenerate when the crystal is undistorted, a splitting will occur when the crystal is distorted. The distortion of the MnO octahedron is `cooperativea since once it occurs in a particular octahed ron, it will a!ect the neighbors. The basic Hamiltonian to describe the interaction between electrons and Jahn}Teller modes was written by Kanamori (1960) and it is of the form #Q i ¹Xi )#(k /2) (Qi #Qi ) , (14) H(2 i "2g(Q i ¹V  i  (2   where g is the coupling constant between the e -electrons and distortions of the MnO octahedron,   Q i and Q i are normal modes of vibration of the oxygen octahedron that remove the degeneracy   between the electronic levels, and k is the spring constant for the Jahn}Teller mode distortions. In (2 the expression of H(2 i , a ¹W i -term does not appear for symmetry reasons, since it belongs to the A representation. The nonzero terms should correspond to the irreducible representations  included in E ;E , namely, E and A . The former representation is expressed by using the     pseudo-spin operators ¹Vi and ¹Xi as discussed here, while the latter, corresponding to the breathing mode, is discussed later in this subsection. For more details the reader should consult Yoshida (1998, p. 40) and the book in preparation by one of the authors (E.D.). Following Kanamori, Q i and Q i are explicitly given by   1 Q i" (X i !X i !> i #> i ) (15)     (2  and 1 Q i" (2Z i !2Z i !X i #X i !> i #> i ) ,        (6

(16)

where X , > , and Z are the displacement of oxygen ions from the equilibrium positions along IH IH IH the x- , y- , and z-direction, respectively. The convention for the labeling of coordinates is shown in Fig. 3.3.1. To solve this Hamiltonian, it is convenient to scale the phononic degrees of freedom as (17) Q i "(g/k )q i , Q i "(g/k )q i , (2   (2   where g/k is the typical energy scale for the Jahn}Teller distortion, which is of the order of 0.1 As , (2 namely, 2.5% of the lattice constant. When the JT distortion is expressed in the polar coordinate as (18) q i "qi sin i , q i "qi cos i ,   the ground state is easily obtained as (!sin[ i /2]diR #cos[ i /2]diR )0 with the use of the phase ?N @N

i . The corresponding eigenenergy is given by !E , where E is the static Jahn}Teller energy, (2 (2 de"ned by E "g/(2k ) . (2 (2

(19)

E. Dagotto et al. / Physics Reports 344 (2001) 1}153

39

Fig. 3.3.1. MnO octahedron at site i. The labeling for oxygen ions is shown. 

Note that the ground state energy is independent of the phase i . Namely, the shape of the deformed isolated octahedron is not uniquely determined in this discussion. In the Jahn}Teller crystal, the kinetic motion of e electrons, as well as the cooperative e!ect between adjacent  distortions, play a crucial role in lifting the degeneracy and "xing the shape of the local distortion. This point will be discussed later in detail. To complete the electron}phonon coupling term, it is necessary to consider the breathing-mode distortion, coupled to the local electron density as H
 i "gQ i i #(1/2)k Qi , 
  where the breathing-mode distortion Q

(20) i

is given by

1 Q i" (X i !X i #> i !> i #Z i !Z i )       (3 

(21)

and k is the associated spring constant. Note that, in principle, the coupling constants of the
 e electrons with the Q , Q , and Q modes could be di!erent from one another. For simplicity,     here it is assumed that those coupling constants take the same value. On the other hand, for the spring constants, a di!erent notation for the breathing mode is introduced, since the frequency for the breathing-mode distortion has been found experimentally to be di!erent from that for the Jahn}Teller mode. This point will be brie#y discussed later. Note also that the Jahn}Teller and breathing modes are competing with each other. As it was shown above, the energy gain due to the Jahn}Teller distortion is maximized when one electron exists per site. On the other hand, the breathing-mode distortion energy is proportional to the total number of e electrons per site, since  this distortion gives rise to an e!ective on-site attraction between electrons. By combining the JT mode and breathing-mode distortions, the electron}phonon term is summarized as H } " (H(2 i #H
 i ) .   i

(22)

This expression depends on the parameter "k /k , which regulates which distortion, the
 (2 Jahn}Teller or breathing mode, play a more important role. This point will be discussed in a separate subsection.

40

E. Dagotto et al. / Physics Reports 344 (2001) 1}153

Note again that the distortions at each site are not independent, since all oxygens are shared by neighboring MnO octahedra, as easily understood by the explicit expressions of Q i , Q i , and    Q i presented before. A direct and simple way to consider this cooperative e!ect is to determine the  oxygen positions X i , X i , > i , > i , Z i , and Z i , by using, for instance, the Monte Carlo       simulations or numerical relaxation methods (see Press et al., 1986, Chapter 10). To reduce the burden on the numerical calculations, the displacements of oxygen ions are assumed to be along the bond direction between nearest-neighboring manganese ions. In other words, the displacement of the oxygen ion perpendicular to the Mn}Mn bond, i.e., the buckling mode, is usually ignored. As shown later, even in this simpli"ed treatment, several interesting results have been obtained for the spin, charge, and orbital ordering in manganites. Rewriting Eqs. (15), (16), and (21) in terms of the displacement of oxygens from the equilibrium positions, it can be shown that 1 (xi #yi #zi ) , Q i "Q#   (3

(23)

1 Q i "Q# (xi !yi ) ,   (2

(24)

1 (2zi !xi !yi ) , Q i "Q#   (6

(25)

where ai is given by (26) ai "uai!uai a \ a with ui being the displacement of oxygen ion at site i from the equilibrium position along the a-axis. The o!set values for the distortions, Q, Q, and Q, are respectively given by    1 Q" (¸x #¸y #¸z ) , (27)  (3 1 Q" (¸x !¸y ) ,  (2

(28)

1 Q" (2¸z !¸x !¸y ) ,  (6

(29)

where ¸a "¸a !¸, the nondistorted lattice constants are ¸a , and ¸"(¸x #¸y #¸z )/3. In the cooperative treatment, the u's are directly optimized in the numerical calculations (see Allen and Perebeinos, 1999a; Hotta et al. 1999). On the other hand, in the noncooperative calculations, Q's are treated instead of the u's. The similarities and di!erences between those two treatments will be discussed later for some particular cases. 3.3.4. Hopping amplitudes Although the t -electrons are assumed to be localized, the e -electrons can move around the   system via the oxygen 2p orbital.

E. Dagotto et al. / Physics Reports 344 (2001) 1}153

41

This hopping motion of e -electrons is expressed as  a (30) H "!  t dRi di a , AAY AN > AYN   ia AAYN where a is the vector connecting nearest-neighbor sites and ta is the nearest-neighbor hopping AAY amplitude between - and -orbitals along the a-direction. The amplitudes are evaluated from the overlap integral between manganese and oxygen ions by following Slater and Koster (1954). The overlap integral between d   - and p -orbitals is given by V V \W (31) Ex (l, m, n)"((3/2) l(l!m) (pd ) ,  where (pd ) is the overlap integral between the d - and p -orbital and (l, m, n) is the unit vector along the direction from manganese to oxygen ions. The overlap integral between d   - and X \P p -orbitals is expressed as V (32) Ex (l, m, n)"l[n!(l#m)/2] (pd ) . 
Thus, the hopping amplitude between adjacent manganese ions along the x-axis via the oxygen 2p -orbitals is evaluated as V !tx "Ex (1, 0, 0);Ex (!1, 0, 0) . (33) AAY A AY Note here that the minus sign is due to the de"nition of hopping amplitude in H . Then, tx is   AAY explicitly given by (34) tx "!(3t x "!(3t x "3t x "3t /4 ,



 where t is de"ned by t "(pd ). By using the same procedure, the hopping amplitude along the   y- and z-axis are given by ty "(3t y "(3t y "3t y "3t /4





(35)

and tz "t , t z "t z "t z "0 , (36)



respectively. It should be noted that the signs in the hopping amplitudes between di!erent orbitals are di!erent between the x- and y-direction, which will be important when the charge-orbital ordered phase in the doped manganites is considered. Note also that in some cases, it is convenient to de"ne t x as the energy scale t, given as t"3t /4.  3.3.5. Heisenberg term Thus far, the role of the e -electrons has been discussed to characterize the manganites. However,  in the fully hole-doped manganites composed of Mn> ions, for instance CaMnO , it is well known  that a G-type antiferromagnetic phase appears, and this property cannot be understood within the above discussion. The minimal term to reproduce this antiferromagnetic property is the Heisenberg-like coupling between localized t -spins, given in the form  H "J  Si ) Sj , (37) $+ $ i j 6 7

42

E. Dagotto et al. / Physics Reports 344 (2001) 1}153

where J is the AFM coupling between nearest-neighbor t spins. The existence of this term is $  quite natural from the viewpoint of the super-exchange interaction, working between neighboring localized t -electrons. As for the magnitude of J , it is discussed later in the text.  $ 3.3.6. Full Hamiltonian As discussed in the previous subsections, there are "ve important ingredients that regulate the physics of electrons in manganites: (i) H , the kinetic term of the e -electrons. (ii) H , the Hund    & coupling between the e -electron spin and the localized t -spin. (iii) H , the AFM Heisenberg   $+ coupling between nearest-neighbor t -spins. (iv) H } , the coupling between the e -electrons and     the local distortions of the MnO octahedron. (v) H } , the Coulomb interactions among the    e -electrons. By unifying those "ve terms into one, the full Hamiltonian H is de"ned as  H"H #H #H #H } #H } . (38)   & $+     This expression is believed to de"ne an appropriate starting model for manganites, but, unfortunately, it is quite di$cult to solve such a Hamiltonian. In order to investigate further the properties of manganites, some simpli"cations are needed. 3.3.7. Free e -electron model  The simplest model is obtained by retaining only the kinetic term. Although this is certainly an oversimpli"cation for describing the complex nature of manganites, it can be a starting model to study the transport properties of these compounds, particularly in the ferromagnetic region in which the static Jahn}Teller distortion does not occur and the e!ect of the Coulomb interaction is simply renormalized into the quasi-particle formation. In fact, some qualitative features of manganites can be addressed in the band picture, as discussed by Shiba et al. (1997) and Gor'kov and Kresin (1998). The kinetic term is rewritten in momentum space as H "  k dRk dk , (39)  k AAY AN AYN AAYN where dk "(1/N)i eGRi  kdi , k "!(3t /2)(C #C ), k "!(t /2)(C #C #4C ), and AN AN  V W

 V W X k "k "((3t /2)(C !C ), with C "cosk ( "x, y, and z). After the diagonalization of

 V W I I k , two bands are obtained as AAY E! (40) k "!t (C #C #C $(C #C#C!C C !C C !C C ) .  V W X V W X V W W X X V Note that the cubic symmetry can be seen clearly in E! k , although the hopping amplitudes at "rst sight are quite anisotropic, due to the choice of a particular basis for the d-orbitals. Other basis certainly lead to the same result. Note also that the bandwidth = is given by ="6t .  3.3.8. One-orbital model A simple model for manganites to illustrate the CMR e!ect is obtained by neglecting the electron}phonon coupling and the Coulomb interactions. Usually, an extra simpli"cation is carried out by neglecting the orbital degrees of freedom, leading to the FM Kondo model or one-orbital double-exchange model, which will be simply referred as the `one-orbital modela

E. Dagotto et al. / Physics Reports 344 (2001) 1}153

43

hereafter, given as (Zener, 1951b; Furukawa, 1994) H "!t  (aRi aj #h.c.)!J  si ) Sj #J  Si ) Sj , (41) N N & i "# $ i j 6i j7N 6 7 where ai is the annihilation operator for an electron with spin at site i, but without orbital index. N Note that H is quadratic in the electron operators, indicating that it is reduced to a one-electron "# problem on the background of localized t spins. This is a clear advantage for the Monte Carlo  simulations, as discussed later in detail. Neglecting the orbital degrees of freedom is clearly an oversimpli"cation, and important phenomena such as orbital ordering cannot be obtained in this model. However, the one-orbital model is still important, since it already includes part of the essence of manganese oxides. For example, recent computational investigations have clari"ed that the very important phase separation tendencies and metal}insulator competition exist in this model. The result will be discussed in detail in the following subsection. 3.3.9. J "R limit & Another simpli"cation without the loss of essential physics is to take the widely used limit J "R, since in the actual material J /t is much larger than unity. In such a limit, the e -electron & &  spin perfectly aligns along the t -spin direction, reducing the number of degrees of freedom. Then,  in order to diagonalize the Hund term, the `spinlessa e -electron operator, ci , is de"ned as A  ci "cos(i /2)di #sin(i /2)e\ (i di . (42) A At As In terms of the c-variables, the kinetic energy acquires the simpler form (43) H "!  Si i a ta cRi ci a ,  > AAY A > AY   ia AAY where Si j is given by  Si j "cos(i /2) cos(j /2)#sin(i /2) sin(j /2)e\ (i \(j  . (44)  This factor denotes the change of hopping amplitude due to the di!erence in angles between t -spins at sites i and j. Note that the e!ective hopping in this case is a complex number (Berry  phase), contrary to the real number widely used in a large number of previous investigations (for details in the case of the one-orbital model see MuK ller-Hartmann and Dagotto, 1996). The limit of in"nite Hund coupling reduces the number of degrees of freedom substantially since the spin index is no longer needed. In addition, the ;- and J-term in the electron}electron interaction within the e -sector are also no longer needed. In this case, the following simpli"ed  model is obtained: H"!  Si i a ta cRi ci a #J  Si ) Sj #;  ni ni #<  ni nj  > AAY A > AY $ i j
ia i 67 6ij7 AAY # E  [2(q i ni #q i  i #q i  i )# qi #qi #qi ] , (2 i   V  X   

(45)

where ni "cRi ci , ni " ni ,  i "ciR ci #cRi ci , and  i "ciR ci !ciR ci . A A A ? @ @ ? X ? ? @ @ A A V Considering the simpli"ed Hamiltonian H, two other limiting models can be obtained. One is the Jahn}Teller model H , de"ned as H "H (;"<"0), in which the Coulomb interactions (2 (2 are simply ignored. Another is the Coulombic model H, de"ned as H"H (E "0), which ! ! (2

44

E. Dagotto et al. / Physics Reports 344 (2001) 1}153

denotes the two-orbital double exchange model in#uenced by the Coulomb interactions, neglecting the phonons. Of course, the actual situation is characterized by ;O0,
E. Dagotto et al. / Physics Reports 344 (2001) 1}153

45

compare against experiments. Then, the numbers quoted below must be taken simply as rough estimations of orders of magnitude. The reader should consult the cited references to analyze the reliability of the estimations mentioned here. Note also that the references discussed in this subsection correspond to only a small fraction of the vast literature on the subject. Nevertheless, the `samplea cited below is representative of the currently accepted trends in manganites. Regarding the largest energy scales, the on-site ; repulsion was estimated to be 5.2$0.3 and 3.5$0.3 eV, for CaMnO and LaMnO , respectively, by Park et al. (1996) using photoemission   techniques. The charge-transfer energy  was found to be 3.0$0.5 eV for CaMnO in the same  study (note that in the models described in previous sections, the oxygen ions were simply ignored). In other photoemission studies, Dessau and Shen (1999) estimated the exchange energy for #ipping an e -electron to be 2.7 eV.  Okimoto et al. (1995) studying the optical spectra of La Sr MnO with x"0.175 estimated \V V  the value of the Hund coupling to be of the order of 2 eV, much larger than the hopping of the one-orbital model for manganites. Note that in estimations of this variety care must be taken with the actual de"nition of the exchange J , which sometimes is in front of a ferromagnetic Heisenberg & interaction where classical localized spins of module 1 are used, while in other occasions quantum spins of value 3/2 are employed. Nevertheless, the main message of the Okimoto et al. paper is that J is larger than the hopping. A reanalysis of the Okimoto et al. results led Millis et al. (1996) to & conclude that the Hund coupling is actually even larger than previously believed. The optical data of Quijada et al. (1998) and Machida et al. (1998) also suggest that the Hund coupling is larger than 1 eV. Similar conclusions were reached by Satpathy et al. (1996) using constrained LDA calculations. The crystal-"eld splitting between the e - and t -states was estimated to be of the order of 1 eV   by Tokura (1999) (see also Yoshida, 1998). Based on the discussion in the previous subsection, it is clear that manganites are in high-spin ionic states due to their large Hund coupling. Regarding the hopping `ta, Dessau and Shen (1999) reported a value of order 1 eV, which is somewhat larger than other estimations. In fact, the results of Bocquet et al. (1992), Arima et al. (1993), and Saitoh et al. (1995) locate its magnitude between 0.2 and 0.5 eV, which is reasonable in transition metal oxides. However, note that fair comparisons between theory and experiment require calculations of, e.g., quasiparticle band dispersions, which are di$cult at present. Nevertheless, it is widely accepted that the hopping is just a fraction of eV. Dessau and Shen (1999) estimated the static Jahn}Teller energy E as 0.25 eV. From the static (2 Jahn}Teller energy and the hopping amplitude, it is convenient to de"ne the dimensionless electron-phonon coupling constant  as (48) "(2E /t"g/(k t . (2 (2 By using E "0.25 eV and t"0.2}0.5 eV,  is estimated as between 1}1.6. Actually, Millis et al. (2 (1996) concluded that  is between 1.3 and 1.5. It can be shown that the recent studies of Allen and Perebeinos (1999b) and Perebeinos and Allen (2000) lead to "1.6, in agreement with other estimations. As for the parameter , it is given by "k /k "( / ), where  and  are the
 (2
 (2
 (2 vibration energies for manganite breathing- and JT modes, respectively, assuming that the reduced masses for those modes are equal. From experimental results and band-calculation data (see Iliev et al., 1998),  and  are estimated as &700 and 500}600 cm\, respectively, leading to +2.
 (2 However, in practice it has been observed that the main conclusions are basically unchanged as

46

E. Dagotto et al. / Physics Reports 344 (2001) 1}153

long as  is larger than unity. Thus, if an explicit value for  is not provided, the reader can consider that  is simply taken to be R to suppress the breathing mode distortion. The value of J is the smallest of the set of couplings discussed here. In units of the hopping, it is $ believed to be of the order of 0.1t (see Perring et al., 1997), namely about 200 K. Note, however, that it would be a bad approximation to simply neglect this parameter since in the limit of vanishing density of e -electrons, J is crucial to induce antiferromagnetism, as it occurs in CaMnO for  $  instance. Its relevance, at hole densities close to 0.5 or larger, to the formation of antiferromagnetic charge-ordered states is remarked elsewhere in this review. Also in mean-"eld approximations by Maezono et al. (1998a) the importance of J has been mentioned, even though in their work this $ coupling was estimated to be only 0.01t. Summarizing, it appears well established that: (i) the largest energy scales in the Mn-oxide models studied here are the Coulomb repulsions between electrons in the same ion, which is quite reasonable. (ii) The Hund coupling is between 1 and 2 eV, larger than the typical hopping amplitudes, and su$ciently large to form high-spin Mn> and Mn> ionic states. As discussed elsewhere in the review, a large J leads naturally to a vanishing probability of e -electron double &  occupancy of a given orbital, thus mimicking the e!ect of a strong on-site Coulomb repulsion. (iii) The dimensionless electron}phonon coupling constant  is of the order of unity, showing that the electron lattice interaction is substantial and cannot be neglected. (iv) The electron hopping energy is a fraction of eV. (v) The AF-coupling among the localized spins is about a tenth of the hopping. However, as remarked elsewhere, this apparent small coupling can be quite important in the competition between FM- and AF-state. 3.3.12. Monte Carlo simulations In this subsection the details related to the Monte Carlo calculations are provided. For simplicity, let us focus here on one-dimensional systems as an example. Generalizations to higher dimensions are straightforward. Also, as a simple example, the case of the one-orbital model will be used, with the two-orbital case left as exercise to the readers. Note that the one-orbital model H is simply denoted by HK in this subsection. Note also that  indicates the inverse temperature, "# i.e., "1/¹, in this subsection. As explained before, the Hamiltonian for the one-orbital model is quadratic in the a, aR operators and thus, it corresponds to a `one-electrona problem, with a density regulated by a chemical potential . For the case of a chain with ¸ sites, the base can be considered as aR 0,2, aR 0, aR 0,2, aR 0, and thus HK is given by a 2¸;2¸ matrix for a "xed t *t s *s con"guration of the classical spins. The partition function in the grand canonical ensemble can be written as




 



* (49) d sin  d Z ( ,  ) . Z" G G G E G G   G Here g denotes conduction electrons and Z ( ,  )"Tr (e\@)K ), where KK "HK ! NK with NK the E G G E number operator and the trace is taken for the mobile electrons in the e -orbital, which are created  and destroyed by the fermionic operators aR and a. It will be shown that Z can be calculated in E terms of the eigenvalues of KK denoted by  ("1,2, 2¸). The diagonalization is performed H numerically using library routines.

E. Dagotto et al. / Physics Reports 344 (2001) 1}153

47

Since KK is an hermitian operator, it can be represented in terms of a hermitian matrix which can be diagonalized by an unitary matrix ; such that




;RK;"



0

2

0

2 \

0



  

 0





.

(50)

2 * The base in which the matrix K is diagonal is given by the eigenvectors uR 0,2, uR 0. De"ning  * uR u "n( and denoting by n the eigenvalues of n( , the trace can be written as K K K K K 0

0

K

*

K

Tr (e\@))"  n 2n e\@)n 2n "   *  * E L 2L* L 2L*

\@H CH LH n 2n  , n 2n e  *  * (51)

since in the uR 0 basis, the operator KK can be replaced by its eigenvalues. The exponential is now K a `ca number and it is equivalent to a product of exponentials given by Z " n e\@C L n 2  n e\@C* L* n  ,   * * E L* L which can be written compactly as

(52)

* Z "  Tr (e\@CH LH ) . (53) E H H Since the particles are fermions, the occupation numbers are either 0 or 1, and the sum in Eq. (52) is restricted to those values, *  * Z "   e\@CH L"  (1#e\@CH ) . E H L H Thus, combining Eqs. (49) and (54), Z is obtained as




(54)

 

*

* Z" d sin  d  (1#e\@CH ) . (55) G G G   G H Note here that the integrand is clearly positive, and thus, `sign problemsa are not present. The integral over the angular variables can be performed using a classical Monte Carlo simulation. The eigenvalues must be obtained for each classical spin con"guration using library subroutines. Finding the eigenvalues is the most time consuming part of the numerical simulation. Calculation of static observables: The equal time or static observables OK (a , aR) are given by G G L 
*

1 * 1 OK "  (56) d sin  d Tr (OK e\@)K )"  d sin  d Z OI  , E G G G G G G E Z Z     G G where OI "Tr (OK e\@)K )/Z . In practice only the Green function has to be calculated, and more E E complicated operators are evaluated using Wick's theorem. The Green function for a given con"guration of classical spins are given by G "a aR . GHNNY GN HNY




 




 

48

E. Dagotto et al. / Physics Reports 344 (2001) 1}153

Let us consider the case in which OK "a aR , relevant for the Green function. In this case, GN HNY K G "Tr (a aR e\@))/Z . (57) GHNNY E GN HNY E Changing to the base in which KK is diagonal through the transformation aR "* uR ;R N , where GN I I IG i "(i, ), it can be shown that N * * * Tr (a aR e\@)K )"   ; N ;R NY Tr u uR  e\@CJ LJ G H EH E H E E GN HNY H E J * * * "   ; N ;R NY Tr  1#(e\@CJ !1)n u uR G H EH E J H E J H E * * "  ; N ;R NY  1#(e\@CJ !1)n (1!n ) G H HH J H J H * *  "  ; N ;R NY   1#(e\@CJ !1)n  G H HH J H JJ$H LJ  * * "  ; N ;R NY  (1#e\@CJ ) . (58) G H HH JJ$H H Thus, the Green function is given by

 










* * * G "  ; N ;R NY  (1#e\@CJ )/  (1#e\@CJ ) GHNNY G H HH H JJ$H J * 1 "  ;N ;R . (59) G H 1#e\@CH HHNY H Let us now consider some examples. The e -electron number is given by  n( " aR a "2¸! G . (60) GN GN GGNN GN GN More complicated operators can be written in terms of Green functions using Wick's theorem (Mahan, 1981, p. 95) which states that a  aR a  aR "a  aR a  aR !a  aR a  aR  . (61) H N H N H N H N H N H N H N H N H N H N H N H N For example, if OK "aR a  aR a  , a combination that appears in the calculation of spin and H N H N H N H N charge correlations, and using Wick's theorem in combination with the fact that aR a "  !a aR "  !G , it can be shown that GN HNY GH NNY HNY GN GH NNY HGNYN OK "aR a  aR a  !aR a  aR a   H N H N H N H N H N H N H N H N "(   !G   ) (   !G   )!(   !G   ) (   !G   ) . H H H H H H H H H H NN H H NN H H NN H H NN (62) Calculation of time-dependent observables: Time-dependent observables are evaluated through the time-dependent Green function which can be readily calculated numerically. The Green

E. Dagotto et al. / Physics Reports 344 (2001) 1}153

49

function is de"ned as G (t)"a (t)aR (0) , GHNN GN HN where

(63)

(64) a (t)"e &K Ra e\ &K R . GN GN Note that HK and KK can be diagonalized by the same basis of eigenvectors uR 0, and the K eigenvalues of HK are denoted by  . Working in this basis it is possible to write a (t) in terms of H GN a as GN   a (t)"e R J MJ LJ  ; N u e\ R J MJ LJ GN G E E E * * "   ; N e\ RMH ;R a , HJ J G H J H where a "a if 4¸ and a "a if '¸. J Jt J J\*s Substituting Eq. (65) in Eq. (63), the time-dependent Green function given by







(65)



* * G (t)"   ; N e\ RMH ;R a aR  . (66) GHNN G H HJ J HN J H In Eqs. (57)}(59), it has been shown that a aR "* ; [1/(1#e\@CH )];R N , where HH J HN H JH  " ! . Thus, replacing Eq. (59) in Eq. (66), H H * e\ RMH ;R . (67) G (t)"  ; N GHNN G H 1#e\@MH \I HHN H Now, as an example, let us calculate the spectral function A(k,), given by 1 A(k, )"! Im G (k, ) , 


(68)

where the retarded Green function G (k,) is given by (see Mahan, 1981, p. 135)   dt e SRG (k, t) G (k, )"   \ and

(69)

G (k, t)"!i(t)  [a (t)aR (0)#aR (0)a (t)]  IN IN IN IN N "!i(t)  (G #G ) . IN IN N Note here that G "a (t)aR (0) and G "aR (0)a (t) are implicitly de"ned. IN IN IN IN IN IN

(70)



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E. Dagotto et al. / Physics Reports 344 (2001) 1}153

Since the measurements are performed in coordinate space, G and G must be expressed in IN IN terms of real space operators using a "(1/(¸) e\ IHa . Then H HN IN 1 G "  e\ IH\Ja (t)aR (0) IN ¸ HN JN HJ 1 "  e\ IH\JG (71) HJNN ¸ HJ and analogously an expression for G can be obtained. Thus, Eq. (70) becomes IN 1 (72) G (k, t)"!i(t)  [e IJ\HG #e\ IJ\HG ] . HJNN HJNN  ¸ HJN Substituting Eq. (72) in Eq. (69) it can be shown that



!i  dt e SR  [e IJ\HG #e\ IJ\HG ] . G (k, )" HJNN HJNN  ¸  HJN The next step is to evaluate the integral. Using Eq. (67), the "rst term in Eq. (73) becomes

(73)



* ; N ;R N  !i H H HJ  e IJ\H  (74) dt e S\MH R . 1#e\@MH \I ¸  HJN H Note that the integral is equal to
(! ). A similar expression is obtained for the second term H and "nally the spectral function can be expressed in terms of the eigenvectors and eigenvalues of HK as

   





; N ;R N ; ;R 1 H H HJ #e\ IJ\H JN H HHN (! ) A(k, )" Im i  e IJ\H H H 1#e\@M \I 1#e@MH \I ¸ HJNH 1 1 1 " Im i  e IH\J; N ;R N (! ) # J H HH H 1#e\@MH \I 1#e@MH \I ¸ HJNH Noticing that the sum on the "nal line is equal to 1, the "nal expression is



1 A(k, )" Re ¸






.

(75)



 e IH\J; N ;R N (! ) . (76) J H HH H HJNH Similar algebraic manipulations allow to express other dynamical observables, such as the optical conductivity and dynamical spin correlation functions, in terms of the eigenvalues and eigenvectors of the Hamiltonian matrix. 3.3.13. Mean-xeld approximation for H Even a simpli"ed model such as H is still di$cult to be solved exactly, except for some special cases. Thus, in this subsection, the mean-"eld approximation (MFA) is developed for H to attempt to grasp its essential physics. Note that even at the mean-"eld level, due care should be paid to the self-consistent treatment to lift the double degeneracy of the e -electrons. The present 

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analytic MFA will be developed based on the following assumptions: (i) The background t -spin  structure is "xed through the calculation by assuming that the nearest-neighbor t -spins (not to be  confused with the full state) can only be in the FM or AF con"guration. (ii) The JT- and breathing-mode distortions are noncooperative. These assumptions will be discussed later in this subsection. First, let us rewrite the electron}phonon term by applying a standard mean-"eld decoupling procedure. In this approximation, a given operator O is written as O"O#O, where O"O!O. In a product of operators O O , terms of order O O are simply discarded.     Applying this trick to our case, it is shown that q i ni +q i ni #q i ni !q i ni  ,     q i  i +q i  i #q i  i !q i  i  ,  V  V  V  V

(77)

q i  i +q i  i #q i  i !q i  i  ,  X  X  X  X qi +2q i q i !q i  ? ? ? ?

("1,2,3) ,

where the bracket denotes the average value using the mean-"eld Hamiltonian described below. By minimizing the phonon energy, the local distortion is determined in the MFA as q i "!ni /, q i "! i , q i "! i  .  V  X 

(78)

Thus, after straightforward algebra, the electron}phonon term in the MFA is given by H+$} "!2 [E ni ni #E ( i  i # i  i )]  
 (2 V V X X i # [E ni #E ( i # i )] ,
 (2 V X i

(79)

where E "E /, as already explained.
 (2 Now let us turn our attention to the electron}electron interaction term. At a "rst glance, it appears enough to make a similar decoupling procedure for H } . However, such a decoupling   cannot be uniquely carried out, since it will be shown below that H } is invariant with respect to   the choice of e -electron orbitals due to the local SU(2) symmetry in the orbital space. Thus, it is  necessary to "nd the optimal orbital set by determining the relevant e -electron orbital self consistently at each site. For this purpose, it is convenient to use the expression Eq. (18) for q i and  q i . Note in the MFA that the amplitude qi and the phase i are, respectively, determined as  qi "( i # i , V X

i "#tan\( i / i ) , V X

(80)

where `a is added to i in the MFA. Originally, i is de"ned as i "tan\(q i /q i ), but in the   MFA, the distortions are given by Eq. (78), in which minus signs appear in front of  i  and  i . V X Thus, due to these minus signs, the additional phase  in i should appear in order to maintain consistency with the previous de"nition, if i is obtained with the use of  i  and  i  in the V X MFA. By using the phase i determined by this procedure, it is convenient to transform ci and

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ci into the `phase-dresseda operators, c i and c i , as

c i cos( i /2) sin( i /2) ci "e Ki  , (81) !sin( i /2) cos( i /2) ci c i

where the 2;2 matrix is SU(2) symmetric. Note that if i is increased by 2
, the SU(2) matrix itself changes its sign. To keep the transformation unchanged upon a 2
-rotation in i , a phase factor e Ki  is needed. In the expression for the ground state of the single JT molecule, namely the single-site problem discussed before, this phase factor has not been added, since the electron does not hop around from site to site and the phases do not correlate with each other. Namely, it was enough to pay attention to the double valuedness of the wave function at a single site. However, in the JT crystal in which e -electrons move in the periodic array of the JT centers, the addition of this  phase factor is useful to take into account the e!ect of the Berry phase arising from the circular motion of e -electrons around the JT center, as has been emphasized in Koizumi et al. (1998a, b). It  could be possible to carry out the calculation without including explicitly this phase factor, but in that case, it is necessary to pay due attention to the inclusion of the e!ect of the Berry phase. The qualitative importance of this e!ect will be explained later in the context of the `band-insulating picturea for the CE-type phase of half-doped manganites. Note also that the phase i determines the electron orbital set at each site. In the previous section, the single-site problem was discussed and the ground state at site i was found to be










(82) &&b''"[!sin( i /2)dRi #cos( i /2)diR ]0 , N
N which is referred to as the `ba-orbital, namely the combination with the lowest energy at a given site. The excited-state or `aa-orbital is simply obtained by requesting it to be orthogonal to `ba as (83) &&a''"[cos( i /2)diR #sin( i /2)diR ]0 . N
N For instance, at i "2
/3, `aa and `ba denote the d   - and d   -orbitals, respectively. In W \X V \P Table 2, the correspondence between i and the local orbital is summarized for several important values of i . In order to arrive to the results of the table, remember that the original orbitals must be normalized such that (x!y)/(2 and (3z!r)/(6 are used. Note also that overall phase factors that may a!ect the orbitals are not included in Table 2. Furthermore, it should be noted that d   and d   never appear as the local orbital set. Sometimes those were treated as an W \P V \P Table 2 Phase i and the corresponding e -electron orbitals. Note that `ba corresponds to the lowest-energy orbital for E O0  (2

i

`aa-orbital

`ba-orbital

0
/3 2
/3
4
/3 5
/3

x!y 3y!r y!z 3z!r z!x 3x!r

3z!r z!x 3x!r x!y 3y!r y!z

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orthogonal orbital set to reproduce the experimental results, but such a treatment is an approximation, since the orbital ordering is not due to the simple alternation of two arbitrary kinds of orbitals. Using the above-described transformations, H+$} and H } can be rewritten after some     algebra as H+$} " E (!2ni n i #ni )#E [2qi (n i !n i )#qi ]  
 (2
i

(84)

and (85) H } "; n i n i #<  n i n j ,
  i ij 67 where n i "c Ri c i and n i "n i #n i . Note that H } is invariant with respect to the choice of i . A A A
  Eq. (85) can be obtained by calculating c Ri c i #c iR c i using Eq. (81). This immediately leads to

n i "ni . Then, from n i "ni and recalling that ni "ni and n i "n i for "a and b, it can be A A A A shown that n i n i "ni ni . Now let us apply the decoupling procedure as

(86) n i n i +n i n i #n i n i !n i n i 



and use the relations n i "(ni !qi )/2, n i "(ni #qi )/2, which arise from
n i !n i "!qi and n i #n i "ni . The former relation indicates that the modulation in

the orbital density is caused by the JT distortion, while the latter denotes the local charge conservation irrespective of the choice of electron basis. Then, the electron}electron interaction term is given in the MFA as H+$} "(;/4)  [2ni n i!ni #2qi (n i!n i )#qi ]#< [ni a n i !(1/2)ni a ni ] , > > ? @   i ia (87) where the vector a has the same meaning as in the hopping term H . For instance, in two   dimensions, it denotes a"($1, 0) and (0,$1), where the lattice constant is taken as unity for simplicity. It should be noted that the type of orbital ordering would be automatically "xed as either x!y or 3z!r, if the original operators c would be simply used for the Hartree}Fock approximation. However, as it was emphasized above, the H } term has a rotational invariance in   orbital space, and there is no reason to "x the orbital only as x!y or 3z!r. In order to discuss properly the orbital ordering, the local e -electron basis, i.e., the phase i should be  determined self-consistently. By combining H+$} with H+$} and transforming c i and c i into the original operators ci and  
  ci , the mean-"eld Hamiltonian is "nally obtained as
H "!  ta cRi ci a #J  Si ) Sj#EI  [!2( i  i# i  i )# i # i ] +$ (2 i V V X X V X AAY A > AY $ i j ia AAY 67 (88) # [(;I /2)ni #<  ni a ](ni !ni /2) , > i a where the renormalized JT energy is given by EI "E #;/4 (2 (2

(89)

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and the renormalized inter-orbital Coulomb interaction is expressed as ;I ";!4E . (90)
 Physically, the former relation indicates that the JT energy is e!ectively enhanced by ;. Namely, the strong on-site Coulombic correlation plays the same role as that of the JT phonon, at least at the mean-"eld level, indicating that it is not necessary to include ; explicitly in the models, as has been emphasized by the present authors in several publications (see for instance Hotta et al., 2000). The latter equation for ;I  means that the one-site inter-orbital Coulomb interaction is e!ectively reduced by the breathing-mode phonon, since the optical-mode phonon provides an e!ective attraction between electrons. The expected positive value of ;I  indicates that e -electrons dislike  double occupancy at the site, since the energy loss is proportional to the average local electron number in the mean-"eld argument. Thus, to exploit the gain due to the static JT energy and avoid the loss due to the on-site repulsion, an e -electron will singly occupy a given site.  Now let us brie#y discuss how to solve the present mean-"eld Hamiltonian on the background of the "xed t -spin arrangement. For some "xed spin pattern, by using appropriate initial values for  the local densities ni ,  i , and  i , the mean-"eld Hamiltonian H is constructed, V X +$ where the superscript number (j) indicates the iteration step. By diagonalizing H on "nite +$ clusters, and usually using a variety of boundary conditions depending on the problem, the improved local densities, ni ,  i , and  i , are obtained. This procedure is simply V X repeated such that in the jth iteration step, the local densities ni H>,  i H>, and  i H> are V X obtained by using the Hamiltonian HH. The iterations can be terminated if +$ ni H>!ni H(,  i H! i H\(, and  i H! i H\( are satis"ed, where V V X X  is taken to be a small number to control the convergence. As for the choice of the cluster, in order to obtain the charge and orbital ordering pattern in the insulating phase, it is usually enough to treat a "nite-size cluster with periodic boundary conditions. Note that the cluster size should be large enough to reproduce the periodicity in the spin, charge, and orbital ordering under investigation. However, to consider the transition to the metallic state from the insulating phase, in principle it is necessary to treat an in"nite cluster. Of course, except for very special cases, it is impossible to treat the in"nite-size cluster exactly, but fortunately, in the present MFA, it is quite e!ective to employ the twisted-boundary condition by introducing the momentum k in the Bloch phase factor e k  N at the boundary, where N"(Nx , Ny , Nz ), and Na is the size of the cluster along the a-direction. Note that if the spin directions are changed periodically, an additional phase factor appears to develop at the boundary, but this is not the case. In the present MFA, the t -spin pattern is "xed from the outset, and the  periodicity due to the spin pattern is already taken into account in the cluster. Finally, here comments on some of the assumptions employed in the present MFA are provided. In the "rst approximation for t -spins, their pattern is "xed throughout the mean-"eld calculation  and the nearest-neighbor spin con"guration is assumed to be only FM or AFM. Note that this assumption does not indicate only the fully FM phase or three-dimensional G-type AFM spin pattern, but it can include more complicated spin patterns such as the CE-type AFM phase. However, under this assumption, several possible phases such as the spin canted phase and the spin #ux phase, in which neighboring spins are neither FM nor AFM, are neglected from the outset. Unfortunately, this assumption for the "xed t spin pattern cannot be justi"ed without extra  tests. Thus, it is unavoidable to con"rm the assumption using other methods. In order to perform

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this check, unbiased numerical calculations such as the Monte Carlo simulations and relaxation techniques are employed to determine the local distortions, as well as the local spin directions. Especially for a "xed electron number, the optimization technique is found to work quite well in this type of problems. Then, our strategy to complete the mean-"eld calculations is as follows: (i) For some electron density and small-size cluster, the mean-"eld calculations are carried out for several "xed con"gurations of t spins. (ii) For the same electron density in the same size of cluster as in (i), both local  distortions and t -spin directions are optimized by using an appropriate computer code. (iii)  Results obtained in (i) are compared to those in (ii). If there occurs a serious disagreement between them, go back to step (i) and/or (ii) to do again the calculations by changing the initial inputs. In this retrial, by comparing the energies between cases (i) and (ii), the initial condition for the case with higher energy should be replaced with that for the lower energy. To save CPU time it is quite e!ective to combine analytic MFA and numerical techniques. (iv) After several iterations, if a satisfactory agreement between (i) and (ii) is obtained, the MFA on a larger-size cluster is used to improve the results in (i). Namely, by combining the MFA and the optimization technique, it is possible to reach physically important results in a rapid and reliable way. As for the assumption made regarding the use of noncooperative phonons in the MFA, it is also checked by comparing the noncooperative mean-"eld results with the optimized ones for cooperative distortions. Note here that, due to the CPU and memory restrictions, the optimization technique cannot treat large-size clusters. However, this numerical technique has the clear advantage that it is easily extended to include the cooperative e!ect by simply changing the coordinates from Q to u, where u symbolically indicates the oxygen displacements, while Q denotes the local distortions of the MnO octahedron. The e!ect of the cooperative phonons will be discussed  separately for several values of the hole density. 3.4. Main results: one orbital model 3.4.1. Phase diagram with classical localized spins Although the one-orbital model for manganites is clearly incomplete to describe these compounds since, by de"nition, it has only one active orbital, nevertheless, it has been shown in recent calculations that it captures part of the interesting competition between ferromagnetic and antiferromagnetic phases in these compounds. For this reason, and since this model is far simpler than the more realistic two-orbital model, it is useful to study it in detail. A fairly detailed analysis of the phase diagram of the one-orbital model has been recently presented, mainly using computational techniques. Typical results are shown in Fig. 3.4.1a}c for D"1,2, and R (D is spatial dimension), the "rst two obtained with Monte Carlo techniques at low temperature, and the third with the dynamical mean-"eld approximation in the large J limit & varying temperature. There are several important features in the results which are common in all dimensions. At e -density n"1.0, the system is antiferromagnetic (although this is not clearly  shown in Fig. 3.4.1). The reason is that at large Hund coupling, double occupancy in the ground state is negligible at e -density n"1.0 or lower, and at these densities it is energetically better to  have nearest-neighbor spins antiparallel, gaining an energy of order t/J , rather than to align & them, since in such a case the system is basically frozen due to the Pauli principle. On the other hand, at "nite hole density, antiferromagnetism is replaced by the tendency of holes to polarize the

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Fig. 3.4.1. Phase diagram of the one-orbital model with classical spins (and without J coupling). (a) Results obtained $ with Monte Carlo methods at low temperature in 1D (Yunoki et al., 1998a; Dagotto et al., 1998). FM, PS, and IC, denote ferromagnetic, phase-separated, and spin incommensurate phases, respectively. Although not shown explicitly, the n"1.0 axis is antiferromagnetic. The dashed lines correspond to results obtained using quantum localized spins. For more details see Yunoki et al. (1998a) and Dagotto et al. (1998). (b) Similar to (a) but in 2D. The gray region denotes the possible location of the PS-IC transition at low Hund coupling, which is di$cult to determine. Details can be found in Yunoki et al. (1998a). (c) Results obtained in the in"nite dimension limit and at large Hund coupling varying the temperature (here in units of the half-width = of the density of states). Two regions with PS were identi"ed, as well as a paramagnetic PM regime. For details see Yunoki et al. (1998a).

spin background to improve their kinetic energy, as discussed in Section 3.1. Then, a very prominent ferromagnetic phase develops in the model as shown in Fig. 3.4.1. This FM tendency appears in all dimensions of interest, and it manifests itself in the Monte Carlo simulations through the rapid growth with decreasing temperature, and/or increasing number of sites, of the zeromomentum spin}spin correlation, as shown in Fig. 3.4.2a and b reproduced from Yunoki et al. (1998a). In real space, the results correspond to spin correlations between two sites at a distance d which do not decay to a vanishing number as d grows, if there is long-range order (see results in Dagotto et al., 1998). In 1D, quantum #uctuations are expected to be so strong that long-range order cannot be achieved, but in this case the spin correlations still can decay slowly with distance following a power law. In practice, the tendency toward FM or AF is so strong even in 1D that issues of long-range order vs power-law decays are not of much importance for studying the

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Fig. 3.4.2. Spin}spin correlations of the classical spins at zero momentum S(q"0) vs. temperature ¹ (units of t) obtained with the Monte Carlo technique, taken from Yunoki et al. (1998a). Density, Hund coupling, and lattice sizes are shown. (a) and (b) correspond to one and two dimensions, respectively. Closed shells and open boundary conditions were used in (a) and (b), respectively. For details see Dagotto et al. (1998).

Fig. 3.4.3. Rough estimation of the Curie temperature ¹ in 3D and in the limit J "R, as reported by Yunoki et al. ! & (1998a). Other calculations discussed in the text produce results in reasonable agreement with these Monte Carlo simulations (see Motome and Furukawa, 1999).

dominant tendencies in the model. Nevertheless, care must be taken with these subtleties if very accurate studies are attempted in 1D. In 3D, long-range order can be obtained at "nite temperature and indeed it occurs in the one-orbital model. A rough estimation of the critical Curie temperature ¹ is shown in Fig. 3.4.3 ! based on small 6 3D clusters (from Yunoki et al., 1998a). ¹ is of the order of just 0.1t, while other !

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estimations predicted a much higher value (Millis et al., 1995). More recent work has re"ned ¹ , ! but the order of magnitude found in the "rst Monte Carlo simulations remains the same (see Calderon and Brey, 1998; Yi et al., 1999b; Motome and Furukawa, 1999; Held and Vollhardt, 1999). If t is about 0.2 eV, the ¹ becomes of the order of 200 K, a value in reasonable agreement ! with experiments. However, remember that this model cannot describe orbital order properly, and thus it remains a crude approximation to manganites. The most novel result emerging from the computational studies of the one-orbital model is the way in which the FM phase is reached by hole doping of the AF phase at n"1.0. As explained before, mean-"eld approximations by de Gennes (1960) suggested that this interpolation should proceed through a so-called `canteda state in which the spin structure remains antiferromagnetic in two directions but develops a uniform moment along the third direction. For many years this canted state was assumed to be correct, and many experiments were analyzed based on such state. However, the computational studies showed that instead of a canted state, an electronic `phaseseparateda (PS) regime interpolates between the FM- and AF-phase. This PS region is very prominent in the phase diagram of Fig. 3.4.1a}c in all dimensions. As an example of how PS is obtained from the computational work, consider Fig. 3.4.4. In the Monte Carlo simulations carried out in this context, performed in the grand-canonical ensemble, the density of mobile e -electrons n is an output of the calculation, the input being the chemical  potential . In Fig. 3.4.4a, the density n vs. is shown for one-dimensional clusters of di!erent sizes at low temperature and large Hund coupling, in part (b) results in two dimensions are presented, and in part (c) the limit D"R is considered. In all cases, a clear discontinuity in the density appears at a particular value of , as in a "rst-order phase transition. This means that there is a "nite range of densities which are simply unreachable, i.e., that they cannot be stabilized regardless of how carefully is tuned. If the chemical potential is "xed to the value where the discontinuity occurs, frequent tunneling events among the two limiting densities are observed (Dagotto et al., 1998). In the inset of Fig. 3.4.4a, the spin correlations are shown for the two densities at the extremes of the discontinuity, and they correspond to FM- and AF-state. Strictly speaking, the presence of PS means that the model has a range of densities which cannot be accessed, and thus, those densities are simply unstable. This is clari"ed better using now the canonical ensemble, where the number of particles is "xed as an input and is an output. In this context, suppose that one attempts to stabilize a density such as n"0.95 (unstable in Fig. 3.4.4), by locating, say, 95 electrons into a 10;10 lattice. The ground state of such a system will not develop a uniform density, but instead two regions separated in space will be formed: a large one with approximately 67 sites and 67 electrons (density 1.0) and a smaller one with 33 sites and 28 electrons (density &0.85). The last density is the lower value in the discontinuity of Fig. 3.4.4b in 2D, i.e., the "rst stable density after n"1.0 when holes are introduced. Then, whether using canonical or grand-canonical approximations, a range of densities remains unstable. The actual spatial separation into two macroscopic regions (FM and AF in this case) leads to an energy problem. In the simulations and other mean-"eld approximations that produce PS, the `taila of the Coulomb interaction was not explicitly included. In other words, the electric charge was not properly accounted for. Once this long-range Coulomb interaction is introduced into the problem, the fact that the FM- and AF-state involved in PS have di!erent densities leads to a huge energy penalization even considering a large dielectric constant due to polarization (charge certainly cannot be accumulated in a macroscopic portion of a sample). For this reason, it is more

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Fig. 3.4.4. Density of e electrons vs. chemical potential . The coupling is J "8t in (a) and (b) and 4= in (c) (= is the  & half-width of the density of states). Temperatures and lattice sizes are indicated. (a) Results in 1D with PBC. The inset contains the spin correlations at the electronic densities 1.00 and 0.72, that approximately limit the density discontinuity. (b) Same as (a) but in 2D. (c) Same as (a) but in D"R. Results reproduced from Yunoki et al. (1998a).

reasonable to expect that the PS domains will break into smaller pieces, as sketched in Fig. 3.4.5 (Moreo et al., 1999a; see also Section 3.9 and Lorenzana et al., 2000). The shape of these pieces remains to be investigated in detail since the calculations are di$cult with long-range interactions (for results in 1D see below), but droplets or stripes appear as a serious possibility. This state would now be stable, since it would satisfy in part the tendency toward phase separation and also it will avoid a macroscopic charge accumulation. Although detailed calculations are not available, the common folklore is that the typical size of the clusters in the mixed-phase state arising from the competition PS vs. 1/r Coulomb will be in the nanometer scale, i.e., just a few lattice spacings since the Mn}Mn distance is about 4 As . This is the electronic `phase separateda state that one usually has in mind as interpolating between FM and AF. Small clusters of FM are expected to be created in the AF background, and as the hole density grows, these clusters will increase in number and eventually overcome the AF clusters. For more details see also Section 3.9, where the e!ort of other authors in the context of PS is also described.

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Fig. 3.4.5. Schematic representation of a macroscopic phase-separated state (a), as well as possible charge inhomogeneous states stabilized by the long-range Coulomb interaction (spherical droplets in (b), stripes in (c)). Reproduced from Moreo et al. (1999). Similar conclusions have been reached before in the context of phase separation applied to models of high-temperature superconductors.

3.4.2. Spin incommensurability and stripes In the regime of intermediate or small J , the one-orbital model does not have ferromagnetism & at small hole densities, which is reasonable since a large J was needed in the discussion of & Section 3.1 to understand the stabilization of a spin polarized phase. Instead, in this regime of J the spin sector develops incommensurability (IC), namely the peak in the Fourier transform of & the real space spin}spin correlations is neither at 0 (FM) nor at
(AF), but at intermediate momenta. This feature is robust and it appears both in 1D and 2D simulations, as well as with both classical and quantum spins (Yunoki et al., 1998a; Dagotto et al., 1998). An example in 2D is presented in Fig. 3.4.6a. Since a regime with IC characteristics had not been found in experiments by the time the initial Monte Carlo simulations were carried out, the spin IC regime was not given much importance, and its origin remained unclear. However, recent neutron scattering results (Adams et al., 2000; Dai et al., 2000; Kubota et al., 2000) suggest that stripes may appear in some compounds, similar to that found in the cuprates. This result induced us to further examine the numerical data obtained in the original Monte Carlo simulations. It turns out that the spin IC structure found in the 2D one-orbital model has its origin in stripes, as shown in Fig. 3.4.6b. These structures correspond to 1D-like regions of the 2D plane that are populated by holes, leaving undoped the area between the stripes, similar to those structures that are believed to occur in some high-temperature superconductors and (t}J)-like models (Tranquada, 1995; Dai et al., 1998; Mook, 1998; Bourges et al., 2000. See also Emery et al., 1997; Zaanen, 1998; White and Scalapino, 1998; Martins et al., 2000). In fact, the results shown in Fig. 3.4.6b are very similar to those found recently by Buhler et al. (2000) in the context of the so-called spin-fermion model for cuprates, with classical spins used for the spins (the spin-fermion model for cuprates and the one-orbital model for manganites only di!er in the sign of the Hund coupling). Stripe formation with hole density close to n"1.0, i.e., electronic density close to 0.0, is natural near phase separation regimes. Stripes have also been identi"ed in the more realistic case of the two-orbital model (see Section 3.5 below). A discussion of the similarities and di!erences between the electronic phase separation scenarios for manganites and cuprates, plus a substantial body of references, can be found in Hotta et al. (2000).

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Fig. 3.4.6. (a) Spin}spin correlation S(q) vs. momentum, for 2D clusters. Couplings, temperature, and densities are indicated. The cluster is 6;6. Reproduced from Dagotto et al. (1998). (b) Snapshot obtained with Monte Carlo techniques applied to the one-orbital model using an 8;8 cluster, J "2.0 and n"0.75, illustrating the existence of & stripes. The area of the circles are proportional to the electronic density at each site. The arrows are proportional to the value of the z-component of the spin. Result courtesy of C. Buhler, using a program prepared by S. Yunoki (unpublished).

3.4.3. Inyuence of J $ The one-orbital model described in Section 3.3 included an antiferromagnetic coupling among the localized spins that is regulated by a parameter J , which was not considered in the previous $ subsections. In principle, this number is the smallest of the couplings in the model according to the estimations discussed in Section 3.3, and one may naively believe that its presence is not important. However, this is incorrect as can be easily understood in the limit of n"0.0 (x"1.0), which is realized in materials such as CaMnO . This compound is antiferromagnetic and it is widely  believed that such magnetic order is precisely caused by the coupling among the localized spins. Then, J cannot be simply neglected. In addition, the studies shown below highlight the $

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(unexpected) importance of this coupling in other contexts: it has been found to be crucial for the stabilization of an A-type AF phase at n"1.0 in the two-orbital model, and also to make stable the famous CE-phase at n"0.5, at least within the context of a two-orbital model with strong electron Jahn}Teller phonon coupling. Then, it is important to understand the in#uence of J starting with the one-orbital model. $ The "rst numerical study that included a nonzero J was reported by Yunoki and Moreo (1998) $ (note that hereafter J will be an alternative notation for J , as used sometimes in previous $ literature). An interesting observation emerging from their analysis is that PS occurs not only near n"1.0 but also near the other extreme of n"0.0, where again a FM}AF competition exists. In this regime, Batista et al. (2000) have shown the formation of ferromagnetic polarons upon electron doping of the n"0.0 AF-state. Considering several of these polarons it is likely that extended structures may form, as in a phase separated state. The 1D phase diagram at low temperature in the (J, n)-plane is in Fig. 3.4.7. Three AF regions and two PS regions are shown, together with a FM regime at intermediate densities already discussed in previous subsections. In addition, a novel phase exists at intermediate values of J and n. This phase has a curious spin arrangement given by a periodically arranged pattern !! of localized spins, namely it has an equal number of FM and AF links, and for this reason interpolates at constant density between FM and AF phases (see also Garcia et al., 2000; Aliaga et al., 2000a). This phase is a precursor in 1D of the CE-phase in 2D, as will be discussed later. Calculations of the Drude weight show that this state is insulating, as expected since it has AF links. 3.4.4. Quantum localized spins An important issue in the context discussed in this section is whether the approximation of using classical degrees of freedom to represent the t spins is su$ciently accurate. In principle, this spin  should be S"3/2, which appears large enough to justify the use of classical spins. Unfortunately, it is very di$cult to study quantum spins in combination with mobile fermions, and the approximation can be explicitly tested only in a few cases. One of them is a 1D system, where the density matrix renormalization group (DMRG) method and Lanczos techniques allow for a fairly accurate characterization of the fully quantum model. The phase diagram obtained in this context by Dagotto et al. (1998) is reproduced in Fig. 3.4.8. Fortunately, the shape and even quantitative aspects of the diagram (with AF, FM, IC and PS regions) are in good agreement with those found with classical spins. The PS regime certainly appears in the study, although "nite values of J are & needed for its stabilization. The study leading to Fig. 3.4.8 was carried out in the canonical ensemble, with "xed number of particles, and the possibility of PS was analyzed by using the compressibility (), criterion where a (0 corresponds to an unstable system, as it is well known from elementary thermodynamic considerations. \ is proportional to the second derivative of the ground state with respect to the number of particles, which can be obtained numerically for N electrons by discretizing the derivative using the ground state energies for N, N#2 and N!2 particles at the "xed couplings under consideration (for details see Dagotto et al., 1998). Following this procedure, a negative compressibility was obtained, indicative of phase separation. Another method is to "nd from the ground state energies at various number of electrons, and plot density vs. . As in Monte Carlo simulations with classical spins, a discontinuity appears in the results in the regime of PS. It is clear that the tendency toward these unstable regimes, or mixed states after proper consideration of the 1/r Coulomb interaction, is very robust and independent of details in

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Fig. 3.4.7. Phase diagram of the one-orbital model for manganites including an antiferromagnetic Heisenberg coupling among the localized spins, here denoted by J (while in other parts of the text it is referred to as J ). The Hund coupling is $ "xed to 8 and t"1. Two PS regions are indicated, three AF regimes, and one FM phase. The `Ia insulating phase is described in more detail in the text. Reproduced from Yunoki and Moreo (1998). Fig. 3.4.8. Phase diagram of the one-orbital model with S"3/2 localized t -spins, obtained with the DMRG and  Lanczos methods applied to the chains of "nite length L indicated. The notation is as in previous "gures. Results reproduced from Dagotto et al. (1998), where more details can be found.

the computational studies. Results for the much simpler case of localized S"1/2 spins can also be obtained numerically. The phase diagram (Dagotto et al., 1998) is still in qualitative agreement with S"3/2 and R, although not quantitatively. PS appears clearly in the computational studies, as well as FM and spin IC phases. 3.4.5. Inyuence of long-range Coulomb interactions As already explained before, it is expected that long-range Coulomb interactions will break the electronic PS regime with two macroscopic FM and AF regions, into a stable state made out of small coexisting clusters of both phases. However, calculations are di$cult in this context. One of the few attempts was carried out by Malvezzi et al. (1999) using a 1D system. On-site ; and nearest-neighbor < Coulomb interactions were added to the one-orbital model. The resulting phase diagram can be found in Fig. 14 of Malvezzi et al. (1999). At <"0, the e!ect of ; is not much important, namely PS is found at both extremes of densities, and in between a chargedisordered FM-phase is present, results in good agreement with those described in previous subsections. This is reasonable since a large Hund coupling by itself suppresses double occupancy even without ; added explicitly to the model. However, when < is switched-on, the PS regime of small hole density is likely to be a!ected drastically due to the charge accumulation. Indeed, this regime is replaced by a charge-density wave with a peak in the spin structure factor at a momentum di!erent from 0 and
(Malvezzi et al., 1999). Holes are spread over a few lattice spacings, rather than being close to each other as in PS. In the other extreme of many holes, the very small electronic density makes < not as important. In

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between, the FM-phase persists up to large value of <, but a transition exists in the charge sector, separating a charge disordered from a charge-ordered state. Certainly more work in this interesting model is needed to fully clarify its properties, and extensions to 2D would be important, but the results thus far are su$cient to con"rm that PS is rapidly destroyed by a long-range Coulomb interaction leading to nontrivial charge density waves (Malvezzi et al., 1999). 3.4.6. Tendencies toward electronic phase separation in t}J-like models for transition metal oxides The "rst indications of a strong tendency toward electronic phase separation in models for manganites were actually obtained by Riera et al. (1997) using computational techniques applied to t}J-like models for Ni- and Mn-oxide. In these models, it was simply assumed that the Hund coupling was su$ciently large that the actual relevant degrees of freedom at low energy are `spinsa (of value 2 and 1, for Mn- and Ni-oxide, respectively) and `holesa (with spin 3/2 and 1/2, for Mnand Ni-oxide, respectively). The Hamiltonian at large J can be perturbatively deduced from the & quantum one-orbital model and its form is elegant, with hole hopping which can take place with rearrangement of the spin components of `spina and `holea. Details can be found in Riera et al. (1997). The phase diagram found with DMRG and Lanczos methods is in Fig. 3.4.9 for the special case of one dimension. The appearance of ferromagnetic and phase-separated regions is clear in this "gure, and the tendency grows as the magnitude of the spin grows. In between PS and FM, regions with hole binding were identi"ed that have not been studied in detail yet. The result in Fig. 3.4.9 has to be contrasted against those found for the standard 1D t}J model for the cuprates (see results in Dagotto, 1994) where the PS regime appears at unphysically large values of J/t, and FM was basically absent. There is a substantial qualitative di!erence between the results found for

Fig. 3.4.9. Phase diagram of t}J-like models in 1D corresponding to (a) nickelates and (b) manganites. J is the coupling between Heisenberg spins at each site in the large Hund coupling limit and x is the hole density. PS, B, and FM, denote phase-separated, hole binding, and ferromagnetic phases. For the meaning of the various symbols used to "nd the boundaries of the phases see Riera et al. (1997). DMRG and Lanczos techniques were used for this result.

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Cu-oxides and those of Ni- and Mn-oxide, mainly caused by the presence of localized spins in the last two. This suggests that cuprates do not share the same physics as other transition metal oxides. In particular, it is already known that cuprates are superconductors upon hole doping, while nickelates and manganites are not. 3.5. Main results: two orbital model 3.5.1. Phase diagram at density x"0.0 The results of the previous section showed that the one-orbital model for manganites contains interesting physics, notably a FM}AF competition that has similarities with those found in experiments. However, it is clear that to explain the notorious orbital order tendency in Mn-oxides, it is crucial to use a model with two orbitals, and in Section 3.3 such a model was de"ned for the case where there is an electron Jahn}Teller phonon coupling and also Coulomb interactions. Under the assumption that both localized t -spins and phonons are classical, the model without  Coulombic terms can be studied fairly accurately using numerical and mean-"eld approximations. Results obtained with both approaches will be presented here. For the case where Coulomb terms are included, unfortunately, computational studies are di$cult but mean-"eld approximations can still be carried out. As in the case of one orbital, let us start with the description of the phase diagram of the two-orbital model. In this model there are more parameters than in the previous case, and more degrees of freedom, thus at present only a fraction of parameter space has been investigated. Consider "rst the case of e -density n"1.0, which is relatively simple to study numerically since  this density is easy to stabilize in the grand canonical simulations. It corresponds to having one electron on average per site, and in this respect it must be related to the physics found in hole undoped compounds such as LaMnO . Carrying out a Monte Carlo simulation in the localized  spins and phonons, and considering exactly the electrons in the absence of an explicit Coulomb repulsion (as is done for the one-orbital case), a variety of correlations have been calculated to establish the n"1.0 phase diagram. Typical results for the spin and orbital structure factors, S(q) and ¹(q), respectively, at the momenta of relevance are shown in Fig. 3.5.1, obtained at a large Hund coupling equal to 8t, J"0.05t, and a small temperature, plotted as a function of the electron}phonon coupling . Results at dimensions 1, 2 and 3 are shown. At small , S(0) is dominant and ¹(q) is not active. This signals a ferromagnetic state with disordered orbitals, namely a standard ferromagnet (note, however, that Khomskii (2000a) and Maezono and Nagaosa (2000) believe that this state in experiments may have complex orbital ordering). The result with FM tendencies dominating may naively seem strange given the fact that for the one-orbital model at n"1.0 an AF-state was found. But here two orbitals are being considered and one electron per site is 1/2 electron per orbital. In this respect, n"1.0 with two orbitals should be similar to n"0.5 for one orbital and indeed in the last case a ferromagnetic state was observed (Section 3.4). Results become much more interesting as  grows beyond 1. In this case, "rst ¹(Q) increases rapidly and dominates (in the spin sector still S(0) dominates, i.e. the system remains ferromagnetic in the spin channel). The momentum Q corresponds to , (, ), and (, , ), in 1D, 2D, and 3D, respectively. It denotes a staggered orbital order, namely a given combination of the a and b original orbitals is the one mainly populated in the even sites of the cluster, while in the odd sites

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Fig. 3.5.1. (a) ¹(q) and S(q), orbital and spin structure factors vs. , working with the two-orbital model at n"1.0, low temperature, J "8, J"0.05, and in 1D chains. The hopping set t "t "2t "2t was used, but qualitatively the &



results are similar for other hoppings. (b) Same as (a) but using a 4;4 cluster and realistic hoppings (Section 3.3). (c) Same as (a) but for a 4 cluster and realistic hoppings, and using J "R. Results reproduced from Yunoki et al. (1998b), where & more details can be found. Fig. 3.5.2. Magnetic and orbital structures discussed in the text to justify the Monte Carlo results for the two-orbital model at electronic density 1.0.

another orbital combination is preferred. These populated orbitals are not necessarily only the two initial ones used in the de"nition of the Hamiltonian, in the same way that in a Heisenberg spin system not only spins up and down in the z-direction (the usual basis) are possible in mean value. Actually, spins can order with a mean-value pointing in any direction depending on the model and couplings, and the same occurs with the orbitals, which in this respect are like `pseudo-spinsa. A particular combination of the original orbitals 1 and 2 could be energetically the best in even sites and some other combination in the odd sites. In Fig. 3.5.1, as  increases further, a second transition was identi"ed this time into a state which is staggered in the spin, and uniform in the orbitals. Such a state is the one in correspondence with the AF-state found in the one-orbital model, namely only one orbital matters at low energies and the spin, as a consequence, is antiferromagnetic. However, from the experimental point of view, the intermediate regime between 1.0 and 2.0 is the most relevant, since staggered orbital order is known to occur in experiments. Then, the one orbital model envisioned to work for manganites, at least close to n"1.0 due to the static Jahn}Teller distortion, actually does not work even there since it misses the staggered orbital order but instead assumes a uniform order. Nevertheless, the model is qualitatively interesting, as remarked upon before. Why does orbital order occur here? This can be easily understood perturbatively in the hopping t, following Fig. 3.5.2 where a single Mn}Mn link is used and the four possibilities (spin FM or AF, orbital uniform or staggered) are considered. A hopping matrix only connecting the same orbitals, with hopping parameter t, is assumed for simplicity. The energy di!erence between e -orbitals at  a given site is E , which is a monotonous function of . For simplicity, in the notation let us refer to (2 orbital uniform (staggered) as orbital `FMa (`AFa). Case (a) in Fig. 3.5.2 corresponds to spin FM and orbital AF: In this case when an electron moves from orbital a on the left to the same orbital on

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the right, which is the only possible hopping by assumption, an energy of order E is lost, but (2 kinetic energy is gained. As in any second-order perturbative calculation the energy gain is then proportional to t/E . In case (b), both spin and orbital FM, the electrons do not move and the (2 energy gain is zero (again, the nondiagonal hoppings are assumed negligible just for simplicity). In case (c), the spin are AF but the orbitals are FM. This is like a one orbital model and the gain in energy is proportional to t/(2J ). Finally, in case (d) with AF in spin and orbital, both Hund and & orbital splitting energies are lost in the intermediate state, and the overall gain becomes proportional to t/(2J #E ). As a consequence, if the Hund coupling is larger than E , then case (a) is & (2 (2 the best, as it occurs at intermediate E values in Fig. 3.5.1. However, in the opposite case (orbital (2 splitting larger than Hund coupling) case (c) has the lowest energy, a result also compatible to that found in Fig. 3.5.1. Then, the presence of orbital order can be easily understood from a perturbative estimation, quite similarly as done by Kugel and Khomskii (1974) in their pioneering work on orbital order. Recently, X-ray resonant scattering studies have con"rmed the orbital order in manganites (Murakami et al., 1998a, b). 3.5.2. A-type AF at x"0.0 The alert reader may have noticed that the state reported in the previous analysis at intermediate 's is actually not quite the same state as found in experiments. It is known that the actual state has A-type AF spin order, while in the analysis of Fig. 3.5.1, such a state was not included. The intermediate  region has FM spin in the three directions in the 3D simulations of that "gure. Something else must be done in order to arrive at an A-type antiferromagnet. Recent investigations by Hotta et al. (1999) have shown that, in the context of the model with Jahn}Teller phonons, this missing ingredient is J itself, namely by increasing this coupling from 0.05 to larger values, $ a transition from a FM to an A-type AF exists (the relevance of JT couplings at n"1.0 has also been remarked by Capone et al., 2000; see also Fratini et al., 2000). This can be visualized easily in Fig. 3.5.3 where the energy vs. J at "xed intermediate  and J is shown. Four regimes were $ & identi"ed: FM, A-AF, C-AF, and G-AF, states that are sketched also in that "gure. The reason is simple: as J grows, the tendency toward spin AF must grow since this coupling favors such an $ order. If J is very large, then it is clear that a G-AF state must be the one that lowers the energy, $ in agreement with the Monte Carlo simulations. If J is small or zero, there is no reason why spin $ AF will be favorable at intermediate  and the density under consideration, and then the state is ferromagnetic to improve the electronic mobility. It should be no surprise that at intermediate J , $ the dominant state is intermediate between the two extremes, with A- and C-type antiferromagnetism becoming stable in intermediate regions of parameter space. It is interesting to note that similar results regarding the relevance of J to stabilize the A-type $ order have been found by Koshibae et al. (1997) in a model with Coulomb interactions (see also Feiner and OleH s, 1999). An analogous conclusion was found by Solovyev et al. (1996) and Ishihara et al. (1997a,b). Betouras and Fujimoto (1999), using bosonization techniques for the 1D oneorbital model, also emphasized the importance of J , similarly as did Yi et al. (1999) based on $ Monte Carlo studies in two dimensions of the same model. The overall conclusion is that there are clear analogies between the strong Coulomb and strong Jahn}Teller coupling approaches, as discussed elsewhere in this review. Actually, in the mean-"eld approximation presented in Section 3.3 it was shown that the in#uence of the Coulombic terms can be hidden in simple rede"nitions of the electron}phonon couplings (see also Benedetti and Zeyher, 1999). In our opinion, both

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Fig. 3.5.3. (a) Total energy vs. J on a 2 cluster at low temperature with J "8t and "1.5. The results were obtained & using Monte Carlo and relaxational techniques, with excellent agreement among them. (b) The four spin arrangements are also shown. (c) Orbital order corresponding to the A-type AF state. For more details the reader should consult Hotta et al. (1999).

approaches (JT and Coulomb) have strong similarities and it is not surprising that basically the same physics is obtained in both cases. Actually, Fig. 2 of Maezono et al. (1998b) showing the energy vs. J in mean-"eld calculations of the Coulombic Hamiltonian without phonons is very $ similar to our Fig. 3.5.3, aside from overall scales. On the other hand, Mizokawa and Fujimori (1995, 1996) states that the A-type AF is stabilized only when the Jahn}Teller distortion is included, namely, the FM phase is stabilized in the purely Coulomb model, based on the unrestricted Hartree}Fock calculation for the d}p model. The issue of what kind of orbital order is concomitant with A-type AF order is an important matter. This has been discussed at length by Hotta et al. (1999), and the "nal conclusion, after the introduction of perturbations caused by the experimentally known di!erence in lattice spacings between the three axes, is that the order shown in Fig. 3.5.3c minimizes the energy. This state has indeed been identi"ed in recent X-ray experiments, and it is quite remarkable that such a complex pattern of spin and orbital degrees of freedom indeed emerges from mean-"eld and computational studies. Studies by van den Brink et al. (1999a) using purely Coulombic models arrived at similar conclusions. 3.5.3. Electronic phase separation with two orbitals Now let us analyze the phase diagram at densities away from n"1.0. In the case of the one-orbital model, phase separation was very prominent in this regime. A similar situation was observed with two orbitals, as Fig. 3.5.4a}b illustrates where n vs. is shown at intermediate , J "R, and low temperature. A clear discontinuity is observed, both near n"1.0, as well as at &

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Fig. 3.5.4. (a) n vs. at the couplings and temperature indicated on a ¸"22 site chain. The discontinuities characteristic of phase separation are clearly shown. (b) Same as (a) but in 2D at the parameters indicated. The two sets of points are obtained by increasing and decreasing , forming a hysteresis loop. (c) Phase diagram of the two orbitals model in 1D, J "8, J"0.05, and using the hopping set t "t "2t "2t . The notation has been explained in the text. &



For more details see Yunoki et al. (1998b), from where this "gure was reproduced.

low density. Measurements of spin and orbital correlations, as well as the Drude weight to distinguish between metallic and insulating behavior, have suggested the phase diagram in one dimension reproduced in Fig. 3.5.4c. There are several phases in competition. At n"1.0 the results were already described in the previous subsection. Away from the n"1.0 phases, only the spin-FM orbital-disordered survives at "nite hole density, as expected due to the mapping at small  into the one-orbital model with half the density. The other phases at 51.0 are not stable, but electronic phase separation takes place. The n(1.0 extreme of the PS discontinuity is given by a spin-FM orbital-FM metallic state, which is a 1D precursor of the metallic orbitally ordered A-type state identi"ed in some compounds precisely at densities close to 0.5. Then, the two states that compete in the n&1.0 PS regime di!er in their orbital arrangement, but not in the spin sector. This is PS triggered by the orbital degrees of freedom, which is a novel concept. On the other hand, the PS observed at low density is very similar to that observed in the one-orbital model involving spin-FM and AF-states in competition. Finally, at n&0.5 and large , charge ordering takes place, but this phase will be discussed in more detail later. Overall, it is quite reassuring to observe that the stable phases in Fig. 3.5.4c all have an analog in experiments. This gives support to the models used and to the computational and mean-"eld techniques employed. In addition, since all stable regions are realistic, it is natural to assume that the rest of the phase diagram, namely the PS regions, must also have an analog in experiments in the form of mixed-phase tendencies and nanometer-size cluster formation, as discussed in the case of the one-orbital model. PS is very prominent in all the models studied, as long as proper manybody techniques are employed. For instance, using accurate mean-"eld approximations, the PS

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tendencies in 1D can also be properly reproduced (see Section 5 of Hotta, Malvezzi and Dagotto, 2000). It is also important that even in purely Coulombic cases (without JT phonons), PS has been found in 1D models in some regions of parameter space (see Hotta et al., 2000), and, thus, this phenomenon is not restricted to Jahn}Teller systems. Kagan et al. (2000) also reported phase separation near x"0.5 without using JT phonons. Guerrero and Noack (2000) reported phase separation in a one-dimensional copper-oxide model with only Coulomb interactions. Varelogiannis (2000) found coexistence and competition of CO-, AF- and FM-phase in a multicomponent mean-"eld theory, without using a particular microscopic mechanism. It is important to remark that plenty of work still remains to be done in establishing the phase diagram of the two-orbitals model. Studies in 2D carried out by our group suggest that the phase diagram is similar to that found in 1D, but details remain to be settled. The 3D diagram is known only in special cases. Although the experience gained in the one-orbital model suggests that all dimensions have similar phase diagrams, this issue remains to be con"rmed in the two-orbital case at large . In addition, also note that the intermediate J regime has not been explored and & surprises may be found there, such as the stripes described for the one-orbital case at intermediate Hund coupling. Work is in progress in this challenging area of research. 3.5.4. Charge ordering at x"0.5 and the CE-state The so-called CE-type AFM phase has been established as the ground state of half-doped perovskite manganites in the 1950s. This phase is composed of zigzag FM arrays of t -spins, which  are coupled antiferromagnetically perpendicular to the zigzag direction. Furthermore, the checkerboard-type charge ordering in the x}y plane, the charge stacking along the z-axis, and (3x!r/3y!r) orbital ordering are associated with this phase. Although there is little doubt that the famous CE-state of Goodenough, reviewed in Section 3.1, is indeed the ground state of x"0.5 intermediate and low bandwidth manganites, only very recently such a state has received theoretical con"rmation using unbiased techniques, at least within some models. In the early approach of Goodenough it was assumed that the charge was distributed in a checkerboard pattern, upon which spin and orbital order was found. But it would be desirable to obtain the CE-state based entirely upon a more fundamental theoretical analysis, as the true state of minimum energy of a well-de"ned and realistic Hamiltonian. If such a calculation can be done, as a bonus one would "nd out which states compete with the CE-state in parameter space, an issue very important in view of the mixed-phase tendencies of Mn-oxides, which cannot be handled within the approach of Goodenough. One may naively believe that it is as easy as introducing a huge nearest-neighbor Coulomb repulsion < to stabilize a charge-ordered state at x"0.5, upon which the reasoning of Goodenough can be applied. However, there are at least two problems with this approach. First, such a large < quite likely will destabilize the ferromagnetic charge-disordered state and others supposed to be competing with the CE-state. It may be possible to explain the CE-state with this approach, but not others also observed at x"0.5 in large bandwidth Mn-oxides. Second, a large < would produce a checkerboard pattern in the three directions. However, experimentally it has been known for a long time (Wollan and Koehler, 1955) that the charge stacks along the z-axis, namely the same checkerboard pattern is repeated along z, rather than being shifted by one lattice spacing from plane to plane. A dominant Coulomb interaction < cannot be the whole story for x"0.5 low-bandwidth manganese oxides.

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The nontrivial task of "nding a CE-state with charge stacked along the z-axis without the use of a huge nearest-neighbors repulsion has been recently performed by Yunoki et al. (2000) using the two-orbital model with strong electron Jahn}Teller phonon coupling. The calculation proceeded using an unbiased Monte Carlo simulation, and as an output of the study, the CE-state indeed emerged as the ground state in some region of coupling space. Typical results are shown in Fig. 3.5.5. In part (a) the energy at very low temperature is shown as a function of J at "xed density $ x"0.5, J "R for simplicity, and with a robust electron}phonon coupling "1.5 using the & two-orbital model of Section 3.3. At small J , a ferromagnetic phase was found to be stabilized, $ according to the Monte Carlo simulation. Actually, at J "0.0 it has not been possible to $ stabilize a partially AF-state at x"0.5, namely the states are always ferromagnetic at least within the wide range of 's investigated (but they can have charge and orbital order). On the other hand, as J grows, a tendency to form AF links develops, as it happens at x"0.0. At large $ J eventually the system transitions to states that are mostly antiferromagnetic, such as the $ so-called `AF(2)a state of Fig. 3.5.5b (with an up}up}down}down spin pattern repeated along one axis, and AF coupling along the other axis), or directly a fully AF-state in both directions. However, the intermediate values of J are the most interesting ones. In this case the energy of $ the 2D clusters become #at as a function of J suggesting that the state has the same number of $ FM and AF links, a property that the CE-state indeed has. By measuring charge correlations it was found that a checkerboard pattern is formed particularly at intermediate and large 's, as in the CE-state. Finally, after measuring the spin and orbital correlations, it was con"rmed that indeed the complex pattern of the CE-state was fully stabilized in the simulation. This occurs in a robust

Fig. 3.5.5. (a) Monte Carlo energy per site vs. J at density x"0.5, "1.5, low temperature ¹"1/100, and J "R, $ & using the two-orbital model in 2D with Jahn}Teller phonons (noncooperative ones). FM, CE, and AF states were identi"ed measuring charge, spin, and orbital correlations. `AF(2)a denotes a state with spins !! in one direction, and antiferromagnetically coupled in the other. The clusters used are indicated. (b) Phase diagram in the plane -J at $ x"0.5, obtained numerically using up to 8;8 clusters. All transitions are of "rst-order. The notation is the standard one (CD"charge disorder, CO"charge order, OO"orbital order, OD"orbital disorder). Results reproduced from Yunoki et al. (2000), where more details can be found.

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portion of the }J plane, as shown in Fig. 3.5.5b. The use of J as the natural parameter to vary $ $ in order to understand the CE-state is justi"ed based on Fig. 3.5.5b since the region of stability of the CE-phase is elongated along the -axis, meaning that its existence is not so much dependent on that coupling but much more on J itself. It appears that some explicit tendency in the $ Hamiltonian toward the formation of AF links is necessary to form the CE-state. If this tendency is absent, a FM state if formed, while if it is too strong an AF-state appears. The x"0.5 CE-state, similar to the A-type AF at x"0.0, needs an intermediate value of J for stabilization. The $ stability window is "nite and in this respect there is no need to carry out a ,ne tuning of parameters to "nd the CE-phase. However, it is clear that there is a balance of AF and FM tendencies in the CE-phase that makes the state somewhat fragile. Note that the transitions among the many states obtained when varying J are all of ,rst $ order, namely they correspond to crossings of levels at zero temperature. The "rst-order character of these transitions is a crucial ingredient of the recent scenario proposed by Moreo et al. (2000) involving mixed-phase tendencies with coexisting clusters with equal density, to be described in more detail below. Recently, "rst-order transitions have also been reported in the one-orbital model at x"0.5 by Alonso et al. (2000a, b), as well as tendencies toward phase separation. Recent progress in the development of powerful techniques for manganite models (Alonso et al., 2000c; Motome and Furukawa, 2000a, b) will contribute to the clari"cation of these issues in the near future. 3.5.5. Charge stacking Let us address now the issue of charge-stacking along the z-axis. For this purpose simulations using 3D clusters were carried out. The result for the energy vs. J is shown in Fig. 3.5.6, with $ J "R and "1.5 "xed. The CE-state with charge stacking was found to be the ground state on & a wide J window. The reason that this state has lower energy than the so-called `Wigner-crystala $ (WC) version of the CE-state, namely with the charge spread as much as possible, is once again the in#uence of J . With a charge stacked arrangement, the links along the z-axis can all be $ simultaneously antiferromagnetic, thereby minimizing the energy. In the WC-state this is not possible. It should be noted that this charge stacked CE-state is not immediately destroyed when the weak nearest-neighbor repulsion < is introduced to the model, as shown in the mean-"eld calculations

Fig. 3.5.6. Monte Carlo energy per site vs. J obtained working on a 4 cube ("1.5, J "R, ¹"1/100). Results $ & with both cooperative and noncooperative phonons are shown, taken from Yunoki et al. (2000). The state of relevance here is the `CE (CS)a one, which is the CE-state with charge stacking.

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by Hotta et al. (2000). If < is further increased for a realistic value of J , the ground state $ eventually changes from the charge stacked CE-phase to the WC version of the CE-state or the C-type AFM phase with WC charge ordering. As explained above, the stability of the charge stacked phase to the WC version of the CE-state is due to the magnetic energy di!erence. However, the competition between the charge-stacked CE-state and the C-type AFM phase with the WC structure is not simply understood by the e!ect of J , since those two kinds of AFM phases have $ the same magnetic energy. In this case, the stabilization of the charge stacking originates from the di!erence in the geometry of the 1D FM-path, namely a zigzag-path for the CE-phase and a straight-line path for the C-type AFM state. As will be discussed later in detail, the energy for e -electrons in the zigzag path is lower than that in the straight-line path, and this energy di!erence  causes the stabilization of the charge stacking. In short, the stability of the charge-stacked structure at the expense of < is supported by `the geometric energya as well as the magnetic energy. Note that each energy gain is just a fraction of t. Thus, in the absence of other mechanisms to understand the charge stacking, another consequence of this analysis is that < actually must be substantially smaller than naively expected, otherwise such a charge pattern would not be stable. In fact, estimations given by Yunoki et al. (2000) suggest that the manganites must have a large dielectric function at short distances (see Arima and Tokura, 1995) to prevent the melting of the chargestacked state. Note also that the mean-"eld approximations by Hotta et al. (2000) have shown that on-site Coulomb interactions ; and ; can also generate a 2D CE-state, in agreement with the calculations by van den Brink et al. (1999b). Then, the present authors believe that strong JT and Coulomb couplings tend to give similar results. This belief "nds partial con"rmation in the mean-"eld approximations of Hotta et al. (2000), where the similarities between a strong  and (;, ;) were investigated. Even doing the calculation with Coulombic interactions, the in#uence of J is still crucial to inducing charge-stacking (note that the importance of this parameter has also $ been recently remarked by Mathieu et al. (2000) based on experimental results). Many other authors carried out important work in the context of the CE-state at x"0.5. For example, with the help of Hartree}Fock calculations, Mizokawa and Fujimori (1997) reported the stabilization of the CE-state at x"0.5 only if Jahn}Teller distortions were incorporated into a model with Coulomb interactions. This state was found to be in competition with a uniform FM-state, as well as with an A-type AF-state with uniform orbital order. In this respect the result are very similar to those found by Yunoki et al. (2000) using Monte Carlo simulations. In addition, using a large nearest neighbor repulsion and the one-orbital model, charge ordering and a spin structure compatible with the zigzag chains of the CE-state was found by Lee and Min (1997) at x"0.5. Also Jackeli et al. (1999) obtained charge-ordering at x"0.5 using mean-"eld approximations and a large <. Charge-stacking was not investigated by those authors. The CE-state in x"0.5 Pr Ca MnO was also obtained by Anisimov et al. (1997) using LSDA#U techniques. \V V  3.5.6. Topological origin of the x"0.5 CE-state The fact that  does not play the most crucial role for the CE-state also emerges from the `topologicala arguments of Hotta et al. (2000), where at least the formation of zigzag ferromagnetic chains with antiferromagnetic interchain coupling, emerges directly for "0 and large J , as & a consequence of the `band-insulatoracharacter of those chains. Similar conclusions as those reached by Hotta et al. (2000), were independently obtained by Solovyev and Terakura (1999),

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Solovyev (2000), and by van den Brink et al. (1999b). The concept of a band insulator in this context was "rst described in Hotta et al. (1998). To understand the essential physics present in half-doped manganites, it is convenient to consider the complicated CE-structure in Hamiltonians simpler than those analyzed in Section 3.3. Based on the concept of `adiabatic continuationa for the introduction of the JT distortion and/or the Coulombic interactions, the following approximations will be made: (i) H"H(E ";"<"0): In the "rst place, this simple Hamiltonian is considered based on the (2 DE mechanism, but even in this situation, qualitative concepts can be learned for the stabilization of the zigzag AFM phase. (ii) H"H(E O0, ;"<"0): In order to consider the charge and (2 orbital ordering, the non-cooperative JT phonons are included in the two-orbital DE model by using the analytic MFA. In particular, the charge-stacked structure is correctly reproduced, and its origin is clari"ed based on a `topologicala framework. (iii) H"H(E O0, ;O0,

 i







i 1 1 " i (2 1 !i

ci

. ci
is rewritten as

(91)

After simple algebra, H   H "!t /2 (iRi a #iRi a #e (a Ri i a #e\ (a iRi a ) , (92)    ia > > > >  where the phase a depends only on the hopping direction, and it is given by x "!, y ", and z "0, with "
/3. Note that the e -electron picks up a phase change when it moves between  di!erent neighboring orbitals. In this expression, the e!ect of the change of the local phase is correctly included in the Hamiltonian.

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Fig. 3.5.7. The unit cell for the zigzag FM chain in the CE-type AFM phase at x"0.5. Note that the hopping direction is changed periodically as 2, x, x, y, y,2.

To introduce the momentum k along the zigzag chain, the Bloch's phase e! I is added to the hopping term between adjacent sites. Then, the problem is reduced to "nding the eigenvalues of an 8;8 matrix, given by



  

 



?



@



?



?

@ ?

@ " @ , (93) I  ? ?   @ @   ? ?   @ @ where  and  are the basis function for - and -electron at the j-site of the unit cell, ?H @H respectively, and the Hamiltonian matrix hK is given by hK






OK

¹K x I OK

OK



¹K yH I OK

¹K x t ¹K xH I hK "!  I . (94) 2 OK ¹K xH OK ¹K y I I OK ¹K yH OK ¹K y I I Here OK is the 2;2 matrix in which all components are zeros, and the hopping matrix ¹K a along the I a-direction is de"ned by




1 ¹K a "e I I e\ (a

e (a 1



,

where note again that !x "y ""
/3.

(95)

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Although it is very easy to solve the present eigenvalue problem by using the computer, it is instructive to "nd the solution analytically. In the process of "nding this solution, several important points will be clari"ed. First, note that there are two eigenfunctions of the 8;8 matrix which have a `localizeda character, satisfying hK ( !e\ ( )"0 ? @

(96)

and hK ( !e> ( )"0 . (97) ? @ As easily checked by simple algebra, those localized basis functions correspond to y!z and z!x orbitals at sites 2 and 4, respectively. By orthogonality, the active orbitals are then "xed as 3x!r and 3y!r at sites 2 and 4, respectively. This fact suggests that if some potential acts over the e -electrons, the (3x!r/3y!r)-type orbital ordering immediately occurs in such  a one-dimensional zigzag path due to the standard Peierls instability. This point will be discussed again later in the context of charge-orbital ordering due to the JT distortion. To "nd the other extended eigenstates, it is quite natural to consider active basis functions at sites 2 and 4, given by ( #e\ ( )/(2 and ( #e> ( )/(2, respectively. Then, the ? @ ? @ bonding and antibonding combinations of those basis are constructed by including appropriate phases such as !"(1/2)[(e\ ( #e ( )$ (e ( #e\ ( )] . (98)  ? @ ? @ By acting with hK over !, it is found that two kinds of 3;3 block Hamiltonians hI ! can be  constructed using new basis functions de"ned as !"( $ )/(2  ? @

(99)

and (100) !"( $ )/(2 .  @ ? As expected, the block Hamiltonian including the ground state is constructed using the bondingtype basis functions





> >   hI > > "> > , I   > >   where the 3;3 matrix hI > is given by




0

cos k \ 0

(101)

cos k > 0



(102) hI >"!(2t cos k  \ cos k 0 0 > with k "k$/2. By solving this eigenvalue problem, it can be easily shown that the eigenener! gies are >"0, $t (2#cos(2k) . I 

(103)

E. Dagotto et al. / Physics Reports 344 (2001) 1}153

77

Note here that the momentum k is restricted to the reduced zone, !
/44k4
/4. As expected, the lowest-energy band has a minimum at k"0, indicating that this block correctly includes the ground state of the one e -electron problem. At a "rst glance, this point appears obvious, but if the  e!ect of the local phase is not treated correctly, an unphysical solution easily appears, as will be discussed below. The block Hamiltonian for the antibonding sector is given by





\ \   hI \ \ "\ \ I   \ \   with




(104)

0

hI \"!(2t !i sin k  \ !i sin k > The eigenenergies are given by

i sin k \ 0 0



i sin k > . 0 0

\"0, $t (2!cos(2k) . I  In summary, eight eigenenergies have been obtained as

(105)

(106)

 "0, $t (2#cos(2k), $t (2!cos(2k) , (107) I   where the #at band k "0 has four-fold degeneracy. The band structure is shown in Fig. 3.5.8. The most remarkable feature is that the system is band-insulating, with a bandgap of value t for quarter  "lling, i.e., x"0.5. This band insulating state, without any explicit potential among the electrons moving along the zigzag chains, is caused by the phase di!erence between t x and t y . Intuitively, IJ IJ such band-insulator originates in the presence of a standing-wave state due to the interference between two traveling waves running along the x- and y-direction. In this interference picture, the nodes of the wavefunction can exist on `the cornera of the zigzag structure, and the probability amplitude becomes larger in the `straighta segment of the path. Thus, even a weak potential can produce the charge and orbital ordering based on this band-insulating phase. Since t is at least of  the order of 1000 K, this band-insulating state is considered to be very robust. In fact, if some potential is included into such an insulating phase, the system maintains its `insulatinga properties, and a modulation in the orbital density appears. The problem in the zigzag one-dimensional chain provided us with a typical example to better understand the importance of the additional factor e Ki  in front of the 2;2 SU(2) unitary matrix to generate the phase dressed operator at each site. As clearly shown above, the `aa and `ba orbitals should be chosen as `aa"y!z and `ba"3x!r at site 2, and `aa"z!x and `ba"3y!r at site 4, respectively. Namely, "2
/3 and "4
/3. The reason for these   choices of i is easily understood due to the fact that the orbital tends to polarize along the hopping direction to maximize the overlap. Thus, to make the Hamiltonian simple, it is useful to "x the orbitals at sites 2 and 4 as "2
/3 and "4
/3. Here, the phase factor e Ki  in the basis   function is essential to reproduce exactly the same solution as obtained in the discussion above. As

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Fig. 3.5.8. Band structure for the zigzag 1D chain (solid curve) in the reduced zone !
/44k4
/4. For reference, the band structure !2t cos k for the straight 1D path is also shown (broken curve). Note that the line at zero energy  indicates the four-fold degenerate #at-band present for the zigzag 1D path.

already mentioned, in a single-site problem, this phase factor can be neglected, since it provides only an additional phase to the whole wave function. However, if the e -electron starts moving  from site to site, the accumulation of the phase di!erence between adjacent sites does not lead just to an additional phase factor to the whole wave function. In fact, if this additional phase is accidentally neglected, the band structure will shift in momentum space as kPk#
, indicating that the minimum of the lowest-energy band is not located at k"0, but at k"
, as already pointed out by Koizumi et al. (1998b). Of course, this can be removed by the rede"nition of k by including `the crystal momentuma, but it is not necessary to rede"ne k, if the local phase factors are correctly included in the problem. Now let us discuss the stabilization of the zigzag structure in the CE-type phase. Although it is true that the zigzag one-dimensional FM chain has a large band gap, this fact does not guarantee that this band-insulating phase is the lowest-energy state. To prove that the CE-type AFM phase composed of these zigzag FM chains is truly the ground-state, at least the following three points should be clari"ed: (i) Does this zigzag structure have the lowest energy compared to other zigzag paths with the same periodicity and compared with the straight one-dimensional path? (ii) Does the periodicity with four lattice spacings produce the global ground state? In other words, can zigzag structures with another periodicity be the global ground state? (iii) Is the energy of the zigzag AFM phase lower than that of the FM or other AFM phases? All these points have been clari"ed in Hotta et al. (2000), and here the essential points are discussed brie#y. The "rst point can be checked by directly comparing the energies for all possible zigzag structures with the periodicity of four lattice spacings. Due to translational invariance, there exist four types of zigzag states which are classi"ed by the sequence of the hopping directions: x, x, x, x, x, y, x, y, x, x, x, y, and x, x, y, y. For quarter-"lling, by an explicit calculation it has been shown that the zigzag pattern denoted by x, x, y, y has the lowest energy among them (see details

E. Dagotto et al. / Physics Reports 344 (2001) 1}153

79

in Hotta et al., 2000), but here an intuitive explanation is provided. The state characterized by x, x, x, x is, of course, the one-dimensional metal with a dispersion relation simply given by !2t cos k. Note that in this straight FM chain, the active orbital at every site is 3x!r, since the  hopping direction is restricted to be along the x-direction. As emphasized in the above discussion, a periodic change in the hopping amplitude produces a band gap, indicating that the metallic state has an energy higher than the band-insulating states. Among them, the state with the largest bandgap will be the ground state. After several calculations at quarter-"lling, the zigzag state with x, x, y, y was found to have the lowest energy, but without any calculation this result can be deduced based on the interference e!ect. As argued above, due to the interference of two traveling waves along the x- and y-direction, a standing-wave state occurs with nodes on the corner sites, and a large probability amplitude at the sites included in the straight segments. To contain two e -electrons in the unit cell with four lattice spacings, at least two sites in the straight segment are  needed. Moreover, the nodes are distributed with equal spacing in the wave function as long as no external potential is applied. Thus, it is clear that the lowest-energy state corresponds to the zigzag structure with x, x, y, y. As for the second point regarding the periodicity, it is quite di$cult to carry out the direct comparison among the energies for all possible states, since there are in"nite possibilities for the combinations of hopping directions. Instead, to mimic the periodic change of the phase a in the hopping process, let us imagine a virtual situation in which a JT distortion occurs in the one-dimensional e -electron system, by following Koizumi et al. (1998a). To focus on the e!ect of  the local phase, it is assumed that the amplitude of the JT distortion q is independent of the site G index, i.e., q "q, and only the phase is changed periodically. For simplicity, the phase is G G uniformly twisted with the period of M lattice spacings, namely, "j;(2
)/M for 14j4M. H Since the periodic change of the hopping direction is mimicked by the phase change of the JT a distortion, t is simply taken as the unit matrix t  to avoid the double counting of the e!ect of IJ  IJ the phase change. If the potential amplitude is written as v"2qE , the Hamiltonian for the (2 present situation is given by H"!t  (cR c #cR c #h.c.)#v [sin (cR c #cR c )#cos (cR c !cR c )] ,  G H G
H
G G G
G
G G G G G
G
6GH7 G (108) where the spinless e -electron operator is used since the one-dimensional FM chain is considered  here, and the potential term for the JT distortion is neglected since it provides only a constant energy shift in this case. By using the transformation Eq. (81), this Hamiltonian is rewritten as H"!t  [e KG \KH (c R c #c R c )#h.c.]#v (c R c !c R c ) . G H G
H
G G G
G
 G 6GH7 The Hamiltonian in momentum space is obtained by the Fourier transform as

(109)

H"  [cos(
/M)(c R c #c R c )#i sin(
/M)(c R c !c R c )] I I I I
I
I I
I
I I (110) # v (c R c !c R c ) , I I I
I
I where  "!2t cos k and the periodic boundary condition (PBC) is imposed. Note that in this I  expression, k is the generalized quasi-momentum, rede"ned as k!
/MPk, to incorporate the

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additional phase
/M which appears to arise from a "ctitious magnetic "eld (see Koizumi et al., 1998b). The eigenenergies are easily obtained by diagonalization as E!" cos(
/M)$(v# sin(
/M) I I I "()[ # $(v#( ! )] .  I>
+ I\
+ I>
+ I\
+

(111)

Since this is just the coupling of two bands,  and  , it is easily understood that the I>
+ I\
+ energy gain due to the opening of the bandgap is the best for the "lling of n"2/M. In other words, when the periodicity M is equal to 2/n, the energy becomes the lowest among the states considered here with several possible periods. Although this is just a proof in an idealized special situation, it is believed that it captures the essence of the problem. Here the e!ect of the local phase factor e Ki  should be again noted. If this factor is dropped, the phase
/M due to the "ctitious magnetic "eld disappears and the eigenenergies are given by the and  , which has been also checked by the computational calculation. coupling of  I>
>
+ I>
\
+ This `
a shift in momentum space appears at the boundary, modifying the PBC to anti-periodic BC, even if there is no intention to use APBC. Of course, this is avoidable when the momentum k is rede"ned as k#
Pk, as pointed out in Koizumi et al. (1998b). However, it is natural that the results for PBC are obtained in the calculation using PBC. Thus, also from this technical viewpoint, it is recommended that the phase factor e Ki  is added for the local rotation in the orbital space. To show the last item of the list needed to show the stability of the CE state (see before), it is necessary to include the e!ect of the magnetic coupling between adjacent t -spins. The appearance  of the AFM phase with the zigzag geometry can be understood by the competition between the kinetic energy of e -electrons and the magnetic energy gain of t spins based on the double  exchange mechanism. Namely, if J is very small, for instance equal to zero, the FM phase best $ optimizes the kinetic energy of the e -electrons. On the other hand, when J is as large as t , the  $  system stabilizes a G-type AFM phase to exploit the magnetic energy of the t -spins. For  intermediate values of J , the AFM phase with a zigzag structure can appear to take advantage at $ least partially of both interactions. Namely, along the FM zigzag chain with alignment of t -spins,  the e -electrons can move easily, optimizing the kinetic energy, and at the same time there is  a magnetic energy gain due to the antiferromagnetic coupling between adjacent zigzag chains. The `windowa in J in which the zigzag AFM phase is stabilized has been found to be around $ J +0.1t in Monte Carlo simulations and the mean-"eld approximation, as discussed elsewhere $  in this review. In summary, at x"0.5, the CE-type AFM phase can be stabilized even without the Coulombic and/or the JT phononic interactions, only with large Hund and "nite J couplings. Of course, $ those interactions are needed to reproduce the charge and orbital ordering, but as already mentioned in the above discussion, because of the special geometry of the one-dimensional zigzag FM chain, it is easy to imagine that the checkerboard-type charge ordering and (3x!r/3y!r) orbital-ordering pattern will be stabilized. Furthermore, the charge con"nement in the straight segment (sites 2 and 4 in Fig. 3.5.7), will naturally lead to charge stacking along the z-axis, with stability caused by the special geometry of the zigzag structure. Thus, the complex spin-chargeorbital structure for half-doped manganites can be understood intuitively simply from the viewpoint of its band-insulating nature.

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3.5.7. Bi-stripe structure at x'0.5 In the previous subsection, the discussion focused on the CE-type AFM phase at x"0.5. Naively, it may be expected that similar arguments can be extended to the regime x'1/2, since in the phase diagram for La Ca MnO , the AFM phase has been found at low temperatures in the \V V  region 0.50(x:0.88. Then, let us try to consider the band-insulating phase for density x"2/3 based on H (as de"ned in Section 3.5), without both the JT phononic and Coulombic interactions, since this doping is quite important for the appearance of the bi-stripe structure, as already discussed in previous sections (see Mori et al., 1998a). Following the discussion on the periodicity of the optimal zigzag path, at x"2/3 it is enough to consider the zigzag structure with M"6. After several calculations for x"2/3, as reported by Hotta et al. (2000), the lowest-energy state was found to be characterized by the straight path, not the zigzag one, leading to the C-type AFM phase which was also discussed in previous sections (for a visual representation of the C-type state see Fig. 4 of Kajimoto et al., 1999). At "rst glance, the zigzag structure, for instance the x, x, x, y, y, y-type path, could be the ground state for the same reason, as it occurs in the case of x"0.5. However, while it is true that the state with such a zigzag structure is a band insulator, the energy gain due to the opening of the bandgap is not always the dominant e!ect. In fact, even in the case of x"0.5, the energy of the bottom of the band for the straight path is !2t , while for the  zigzag path, it is !(3t . For x"1/2, the energy gain due to the gap opening overcomes the  energy di!erence at the bottom of the band, leading to the band-insulating ground state. However, for x"2/3 even if a band-gap opens the energy of the zigzag structure cannot be lower than that of the metallic straight-line phase. Intuitively, this point can be understood as follows: An electron can move smoothly along the one-dimensional path if it is straight. However, if the path is zigzag, `re#ectiona of the wave function occurs at the corner, and then a smooth movement of one electron is no longer possible. Thus, for small numbers of carriers, it is natural that the ground state is characterized by the straight path to optimize the kinetic energy of the e -electrons.  However, in neutron scattering experiments a spin pattern similar to the CE-type AFM phase has been suggested (Radaelli et al., 1999). In order to stabilize the zigzag AFM phase to reproduce those experiments it is necessary to include the JT distortion e!ectively. As discussed in Hotta et al. (2000), a variety of zigzag paths could be stabilized when the JT phonons are included. In such a case, the classi"cation of zigzag paths is an important issue to understand the competing `bi-stripea vs. `Wigner-crystala structures. The former has been proposed by Mori et al. (1998b), while the latter was claimed to be stable by Radaelli et al. (1999). In the scenario by Hotta et al. (2000), the shape of the zigzag structure is characterized by the `winding numbera w associated with the Berry-phase connection of an e -electron parallel-transported through Jahn}Teller centers,  along zigzag one-dimensional paths. Namely, it is de"ned as



w"

dr  . 2


(112)

This quantity has been proven to be an integer, which is a topological invariant (see Hotta et al., 1998). Note that the integral indicates an accumulation of the phase di!erence along the onedimensional FM path in the unit length M. This quantity is equal to half of the number of corners included in the unit path, which can be shown as follows. The orbital polarizes along the hopping direction, indicating that i "2
/3(4
/3) along the x-(y-)direction, as was pointed out above. This

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is simply the double exchange mechanism in the orbital degree of freedom. Thus, the phase does not change in the straight segment part, indicating that w"0 for the straight-line path. However, when an e -electron passes a corner site, the hopping direction is changed, indicating that the phase  change occurs at that corner. When the same e -electron passes the next corner, the hopping  direction is again changed. Then, the phase change in i after moving through a couple of corners should be 2
, leading to an increase of unity in w. Thus, the total winding number is equal to half of the number of corners included in the zigzag unit path. Namely, the winding number w is a good label to specify the shape of the zigzag one-dimensional FM path. After several attempts to include e!ectively the JT phonons, it was found that the bi-stripe phase and the Wigner crystal phase universally appear for w"x/(1!x) and w"1, respectively. Note here that the winding number for the bi-stripe structure has a remarkable dependence on x, re#ecting the fact that the distance between adjacent bi-stripes changes with x. This x-dependence of the modulation vector of the lattice distortion has been observed in electron microscopy experiments (Mori et al., 1998). The corresponding zigzag paths with the charge and orbital ordering are shown in Fig. 3.5.9. In the bi-stripe structure, the charge is con"ned in the short straight segment as in the case of the CE-type structure at x"0.5. On the other hand, in the Wigner-crystal structure, the straight segment includes two sites, indicating that the charge prefers to occupy either of these sites. Then, to minimize the JT energy and/or the Coulomb repulsion, the e electrons are distributed with equal spacing. The corresponding spin structure is shown in Fig.  3.5.10. A di!erence in the zigzag geometry can produce a signi"cant di!erent in the spin structure. Following the de"nitions for the C- and E-type AFM structures (see Wollan and Koehler (1955) and introductory section of this review), the bi-stripe and Wigner crystal structure have C E \V V type and C E -type AFM spin arrangements, respectively. Note that at x"1/2, half of the plane V \V is "lled by the C-type, while another half is covered by the E-type, clearly illustrating the meaning of `CEa in the spin structure of half-doped manganites. As for the charge structure along the z-axis for x"2/3 shown in Fig. 3.5.11, a remarkable feature can be observed. Due to the con"nement of charge in the short straight segment for the bi-stripe phase, the charge stacking is suggested from our topological argument. On the other hand, in the Wigner-crystal-type structure, charge is not stacked, but it is shifted by one lattice constant to avoid the Coulomb repulsion. Thus, if the charge stacking is also observed in the experiment for

Fig. 3.5.9. (a) Path with w"1 at x"1/2. Charge and orbital densities are calculated in the MFA for E "2t. At each (2 site, the orbital shape is shown with its size in proportion to the orbital density. (b) The BS-structure path with w"2 at x"2/3. (c) The BS-structure path with w"3 at x"3/4. (d) The WC-structure path with w"1 at x"2/3. (e) The WC-structure path with w"1 at x"3/4.

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83

Fig. 3.5.10. (a) C- and E-type unit cell (Wollan and Koehler, 1955). (b) The spin structure in the a}b plane at x"1/2. Open and solid circle denote the spin up and down, respectively. The thick line indicates the zigzag FM path. The open and shaded squares denote the C- and E-type unit cells. At x"1/2, C-type unit cell occupies half of the 1D plane, clearly indicating the `CEa-type phase. (c) The spin structure at x"2/3 for Wigner-crystal type phase. Note that 66% of the 2D lattice is occupied by C-type unit cell. Thus, it is called `C Ea-type AFM phase. (d) The spin structure at x"2/3 for bi stripe-type phase. Note that 33% of the 2D lattice is occupied by C-type unit cell. Thus, it is called `CE a-type AFM  phase.

Fig. 3.5.11. Schematic "gures for spin, charge, and orbital ordering for (a) WC and (b) BS structures at x"2/3. The open and solid symbols indicate the spin up and down, respectively. The FM 1D path is denoted by the thick line. The empty sites denote Mn> ions, while the robes indicate the Mn> ions in which 3x!r or 3y!r orbitals are occupied.

x"2/3, our topological scenario suggests the bi-stripe phase as the ground state in the lowtemperature region. To "rmly establish the "nal `winnera in the competition between the bi-stripe and Wigner-crystal structure at x"2/3, more precise experiments, as well as quantitative calculations, will be needed in the future.

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3.5.8. Charge order at x(0.5 Regarding densities smaller than 0.5, the states at x"1/8,1/4 and 3/8 have received considerable attention recently (see Mizokawa et al., 1999; Korotin et al., 1999; Hotta and Dagotto, 2000). These investigations are still in a `#uida state, and the experiments are not quite decisive yet, and for this reason, this issue will not be discussed in much detail here. However, without a doubt, it is very important to clarify the structure of charge-ordered states that may be in competition with the ferromagnetic states in the range in which the latter is stable in some compounds. `Stripesa may emerge from this picture, as recently remarked in experiments (Adams et al., 2000; Dai et al., 2000; Kubota et al., 2000. See also Vasiliu-Doloc et al., 1999) and calculations (Hotta et al., 2000c), and surely the identi"cation of charge/orbital arrangements at x(0.5 will be an important area of investigations in the very near future. Here a typical result for this stripe-like charge ordering is shown in Fig. 3.5.12, in which the lower-energy orbital at each site is depicted, and its size is in proportion to the electron density occupying that orbital. This pattern is theoretically obtained by the relaxation technique for the optimization of oxygen positions, namely including the cooperative JT e!ect. At least in the strong electron}phonon coupling region, the stripe charge ordering along the diagonal direction in the x}y plane becomes the global ground state. Note, however, that many meta-stable states can appear very close to this ground state. Thus, the shape of the stripe is considered to #uctuate both in space and time, and in experiments it may occur that only some fragments of this stripe can be detected. It should also be emphasized that the orbital ordering occurs concomitant with this stripe charge ordering. In the electron-rich region, the same antiferro orbital-order exists as that corresponding

Fig. 3.5.12. Orbital densities in the FM phase for (a) x"1/2, (b) 1/3, and (c) 1/4. The charge density in the lower-energy orbital is shown, and the size of the orbital is in proportion to this density. The broken line indicates one of the periodic paths to cover the whole 2D plane.

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to x"0.0. On the other hand, the pattern around the diagonal array of electron-poor sites is quite similar to the building block of the charge/orbital structure at x"0.5. If these "gures are rotated by 453, the same charge and orbital structure is found to stack along the b-axis. Namely, it is possible to cover the whole 2D plane by some periodic charge-orbital array along the a-axis (see, for instance, the broken-line path). If this periodic array is taken as the closed loop C in Eq. (112), the winding numbers are w"1, 2, and 3, for x"1/2, 1/3, and 1/4, respectively. Note that in this case w is independent of the path along the a-axis. A relation w"N /2 holds only  when the 1D FM path is "xed in the AFM spin arrangement. The results imply a general relation w"(1!x)/x for the charge-orbital stripe in the FM phase, re#ecting the fact that the distance between the diagonal arrays of holes changes with x. Our topological argument predicts stable charge-orbital stripes at special doping such as x"1/(1#w), with w an integer. This orbital ordering can be also interpreted as providing a `
a-shift in the orbital sector, by analogy with the dynamical stripes found in cuprates (see, for instance, Buhler et al., 2000), although in copper oxides the charge/spin stripes mainly appear along the x- or y-direction. The study of the similarities and di!erences between stripes in manganites and cuprates is one of the most interesting open problems in the study of transition metal oxides, and considerable work is expected in the near future. Finally, a new zigzag AFM spin con"guration for x(0.5 is here brie#y discussed (Hotta et al., 2000c). In Fig. 3.5.13, a schematic view of this novel spin-charge-orbital structure on the 8;8 lattice at x"1/4 is shown, deduced using the numerical relaxation technique applied to cooperative Jahn}Teller phonons in the strong-coupling region. This structure appears to be the global ground state, but many excited states with di!erent spin and charge structures are also found with small excitation energy, suggesting that the AFM spin structure for x(0.5 in the layered manganites is

Fig. 3.5.13. Schematic representation of the spin-charge-orbital structure at x"1/4 in the zigzag AFM phase at low temperature and large electron}phonon coupling. The symbol convention is the same as in Fig. 3.5.11. This "gure was obtained using numerical techniques, and cooperative phonons, for J "R and J "0.1t. For the noncooperative & $ phonons, basically the same pattern can be obtained. Reproduced from Hotta et al. (2000c).

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easily disordered due to this `quasi-degeneracya in the ground state. This result may be related to the `spin-glassa nature of the single layer manganites reported in experiments (see Moritomo et al., 1995). It should be noted that the charge-orbital structure is essentially the same as that in the 2D FM-phase (see Fig. 3.5.12). This suggests the following scenario for the layered manganites: When the temperature is decreased from the higher-temperature region, "rst charge ordering occurs due to the cooperative Jahn}Teller distortions in the FM (or paramagnetic) region. If the temperature is further decreased, the zigzag AFM spin arrangement is stabilized, adjusting itself to the orbital structure. Thus, the separation between the charge ordering temperature ¹ and the NeH el !temperature ¹ occurs naturally in this context. This is not surprising, since ¹ is due to the , !electron}lattice coupling, while ¹ originates in the coupling J . However, if the electron}phonon , $ coupling is weak, then ¹ becomes very low. In this case, the transition to the zigzag AFM phase !may occur prior to the charge ordering. As discussed above, the e -electron hopping is con"ned to  one-dimensional structures in the zigzag AFM environment. Thus, in this situation, even a weak coupling electron}phonon coupling can produce the charge-orbital ordering, as easily understood from the Peierls instability argument. Namely, just at the transition to the zigzag AFM phase, the charge-orbital ordering occurs simultaneously, indicating that ¹ "¹ . Note also that in the !, zigzag AFM phase, there is no essential di!erence in the charge-orbital structures for the noncooperative and cooperative phonons, due to the one dimensionality of those zigzag chains. 3.6. Pseudo-gap in mixed-phase states Recent theoretical investigations suggest that the density of states (DOS) in mixed-phase regimes of manganites may have `pseudo-gapa characteristics, namely a prominent depletion of weight at the chemical potential. This feature is similar to that extensively discussed in copper oxides. The calculations in the Mn-oxide context have been carried out using both the one- and two-orbital models, with and without disorder (see Moreo et al., 1999b; Moreo et al., 2000). Typical results are shown in Fig. 3.6.1 Part (a) contains the DOS of the one-orbital model on a 2D cluster varying the electronic density slightly below n"1.0, as indicated in the caption. At zero temperature, this density regime is unstable due to phase separation, but at the temperature of the simulation those densities still correspond to stable states, but with a dynamical mixture of AF and FM features (as observed, for instance, in Monte Carlo snapshots of the spin con"gurations). A clear minimum in the DOS at the chemical potential can be observed. Very similar results appear also in 1D simulations (Moreo et al., 1999b). Part (b) contains results for two-orbitals and a large electron}phonon coupling, this time at a "xed density and changing temperature. Clearly a pseudogap develops in the system as a precursor of the phase separation that is reached as the temperature is further reduced. Similar results have been obtained in other parts of parameter space, as long as the system is near unstable phase-separated regimes. A pseudo-gap appears also in cases where disorder is added to the system. In Fig. 3.6.1c, taken from Moreo et al. (2000), results can be found for the case where a random on-site energy is added to the one-orbital model. A tentative explanation of this phenomenon for the case without disorder was described by Moreo et al. (1999b), and it is explained in Fig. 3.6.2. In part (a) a typical mixed-phase FM}AF state is sketched. Shown are the localized spins. In the FM regions, the e -electrons improve their kinetic  energy, and thus they prefer to be located in those regions as shown in (b). The FM domains act as e!ective attractive potentials for electrons, as sketched in part (c). When other electrons are added,

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Fig. 3.6.1. (a) DOS of the one-orbital model on a 10;10 cluster at J "Rand temperature ¹"1/30 (hoping t"1). The & four lines from the top correspond to densities 0.90, 0.92, 0.94, and 0.97. The inset has results at n"0.86, a marginally stable density at ¹"0. (b) DOS of the two-orbital model on a 20-site chain, working at n"0.7, J "8, and "1.5. & Starting from the top at - "0, the three lines represent temperatures 1/5, 1/10, and 1/20, respectively. Here the hopping along x between orbitals a is the unit of energy. Both, (a) and (b) are taken from Moreo et al. (1999b). (c) DOS using a 20-site chain of the one-orbital model at ¹"1/75, J "8, n"0.87, and at a chemical potential such that the & system is phase separated in the absence of disorder. = regulates the strength of the disorder, as explained in Moreo et al. (2000) from where this "gure was taken.

FM clusters are created and new occupied levels appear below the chemical potential, creating a pseudogap (part (d) of Fig. 3.6.2). The DOS is clearly nonrigid. These results are compatible with the photoemission experiments by Dessau et al. (1998) for bilayer manganites. Other features of the experiments are also reproduced such as the large width of the peaks, and the momentum independence of the results. This agreement adds to the notion pursued in this review that mixed-phase states are important to understand the behavior of manganese oxides. The reduction of the DOS at the chemical potential is also compatible with the insulating characteristics of the bilayers in the regime of the photoemission experiments. It is conceivable that the manganites present a pseudogap regime above their Curie and NeH el temperatures, as rich as that found in the cuprates. More details are given in the Discussion section.

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Fig. 3.6.2. Schematic explanation of pseudogap formation at low electronic density (taken from Moreo, et al., 1999b). In (a) a typical Monte Carlo con"guration of localized spins is shown. In (b), the corresponding electronic density is shown. In (c), the e!ective potential felt by electrons is presented. A populated cluster band (thick line) is formed. In (d), the resulting DOS is shown. Figure taken from Moreo et al. (1999b).

3.7. Phase separation caused by the inyuence of disorder on xrst-order transitions Although it is frequently stated in the literature that a variety of chemical substitutions in manganites lead to modi"cations in the bandwidth due to changes in the `averagea A-site cation radius r , this statement is only partially true. Convincing analysis of data and experiments by  Rodriguez-Martinez and Att"eld (1996) have shown that the disorder introduced by chemical replacements in the A-sites is also crucially important in determining the properties of manganites. For instance, Rodriguez-Martinez and Att"eld (1996) found that the critical temperature ¹ can be ! reduced by a large factor if the variance  of the ionic radii about the mean r  is modi"ed,  keeping r  constant. Rodriguez-Martinez and Att"eld (1996) actually observed that maximum  magnetoresistance e!ects are found in materials not only with a low value of r  (small  bandwidth) but also a small value of . A good example is Pr Ca MnO since the Pr> and \V V  Ca> ions are similar in size (1.30 and 1.34 As , respectively, according to Tomioka and Tokura (1999)). Disorder, as described in the previous paragraph, is important for the phase separation scenario. The recent experimental results showing the existence of micrometer size coexisting clusters in (La Pr )Ca MnO (LPCMO) by Uehara et al. (1999), to be reviewed in detail later, \W W   highlights a property of manganites that appears universal, namely the presence of intrinsic inhomogeneities in the system, even in single crystals. This issue is discussed at length in various sections of this review. In the theoretical framework described thus far, the scenario that is the closest to predicting such inhomogeneous state is the one based on electronic phase separation. However, the analysis presented before when considering the in#uence of long-range Coulomb interactions over a phase separated state, led us to believe that only nanometer size coexisting clusters are to be expected in this problem. Those found in LPCMO are much larger, suggesting that there must be another mechanism operative in manganites to account for their formation.

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A possible explanation of the results of Uehara et al. (1999) has been recently proposed by Moreo et al. (2000), and it could be considered as a form of `disorder-induceda or `structurala phase separation, rather than electronic. The idea is based on the in#uence of disorder over the "rst-order metal}insulator (or FM}AF) transition found in models where the interactions are translationally invariant (without disorder), as it was described in Sections 3.4 and 3.5 When such a transition occurs, abruptly a metal changes into an insulator, as either concentrations or couplings are suitably changed. Unless metastable states are considered, there is no reason to assume that in the actual stable ground state of this system coexisting clusters will be found, namely the state is entirely FM or AF depending on parameters. However, di!erent is the situation when disorder is considered into the problem. The type of disorder taken into account by Moreo et al. (2000) is based on the in#uence of the di!erent ionic radius of the various elements that compose the manganites, as discussed at the beginning of this section. Depending on the environment of A-type ions (which in LPCMO involve La, Pr or Ca) a given Mn}O}Mn bond can be straight (1803) or distorted with an angle less than 1803. In the latter, the hopping across the bond under study will be less than the ideal one. For a schematic representation of this idea see Fig. 3.7.1. The random character of the distribution of A ions, leads to a concomitant random distribution of hoppings, and also random exchange between the localized spins J since this quantity is also in#uenced by $ the angle of the Mn}O}Mn bond. To account for this e!ect, Moreo et al. (2000) studied the one- and two-orbital models for manganites described before, including a small random component to both the hoppings and J . $ This small component did not in#uence the FM- and AF-phase much away from their transition boundary, but in the vicinity of the "rst-order transition its in#uence is important. In fact, numerical studies show that the transition now becomes continuous, with FM and AF clusters in coexistence in a narrow region around the original transition point. Typical results are shown in Fig. 3.7.2a}f, using one-dimensional clusters as an example. In the two upper frames, the energy versus J (or J) is shown at "xed values of the other couplings such $ as J and , in the absence of disorder and at a "xed density x"0.5. The abrupt change in the & slope of the curves in (a) and (d) clearly shows that the transition is indeed "rst order. This is a typical result that appears recurrently in all Monte Carlo simulations of manganite models, namely FM and AF are so di!erent that the only way to change from one to the other at low temperature is abruptly in a discontinuous transition (and spin-canted phases have not been found

Fig. 3.7.1. Schematic representation of the in#uence of the A-site ionic size on the hopping amplitude `ta between two Mn ions.

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Fig. 3.7.2. Results that illustrate the generation of `gianta coexisting clusters in models for manganites (taken from Moreo et al., 2000). (a}c) are Monte Carlo results for the two-orbital model with n"0.5, ¹"1/100, J "R, "1.2, & t"1, PBC, and using a chain with ¸"20 sites. (a) is the energy per site vs. J /t for the nondisordered model, with level $ crossing at 0.21. (b) MC averaged nearest-neighbor t -spins correlations vs. position along the chain (denoted by i) for  one set of random hoppings t? and J couplings (J /t at every site is between 0.21! and 0.21# with "0.01).
$ $ FM and AF regions are shown. For more details see Moreo et al. (2000). (c) Same as (b) but with "0.05. (d}f): results for the one-orbital model with n"0.5, ¹"1/70, J "R, t"1, open boundary conditions, and ¸"64 (chain). (d) is & energy per site vs. J for the nondisordered model, showing the FM}AF states level crossing at J &0.14. (e) are the $ $ MC averaged nearest-neighbor t -spin correlations vs. position for one distribution of random hoppings and  t exchanges, such that J /t is between 0.14- and 0.14# with "0.01. (f) Same as (e) but with "0.03.  $

in our analysis in the absence of magnetic "elds, as possible intermediate phases between FM and AF). These results are drastically changed upon the application of disorder, as shown in frames (b, c, e, and f ) of Fig. 3.7.2, where the mean couplings have been "xed such that the model is located exactly at the "rst-order transition of the nondisordered system. In these frames, the nearestneighbor spin correlations along the chain are shown. Clearly, this correlation is positive in some portions of the chain, while it alternates from positive to negative in others. This alternation is compatible with an AF state, with an elementary unit cell of spins in the con"guration up}up}down}down, but the particular form of the AF state is not important in the following; only its competition with other ordered states, such as the FM one is signi"cant. The important point is that there are coexisting FM and AF regions. The cluster size is regulated by the strength of the disorder, such that the smaller the disorder, the larger the cluster size. Results such as those in Fig. 3.7.2 have appeared in all simulations carried out in this context, and in dimensions larger than one (see Moreo et al., 2000). The conclusions appear independent of the particular type of AF insulating state competing with the FM-state, the details of the distribution of random numbers used, and the particular type of disorder considered which could also be in the form of a random on-site energy in some cases (Moreo et al., 2000). Note that the coexisting clusters have the same density, namely these are FM- and AF-phase that appear at a "xed hole concentration in the nondisordered

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models, for varying couplings. Then, the problem of a large penalization due to the accumulation of charge is not present in this context. What is the origin of such a large cluster coexistence with equal density? There are two main opposing tendencies acting in the system. On one hand, energetically it is not convenient to create FM}AF interfaces and from this perspective a fully homogeneous system is preferable. On the other hand, locally at the level of the lattice spacing the disorder in t and J alter the couplings $ such that the system prefers to be either on the FM- or AF-phase, since these couplings #uctuate around the transition value. From the perspective of the disorder, the clusters should be as small as possible such that the local di!erent tendencies can be properly accounted for. From this competition emerges the large clusters of Fig 3.7.2, namely by creating large clusters, the number of interfaces is kept small while the local tendencies toward one phase or the other are partially satis"ed. `Largea here means substantially larger in size than the lattice spacing. A region where accidentally the distribution of random couplings favors the FM- or AF-state on average, will nucleate such a phase in the form of a bubble. These simple ideas can be made more elegant using the well-known arguments by Imry and Ma (1975), which were applied originally to the random "eld Ising model (RFIM) (see contributions on the subject in the book of Young, 1998), namely a model with a FM Ising interaction among spins of a lattice in the presence of a magnetic "eld in the z-direction which changes randomly from site to site. This local "eld is taken from a distribution of random numbers of width 2=. In the context of manganites it can be imagined that the spin up and down of the RFIM represent the two states in competition (metal and insulator) in the real compounds. The random "eld represents the local tendency to prefer either a metal or an insulator, due to the #uctuations in the disorder of the microscopic models. As a function of an external uniform magnetic "eld, the RFIM at zero temperature has a "rst-order transition at zero external "eld in the absence of random "elds (between all spins up and all down), which turns continuous as those random "elds are added, quite similar to the case described above in the FM}AF competition. Then, the RFIM captures at least part of the physics of Mn-oxides that emerged from the study of realistic Hamiltonians in the presence of disorder, as shown above. For this reason it is instructive to study this simple spin model, which can be analyzed on lattices much larger than those that can be reached with the one or two orbital models of Section 3.3. However, note that the use of the RFIM is only to guide the intuition, but it is not claimed that this model belongs to exactly the same universality class as the microscopic Hamiltonians for Mn-oxides used here. The study of universality is very complex and has not been addressed in this context yet. Nevertheless, it is expected that the RFIM will at least provide some intuition as to how real manganites behave. Typical results are shown in Fig. 3.7.3. In part (a), the data corresponding to a simulation at low temperature on a 100;100 cluster for a "xed set of random "elds is shown. The clusters are basically frozen, namely the result is representative of the ground state. The presence of coexisting clusters of spins up and down is clear. Their distribution is certainly random, and their shape fractalic, similar to that observed in experiments for LPCMO. Upon reduction of =, in frame (b) results now for a 500;500 cluster show that the typical size of the clusters grow and can easily involve a few hundred lattice spacings. When an external "eld is applied, a percolation among disconnected clusters emerges. This is a very important point, in agreement with the expectations arising from several experiments, namely percolative characteristics should appear in real manganites to the extend that the theoretical investigations presented in this section are correct. Uehara

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Fig. 3.7.3. Results of a Monte Carlo simulation of the random "eld Ising model at ¹"0.4 (J"1), with PBC, taken from Moreo et al. (2000). The dark (white) small squares represent spins up (down). At ¹"0.4 the thermal #uctuations appear negligible, and the results shown are those of the lowest energy con"guration. (a) was obtained for a random "eld with strength ="3 taken from a box distribution [!=, =], external "eld H "0, using a 100;100 cluster, and one set of  random "elds hi . (b) Results using a 500;500 cluster with ="1.2 and for one "xed con"guration of random "elds. The dark regions are spins up in the H "0 case, the grey regions are spins down at zero "eld that have #ipped to up at  H "0.16, while the white regions have spins down with and without the "eld. The percolative-like features of the giant  clusters are apparent in the zero-"eld results. Special places are arrow-marked where narrow spin-down regions have #ipped linking spin-up domains. For more details see Moreo et al. (2000).

et al. (1999) and other experimentalists intuitively concluded that indeed percolation is important in the study of Mn-oxides, and in the following section it will be shown that it plays a key role in rationalizing the dc resistivity of these compounds. Gor'kov and Kresin (1998) also brie#y discussed a possible percolation process at low temperature.

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Summarizing, phase separation can be driven by energies other than purely electronic. In fact, it can also be triggered by the in#uence of disorder on "rst-order transitions. In this case the competing clusters have the same density and for this reason can be very large. Micrometer size clusters, such as those found in the RFIM, are possible in this context, and have been observed in experiments. This result is very general, and should apply to a variety of compounds where two very di!erent ordered states are in competition at low temperatures. The remarkable phenomenological results of Rodriguez-Martinez and Att"eld (1996) appear to be in qualitative agreement with the theoretical calculations. As explained above, Moreo et al. (2000) found that the size of the clusters induced by disorder near a transition, such as those produced by chemical substitutions in real manganites, which would be of "rst order in the clean limit, can be controlled by the `strengtha of that disorder. In practice, this strength is monotonically related to  (in the limit "0 there is no disorder). At small (but not vanishing) or disorder in the calculations of Moreo et al. (2000), the coexisting clusters are large. As the disorder grows, the clusters reduce their size. To the extent that the size of the coexisting clusters is directly proportional the strength of the CMR e!ect, then weak disorder is associated with large magnetoresistance changes with the composition, magnetic "elds or pressure, a somewhat counter-intuitive result since naively strong disorder could have been expected to lead to larger modi"cations in the resistivity. 3.8. Resistivity of manganites in the mixed-phase regime One of the main lessons learned from the previous analysis of models for manganites is that intrinsic inhomogeneities are very important in this context. It is likely that the real Mn-oxides in the CMR regime are in such a mixed-phase state, a conclusion that appears inevitable based on the huge recent experimental literature, to be reviewed in the next section, reporting phase separation tendencies in some form or another in these compounds. However, note that until recently estimations of the dc resistivity  in such a mixed-phase regime were not available. This was  unfortunate since the interesting form of the  vs. temperature curves, parametric with magnetic  "elds, is one of the main motivations for the current huge e!ort in the manganite context. However, the lack of reliable estimations of  is not accidental: it is notoriously di$cult to calculate  transport properties in general, and even more complicated in regions of parameter space that are expected to be microscopically inhomogeneous. Although there have been some attempts in the literature to calculate  , typically a variety of approximations that are not under control have  been employed. In fact, the micrometer size of some of the coexisting clusters found in experiments strongly suggest that a fully microscopic approach to the problem will likely fail since, e.g., in a computational analysis it would be very di$cult to study su$ciently large clusters to account for such large scale structures. It is clear that a more phenomenological approach is needed in this context. For all these reasons, recently Mayr et al. (2000) carried out a study of  using a random resistor  network model (see Kirkpatrick, 1973), and other approximations. This model was de"ned on square and cubic lattices, but with a lattice spacing much larger than the 4 As distance between nearest-neighbor Mn ions. A schematic representation is presented in Fig. 3.8.1. Actually, the new lattice spacing is a fraction of micrometer, since the random network tries to mimic the complicated fractalic-like structure found experimentally. At each link in this sort of e!ective lattice, randomly

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Fig. 3.8.1. Schematic representation of the random resistor network approximation. On the left is a sketch of the real system with metallic and insulating regions. On the right is the resistor network where dark (light) resistances represent the insulator (metal). `aa is the Mn}Mn lattice spacing, while ¸ is the actual lattice spacing of the resistor network. Fig. 3.8.2. Net resistivity  of a 100;100 random resistor network cluster vs. temperature, at the indicated metallic  fractions p (result taken from Mayr et al., 2000). Inset: Results for a 20 cluster with (from the top) p"0.0,0.25,0.3,0.4 and 0.5. In both cases, averages over 40 resistance con"gurations were made. The p"1 and 0 limits are from the experiments corresponding to LPCMO (see Uehara et al., 1999). Results on 200;200 clusters (not shown) indicate that size e!ects are negligible.

either a metallic or insulating resistance was located in such a way that the total fraction of metallic component was p, a number between 0 and 1. The actual values of these resistances as a function of temperature were taken from experiments. Mayr et al. (2000) used the  (¹) plots obtained by Uehara et al. (1999) corresponding to  (La Pr )Ca MnO (LPCMO), one of the compounds that presents the coexistence of giant \W W   FM and CO clusters at intermediate values of the Pr concentration. More speci"cally, using for the insulating resistances the results of LPCMO at y"0.42 (after the system becomes a CO state with increasing Pr doping) and for the metallic ones the results at y"0.0 (which correspond to a metallic state, at least below its Curie temperature), the results of a numerical study on a 100;100 cluster are shown in Fig. 3.8.2 (the Kircho! equations were solved by a simple iterative procedure). It is interesting to observe that, even using such a simple phenomenological model, the results are already in reasonable agreement with the experiments, namely, (i) at large temperature insulating behavior is observed even for p as large as 0.65 (note that the classical percolation is expected to occur near p"0.5; see Kirkpatrick, 1973); (ii) at small temperature a (`bada) metallic behavior appears; and (iii) a broad peak exists in between. Results in both 2D and 3D lead to similar conclusions. It is clear that the experimental results for manganites can be at least partially accounted for within the mixed-phase scenario. The results of Fig. 3.8.2 suggest a simple qualitative picture to visualize why the resistivity in Mn-oxides has the peculiar shape it has. The relevant state in this context should be imagined as percolated, as sketched in Fig. 3.8.3a as predicted by the analysis of the previous section. Metallic "laments from one side of the sample to the other exist in the system. At low temperature,

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conduction is through those "laments. Necessarily,  at ¹"0 must be large, in such a perco lative regime. As temperature increases, the  of the "laments grows as in any metal. However, in  the other limit of large or room temperature, the resistance of the percolated metallic "lament is expected to be much larger than that corresponding to one of the insulator paths. Actually, near room temperature in many experimental graphs, it can be observed that  in the metallic and  insulating regimes are quite similar in value, even comparing results away from the percolative region. Then, at room temperature it is more likely that conduction will occur through the insulating portions of the sample, rather than through the metallic "laments. Thus, near room temperature insulating behavior is expected. In between low and high temperatures, it is natural that  will present a peak. Then, a simple `two resistances in parallela description appears  appropriate (see Fig. 3.8.3b). The insulating resistance behaves like any insulator, while the metallic one starts at ¹"0 at a high value and then it behaves like any metal. The e!ective resistance shown in Fig. 3.8.3b properly reproduces the experiments at least qualitatively. Note, however, that many experimental results suggest that  has an intermediate temperature  peak sharper than shown in Fig. 3.8.2. In some compounds this is quite notorious, while in others the peak is fairly broad as in Fig. 3.8.2. Nevertheless, it is important to "nd out alternative procedures to sharpen the  peak to better mimic experiments. One possible solution to this  problem is to allow for the metallic fraction p to vary with temperature. This is a reasonable assumption since it is known that the metallic portions of the sample in mixed-phase manganites originate in the ferromagnetic arrangement of spins that improves conduction. The polarization of the spins deteriorates as the temperature increases, and it is reasonable to imagine that the FM islands decrease in size as the temperature grows. Then, a pattern of FM clusters that are connected at low temperature leading to a metallic behavior may become disconnected at higher temperatures. The tendencies toward a metallic percolation decrease with increasing temperature. Such a conjecture was studied qualitatively by Mayr et al. (2000) using the random "eld Ising model and the one-orbital model for Mn-oxides. In both cases, indications of the disappearance of percolation with increasing temperature were indeed found. Then, assuming that p decreases with increasing temperature, approximately following the magnetization, seems a reasonable assumption. Results with a temperature dependent p are shown in Fig. 3.8.4. The actual values of p are indicated, at least in part, in the "gure. Certainly the peak is now sharper than in Fig. 3.8.2, as expected, and the results indeed resemble those found in a variety of experiments. Note that the function p"p(¹) has not been "ne-tuned, and actually a variety of functions lead to similar conclusions as those in Fig. 3.8.4. Note also that obtaining such a result from a purely microscopic approach would have been quite di$cult, although Mayr et al. (2000) showed that data taken on small clusters using the one-orbital model are at least compatible with those of the phenomenological approach. To evaluate the conductance of these clusters, the approach of Datta (1995) and VergeH s (1999) were used. Also note that calculations using a cubic cluster with either metallic or insulating `hoppinga (Avishai and Luck, 1992), to at least partially account for quantum e!ects, lead to results similar to those found in Fig. 3.8.4. The success of the phenomenological approach described above leads to an interesting prediction. In the random resistor network, it is clear that above the peak in the resistivity, the mixed-phase character of the system remains, even with a temperature dependent metallic fraction p. Then, it is conceivable to imagine that above the Curie temperature in real manganites, a substantial fraction of the system should remain in a metallic FM-state (likely not percolated, but

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Fig. 3.8.3. (a) Schematic representation of the mixed-phase state near percolation. The arrows indicate conduction either through the insulating or metallic regions depending on temperature (see text). (b) Two resistances in parallel model for Mn-oxides. The (schematic) plot for the e!ective resistance R vs. ¹ arises from the parallel connection of metallic  (percolative) R and insulating R resistances. Figure taken from Mayr et al. (2000). + ' Fig. 3.8.4. Net resistivity  of the 100;100 random resistor network used in the previous "gure, but with a metallic  fraction p changing with ¹. Representative values of p are indicated. Results averaged over 40 resistance con"gurations are shown (taken from Mayr et al., 2000).

forming disconnected clusters). A large variety of experiments reviewed in the next section indeed suggest that having FM clusters above ¹ is possible. As a consequence, this has led us to ! conjecture that there must exist a temperature ¹H at which those clusters start forming. This de"nes a new temperature scale in the problem, somewhat similar to the famous pseudo-gap ¹H scale of the high-temperature superconducting compounds. In fact, in mixed phase FM}AF states it is known that a pseudo-gap appears in the density of states (Sections 3.6 and 3.7; Moreo et al., 1999b, 2000), thus increasing the analogy between these two materials. In our opinion, the experimental veri"cation that indeed such a new scale ¹H exists in manganites is important to our understanding of these compounds. In fact, recent results by Kim et al. (2000) for La Ca MnO \V V  at various densities have been interpreted as caused by small FM segments of the CE-type CO state, appearing at hole densities smaller than x"1/2 and at high temperature. This result is in qualitative agreement with the theoretical analysis presented here. The study of e!ective resistivities and conductances has also been carried out in the presence of magnetic "elds (Mayr et al., 2000), although still mainly within a phenomenological approach. From the previous results Figs. 3.8.2}4, it is clear that in the percolative regime `smalla changes in the system may lead to large changes in the resistivity. For instance, if p changes by only 5% from 0.45 to 0.5 in Fig. 3.8.4,  is modi"ed by two orders of magnitude! It is conceivable that small 

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Fig. 3.8.5. (a) Inverse conductivity of the half-doped one-orbital model on a 64-site chain in the regime of coexisting clusters, with J "R, AF coupling among localized spins J"0.14, t"1, and "0.03, varying a magnetic "eld as & indicated. The data shown corresponds to a particular disorder con"guration, but results with other con"gurations are similar. (b) E!ective resistivity of a 100;100 network of resistances. Results at  "0.0 (full circles, open triangles, and open squares starting at ¹"0 with p"0.45,0.5 and 0.7, respectively) are the same as found in Fig. 3.8.4. Full triangles, inverse open triangles, and diamonds, correspond to the same metallic fractions, but with a small addition to the insulating conductivity ( "0.1 ( cm)\), to simulate the e!ect of magnetic "elds (see text). Results taken from Mayr et al. (2000).

magnetic "elds could induce such small changes in p, leading to substantial modi"cations in the resistivity. Experiments by Parisi et al. (2000) indeed show a rapid change of the fraction of the FM-phase in La Ca MnO upon the application of magnetic "elds. In addition, studies of the      one-orbital model carried out in one dimension (Mayr et al., 2000) also showed that other factors may in#uence the large  changes upon the application of external "elds. For instance, in  Fig. 3.8.5, the inverse conductance C\ of a 64-site chain is shown in the presence of small magnetic "elds (in units of the hopping), in the regime of FM}AF cluster coexistence, which is achieved by the introduction of disorder where a "rst-order FM}AF transition occurs, as discussed in the previous subsection. The results of Fig. 3.8.5 clearly indicate that C\ can indeed change by several orders of magnitude in the presence of small "elds even in a 1D system that certainly cannot have a percolation. There must be some other mechanism at work in this context. Mayr et al. (2000) believe this alternative mechanism is caused by small modi"cations in the conductivity of the insulating portions of the sample, independent of what occurs in the metallic clusters. It is possible that in an AF region, with zero conductivity at large Hund coupling due to the perfect antialignment of the nearest-neighbor t -spins, the small "elds may induce a small canting e!ect that leads  to a nonzero conductivity. While this e!ect should be negligible if the AF-phases is totally dominating, it may become more important if small AF clusters separate FM ones. A sort of `valvea e!ect may occur, in other words magnetic "elds can induce a small connection between metallic states leading to a substantial change in the resistivity. This idea can be studied qualitatively by simply altering by a small amount the conductivity of the insulating regions in the random-resistor network. Results are shown in Fig. 3.8.5b, using the same functions p"p(¹) employed before in Fig. 3.8.4. As anticipated, small conductivity changes lead to large resistivity modi"cations, comparable to those observed in experiments upon the application of magnetic "elds. Although the analysis discussed above is only semi-quantitative and further studies in

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magnetic "elds should actively continue in this context, Mayr et al. (2000) have shown that in the percolative regime two mechanisms (described above) can lead to a large MR, leading to at least a possible framework for describing how the famous CMR e!ect can occur. 3.9. Related theoretical work on electronic phase separation applied to manganites The possibility of `electronica phase separation was already discussed by Nagaev (1967, 1968, 1972) well before it became a popular subject in the context of compounds such as the hightemperature superconductors. Its original application envisioned by Nagaev was to antiferromagnetic semiconductors, where the doping of electrons creates ferromagnetic-phase regions embedded in an AF matrix. Nagaev (1994, 1995) remarked that if the two phases have opposite charge, the Coulombic forces will break the macroscopic clusters into microscopic ones, typically of nanometer-scale size, as remarked in this review before. When the number of these FM clusters is small, the system resembles a regular array of charge sort of a Wigner crystal, as found also in the simulations of Malvezzi et al. (1999), and the system remains an insulator. However, as the density grows, a transition will be found where the clusters start overlapping, and a metal is formed. Although it may seem tempting to assign to this transition percolative properties, as Nagaev does, note that at least without incorporating disorder the clusters are regularly spaced and thus the transition does not correspond to the usual percolative ones described in textbooks and in the previous subsection where the random position of the clusters play a key role. In particular, the critical density at which regularly spaced clusters begin overlapping triggers a process that occurs in all clusters at the same time, di!erent from the notion of a percolative "lament with fractalic shape which is crucial in percolative theories. For this reason it is unclear to these authors to what extend electronic phase separation can describe percolative physics in the absence of disorder. It appears that only when randomly distributed clusters of two phases are stabilized, as described in Section 3.8, can true percolation occur. The calculations of Nagaev (1994, 1995, 1996) have been carried out for one orbital models and usually in the limit where the hopping t of the conduction electrons is much larger than the Hund coupling (although Nagaev expects the results to qualitatively hold even in the opposite limit J 't). Also a low density of carriers was assumed, and many calculations were performed mainly & for the one-electron problem (magneto polaron), and then rapidly generalized to many electrons. The formation of lattice polarons is not included in the approach of Nagaev. These parameters and assumptions are reasonable for AF semiconductors, and Nagaev (1995) argued that his results can explain a considerable body of experimental data for EuSe and EuTe. However, note that Mauger and Mills (1984, 1985) have shown that self-trapped FM polarons (ferrons) are not stable in three dimensions. Instead, Mauger and Mills (1984, 1985) proposed that electrons bound to donor sites induce a ferromagnetic moment, and they showed that those bound magnetic polarons can account for the FM clusters observed in EuTe. Free carriers appear `frozena at low temperatures in these materials, and there are no ferron-like solutions of the underlying equations in the parameter range appropriate to Eu chalcogenides. In addition, note that the manganites have a large J and a large density of electrons, and in & principle calculations such as those described above have to be carried out for more realistic parameters, if the results can indeed apply to manganites. These calculations are di$cult without the aid of computational techniques. In addition, it is clearly important to consider two orbitals to

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address the orbital ordering of the manganites, the possibility of orbital phase separation, and the in#uence of Jahn-Teller or Coulombic interactions that lead to charge-order AF-states. Disorder also appears to play a key role in manganites. The unstable character of the low hole-density region of the phase diagram corresponding to the one-orbital model for manganites has also been analyzed by other authors using mostly analytic approximate techniques. In fact, Arovas and Guinea (1998) found an energy convex at small hole concentration, indicative of phase separation, within a mean-"eld treatment of the one-orbital model using the Schwinger formalism (see also Mishra et al., 1997; Arovas et al., 1999; Guinea et al., 1999; Yi and Lee, 1999; Chattopadhyay et al., 2000; Yuan et al., 2000). Nagaev (1998) using the one-orbital model also arrived at the conclusion that the canted AF-state of the small hole density region is unstable. The same conclusion was obtained in the work of Kagan et al. (1999) where the dominance of phase separation in the small hole-density region was remarked upon, both using classical and quantum spins. Ferromagnetic polarons embedded into an AF surrounding were also discussed by those authors. Polarons in electron-doped one-orbital models where also analyzed by Batista et al. (1998). Nagai et al. (1999) using the dynamical mean-"eld approximation (exact in in"nite dimension) studied the one-orbital model with S"1/2 localized spins. Nagai et al. (1999) (see also Momoi and Kubo, 1998) identi"ed FM-, AF-, and PM-phase. Regimes of phase separation were observed involving the AF- and PM-phase, as well as the PM and FM ones. A representative density vs. chemical potential plot is shown in Fig. 3.9.1. The results obtained by those authors are qualitatively similar to those found using the DMRG in one dimension (Dagotto et al., 1998) for S"1/2 localized spins, and also similar to results obtained in higher dimensions with classical localized spins (Yunoki et al., 1998a). An AF}PM phase separation was also detected in in"nite dimension calculations (Yunoki et al., 1998a), showing that not only AF}FM coexistence is possible. Calculations using (t}J)-like models, derived at large J starting with the one-orbital & model, also reveal phase separation, as shown by Shen and Wang (1998). Overall, it can be safely concluded that using a variety of numerical and analytical techniques, convincing evidence has accumulated that the canted AF-state of deGennes (1960) is simply not stable in models believed to be realistic for manganese oxides. This state is replaced by a mixed-phase or phase-separated regime. The importance of heterogeneity in manganites was also remarked upon by von Molnar

Fig. 3.9.1. Electron density vs. chemical potential in the ground state of the one-orbital model with S"1/2, and a large Hund coupling. AF, P, and F, denote antiferromagnetic, paramagnetic, and ferromagnetic states, respectively. The result is taken from Nagai et al. (1999), where more details can be found.

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and Coey (1998), based on an analysis of several experiments. Also Khomskii (1999) remarked upon the importance of phase-separation and percolation. Other calculations have also shown tendencies to phase separation. For instance, Yamanaka et al. (1998) studied the one-orbital model in two and three dimensions and found phase separation between a #ux and antiferromagnetic states (see also Agterberg and Yunoki, 2000). Working with the one-orbital model, computational studies by Yi and Yu (1998) arrived to the same conclusions, previously presented by Yunoki and Moreo (1998), regarding the presence of PS at both small and large hole density once the direct Heisenberg coupling among the t spins is considered. Golosov  et al. (1998), using a mean-"eld approximation for the one-band model, also found that the spin canted state was unstable, and indications of phase separation were reported. Schlottmann (1999) using a simple alloy-analogy model showed that the system is unstable to phase separation. Symmetry arguments discussed by Zhong and Wang (1999) also led to PS at low hole doping. In the continuum model, PS has also been found (RomaH n and Soto, 1998). Even for the two-orbital model, evidence has accumulated that phase separation is present, particularly at low- and high density of holes. Besides the already described robust computational evidence for the case where the orbital degree of freedom plays the key triggering role for this e!ect (Yunoki et al., 1998b), mean-"eld approximations presented by Okamoto et al. (1999) also detected phase separation involving two phases with the same spin characteristics (ferromagnetic), but di!ering orbital arrangement. A representative result is reproduced in Fig. 3.9.2, where the orbital states are also shown. 3.10. On-site Coulomb interactions and phase separation What happens with phase separation when the on-site Coulomb ; interaction is dominant over other interactions? This question does not have an easy answer due to the technical complications of carrying out reliable calculations with a nonzero ;. In fact, the one-band Hubbard model has been studied for a long time as a model of high temperature superconductors and after more than 10 years of work it is still unclear whether it phase separates in realistic regimes of parameters. Thus, it is not surprising that similar uncertainties may arise in the context of models for manganites. As remarked before, studies of the one-dimensional one-orbital model including a nonzero ; were carried out by Malvezzi et al. (1999). In this study, a region of phase separation was identi"ed in a similar location as obtained in the Monte Carlo simulations without ; (Yunoki et al., 1998). Then, certainly switching on ; `slowlya starting in the phase separated regime of the one-orbital model does not alter the presence of this regime. On the other hand, Shen and Wang (1999a) claimed that if ; is made larger than J , the model does not lead to phase separation & according to their calculations (see also Gu et al., 1999). This issue is somewhat complicated by the well-known fact that pure Hubbard-like models tend to present a large compressibility near half-"lling, namely the slope of the curve density vs. chemical potential is large at that density (Dagotto, 1994). This may already be indicative of at least a tendency to phase separation that could be triggered by small extra terms in the Hamiltonian. More recently, it has been shown that a two-orbital 1D model with a form resembling those studied in manganites (but without localized spins) indeed presents phase separation when studied using the DMRG technique (Hotta et al., 2000). This is in agreement with the results of Shen and Wang (1999b) using the two-orbital model at both large ; and J , where it was concluded that having both couplings leads to a rich phase &

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Fig. 3.9.2. Phase diagram at zero temperature in the plane of AF interaction J and hole concentration x, using $ a two-orbital model with Coulomb interactions. F and F are the ferromagnetic phases with di!erent types of orbital   ordering (indicated). PS(F /F ) is the phase-separated state between the F and F phases. Results taken from Okamoto     et al. (2000) where more details, including couplings, can be found.

diagram with phase-separated and charge-ordered states. It is likely that this conclusion is correct, namely phase separation may be weak or only incipient in the purely Coulombic models, but in order to become part of the phase diagram, the Hund coupling to localized spins may play a key role. More work is needed to clarify these issues. Finally, the reader should recall the discussion of Section 3.3, where at least within a mean-"eld approximation it was shown that a large electron-JT phonon coupling or large Coulombic couplings are qualitatively equivalent. This is especially true when issues such as phase separation induced by disorder are considered, in which the actual origin of the two competing phases is basically irrelevant. Note also that Motome et al. (1998) have found phase separation in a two orbital model for manganites when a combination of Coulombic and Jahn}Teller interactions is considered. Recently, Laad et al. (2000) have also investigated a model including both Coulombic and JT-phononic couplings, analyzing experiments at x"0.3 La Sr MnO . \V V  3.11. Theories based on Anderson localization There is an alternative family of theories which relies on the possibility of electron localization induced by two e!ects: (1). o!-diagonal disorder caused by the presence of an e!ective complex

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electron hopping in the large Hund-coupling limit (see for instance MuK ller-Hartmann and Dagotto, 1996; Varma, 1996), and (2). non-magnetic diagonal disorder due to the di!erent charge and sizes of the ions involved in manganese oxides, as discussed before. Calculations in this context by Sheng et al. (1997), using scaling theory and a mean-"eld distribution for the spin orientations (one orbital model, J "R), were claimed to reproduce quantitatively the magnetoresistance & e!ect of real materials. Related calculations have been presented by Allub and Alascio (1996, 1997) and Aliaga et al. (1998). In these calculations, the electrons are localized above ¹ due to strong ! disorder, while at low temperature the alignment of the spins reduce the spin disorder and the electrons are delocalized. In this framework, also Coey et al. (1995) argued that the e -electrons,  while delocalized at the Mn}Mn scale, are localized at larger scales. There are some problems with approaches based on simple Anderson localization. For example, the phases competing with ferromagnetism are in general of little importance, and the mixed-phase tendencies of manganites, which are well established from a variety of experiments as shown in Section 4, are not particularly relevant in this context. The "rst-order-like nature of the transitions in these compounds is also not used. Note also that recently Smolyaninova et al. (1999) have experimentally shown that the metal}insulator transition of La Sr MnO at x"0.33 is not an \V V  Anderson localization transition. In addition, the x"0.5 CO state, crucial in real manganites to drive the strong CMR e!ect near this density, plays no important role in this context. It appears also somewhat unnatural to deal with an on-site disorder with such a large strength, typically =&12t (the random energies  are taken from the distribution [!=/2, =/2], and t is the G one-orbital hopping amplitude). However, it may occur that this strong disorder is a way to e!ectively mimic, in a sort of coarse-grained lattice, the disorder induced by cluster formation, similar to the calculation of the resistivity in Section 3.8. For instance, Sheng et al. (1997b) noticed the relation between the ¹"0 residual resistivity and the presence of a peak in the same quantity at ¹ . However, instead of assigning the large  (¹"0) to the percolative process described in !  Section 3.8, nonmagnetic randomness was used, and naturally a large = was needed to arrive at the large resistivities that appear near percolative transitions. The authors of this review believe that theories based on electron localization ideas, although they appear at "rst sight not directly related to the ubiquitous clustering tendencies of real manganites, may e!ectively contain part of the answer to the manganite puzzle, and further work in this context should be encouraged, if possible including in the approach a description of how localization phenomena relates to the phase separation character of manganites. Steps in this direction were recently taken by Sheng et al. (1999), in which calculations with JT phonons were carried out, and phase separation tendencies somewhat similar to those reported by Yunoki et al. (1998) were observed.

4. Experimental evidence of inhomogeneities in manganites 4.1. ¸a Ca MnO at density 0.04x(0.5 \V V  The regime of intermediate and low hole densities of La Ca MnO the former being close to \V V  the AF CE-type state at x"0.5 and the latter to the antiferromagnetic A-type state at x"0, is complex and interesting. In this region, the FM metallic state believed to be caused by double exchange is in competition with other states, notably AF ones, leading to the mixed-phase

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tendencies that are the main motive of this review. The special density x"0.33 in La Ca MnO \V V  has received considerable experimental attention, probably caused by the peak in the Curie temperature which occurs near this hole concentration (see phase diagram in Section 2.2). For potential technological applications of manganites it is important that the FM transition temperature be as high as possible, and thus it is important to understand this particular composition. However, although this reason for focusing e!orts at x"0.33 is reasonable, recent experimental and theoretical work showed that it is convenient to move away from the optimal density for ferromagnetism to understand the behavior of manganites, since many of the interesting e!ects in these compounds are magni"ed as ¹ decreases. Nevertheless, the information gathered at the hole ! density x"0.33 is certainly important, and analyzed together with the results at other densities, illustrates the inhomogeneous character of manganites. Historically, the path followed in the study of data at low and intermediate densities of LCMO is fairly clear. Earlier works focused on ideas based on polarons, objects assumed to be usually small in size, and simply represented as a local distortion of the homogeneous background caused by the presence of a hole. The use of polarons was understandable due to the absence of theoretical alternatives until a few years ago, and it may still be quite appropriate in large regions of parameter space. However, recent experimental work has shifted toward the currently more widely accepted mixed-phase picture where the ferromagnetic regions are not small isolated polarons but substantially larger clusters, at least in the important region in the vicinity of ¹ (polaronic descriptions ! may still be realistic well above ¹ ). Note that the various e!orts reporting polarons usually ! employed techniques that obtained spatially averaged information, while only recently, real-space images of the local electronic properties have been obtained that clearly illustrates the mixed-phase character of the manganite states. Below follows a summary of the main experimental results addressing mixed-phase characteristics in La Ca MnO at densities between x"0.0 and 0.5 \V V  (excluding the latter which will be analyzed separately). These results are not presented in historical order, but are mainly grouped by technique. Although the list is fairly complete, certainly it is not claimed that all reports of mixed-phase tendencies are described here as other e!orts in this direction may have escaped our attention. 4.1.1. Electron microscopy Among the most important experimental results that have convincingly shown the presence of intrinsic mixed-phase tendencies in manganites are those recently obtained by Uehara et al. (1999) in their study of La Pr Ca MnO using transport, magnetic, and electron microscopy \W W   techniques (see also Kiryukhin et al., 2000; Zuo and Tao, 2000). The results reported by those authors for the resistivity vs. temperature at several Pr compositions are reproduced in Fig. 4.1.1a. Note the rapid reduction with increasing y of the temperature at which the peak occurs, which correlates with the Curie temperature. Note also the hysteretic behavior of the resistivity, signalling the presence of "rst-order-like characteristics in these compounds. Another features of Fig. 4.1.1a is the presence of an abnormally large residual resistivity at low temperatures in spite of the fact that d/d¹'0 suggests metallic behavior. The magnetoresistance factor shown in Fig. 4.1.1b is clearly large and increases rapidly as ¹ is reduced. This factor is robust even at low temperatures where ! the resistivity is #at, namely the large MR e!ect does not happen exclusively at ¹ . ! The results of Uehara et al. (1999) have been interpreted by those authors as evidence of two-phase coexistence, involving a stable FM state at small y, and a stable CO state in the large

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Fig. 4.1.1. Transport and magnetic properties of La Pr Ca MnO as a function of temperature and y, reproduced \W W   from Uehara et al. (1999). (a) contains the temperature dependence of the resistivity. Both cooling (solid lines) and heating (dotted lines) curves are shown. (b) Magnetoresistance of representative specimens at 4 kOe. (c) Phase diagram of La Pr Ca MnO as a function of the ionic radius of (La, Pr, Ca). ¹ and ¹ are shown as "lled circles (or \W W   ! !triangles) and open circles, respectively. For more details, see Uehara et al. (1999) from where this "gure was taken. (d) Generic spectroscopic images reported by FaK th et al. (1999) using scanning tunneling spectroscopy applied to a thin "lm of La Ca MnO with x close to 0.3, and the temperature just below ¹ . The size of each frame is 0.61 m;0.61 m. \V V  ! From left to right and top to bottom the magnetic "elds are 0, 0.3, 1, 3, 5, and 9 T. The light (dark) regions are insulating (metallic).

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y PCMO compound. A percolative transition in the intermediate regime of compositions was proposed. The phase diagram is in Fig. 4.1.1c and it contains at low temperatures and a small range of Pr densities a phase labeled `CO#FMa which corresponds to the two-phase regime. In other regions of parameter space, short-range `s-ra FM or CO order has been observed. Uehara et al. (1999) substantiated their claims of phase separation using electron microscopy studies. Working at y"0.375 and at low temperature of 20 K, coexisting domains having sizes as large as 500 nm were found. At 120 K, the clusters become nanometer in size. Note that these low-temperature large clusters appear at odds with at least one of the sources of inhomogeneities discussed in the theoretical review (electronic phase separation), since 1/r Coulomb interactions are expected to break large clusters into smaller ones of nanometer size. In fact, Uehara et al. (1999) remarked it is reasonable to assume that the competing phases are of the same charge density. However, the experimental results for La Pr Ca MnO are in excellent agreement with the other proposed source of \W    mixed-phase tendencies, namely the ideas presented by Moreo et al. (2000), where "rst-order transitions are transformed into regions of two-phase coexistence by the intrinsic chemical disorder of the manganites (Section 3). This e!ect is called `disorder-induced phase separationa. 4.1.2. Scanning tunneling spectroscopy Another remarkable evidence of mixed-phase characteristics in La Ca MnO with x&0.3 \V V  has been recently reported by FaK th et al. (1999) using scanning tunneling spectroscopy. With this technique, a clear phase-separated state was observed below ¹ using thin "lms. The clusters ! involve metallic and insulating phases, with a size that is dependent on magnetic "elds. FaK th et al. (1999) believe that ¹ and the associated magnetoresistance behavior is caused by a percolation ! process. In Fig. 4.1.1d, a generic spectroscopic image is shown. A coexistence of metallic and insulating `cloudsa can be observed, with a variety of typical sizes involving tens to hundreds of nanometers. FaK th et al. (1999) remarked that it is clear that such length scales are not compatible with a picture of homogeneously distributed small polarons. The authors of this review agree with that statement. The results of FaK th et al. (1999) suggest that small changes in the chemical composition around La Ca MnO at x"0.25 can lead to dramatic changes in transport properties. This is \V V  compatible with results by other groups. For example, Ogale et al. (1998) reported transport measurements applied to La Ca Mn Fe O , i.e., with a partial replacement of Mn by Fe,     \V V  the latter being in a Fe> state. In this case, just a 4% Fe doping (x"0.04) leads to an instability of the low-temperature ferromagnetic metallic phase of the x"0.0 compound toward an insulating phase. The results for the resistivity vs. temperature are shown in Fig. 4.1.2. The shape of these curves is quite similar to the results observed in other compounds, such as those studied by Uehara et al. (1999), and they are suggestive of a percolative process leading eventually to a fully insulating state as x grows. Note the similarities of these curves with the theoretical calculations shown in Figs. 3.8.2 and 4. 4.1.3. Small-angle neutron scattering Small-angle neutron scattering combined with magnetic susceptibility and volume thermal expansion measurements by De Teresa et al. (1997b) (see also Ibarra and De Teresa, 1998a) applied to La Ca MnO with x"1/3 provided evidence for small magnetic clusters of size 12 As above \V V  ¹ . Although to study their data De Teresa et al. (1997b) used the simple picture of small !

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Fig. 4.1.2. Resistivity vs. temperature at several densities for La Ca Mn Fe O , taken from Ogale et al. (1998).     \V V  The results for the undoped sample are shown on an expanded scale (right) still using -cm as unit.

lattice/magneto polarons available by the time of their analysis, by now it is apparent that individual small polarons may not be su$cient to describe the physics of manganites near the Curie temperature. Nevertheless, leaving aside these interpretations, the very important results of De Teresa et al. (1997b) clearly experimentally showed the presence of an inhomogeneous state above ¹ early in the study of manganese oxides. The coexisting clusters were found to grow in size with ! a magnetic "eld and decrease in number. Ibarra and De Teresa (1998c), have reviewed their results and concluded that electronic phase segregation in manganites emerges from their data. Even percolative characteristics were assigned by Ibarra and De Teresa (1998a) to the metal}insulator transition, in excellent agreement with theoretical calculations (Moreo et al., 2000; Mayr et al., 2000). Hints of the mixed-phase picture (involving FM clusters larger than the size of a single ferro polaron) are also contained in the comment on the De Teresa et al. results presented by Zhou and Goodenough (1998b). Using neutron di!raction, muon-spin relaxation, and magnetic techniques, studies of (La Tb ) Ca MnO were also reported by De Teresa et al. (1996, 1997a). At low temper\V V    atures, an evolution from the FM metallic state at x"0 to the antiferromagnetic insulating state at x"1 was reported, involving an intermediate regime between x"0.33 and 0.75 with spin-glass insulating characteristics. The phase diagram is in Fig. 4.1.3a. Static local "elds randomly oriented were identi"ed at, e.g., x"0.33. No long-range ferromagnetism was found in the intermediate density regime. In view of the recent theoretical and experimental reports of giant cluster coexistence in several manganites, it is natural to conjecture the presence of similar phenomena in the studies of De Teresa et al. (1996, 1997a). In fact, the plots of resistance vs. temperature (see Fig. 4.1.3b, taken from Blasco et al., 1996) between x"0.0 and 0.5 have a shape very similar to those found in other manganites that were described using percolative ideas, such as La Pr Ca MnO (Uehara et al., 1999). \W W   Another interesting aspect of the physics of manganites that has been emphasized by Ibarra et al. (1995), Ibarra and De Teresa (1998c) and others, is the presence in the paramagnetic regime above

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Fig. 4.1.3. (a) Electronic and magnetic phase diagram of (La Tb ) Ca MnO as a function of x, reproduced from \V V    De Teresa et al. (1997a), where more details can be found. SGI is a `spin-glassa insulating state. (b) Resistance vs. temperature corresponding to (La Tb ) Ca MnO at the densities indicated, reproduced from Blasco et al. (1996). \V V    The similarities with analogous plots for other manganites described by a percolative process are clear.

¹ of a large contribution to the volume thermal expansion that cannot be explained by the ! GruK neisen law. Those authors assign this extra contribution to polaron formation. Moreover, the results for the thermal expansion vs. temperature corresponding to several manganites at x approximately 0.30 can be collapsed into a universal curve (Fig. 57 of Ibarra and De Teresa, 1998c) showing that the phenomenon is common to all compounds even if they have di!erent Curie temperatures. Above ¹ , there is a coexistence of a high-volume region associated with localized ! carriers and a low-volume region associated with delocalized carriers. The spontaneous or "eld-induced metal to insulator transition is associated with a low-to-high-volume transition. From this analysis it was concluded that there are two states in close competition and that the transition should be of "rst order, in excellent agreement with the recent simulations of Yunoki et al. (2000). The analysis of elastic neutron scattering experiments by Hennion et al. (1998) (see also Moussa et al., 1999) has provided very useful information on the behavior of La Ca MnO at low values \V V  of x. While previous work by the same authors (Hennion et al., 1997) was interpreted using a description in terms of simple magnetic polarons, Hennion et al. (1998) reinterpreted their results

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as arising from a liquid-like spatial distribution of magnetic droplets. The radius of these droplets was estimated to be 9 As and their number was found to be substantially smaller than the number of holes (ratio droplets/holes"1/60 for x"0.08), leading to a possible picture of hole-rich droplets within a hole-poor medium, if spin polarized regions are induced by carriers. Note the use of the word droplet instead of polaron in this context: polarons are usually associated with only one carrier, while droplets can contain several. It is quite remarkable that recent analysis by the same group (Hennion et al., 1999) of the compound La Sr MnO at x"0.06 has lead to very similar \V V  results: ferromagnetic clusters were found in this `largea bandwidth manganite and the number of these clusters is larger by a factor 25 than the number of holes. Hennion et al. (1999) concluded that phase separation between hole-rich and hole-poor regions is a general feature of the low doping state of manganites. These authors believe that this phenomenon likely occurs even at higher concentrations close to the metal}insulator transition. 4.1.4. Neutron scattering Early in the study of manganites, results of neutron scattering experiments on La Ca MnO \V V  for a wide range of compositions were interpreted by Lynn et al. (1996, 1997) in terms of a competition between ferromagnetic metallic and paramagnetic insulating states, leading to a state consisting of two coexisting phases. The relative fraction of these two phases was believed to change as the temperature was reduced to ¹ . This occurs even at the optimal composition for ! ferromagnetism close to x"1/3. A typical result of their measurements is presented in Fig. 4.1.4a where the inelastic spectrum is shown at two temperatures and small momentum transfer, for the x"1/3 compound which has a ¹ "250 K. The two peaks at nonzero energy are interpreted as ! spin waves arising from the ferromagnetic regions while the central peak is associated with the paramagnetic phase. Even at temperature as low as 200 K the two features can be observed. Fernandez-Baca et al. (1998) extended the analysis of Lynn et al. (1996) to other compounds with a similar hole concentration x&0.33. Their conclusions are very similar, i.e., a central component near ¹ is found in all the compounds studied and those authors concluded that `magnetism alone ! cannot explain the exotic spin dynamical propertiesa of manganites (Fig. 4.1.4b contains their results for Pr Sr MnO and Nd Sr MnO at x&0.3). Even the compound La Sr MnO \V V  \V V  \V V  at x"0.15 and 0.30 show a similar behavior (Fig. 4.1.4c and d), in spite of the fact that Sr-based manganites are usually associated with more conventional behavior than Ca-based ones. Overall, the neutron scattering experimental results are in good qualitative agreement with the conclusions reached by other experimental techniques, such as tunneling measurements at similar compositions which were reviewed before, and with theoretical calculations (already reviewed in Section 3). Lynn et al. (1996) also noticed the presence of irreversibilities in the transitions, and they remarked that these transitions are not of second order. These early results are also in agreement with the more recent theoretical ideas of Yunoki et al. (2000) and Moreo et al. (2000) where "rst-order transitions are crucial for the coexistence of giant clusters of the competing phases. 4.1.5. PDF techniques Using pair-distribution-function (PDF) analysis of neutron powder-di!raction data, Billinge et al. (1996) studied La Ca MnO at small and intermediate densities x. They explained their \V V  results in terms of lattice polaron formation associated with the metal}insulator transition in these materials. Below ¹ , Billinge et al. (1996) believe that the polarons can be large, dynamic, and !

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Fig. 4.1.4. (a) Inelastic spectrum at the two temperatures indicated and for q"0.07 As \, reported by Lynn et al. (1996) in their study of La Ca MnO at x"0.33. The left and right peaks are associated with spin waves in FM portions of the \V V  sample, while the central peak is attributed to paramagnetic regions. (b) Similar as (a) but for Pr Sr MnO (x"0.37) \V V  and Nd Sr MnO (x"0.30) at the temperatures and momenta indicated (reproduced from Fernandez-Baca et al., \V V  1998). (c,d) Similar as (a) but for La Sr MnO at the compositions, temperatures, and momenta indicated. (c) is \V V  reproduced from Vasiliu-Doloc et al. (1998a), while (d) is from Vasiliu-Doloc et al. (1998b).

spread over more than one atomic site. Note, however, that these authors use a polaronic picture due to the presence in their data of a mixture of short and long Mn}O bonds, implying distorted and undistorted MnO clusters. Whether the distorted octahedra are randomly distributed,  compatible with the polaronic theory, or gathered into larger structures, compatible with the phase separation theory, has not been analyzed. More recent studies by Billinge et al. (1999), using the same technique, produced the schematic phase diagram shown in Fig. 4.1.5. Note the light shaded region inside the FM-phase: in this regime Billinge et al. (1999) believe that localized and delocalized phases coexist. The white region indicates the only regime where an homogeneous FM

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Fig. 4.1.5. Schematic phase diagram of La Ca MnO , from Billinge et al. (1999). The solid lines are transport and \V V  magnetic transitions taken from Ramirez et al. (1996). The notation is standard. The small insets are PDF peaks (for details see Billinge et al., 1999). The dark shaded regions are claim to contain fully localized polaronic phases. The light shaded region denotes coexistence of localized and delocalized phases, while the white region is a FM homogeneous phases. The boundaries between the three regimes are di!use and continuous, and are only suggestive.

phase was found. This result is remarkable and it illustrates the fact that the simple doubleexchange ideas, that lead to an homogeneous FM-state, are valid in La Ca MnO in such \V V  a narrow region of parameter space that they are of little value to describe narrow band manganites in the important CMR regime. As remarked before, it appears that it is the competition between DE and the other tendencies dominant in manganites that produces the interesting magneto-transport properties of these compounds. 4.1.6. X-ray absorption, transport and magnetism Similar conclusions as those found by Billinge et al. (1996, 1999) were reached by Booth et al. (1998a, b) using X-ray absorption measurements applied to La Ca MnO at several hole \V V  concentrations. This technique provides information about the distribution of Mn}Mn bond lengths and the Mn}O environment. The results, obtained at several densities, favor a picture similar to that described in the previous subsection, namely there are two types of carriers: localized and delocalized. The number of delocalized holes grows exponentially with the magnetization below ¹ . These results clearly show that, even in the ferromagnetic regime, there are two types of ! phases in competition. In agreement with such conclusions, the presence of large polarons below ¹ at x"0.25 was also obtained by Lanzara et al. (1998) using X-ray techniques. Near ¹ those ! ! authors believe that small and large polarons coexist and a microscopic phase separation picture is suitable to describe their data. Early work using X-ray absorption for La Ca MnO at x"0.33 by Tyson et al. (1996) \V V  showed the presence of a complex distribution of Mn}O bond lengths, with results interpreted as generated by small polarons. Hundley et al. (1995) studied the same compound using transport

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techniques and, due to the observation of exponential behavior of the resistivity with the magnetization, they concluded that polaron hopping could explain their data. As remarked before, it is not surprising that early work used polaronic pictures to analyze their results, since by that time it was the main theoretical possibility available for manganites. However, Hundley et al. (1995) already noticed that the polarons could form superlattices or domains, a conjecture that later experimental work contained in this section showed to describe experiments more properly. 4.1.7. Nuclear magnetic resonance The coexistence of FM and AF resonances in NMR data obtained for La Ca MnO at \V V  several small hole densities was reported by Allodi et al. (1997) and Allodi et al. (1998a, b) using ceramic samples. No indications of a canted phase were observed by these authors, compatible with the conclusions of theoretical work showing that indeed the canted phase is unstable, at least within the models studied in Section 3. The NMR results showing the FM}AF coexistence contain a peak at &260 MHz which corresponds to AF, and another one slightly above 300 MHz which is believed to be FM in origin, according to the analysis of Allodi et al. (1997). A study of dynamic and static magnetic properties of La Ca MnO in the interval between \V V  x"0.1 and 0.2 by Troyanchuk (1992) also showed indications of a mixed-state consisting of ferromagnetic and antiferromagnetic clusters. Troyanchuk (1992) remarked very early on that his data was not consistent with the canted structure of deGennes (1960). 4.1.8. Muon spin relaxation The observation of two time scales in La Ca MnO at x&0.3 using zero-"eld muon spin \V V  relaxation were explained by He!ner et al. (1999) in terms of a microscopically inhomogeneous FM phase below ¹ , caused by the possible overlapping of growing polarons as the temperature is ! reduced. He!ner et al. (1999) concluded that a theoretical model mixing disorder and coupled JT-modes with the spin degrees of freedom may be necessary to explain their results, in agreement with the more recent theoretical calculations presented by Moreo et al. (2000) which used a mixture of disorder and strong JT correlations. Evidence for spatially inhomogeneous states using muon spin relaxation methods were also discussed by the same group in early studies (He!ner et al., 1996) where glassy spin dynamics was observed. Nonhomogeneous states for manganites were mentioned in that work as a possible alternative to the polaronic picture. 4.1.9. Photoemission Recently, Hirai et al. (2000) applied photoemission techniques to La Ca MnO with \V V  x"0.3,0.4 and 0.5, measuring the photoabsorption and magnetic circular dichroism. Interesting systematic changes in the core level edges of Ca 2p, O 1s and Mn 2p were observed as temperature and stoichiometry were varied. The results were interpreted in terms of a phase-separated state at room temperature, slightly above the Curie temperature. The metallic regions become larger as the temperature is reduced. These results are in excellent agreement with several other experiments describing the physics above ¹ as caused by a mixed-phase state, and with the ! theoretical calculations reviewed in Section 3. Based on the results of Hirai et al. (2000), it is conceivable that photoemission experiments may play a role as important in manganites as they do in the cuprates.

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4.1.10. Hall ewect Recent studies of the Hall constant of La Ca MnO at x"0.3 by Chun et al. (1999b) \V V  provided evidence that the picture of independent polarons believed in earlier studies to be valid in this compound above ¹ is actually valid only for temperatures larger than 1.4¹ i.e. well above the ! ! region of main interest from the point of view of the CMR phenomenon. In the temperature regime between ¹ and 1.4¹ , Chun et al. (1999b) describe their results as arising from a two-phase state, ! ! with percolative characteristics at ¹ . Once again, from this study it is clear that the insulating state ! of manganites above ¹ is not a simple gas of independent lattice/spin polarons (or bipolarons, see ! Alexandrov and Bratkovsky, 1999). This is compatible with the phenomenological two-#uid picture of localized and itinerant carriers near ¹ which was envisioned by Jaime et al. (1996, 1999) ! early in the study of manganites, and it is expected to apply to the x"0.3 La Ca MnO \V V  material. 4.1.11. Studies with high pressure The properties of manganites are also very sensitive to pressure, as explained in the Introduction. As an example, consider the results of Zhou et al. (1998a) Zhou and Goodenough (1998b) obtained analyzing (La Nd ) Ca MnO as a function of pressure (see also Zhou et al., 1996).          This compound appears to have a tolerance factor slightly below the critical value that separates the ferromagnetic regime from the antiferromagnetic one. While Zhou and Goodenough (1998) emphasized in their work the giant isotope e!ect that they observed in this compound upon oxygen isotope substitution, a very interesting feature indeed, here our description of their results will mainly focus on the resistivity vs. temperature plots at various pressures shown in Fig. 4.1.6. In view of the recent experimental results observed in similar materials that are also in the region of competition between FM- and AF-state, it is natural to contrast the results of Fig. 4.1.6 with those of, e.g., Uehara et al. (1999). Both sets of data, one parametric with pressure at "xed Nd-density and the other (Fig. 4.1.1a) parametric with Pr-density at ambient pressure, are similar and also in agreement with the theoretical calculations (Moreo et al., 2000; Mayr et al., 2000). The shape of the curves Fig. 4.1.6 reveal hysteretic e!ects as expected in "rst-order transitions, #at

Fig. 4.1.6. Resistivity vs. temperature for (La Nd ) Ca MnO reproduced from Zhou and Goodenough (1998).          Pressures are indicated.

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resistivities at low temperatures, a rapid change of (¹"0) with pressure, and the typical peak in the resistivity at "nite temperature that leads to CMR e!ects. All those features exist also in Fig. 4.1.1a. Similar pressure e!ects in (La Nd ) Ca MnO were reported by Ibarra et al. (1998b).        Those authors concluded that, at low temperatures, insulating CO and metallic FM regions coexist, and that this is an intrinsic feature of the material. The interpretation of their results appears simple: a "rst-order transition smeared by the intrinsic disorder in manganites can be reached by compositional changes or by changes in the couplings induced by pressure. But the overall physics is similar. In view of this interpretation, it is natural to conjecture that the material (La Nd ) Ca MnO discussed in the previous paragraph should also contain regions with \V V      a coexistence of giant clusters of FM- and AF-phase. Zhou et al. (1998a), Zhou and Goodenough (1998b) indeed mentioned the possibility of phase segregation between hole-rich and hole-poor regions in the paramagnetic state, but the low-temperature regime may have mixed-phase properties as well. In particular, the `canted-spin ferromagnetisma below ¹ reported by Zhou and , Goodenough (1998) could be induced by phase coexistence. 4.1.12. Related work Several other studies have shed light on the behavior of ferromagnetically optimally doped manganites. For instance, studies of thin "lms of La (Ca Sr ) MnO by Broussard et al.   V \V    (1999a, b) showed that the value of the magnetoresistance decreases rapidly as x is reduced from 1, namely as the system moves from a low to a large bandwidth manganite at a "xed hole density of 0.33. This interesting material should indeed present a transition from a mixed-phase state near ¹ for x"1 (all Ca), in view of the tunneling results of FaK th et al. (1999) and several others, to ! a more standard metal at x"0 (all Sr). In addition, Zhao et al. (1998) found a two component signal in the pulsed laser-excitation-induced conductance of La Ca MnO at x"0.3. The \V V  results can also be interpreted as a two-phase coexistence. Recently, Wu et al. (2000) reported the presence of colossal electroresistance (CER) e!ects in La Ca MnO with x"0.3, an interesting \V V  e!ect indeed, and they attributed its presence to phase separation tendencies. Kida et al. (2000a) estimated the complex dielectric constant spectrum of La Ca MnO with x"0.3, concluding \V V  that the results are compatible with a mixed-phase state. Belevtsev et al. (2000) reported studies in La Ca MnO x"0.33 "lms, where upon the application of a small dose of irradiation, large \V V  changes in the "lm resistivity were obtained. This is natural in a percolative regime, where small changes can lead to important modi"cations in transport. Complementing the previous studies, recently Smolyaninova et al. (1999) have shown that the metal}insulator transition of La Ca MnO at x"0.33 is not an Anderson localization \V V  transition, since scaling behavior was clearly not observed in resistivity measurements of thin "lms. This important study appears to rule out simple theories based on transitions driven by magnetic disorder, such as those proposed by MuK ller-Hartmann and Dagotto (1996), Varma (1996), and Sheng et al. (1997). A similar conclusion was reached by Li et al. (1997) through the calculation of density of states with random hopping (for a more recent density of states and localization study of the one-orbital model at J "R see Cerovski et al., 1999). It was observed that this randomness & was not su$cient to move the mobility edge, such that at 20% or 30% doping there was localization. It appears that both Anderson localization and the simple picture of a gas of independent small polarons are ruled out in manganites.

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4.2. ¸a Ca MnO at x&0.5 \V V  After a considerable experimental e!ort, the evidence for mixed-phase FM}CO characteristics in La Ca MnO near x"0.5 is simply overwhelming. The current theoretical explanation of \V V  experimental data at this density appears simple. According to computer simulations and mean"eld approximations the FM- and CO-phase are separated by "rst-order transitions when models without disorder are studied. This abrupt change is due to the substantial di!erence between these phases that makes it di$cult a smooth transition from one to the other. Intrinsic disorder caused by the slightly di!erent ionic sizes of La and Ca can induce a smearing of the "rst-order transition, transforming it into a continuous transition with percolative characteristics. Coexistence of large clusters with equal density is possible, as described in the theoretical section of this review. In addition, intrinsic tendencies to electronic phase separation, which appear even without disorder, may contribute to the cluster formation. It is important to remark that although in this review the cases of x(0.5 and x&0.5 are treated separately, it is expected that a smooth connection between the two types of mixed-phase behavior exists. Hopefully, future theoretical and experimental work will clarify how the results at, say, x&0.3 and x&0.5 can evolve one into the other changing the hole density. 4.2.1. Experimental evidence of inhomogeneities Early work by Chen and Cheong (1996) and Radaelli et al. (1997) using electron and X-ray di!raction experiments found the surprising coexistence of ferromagnetism and charge ordering in a narrow temperature window of La Ca MnO . Further studies by Mori et al. (1998b) showed \V V  that the x"0.5 mixture of FM and CO states arises from an inhomogeneous spatial mixture of incommensurate charge-ordered and ferromagnetic charge-disordered microdomains, with a size of 20}30 nm. Papavassiliou et al. (1999a, b) (see also Belesi et al., 2000) observed mixed-phase tendencies in La Ca MnO using Mn NMR techniques. Fig. 4.2.1a shows the NMR spectra for \V V  La Ca MnO at several densities and low temperature ¹"3.2 K obtained by those authors. \V V  The appearance of coexisting peaks at x"0.1,0.25, and 0.5 is clear from the "gure, and these peaks correspond to either FM metal, FM insulator, or AF-states according to the discussion presented in Papavassiliou et al. (1999a, b). The results at x"0.5 are in agreement with previous results reported by the same group (Papavassiliou et al., 1997). The revised phase diagram proposed by those authors is shown in Fig. 4.2.1b. In agreement with the conclusions of other groups, already reviewed in the previous subsection, the region in the vicinity of ¹ corresponds to a mixed-phase ! regime. The same occurs at low temperatures in the region between the CO- and FM-state of x"0.5. The coexistence of FM- and AF-phase was also observed in La Ca MnO by Allodi et      al. (1998) using similar NMR techniques. First-order characteristics in the FM}AF transition were found, including an absence of critical behavior. Their spectra is shown in Fig. 4.2.1c. As in Fig. 4.2.1a, a clear two signal spectra is observed in the vicinity of x"0.5 and low temperatures. The presence of mixed-phase characteristics in NMR data was also observed by Dho et al. (1999a, b) in their studies of La Ca MnO . Their results apply mainly near the phase boundaries of the \V V  ferromagnetic regime at a "xed temperature, or near ¹ at a "xed density between 0.2 and 0.5. ! It can be safely concluded, overall, that the NMR results described here are in general agreement, and also in agreement with the phase separation scenario which predicts that all around the FM

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Fig. 4.2.1. (a) Mn NMR spectra of La Ca MnO at ¹"3.2 K for the densities shown, reproduced from \V V  Papavassiliou et al. (1999b). Coexistence of features corresponding to two phases appear in the data. (b) Revised temperature}density phase diagram proposed by Papavassiliou et al. (1999b). The circles denote the NMR results presented in that reference. The notation is standard. (c) Mn NMR spectra at ¹"1.3 K of La Ca MnO with \V V  x"0.5 in zero and applied "eld. FM lines are marked with "lled symbols, while AF ones are marked with open symbols. Figure reproduced from Allodi et al. (1998).

metallic phase in the temperature}density plane there are regions of mixed-phase characteristics due to the competition between metallic and insulating states. Magnetization, resistivity, and speci"c heat data analyzed by Roy et al. (1998, 1999, 2000a, b) led to the conclusion that in a narrow region of hole densities centered at x"0.5, two types of carriers coexisted: localized and free. The evidence for a rapid change from the FM to the CO phases as x was varied is clear (see Fig. 3 of Roy et al., 1998). This is compatible with the fact that La> and Ca> have a very similar ionic radius and, as a consequence, the disorder introduced by their mixing is `weaka. The theoretical scenario described before (Moreo et al., 2000) suggests that, at zero temperature and for weak disorder, the density window with large cluster coexistence should be narrow (conversely in this region large cluster sizes are expected). It is also to be expected that the magnetoresistance e!ect for low values of magnetic "elds will appear only in the same narrow

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region of densities. Actually, Roy et al. (1999) showed that at x"0.55, a "eld of 9 T is not enough to destabilize the charge-ordered state. Very recently, Roy et al. (2000a) studied, among other quantities, the resistivity vs. temperature for magnetic "elds up to 9 T. The result is reproduced in Fig. 4.2.2. This "gure clearly resembles results found by Uehara et al. (1999) in their study of La Pr Ca MnO (see Fig. 4.1.1a) varying the Pr concentration. In both cases, the curves are \W W   similar to those that appear in the percolative process studied by Mayr et al. (2000) (see Figs. 3.8.2 and 3.8.4). Percolation between the CO and FM states appears to occur similarly both by changing chemical compositions and also as a function of magnetic "elds, a very interesting result. Phase separation in x"0.5 polycrystalline samples obtained under di!erent thermal treatments was also reported by Levy et al. (2000a). LoH pez et al. (2000) also found results compatible with FM droplets immersed in a CO background. Kallias et al. (1999) using magnetization and MoK ssbauer measurements also reported coexisting FM and AF components in x"0.5 La Ca MnO . \V V  It is also important to remark that experimentally it is very di$cult to make reproducible La Ca MnO samples with x&0.5 (see for instance Roy et al., 1998, 1999, 2000a, b). \V V  Samples with the same nominal Ca content can actually present completely di!erent behavior. This is compatible with a phase separated state at this density, which is expected to be very sensitive to small chemical changes. Another result compatible with phase separation can be found in the magnetization curves (Fig. 4.2.2), which are well below the expected saturation value for a ferromagnet, even in several Tesla "elds where the magnetization is not increasing rapidly.

Fig. 4.2.2. Upper panel: resistivity vs. temperature on cooling (solid) and warming (open) at "elds between 0 and 9 T, in steps of 1 T starting from the top. The material is La Ca MnO and the results are reproduced from Roy et al.      (2000a). Bottom panel: magnetization versus temperature for "elds between 1 T and 7 T in steps of 2 T, starting from the bottom. The inset illustrates two distinct features in the resistivity associated with the coexistence of two states. For more details see Roy et al. (2000a).

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Neutron powder di!raction studies by Radaelli et al. (1995) of La Ca MnO at x"0.5 \V V  revealed peak broadening e!ects that were explained assuming multiple phases simultaneously present in the sample. Rhyne et al. (1998) studied La Ca MnO with x"0.47 using elastic and \V V  inelastic neutron scattering. Coexisting ferromagnetic and antiferromagnetic phases were found at low temperatures. Similar conclusions were also reached by Dai et al. (1996). Further con"rmation that near x"0.5 in La Ca MnO the material has mixed-phase characteristics has been \V V  recently provided by neutron powder di!raction measurements by Huang et al. (1999). Discontinuous features in the results discussed by those authors also indicate that the competing phases are likely separated by "rst-order transitions in the absence of intrinsic disorder, as found in the theoretical calculations (Yunoki et al., 2000). Infrared absorption studies by Calvani et al. (1998) were also described in terms of a phase separation scenario. It is also very interesting to test materials with the hole density x"0.5 but allowing for slight deviations away from the La Ca MnO chemical composition. Among these investigations are      the transport and X-ray experiments on R Ca Mn Cr O , with R"La, Nd, Sm and Eu,        carried out by Moritomo et al. (1999a). Their study allowed for a systematic analysis of the charge-ordered state when the ionic radius of the rare-earth ion was changed. Due to the small presence of Cr, this material with R"La has a purely ferromagnetic state while the other rare-earths leads to a CE-type CO state. The main result obtained by Moritomo et al. (1999) is quite relevant to the subject of this review and is summarized in Fig. 4.2.3a. Moritomo et al. (1999) concluded that the region between the CO and FM phases has mixed-phase characteristics involving the two competing states. This hypothesis was con"rmed by the use of electron microscopy, which showed microdomains of size 20}50 nm, a result similar to those observed by other authors in other compounds. Then, once again, mixed-phase tendencies are clear in materials with x"0.5 (see also Oshima et al., 2000). Results for the Fe-doped x"0.5 LCMO compound by Levy et al. (2000b) likely can be rationalized in a similar way. Other very interesting results in the context of Cr doping have been obtained by Kimura et al. (1999, 2000). Moreover, the study of Cr-doped compounds at many Ca densities shows that this type of doping with impurities has an e!ect similar to that of a magnetic "eld, namely a small Cr percentage is enough to destabilize the CO-state into a FM-state. This result is surprising, since impurities are usually associated with a tendency to localize charge, and they are not expected to generate a metallic state. In Fig. 4.2.4a-c, the phase diagrams presented by Katsufuji et al. (1999) for three compounds are shown to illustrate this point. In Pr Ca MnO , Cr-doping destabilizes the CO-state in a wide \V V  range of densities, as a magnetic "eld does, while for La Ca MnO and Nd Sr MnO , it is \V V  \V V  e!ective only near x"0.5. The resistivity plots in Fig. 4.2.4d show that the shape of the curves are very similar to all the previous ones analyzed in this review, indicative of a percolative process. Results similar to those of Moritomo et al. (1999) were obtained using transport techniques by Mallik et al. (1998) studying La Ca Ba MnO with x between 0, where the sample they used    \V V  is in a charge-ordered insulating state, and x"0.5, where a ferromagnetic metallic compound is obtained. The resistivity vs. temperature at several compositions is shown in Fig. 4.2.3b. The results certainly resemble those obtained by Uehara et al. (1999) and other authors, especially regarding the presence of a #at resistivity in a substantial low-temperature range and a rapid variation of (¹"0) with Ba concentration. Mallik et al. (1998) observed that the di!erence in ionic sizes between Ca and Ba plays a crucial role in understanding the properties of this compound. They also found hysteretic behavior and "rst-order characteristics in their results, results all compatible

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Fig. 4.2.3. (a) Phase diagram of a 3% Cr-doped manganites, R Ca (Mn Cr )O , against averaged ionic radius        r of the rare-earth ion. Closed circles and squares are Curie temperatures and critical temperatures for the charge0 ordering transition, respectively. PS is the region of phase separation. Open symbols represent the data for the Cr-undoped compounds. Figure reproduced from Moritomo et al. (1999). (b) Electrical resistivity versus temperature for the compounds La Ca Ba MnO , reproduced from Mallik et al. (1998a, b).    \V V 

with the theoretical scenario described before (Yunoki et al., 2000; Moreo et al., 2000). The critical concentration for percolation in Fig. 4.2.3b appears to be near x"0.1, where the ¹ was found to ! be the smallest in this compound. Finally, it is also interesting to remark that magnetic-"eld-dependent optical conductivity studies by Jung et al. (1999) applied to Nd Sr MnO at x"0.5 have also found indications of \V V  a percolative transition in the melting of the charge ordered state.

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Fig. 4.2.4. Phase diagrams of (a) Pr Ca Mn Cr O , (b) La Ca Mn Cr O , and (c) \V V      \V V      Nd Sr Mn Cr O , taken from Katsufuji et al. (1999). The grey regions are the FM phases in the absence of Cr, \V V      while the dark regions are FM metallic phases stabilized by Cr doping. The rest of the notation is standard. (d) Resistivity vs. temperature for Pr Ca Mn Cr O . The inset contains results at x"0.5 with (y"0.03) and without (y"0.0) \V V      Cr. Results taken from Katsufuji et al. (1999).

4.3. Electron-doped manganites Neutron scattering studies of Bi Ca MnO single crystals in the range between x"0.74 and \V V  0.82 were presented by Bao et al. (1997). It is expected that Bi Ca MnO will have properties \V V  very similar to those of La Ca MnO in the range of densities studied by those authors, and for \V V  this reason the analysis of a Bi-based compound is discussed in this subsection. One of the most interesting results reported by Bao et al. (1997) is the presence of ferromagnetic correlations at high temperatures, which are replaced by antiferromagnetic ones as the temperature is reduced. Fig. 4.3.1a taken from Bao et al. (1997) show the intensity of the FM and AF peaks as a function of temperature at x"0.82. It is clear from the "gure that in the intermediate regime, roughly between 150 and 200 K, there is a coexistence of FM and AF features, as in a mixed-phase state. In a related study, Bao et al. (1998) concluded that manganites have only two important generic states: metallic ferromagnetic and localized antiferromagnetic. This is in agreement with theoretical results, although certainly combinations such as charge-ordered ferromagnetic states are also possible at least in 2D (Yunoki et al., 2000). Subsequent studies of Bi Ca MnO single-crystals performed \V V  by Liu et al. (1998) reported optical re#ectivity results in the same compositional range (i.e. between x"0.74 and 0.82). The main result of this e!ort is reproduced in Fig. 4.3.1b. Liu et al. (1998) concluded that in the intermediate range ¹ (¹(¹ the coexistence of a polaron-like response , !-

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Fig. 4.3.1. (a) Neutron scattering results of Bao et al. (1997) corresponding to Bi Ca MnO at x"0.82. The solid \V V  circles correspond to the AF response, while the open circles are the FM response. The dotted line is the background. A region of FM}AF coexistence is observed. For more details the reader should consult Bao et al. (1997). (b) Real part of the optical conductivity at the three temperatures indicated, from Liu et al. (1998) where the details of the "tting results (dashed and dot}dashed lines) are explained. The upper inset contains the temperature dependence of the energy gap ("led squares) and the polaron oscillator strength (open circles). The lower inset is the e!ective number of carriers. Peak B evolves into a clean charge-gap as ¹ decreases, while A corresponds to polarons.

together with a charge-gap structure signi"es two-phase behavior characterized by domains of both FM and AF spin correlations. Recently, studies of Bi Ca MnO at x"0.81 and 0.82 were \V V  interpreted in terms of spin or charge `chirala #uctuations (Yoon et al., 2000), showing that exotic physics may occur in this electron doped compound. The range of hole densities above 0.8 for Bi Ca MnO was analyzed by Chiba et al. (1996) \V V  using magnetic and transport techniques. They observed that large magnetoresistance e!ects are found even at low ¹ , which is compatible with a mixed-phase state in the ferromagnetic regime, !

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quite di!erent from a spin-canted state. Actually, it is important to remark that there are previous studies of the electron-doped materials (not reviewed here) that have labeled the small x region as `spin canteda due to the observation of coexisting FM and AF features. The conclusions of those papers may need revision in view of the new results described in this section. Studies of Ca Sm MnO by Maignan et al. (1998), using magnetic and transport techniques \W W  in the range from y"0.0 to 0.12, reported results compatible with a `cluster glassa (see also Martin et al., 1999). As y increases from zero, the system rapidly becomes ferromagnetic and metallic. However, those authors remark that no true long-range order exists, and thus the FM-state is unusual. The resistivity is shown in Fig. 4.3.2a. The metallic character at y"0.0 and high temperature is caused by oxygen de"ciency and should not be considered as really representing the electron undoped compound, which is actually antiferromagnetic (G-type).

Fig. 4.3.2. (a) Temperature dependence of the resistivity of Ca Sm MnO for several values of x (shown). For more \V V  details see Maignan et al. (1998). (b) Magnetization M vs. temperature of Ca La MnO (x shown). In the inset M vs. \V V  the magnetic "eld H is plotted. (c) Upper panel: Magnetic saturation moment at 5 K vs. x. Region I is a G-type AF with local ferrimagnetism. Region II has local FM regions and G-type AF. Region III contains C- and G-type AF, as well as local FM. Region IV is a C-type AF. Lower panel: Electrical conductivity at ¹"5 K vs. x. All the results in (b) and (c) are taken from Neumeier and Cohn (2000).

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More recently, a careful and systematic study of Ca La MnO has been carried out by \V V  Neumeier and Cohn (2000) using magnetic and transport techniques. These authors concluded that the addition of electrons to the x"0.0 antiferromagnetic state promotes phase segregation. Representative magnetization vs. temperature data are shown in Fig. 4.3.2b. The saturated moment and conductivity versus density are reproduced in Fig. 4.3.2c. Neumeier and Cohn (2000) reported multiple magnetic phases emerging from the analysis of their data, and remarked that the long-accepted existence of canted AF is supplanted by phase coexistence. In addition, recent NMR studies of La Ca MnO for x"0.65 at low temperature by Kapusta \V V  et al. (2000) reported the existence of electronic phase separation, with FM regions detected over a CO/AF background. This interesting result leads us to believe that it may be possible that the widely accepted phase diagram of La Ca MnO may still need further revision, since a phase with \V V  coexisting FM and AF features may exist at low temperature and x around 0.65, with a shape similar to the `canted statea that appears in the phase diagram of Pr Ca MnO and the bilayer \V V  compounds (see Figs. 2.3.1 and 4.6.1). This conjecture could be tested experimentally with NMR techniques. Results for Ca Y MnO can be found in Aliaga et al. (2000b). \V V  4.4. Large bandwidth manganites and inhomogeneities: the case of ¸a

\V

Sr MnO V 

A compound as much scrutinized as the Ca-based manganites of the previous sections is the Sr-based La Sr MnO , which has a larger bandwidth. In spite of this property, the \V V  La Sr MnO material presents a very complex phase diagram, especially at low Sr-density, with \V V  a behavior in many respects qualitatively similar to that of the Ca-based compound. The main experimental evidence that leads to this conclusion is reviewed below. In the other regime of large densities, the Sr-based material is metallic both at low and high temperatures (see phase diagram Fig. 2.1.1b) and its magnetoresistance e!ect is relatively small. In this density regime, studies using mainly dynamical mean-"eld approaches (D"R) have provided evidence that the simple double-exchange ideas are enough to understand the main properties of La Sr MnO \V V  (Furukawa, 1994, 1995a}c, 1998), especially concerning the interplay between ferromagnetism and transport. This is a reasonable conclusion, and illustrates the fact that materials whose couplings and densities locate them in parameter space far away from insulating instabilities tend to present canonical properties. A review of the status of the theoretical approach based on the doubleexchange ideas and its application to large-bandwidth manganites has been recently presented (Furukawa, 1998). Additional results for the FM Kondo model have been discussed by Zang et al. (1997), and several other authors. However, it must be kept in mind that the more canonical, and governed by double exchange, the behavior of a compound is, the smaller is the magnetoresistance e!ect. For this reason, in the description of experimental results for La Sr MnO the e!ort is \V V  here mainly focused into the low-density regime where e!ects other than canonical double exchange seem to dominate in this material. 4.4.1. ¸a Sr MnO at low density \V V  Among the "rst papers to report inhomogeneities in Sr-based manganites are those based on atomic pair-density-functional (PDF) techniques. In particular, Louca et al. (1997) studied La Sr MnO in a wide range of densities between x"0.0 and 0.4, and interpreted their results \V V  as indicative of small one-site polarons in the paramagnetic insulating phase. Those authors found

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that the local atomic structure deviates signi"cantly from the average. At lower temperatures their polarons increase in size, typically involving three sites according to their analysis. These e!ects were found even in the metallic phase. Based on such results, Louca et al. (1997) questioned the at-that-time prevailing homogeneous picture of the metallic state of manganites, and based their analysis mainly on a small polaron picture rather than large droplets or phase separation ideas. Nevertheless, they envisioned that increasing the density of polarons would lead to larger structures, and in more recent work (Louca and Egami, 1999) they also presented microscopic separation of charge-rich and charge-poor regions as a possible scenario to describe their results. In addition, they conjectured that the conductivity could be determined by some kind of dynamic percolative mechanism, which is the current prevailing view (see also Egami, 1996; Egami et al., 1997). The possible percolative nature of the metal}insulator transition close to x "0.16 in  La Sr MnO was also proposed by Egami and Louca (1998). Tendencies toward a two-phase \V V  regime in low hole-density doped (La, Sr)-based manganites were also reported by Demin et al. (1999) using a variety of techniques. Recently, Endoh et al. (1999a, 1999b) and Nojiri et al. (1999), using transport and resonant X-ray scattering, have studied in detail the region near x&1/8 of La Sr MnO . Interesting results \V V  were observed in this regime, especially a "rst-order transition from a ferromagnetic metal to a ferromagnetic insulator. This ferromagnetic insulator was reported in previous work by Yamada et al. (1996) using neutron scattering techniques. Those authors interpreted their results using a state with charge ordering, which they refer to as `polaron orderinga with polarons involving only one site [note, however, that other authors could not reproduce the Yamada et al. (1996) results. See Vasiliu-Doloc et al. (1998a)]. Endoh et al. (1999a, b) reported huge changes in resistivity upon the application of a magnetic "eld close to the above metal}insulator transition in this compound. Regions with phase-separation characteristics were identi"ed by Endoh et al. (1999b). The key di!erence between the two competing states is the orbital ordering, as revealed by the X-ray experiments. The reported phase diagram is in Fig. 4.4.1a. Similar conclusions were reached by Paraskevopoulos et al. (2000a) and previously by Zhou and Goodenough (1998b) through measurements of resistivity and thermoelectric power. The last authors reported a dynamic phase segregation into hole-rich and hole-poor phases in the region of x"0.12 between the charge-ordered transition temperature and the Curie temperature. Their phase diagram resembles that of Endoh et al. (1999b) (see Fig. 4.4.1b). Overall, these experimental results are in good agreement with mean-"eld calculations using purely Coulombic models (Endoh et al., 1999a) and with Monte Carlo simulations using JT phonons (Yunoki et al., 1998b). In both cases, phase separation triggered by the orbital degree of freedom, instead of the spin, were found. It is clear once again that simple double-exchange ideas or even the proposal of small polarons are not su$cient to explain the physics of manganites, particularly in the most interesting regions of parameter space where the CMR e!ect occurs. The results of Endoh et al. (1999a, b) and Nojiri et al. (1999) have characteristics similar to those of the theoretical scenario described in Section 3, namely a competition between two states which are su$ciently di!erent to generate a "rst-order transition between them. The results of Moreo et al. (2000) suggest that the small ionic radii di!erences between La> and Sr> induces weak disorder that a!ects the "rst-order transition, inducing a narrow region of coexistence of cluster of both phases. Percolative properties are predicted in this regime based on the results of Moreo et al. (2000). It would be quite interesting to search for such properties in x&1/8 La Sr MnO \V V  experiments.

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Fig. 4.4.1. (a) Magnetic and structural phase diagram of La Sr MnO determined by neutron di!raction data, \V V  reproduced from Endoh et al. (1999a). The notation is standard. Note that at densities roughly between 0.10 and 0.15, a FM metallic phase can be identi"ed in a narrow temperature region upon changing the temperature. (b) Phase diagram of La Sr MnO according to Zhou and Goodenough (1998b). Most of the notation is standard. The FMP region \V V  corresponds to ferromagnetic polarons in the analysis of Zhou and Goodenough (1998b), where more details can be found.

In fact, the theoretical calculations are already in qualitative agreement with a recent experimental e!ort. Independent of the previously described results by Endoh et al. (1999a, b) and Nojiri et al. (1999), Kiryukhin et al. (1999) studied x"1/8 La Sr MnO using synchrotron X-ray \V V  scattering. At low temperatures, they observed an X-ray-induced transition from a charge-ordered phase to a charge-disordered state. These results are qualitatively similar to those reported by Kiryukhin et al. (1997) applied to Pr Ca MnO . Kiryukhin et al. (1999) suggest that their results \V V  can be explained within a phase-separation scenario with charge-ordered regions as large as 500 As , sizes similar to those observed in half-doped La Ca MnO , as described before in this review \V V  (see also Baran et al., 1999). Wagner et al. (1999), using transport and magnetic techniques applied to x"1/8 La Sr MnO , also found evidence of a "rst-order transition as a function of \V V  temperature. The possibility of phase separation was brie#y mentioned in that work. Finally, the optical conductivity spectra obtained by Jung et al. (1998) in their study of x"1/8 La Sr MnO has also been explained in terms of a phase-separated picture by \V V 

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comparing results with those of Yunoki et al. (1998b), which were obtained at temperatures such that dynamical clustering was present in the Monte Carlo simulations. It is interesting to remark that a large number of optical experiments have been analyzed in the near past as arising from coexisting metallic (Drude) peaks and mid-infrared bands that were usually assigned to polaronic features (see for instance Kaplan et al., 1996). In view of the novel experimental evidence pointing toward coexisting metallic and insulating clusters, even in optimal regimes for FM such as x"0.33 in La Ca MnO , the previous optical conductivity may admit other interpretations perhaps \V V  replacing polarons by larger droplets. Finally, note that recent optical studies at x"0.175 by Takenaka et al. (1999) have been interpreted as arising from a FM metallic phase below ¹ which ! can have either coherent or incoherent characteristics, and a mixture of them is possible. The anomalous metallic state of Sr-doped manganites has been theoretically addressed recently by Ferrari and Rozenberg (1999) using dynamical mean-"eld calculations. Motome and Imada (1999) and Nakano et al. (2000) also studied this material and concluded that to reproduce the small Drude weight of experiments a mixture of strong Coulomb and electron-phonon (JT) interactions is needed. For completeness, some remarks about related compounds are here included. For instance, LaMnO was studied (see Ritter et al., 1997; Ibarra and De Teresa, 1998b) and at &0.15 >B a large magnetoresistance e!ect was observed. The magnetic and transport properties of La MnO were analyzed by De Brion et al. (1998). In their study, they concluded that a canted \B  state was observed, but magnetization measurements cannot distinguish between FM}AF phase separation and spin canting. In fact, recent studies by Loshkareva et al. (2000, 1999) of optical, electrical, and magnetic properties of the same compound and x"0.1 La Sr MnO were \V V  interpreted in terms of phase-separation. In addition, `cluster-glassa features were reported for this compound by Ghivelder et al. (1999). On the other hand, susceptibility, magnetization, MR and ultrasonic studies of La Sr MnO at low doping x(0.1 by Paraskevopoulos et al. (2000b) \V V  where interpreted as compatible with a canted state, rather than a phase-separated state (see also Pimenov et al., 2000; Mukhin et al., 2000). However, those authors remark that the canting does not arise from DE interactions because the carriers are localized near the Sr-ions. These trapped holes can polarize the Mn-ions in their vicinity leading to FM clusters in a PM matrix. This interesting proposal merits theoretical studies. It is safe to conclude that at very low hole density in La Sr MnO it is still unclear what kind of state dominates the low-temperature behavior, \V V  namely whether it is homogeneous (canted) or inhomogeneous as predicted by phase-separation scenarios. 4.4.2. ¸a Sr MnO at intermediate density \V V  Although some features of La Sr MnO at intermediate densities are well described by the \V V  double-exchange ideas, experiments have revealed mixed-phase tendencies in this region if the study is carried out close to instabilities of the FM metallic phase. For instance, working at x"0.17 in La Sr MnO , Darling et al. (1998) reported measurements of the elastic moduli \V V  using resonant ultrasound spectroscopy. Those authors noticed that their results suggest the existence of very small microstructures in their single crystals. Studies by Tkachuk et al. (1998) of La Sr Mn Fe O also led to the conclusion that the paramagnetic phase contains          ferromagnetic clusters. Recent ESR studies by Ivanshin et al. (2000) have also contributed interesting information to the study of La Sr MnO at hole densities between x"0.00 and \V V 

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0.20. Small-angle polarized neutron scattering measurements by Viret et al. (1998) for La Sr MnO at x"0.25 indicated the presence of nanometer size inhomogeneities of magnetic \V V  origin in the vicinity of the Curie temperature. Approximately at this density occurs the metal}insulator transition above ¹ , and as a consequence, mixed-phase features as observed in ! La Ca MnO (which at all densities presents an insulating state above ¹ ) are to be expected \V V  ! below x"0.25. Machida et al. (1998) studied the absorption spectra of thin-"lms of R Sr MnO with R"Sm, (La Nd ), (Nd Sm ), and (Nd Sm ). They concluded                  that cluster states were formed in these compounds. 4.4.3. Sr-based compounds at high-hole density: the cases of Pr Sr MnO and Nd Sr MnO \V V  \V V  The antiferromagnetic manganite Pr Sr MnO at x"0.5 has been recently studied using \V V  NMR techniques by Allodi et al. (1999). This material has a magnetic-"eld-induced transition to a ferromagnetic state and a CMR e!ect. The NMR results show that the transition proceeds through the nucleation of microscopic ferromagnetic domains, with percolative characteristics. Allodi et al. (1999) believe that the size of the clusters in coexistence is on the nanometer scale, to be compared with the micrometer scale found in other manganites. Kajimoto et al. (1999) studied Nd Sr MnO in a range of densities from x"0.49 to 0.75 \V V  using neutron di!raction techniques. Four states were observed: FM metallic, CE-type insulating, A-type metallic, and a C-type AF insulator. The latter may be charge-ordered. At x&0.5, Kajimoto et al. (1999) reported a possible mixed-phase state involving the CE- and A-type orderings. Other groups arrived at similar conclusions: Woodward et al. (1999) found coexisting macroscopic FM, A-type and CE-type phases, while Fukumoto et al. (1999) reported microscopic scale electronic phase separation in this compound. All these results are compatible with the recent theoretical work of Moreo et al. (2000) and Yunoki et al. (2000), since computer simulations of models with JT phonons at x"0.5 have found "rst-order transitions separating the many possible states in manganites, including one between the A- and CE-type states. The addition of weak disorder would smear this sharp "rst-order transition into a rapid crossover. CMR e!ects are to be expected in this regime. 4.5. Pr Ca MnO \V V  It is interesting to observe that the low-bandwidth compound Pr Ca MnO with x"0.30 \V V  undergoes an unusual insulator}metal transition when it is exposed to an X-ray beam. Without X-rays, the material is in a charge-ordered insulator state below 200 K. However, below 40 K, X-rays convert the insulating state into a metallic state which persists when the X-ray beam is switched o! (Kiryukhin et al., 1997; Cox et al., 1998). A similar transition occurs upon the application of a magnetic "eld. The authors of these experiments interpreted their results as arising from a phase-segregation phenomenon induced by the X-rays, with ferromagnetic droplets coalescing into larger aggregates. Note that x"0.30 is at the border between the CO-state and a FM-insulating state in this compound, and thus unusual behavior is to be expected in such a regime. Recently, transport, optical and speci"c heat results at x"0.28 by Hemberger 2000b, Hemberger et al. (2000a) have been interpreted as a percolative metal}insulator transition induced by a magnetic "eld, with coexisting metallic and insulating clusters below 100 K at zero external "eld. Using neutron di!raction techniques applied to x"0.3 PCMO, Katano et al. (2000) recently

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found evidence of a phase-separated state with percolative characteristics in the metal}insulator transition induced by magnetic "elds. Recent analysis, again using X-rays, of the related material Pr (Ca Sr ) MnO showed \V \W W V  that the metal}insulator transition present in this compound is not caused by a conventional change in the electron density, but by a change in the couplings of the system which a!ect the mobility of the carriers (Casa et al., 1999). It is believed that the X-rays can help connecting adjacent preformed metallic clusters which originally are separated by an insulating barrier. In other words, the picture is similar to that of the percolation process described in other manganites and also in the theoretical analysis of the in#uence of a magnetic "eld on, e.g., the random "eld Ising model as a toy model for cluster coexistence near "rst-order transitions (Moreo et al., 2000). Studies of thermal relaxation e!ects by Anane et al. (1999a) applied to Pr Ca MnO with \V V  x"0.33 are also in agreement with a mixed-phase tendency and percolative characteristics description of this compound. Anane et al. (1999a) focused their e!ort into the hysteresis region that separates the metallic and insulating phases upon the application of a magnetic "eld. More recently, Anane et al. (1999b) studied the low-frequency electrical noise for the same compound, at similar temperatures and "elds. Their conclusion is once again that mixed-phase behavior and percolation are characteristics of this material. More recently, Raquet et al. (2000), studying La Ca MnO (x"0.33), observed a giant and random telegraph noise \V V  in the resistance #uctuations of this compound. They attribute the origin of this e!ect to a dynamic mixed-phase percolative conduction process involving two phases with di!erent conductivities and magnetizations. These important experimental results are compatible with the theoretical expectations described earlier: if it were possible to switch o! the intrinsic disorder of manganites, the transition would be "rst order with more standard hysteresis e!ects (Yunoki et al., 2000; Moreo et al., 2000). But the in#uence of intrinsic disorder produces a distribution of critical "elds which causes mixed-phase characteristics, which themselves induce colossal relaxation e!ects. Oxygen isotope substitution on a material at the verge of a metal}insulator transition, such as (La Pr ) Ca MnO , leads to indications of phase segregation involving AF-insulating          and FM-metallic phases according to neutron powder di!raction studies by Balagurov et al. (1999) (see also Babushkina et al., 1998; Voloshin et al., 2000). The results for the resistivity vs. temperature shown in those papers are quite similar to those observed in other materials where percolation seems to occur. Then, once again it is observed that near a metal}insulator transition it is easy to alter the balance by small changes in the composition. Finally, neutron scattering studies of Pr Ca MnO by Kajimoto et al. (1998) have shown that \V V  in the temperature regime between ¹ and ¹ , ferromagnetic spin #uctuations have been !, observed. In addition, antiferromagnetic #uctuations appear to be present also in the same temperature regime (see Fig. 2 of Kajimoto et al., 1998), and thus a coexistence of FM and AF correlations exist in a "nite window of temperatures. This result is similar to that observed in the same temperature window ¹ (¹(¹ for Bi Ca MnO with large x (see Bao et al., 1997; , !\V V  Liu et al., 1998), and adds to the mixed-phase tendencies of these compounds. Very recently, neutron di!raction and inelastic neutron scattering results by Radaelli et al. (2000) obtained in Pr Ca MnO (x"0.30) indicated mesoscopic and microscopic phase segregation at di!erent \V V  temperatures and magnetic "elds.

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4.6. Mixed-phase tendencies in bilayered manganites Early neutron scattering experiments by Perring et al. (1997) reported the presence of long-lived antiferromagnetic clusters coexisting with ferromagnetic critical #uctuations in La Sr Mn O ,       which has a nominal hole density of x"0.4. Fig. 4.6.1a contains the intensity of their signal vs. momenta. The peaks at 0.0, 1.0 and 2.0 in the horizontal axis correspond to ferromagnetism. The relatively small peak at 0.5 corresponds to an antiferromagnetic signal. In view of their results, Perring et al. (1997) concluded that a simple mean-"eld approach where a given typical site interacts with other typical sites cannot be valid in the bilayered material, a conclusion that the authors of this review fully agree with. Note, however, that other authors disagree with the mixed-phase interpretation of the neutron results [see Millis (1998a, b), and the reply contained in Perring et al. (1998)] and with the data itself [Osborn et al. (1998) believes that the AF signal is smaller than it appears in Fig. 4.6.1a, although they agree with the notion that FM and AF interactions are "nely balanced in this compound]. Nevertheless, regardless of the actual intensities and in view of the overwhelming amount of experimental information pointing toward mixedphase tendencies in 3D manganites, these authors believe that Perring et al. (1997) have provided reasonable evidence that bilayers could also support mixed-phase states. Kubota et al. (1999a) studying the x"0.5 bilayered manganite, concluded that here the CE-type insulating and the A-type metallic phases coexist. Battle et al. (1996a, b) and Argyriou et al. (2000) arrived at similar conclusions. This is qualitatively compatible with the Monte Carlo simulation results described in Section 3 that showed "rst-order transitions between many phases in the limit of a large electron}phonon coupling. In particular, in Section 3 it was shown, based on theoretical calculations, that the A- and CE-type phases are in competition, and their states cross as a function of the t spin coupling J (Yunoki et al., 2000). Weak disorder transforms the "rst-order  $ transition into a second-order one with cluster coexistence in the vicinity of the critical point. This is an interesting detail that deserves to be reemphasized: the phenomenon of mixed-phase formation and percolation is expected to occur whenever a "rst-order transition separates two competing states, and whenever some sort of disorder a!ects the system. There is no need for one of the phases to be the 3D FM metallic state, which usually appears prominently in materials that show the CMR e!ect in manganites. This also shows that the DE mechanism is not needed to have a large magnetoresistance. This is in agreement with the conclusions of the work by Hur et al. (1998), where CMR e!ects for x"0.3 bilayered manganites were presented even without long-range ferromagnetism. Hur et al. (1998) discussed the possibility of nonhomogeneous states at low temperature. Chauvet et al. (1998), using ESR techniques applied to the x"0.325 bilayered system, also arrived at the conclusion that polarons or mixed-phase tendencies are possible in this compound. Based on powder neutron-di!raction studies for bilayered manganites in a wide range of densities, Kubota et al. (1999b, c) reported the phase diagram shown in Fig. 4.6.1b (see also Hirota et al., 1998. For results at x larger than 0.5 see Ling et al., 2000). The AFM-I and -II phases are A-type AF phases with di!erent spin periodicities along the direction perpendicular to the FM planes. The FM-I and -II phases are ferromagnetic states with the spins pointing in di!erent directions (for more details see Kubota et al., 1999b). For our purposes, the region of main interest is the one labeled as `Canted AFMa which arises from the coexistence of AF and FM features in the neutron di!raction signal. However, as repeatedly stressed in this review, a canted state is

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Fig. 4.6.1. (a) Intensity of neutron scattering experiments by Perring et al. (1997) performed on La Sr Mn O \V >V   with x"0.4. The main "gure shows the dependence with Q , while the inset contains a Q dependence (for details the V X reader should consult the original reference). At 150 K and 0.5 in the horizontal axis, a weak peak is observed corresponding to AF correlations, while the most dominant peaks denote ferromagnetism. (b) Magnetic phase diagram of La Sr MnO O reproduced from Kubota et al. (1999a). Most of the notation is standard, but a more detailed \V >V   explanation of the various phases can be found in the text or in the original reference. Note the prominent `Canted AFMa phase, which the authors of this review believe may have mixed-phase characteristics.

indistinguishable from a mixed FM}AF phase if the experimental techniques used average over the sample (see also Battle et al., 1999, and reply by Hirota et al., 1999). Further work, such as NMR studies, is needed to address the canted vs. mixed-phase microscopic nature of this state. Such a study would be important for clarifying these matters. Since the neutron scattering peaks observed by Kubota et al. (1999b, c) are sharp, the FM and AF clusters, if they exist, will be very

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Fig. 4.6.2 . (a) In-plane resistivity component  of (La Nd ) Sr Mn O (single crystals). The arrows indicate the
\X X       Curie temperature. Reproduced from Moritomo et al. (1997). (b) Resistivity of the electron-doped manganite La Y Ca Mn O versus temperature for x"0.0,0.3, and 0.5, reproduced from Raychaudhuri et al. (1998).  \V V    

large as in other manganites that have shown a giant cluster coexistence. The resistivity vs. temperature of x"0.40 and 0.45 already show features (Kubota, 1999d) somewhat similar to those that appeared in related experiments, namely dirty metallic behavior at low temperature with a (¹&0) increasing as x grows toward 0.5 (insulating phase). Very recently, Tokunaga et al. (2000) have observed with magneto-optical measurements a spatial variation of the magnetization in the region of `spin cantinga. Those authors produced clear images of the x"0.45 bilayer compound, and also of Pr Ca MnO at x"0.30, showing domains with typical length scale \V V  exceeding one micrometer. Tokunaga et al. (2000) concluded that phase separation occurs in the region that neutron scattering experiments labeled before as spin canted, in excellent agreement with the theoretical calculations (on the other hand, above ¹ Osborn et al. 1998 reported the !

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presence of canted-spin correlations). In addition,Zhou et al. (1998) also believe that the x"0.4 compound has polaron formation that condenses into clusters as the temperature is reduced. Also Vasiliu-Doloc et al. (1999), using X-ray and neutron scattering measurements for the x"0.4 bilayered manganite, concluded that there are polarons above ¹ (see also Argyriou et al., 1999). ! More recently, Campbell et al. (2000) found indications of micro-phase separation on the x"0.4 bilayer compound based on neutron scattering results. Chun et al. (2000) reported a spin-glass behavior at x"0.4 which is interpreted as caused by FM}AF phase-separation tendencies. The x"0.4 low-temperature phase of double-layer manganites, which appears to be a metal according to Figs. 2.5.2 and 3, can be transformed into a charge-ordered state by chemical substitution using (La Nd ) Sr Mn O . Data for several z's are shown in Fig. 4.6.2a. The \X X       shape of the  vs. temperature curves resemble results found for other materials where clear  indications of inhomogeneities were found using electron microscopy techniques. These authors believe that Fig. 4.6.2a may be indicative of a percolative transition between the FM- and CO-state at low temperature, where clusters of one phase grow in a background of the other until a percolation occurs. Moreover, recent theoretical work in this context (Moreo et al., 2000) allows for CMR e!ects involving two insulators, since apparently the most important feature of the compounds that present these e!ects is (i) a "rst-order-like transition between the competing phases and (ii) the presence of intrinsic disorder in the material. Thus, it is very interesting to note that in the bilayer system Sr Nd Mn O with x"0.0 and 0.10 a colossal MR e!ect has also \V >V   been reported involving two insulators (Battle et al., 1996), showing that it is not necessary to have a double-exchange-induced ferromagnetic metallic phase to observe this e!ect, as remarked before. Layered electron-doped compounds are also known. In fact, Raychaudhuri et al. (1998) reported transport, magnetic and speci"c heat studies of La Y Ca Mn O with x"0.0,0.3, and 0.5.  \V V     For x"0.0 the material is a FM insulator. As x grows, a transition to a metallic state at low temperature was observed. The resistivity vs. temperature results are reproduced in Fig. 4.6.2b. The similarities with the behavior of other materials is clear. Raychaudhuri et al. (1998) concluded that the x"0.0 compound may correspond to a FM}AF mixture involving unconnected ferromagnetic clusters embedded in an antiferromagnetic matrix. Additional, although indirect, evidence for mixed-phase tendencies in bilayer compounds can be obtained from photoemission experiments. In fact, the "rst set of high-energy resolution angleresolved photoemission (ARPES) measurements in the context of manganites was reported by Dessau et al. (1998) and the compound used was precisely La Sr Mn O with x"0.4 \V >V   (high-resolution photoemission results for La Sr Mn O and La Ca Mn O were pre\V V  \V V  viously reported by Park et al., 1996. Dessau and Shen (1999) also presented results for La Sr MnO ). In this experiment it was observed that the low-temperature ferromagnetic state \V V  was very di!erent from a prototypical metal. Its resistivity is unusually high, the width of the ARPES features are anomalously broad, and they do not sharpen as they approach the Fermi momentum. Single Fermi-liquid-like quasiparticles cannot be used to describe these features. In addition, the centroids of the experimental peaks never approach closer than approximately 0.65 eV to the Fermi energy. This implies that, even in the expected `metallica regime, the density of states at the Fermi energy is very small. Dessau et al. (1998) refers to these results as the formation of a `pseudogapa (see Fig. 4.6.3). Those authors found that the e!ect is present both in the FM and paramagnetic regimes, namely below and above ¹ . The pseudogap a!ects the entire Fermi surface, ! i.e., there is no important momentum dependence in its value, making it unlikely that it is caused by

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Fig. 4.6.3. Low-temperature (10 K) ARPES spectra corresponding to (La Nd )Sr Mn O along various high \X X     symmetry directions. Results reproduced from Dessau et al. (1998).

charge, spin, or orbital ordering. Dessau et al. (1998) and Dessau and Shen (1999) argued that the origin of this pseudogap cannot simply be a Mott}Hubbard e!ect since the density is x"0.4. The e!ect cannot arise from the simple DE mechanism which does not predict a pseudogap, and also cannot be caused by Anderson localization due to disorder, which is not expected to signi"cantly a!ect the density of states. In other words, it is not the mobility that appears to lead to large resistivities but the lack of states at the Fermi energy. Recent photoemission studies for bilayers and La Sr MnO with x"0.18 led to similar conclusions (Saitoh et al., 1999). For theoretical \V V  results at x"0 see van den Brink et al. (2000) and Yin et al. (2000). These ARPES results are in qualitative agreement with recent calculations by Moreo et al. (1999b) and Moreo et al. (2000), described in detail elsewhere in this review, where a pseudogap in the density of states was shown to appear naturally in mixed-phase regimes, either those created by electronic phase separation or by the in#uence of disorder on "rst-order transitions that leads to giant cluster formation. In both cases the conductivity was shown to be very small in these regimes (Moreo et al., 1999b), and a pseudogap appears in the density of states. It is possible that the low temperature region of the x"0.4 bilayer can be described in terms of a percolative process, and its reported `spin-canteda character is simply caused by mixing AF- and FM-phase. This rationalization also explains the large value of the resistivity even at low temperature. The photoemission results are consistent with scanning tunneling microscopy data (Biswas et al., 1998), gathered for single crystals and thin "lms of hole-doped manganites. This study showed a rapid variation in the density of states for temperatures near the Curie temperature, such that below ¹ a "nite density of states is observed at the Fermi energy while above ¹ a hard gap opens ! ! up. This result suggests that the presence of a gap or pseudogap is not just a feature of bilayers, but it appears in other manganites as well. In addition, the work of Biswas et al. (1998) suggest that the insulating behavior above ¹ is caused by a depletion in the density of states, rather than by !

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a change in the mobility. As in the photoemission work just described, it appears that Anderson localization is not the reason for the insulating behavior, since this mechanism is not expected to induce a gap in the density of states. 4.7. Mixed-phase tendencies in single-layered manganites Bao et al. (1996) reported the presence of macroscopic phase separation in the planar manganite Sr La MnO in the range between x"0.0 and 0.38. At x"0.0 the material is a 2D AF \V V  insulator, with no carriers in the e -band. As x grows, carriers are introduced and they polarize the  t -spins leading to spin polaron formation, as in other compounds at low electronic density. These  polarons attract each other and form macroscopic ferromagnetic regions. This result is in agreement with the theoretical discussion of Section 3 where it was found, both for one and two orbital models, that the region of small density of e -electrons has phase separation characteristics. The  conclusions of Bao et al. (1996) are also in excellent agreement with the studies discussed in this review in the context of La Ca MnO at large hole density concentration. \V V  4.8. Possible mixed-phase tendencies in nonmanganite compounds There are several other nonmanganite compounds that present a competition between FM and AF regions, states which in clean systems should be separated by "rst-order transitions, at least according to theoretical calculations. One of these compounds is La Y TiO . As y is varied, the \W W  average bandwidth = of the mobile electrons changes, and experiments have shown that a FM}AF transition appears (Tokura et al., 1993). This material may be a candidate for percolative FM}AF transitions, as in the manganites (see also Hays et al., 1999, for results on La Sr Ti O \V V  with phase-separation characteristics). Also Tb PdSi and Dy PdSi present properties that have     been interpreted as indicative of magnetic polaron formation (Mallik et al., 1998). Large MR e!ects have been found in Gd PdSi by Saha et al. (1999). In addition, simply replacing Mn by Co has   been shown to lead to physics somewhat similar to that found in manganites. For instance, results obtained for La Sr CoO using a variety of techniques have been interpreted as mixed-phase or \V V  cluster-glass states (see Caciu!o et al., 1999; Nam et al., 1999, and references therein). Also Se Te CuO presents a FM}AF competition with spin-glass-like features (Subramanian et al., \V V  1999), resembling the mixed-phase states discussed in this review. First-order FM}AF transitions have also been reported in Ce Fe -based pseudobinary systems (Manekar et al., 2000). Even results  obtained in "lms of vinylidene #uoride with tri#uoroethylene (Borca et al., 1999) have been interpreted in terms of a compressibility phase transition similar to those discussed by Moreo et al. (1999a), reviewed in Section 3. In addition, Ni S Se also presents some characteristics similar to \V V those of the materials described here, namely a metal}insulator transition which is expected to be of "rst-order, random disorder introduced by Se substitution, and an antiferromagnetic state (see Husmann et al., 1996; Matsuura et al., 2000; and references therein). Very recently, some ruthenates have been shown to present characteristics similar to those of electron-doped CaMnO (as discussed for example by Neumeier and Cohn, 2000), including  a tendency to phase separation. Transport and magnetic results by Cao et al. (2000) indicate that in the region between x"0.0 and 0.1 of Ca La RuO , the material changes rapidly from an \V V  antiferromagnetic insulator to a ferromagnetic metal. The behavior of the magnetic susceptibility

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vs. temperature is shown in Fig. 4.8.1a. The shape of the M vs. H curve (Fig. 4.8.1b) is quite signi"cant. On one hand, at "nite density x there appears to be a "nite moment as the "eld is removed, characteristic of FM samples. On the other hand, the linear behavior with H is indicative of AF behavior, namely for antiferromagnetically ordered spins the canting that occurs in the presence of a magnetic "eld leads to a linearly growing moment. A mixed-phase FM}AF is probably the cause of this behavior. The curve resistivity vs. temperature (also shown in Cao et al., 2000) indeed appears to have percolative characteristics, as found in many manganites. Also perovskites such as CaFe Co O have an interesting competition between AF and FM states as \V V  x is varied. In Fig. 4.8.2 the resistivity in the range of Co densities where the transition occurs is shown, reproduced from Kawasaki et al. (1998). The similarities with other results described in this review are clear. It is also important to mention here the large MR found in the pyrochlore compound Tl Sc Mn O , although it is believed that its origin maybe di!erent from the analogous e!ect \V V  

Fig. 4.8.1. (a) Magnetic susceptibility de"ned as M/H (M"magnetization, H"magnetic "eld) vs. temperature ¹ for the densities indicated of Ca La RuO (from Cao et al., 2000). Inset: Magnetization vs. temperature. (b) Magnetization \V V  M as a function of magnetic "eld for the densities indicated.

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Fig. 4.8.2. Resistivity (R) vs. temperature for SrFe Co O and CaFe Co O , reproduced from Kawasaki et al. \V V  \V V  (1998). For the meaning of the arrows the reader should consult the original reference.

Fig. 4.8.3. Resistivity vs. temperature of Tl Sc Mn O for various values of x. The upper, middle, and lower curves \V V   for each x correspond to applied "elds of H"0,3, and 6 T, respectively. Result reproduced from Ramirez and Subramanian (1997).

found in manganites (Ramirez and Subramanian, 1997 and references therein. See also Shimakawa et al., 1996; Cheong et al., 1996). The behavior of the resistivity with temperature, parametric with the Sc concentration and magnetic "elds is shown in Fig. 4.8.3. The similarities with the analogous plots for the manganites presented in previous sections is clear. More work should be devoted to clarify the possible connection between pyrochlore physics and the ideas discussed in this review.

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Diluted magnetic semiconductors also present characteristics of phase-separated states. Ohno (1998) has recently reviewed part of the work in this context. The physics of magneto-polarons has also been reviewed before by Kasuya and Yanase (1968). The reader should consult these publications and others to "nd more references and details about this vast area of research. Diluted semiconductors have mobile carriers and localized moments in interaction. At low temperatures the spins are ferromagnetically aligned and the charge appears localized. It is believed that at these temperatures large regions of parallel spins are formed. The cluster sizes are of about 100 As , a large number indeed (see Ohno et al., 1992). At a relatively small polaron density, their overlap will be substantial. Important experimental work in this context applied to Eu Gd Se can be found in \V V von Molnar and Methfessel (1967). The resistivity vs. temperature at several magnetic "elds of EuSe is shown in Fig. 4.8.4, reproduced from Shapira et al. (1974). The similarity with results for manganites is clear. Other compounds of this family present interesting FM}AF competitions. For instance, the phase diagram of EuB C presented by Tarascon et al. (1981) contains an intermediate region \V V labeled with a question mark between the FM and AF phases. This intermediate phase should be analyzed in more detail. Already Tarascon et al. (1981) favored an interpretation of this unusual region based on mixed-phase states. Recently, two magnetically similar but electronically inequivalent phases were detected with NMR applied to EuB by Gavilano et al. (1998). Also  Gavilano et al. (1995) reported a two-component NMR signal in CeAl , signalling inhomogeneities  in the material. Clearly, other compounds seem to present physics very similar to that found in

Fig. 4.8.4. Resistivity vs. temperature of EuSe for several magnetic "elds. The inset contains the zero-"eld resistivity vs. temperature in a di!erent scale. Results reproduced from Shapira et al. (1974).

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manganites, at least regarding the FM}AF competition. The diluted magnetic semiconductors have been rationalized in the past as having physics caused by magneto-polaron formation. However, larger clusters, inhomogeneities, and percolative processes may matter in these compounds as much as in manganites. Actually, optical experiments by Yoon et al. (1998) have already shown the existence of strong similarities between manganites and EuB . More recently, Snow et  al. (2000) presented inelastic light scattering measurements of EuO and Eu La B , as a function \V V  of doping, magnetic "elds, and temperature. A variety of distinct regimes were observed, including a magnetic polaron regime above the Curie temperature and a mixed FM/AF regime at La density x larger than 0.05. These Eu-based systems do not have strong electron}lattice e!ects associated with Jahn}Teller modes. Then, the existence of physical properties very similar to those of manganites show that the key feature leading to such behavior is the competition between di!erent tendencies, rather than the origin and detailed properties of those competing phases. It is clear that further experimental work should be devoted to clarify these interesting issues. The authors of this review "rmly believe that mixed-phase tendencies and percolation are not only interesting properties of manganites, but should be present in a large variety of other compounds as well.

5. Discussion, open questions, and conclusions In this review, the main results gathered in recent years in the context of theoretical studies of models for manganites have been discussed. In addition, the main experiments that have helped clarify the physics of these interesting compounds have also been reviewed. Several aspects of the problem are by now widely accepted, while others still need further work to be con"rmed. Intrinsic inhomogeneities exist in models and experiments and seem to play a key role in these compounds. Among the issues related with inhomogeneities that after a considerable e!ort appear well established are the following: (1) Work carried out by several groups using a variety of techniques have shown that electronic phase separation is a dominant feature of models for manganites, particularly in the limits of small and large hole doping. This type of phase separation leads to nanometer size coexisting clusters once the long-range Coulombic repulsion is incorporated into the models. (2) Working at constant density, the transitions between metallic (typically FM) and insulating (typically CO/AF) states are of xrst order at zero temperature. No counter-example has been found to this statement thus far. (3) A second form of phase separation has been recently discussed. It is produced by the in#uence of disorder on the "rst-order metal}insulator transitions described in the previous item. A simple intuitive explanation is given in Fig. 5.1. If couplings are "xed such that one is exactly at the "rst-order transition in the absence of disorder, the system is `confuseda and does not know whether to be metallic or insulating (at zero disorder). On the other hand, if the couplings are the same, but the strength of disorder is large in such a way that it becomes dominating, then tiny clusters of the two competing phases are formed with the lattice spacing as the typical length scale. For nonzero but weak disorder, an intermediate situation develops where #uctuations in the disorder pin either one phase or the other in large regions of space. This form of phase separation is even more promising than the electronic one for explaining the physics of manganites for a variety of reasons: (i) it involves phases with the same density, thus there

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Fig. 5.1. Sketch of the competition metal}insulator in the presence of disorder, leading to equal-density coexisting large clusters in the `disorder-induceda phase separation scenario.

Fig. 5.2. Sketch of the expected resistivity vs. temperature in the percolative picture. For more details see text.

are no constraints on the size of the coexisting clusters which can be as large as a micrometer in scale, as found in experiments. (ii) The clusters are randomly distributed and have fractalic shapes, leading naturally to percolative transitions from one competing phase to the other, as couplings or densities are varied. This is in agreement with many experiments that have reported percolative features in manganites. (iii) The resistivity obtained in this context is similar to that found in experiments, as sketched in Fig. 5.2: Near the critical amount of metallic fraction for percolation, at room temperature the charge conduction can occur through the insulating regions since their resistivity at that temperature is very similar to that of the metallic state. Thus, the system behaves as an insulator. However, at low temperatures, the insulator regions have a huge resistivity and, thus, conduction is through the percolative metallic "laments which have a large intrinsic

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resistivity. The system behaves as a bad metal, and  (¹"0) can be very large. (iv) Finally, it is  expected that in a percolative regime there must be a high sensitivity to magnetic "elds and other naively `smalla perturbations, since tiny changes in the metallic fraction can induce large conductivity modi"cations. This provides the best explanation of the CMR e!ect of which these authors are aware. (4) The experimental evidence for inhomogeneities in manganites is by now simply overwhelming. Dozens of groups, using a variety of techniques, have converged to such a conclusion. It is clear that homogeneous descriptions of manganites in the region of interest for the CMR e!ect are incorrect. These inhomogeneities appear even above the Curie temperature. In fact, the present authors believe that a new scale of temperature ¹H should be introduced, as very roughly sketched in Fig. 5.3. There must be a temperature window where coexisting clusters exist above the temperatures where truly long-range order develops. Part of the clusters can be metallic, and their percolation may induce long-range order as temperature decreases. The region below ¹H can be as interesting as that observed in high-temperature superconductors, at temperatures higher than the critical values. It is likely that it contains pseudogap characteristics, due to its low conductivity in low-bandwidth manganites. The search for a characterization of ¹H should be pursued actively in experiments. (5) The famous CE-state of half-doped manganites has been shown to be stable in mean-"eld and computational studies of models for manganites. Although such a state was postulated a long time ago, it is only recently that it has emerged from unbiased studies. The simplest view to understand the CE-state is based on a `band insulatinga picture: it has been shown that in a zigzag FM chain a gap opens at x"0.5, reducing the energy compared with straight chains. Thus, elegant geometrical arguments are by now available to understand the origin of the naively quite complicated CE-state of manganites. Its stabilization can be rationalized based simply on models of non-interacting spinless fermions in 1D geometries. In addition, theoretical studies have allowed one to analyze the properties of the states competing with the CE at x"0.5. In order to arrive at the CE-state, the use of a strong long-range Coulomb interaction to induce the staggered charge

Fig. 5.3. Illustration of a conjectured new temperature scale ¹H in manganites. Above the ordering temperatures ¹ , !¹ , and ¹ , a region with coexisting clusters could exist, in view of the theoretical ideas described in this review and the , ! many experiments that are in agreement. It is possible that this region may have pseudogap characteristics, as in the high-temperature superconductors. The sketch shown here tries to roughly mimic the phase diagram of LCMO. The doping independence of ¹H in the "gure is just to simplify the discussion. Actually, a strong hole density dependence of ¹H is possible.

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pattern is not correct, since by this procedure the experimentally observed charge stacking along the z-axis could not be reproduced, and in addition the metallic regimes at x"0.5 found in some manganites would not be stable. Manganese oxides are in the subtle regime where many di!erent tendencies are in competition. (6) Contrary to what was naively believed until recently, studies with strong electron Jahn}Teller phonon coupling or with strong on-site Coulomb interactions lead to quite similar phase diagrams. The reason is that both interactions disfavor double occupancy of a given orbital. Thus, if the goal is to understand the CMR e!ect, the important issue is not whether the material is Jahn}Teller or Coulomb dominated, but how the metallic and insulating phases, of whatever origin, compete. Calculations with Jahn}Teller phonons are the simplest in practice, and they have led to phase diagrams that contain most of the known phases found experimentally for manganites, such as the A-type AF insulating state at x"0, the A-type AF metallic state at x"0.5, the CE-state at x"0.5, etc. Such an agreement theory-experiment is quite remarkable and encouraging. (7) Also contrary to naive expectations, the smallest parameter in realistic Hamiltonians for Mn-oxides, namely `J a between localized t spins, plays an important role in stabilizing the $  experimentally observed phases of manganites, including the CE-state. Modi"cations of this coupling due to disorder are as important as those in the hopping amplitudes for e -electron  movement. In short, it appears that some of the theories proposed in early studies for manganites can already be shown to be incorrect. This includes (i) simple double-exchange ideas where the high resistivity above ¹ is caused by the disordered character of the localized spins that reduce the ! conductivity in the e band. This is not enough to produce an insulating state above ¹ , and does  ! not address the notorious inhomogeneities found in experiments. It may be valid in some large-bandwidth compounds away from the region of competition between metal and insulator. (ii) Anderson localization also appears unlikely to explain the experimental data. An unphysically large value of the disorder strength is needed for this to work at high temperature, the pseudogap found in photoemission experiments cannot be rationalized in this context where the density of states is not a!ected by disorder, and large inhomogeneities, once again, cannot be addressed in this framework. However, note that once a percolative picture is accepted for manganites, then some sort of localization in such a fractalic environment is possible. (iii) Polaronic ideas can explain part of the experimental data at least at high temperatures, far from the Curie temperature. However, the region where CMR is maximized cannot be described by a simple gas of heavy polarons or bipolarons (see experimental results in Section 4). There is no reason in the polaronic framework for the creation of micrometer size coexisting clusters in these compounds. Actually, note that theories based on small polarons and phase separation do not di!er only on subtle points if the phase separation involves microdomains. It may happen that nanometer phase separation leads to physics similar to that created by polaronic states, but certainly not when much larger clusters are formed. As a conclusion, it is clear that the present prevailing paradigm for manganites relies on a phase-separated view of the dominant state, as suggested by dozens of experiments and also by theoretical calculations once powerful many-body techniques are used to study realistic models. Although considerable progress has been achieved in recent years in the analysis of manganites, both in theoretical and experimental aspects, there are still a large number of issues that require

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further work. Here a partial list of open questions is included: (a) The phase-separation scenario needs further experimental con"rmation. Are there counterexamples of compounds where CMR occurs but the system appears homogeneous? (b) On the theory front, a phase-separated percolative state is an important challenge to our computational abilities. Is it possible to produce simple formulas with a small number of parameters that experimentalists can use in order to "t their, e.g., transport data? The large e!ort needed to reproduce the zero magnetic "eld resistivity vs. temperature results (reviewed here) suggests that this will be a hard task. (c) It is believed that at zero temperature the metal}insulator transition is of "rst order and upon the introduction of disorder it becomes continuous, with percolative characteristics. A very important study that remains to be carried out is the analysis of the in#uence of temperature on those results. These authors believe that the generation of a `quantum critical pointa (QCP) is likely in this context, and preliminary results support this view (Burgy et al., 2000). The idea is sketched in Fig. 5.4. Without disorder (part (a)), the "rst-order transition survives the introduction of temperature, namely in a "nite temperature window the transition between the very di!erent FM and AF states remains "rst order. However, introducing disorder (part (b)), a QCP can be generated since the continuous zero-temperature transition is unlikely to survive at "nite temperature at "xed couplings. The presence of such QCP would be a conceptually important aspect of the competition between FM and AF phases in manganites. Experimental results showing that the generation of such QCP is possible have already been presented (Tokura, 2000). (d) There is not much reliable theoretical work carried out in the presence of magnetic "elds addressing directly the CMR e!ect. The reason is that calculations of resistivity are notoriously

Fig. 5.4. Illustration of how a quantum critical point can be generated in models for manganites. In (a) the "rst-order FM}AF transition is shown as a function of temperature, without disorder ("0). In (b), the expected behavior with disorder is shown. In both cases `ga is a coupling or hole density that allows the system to change from a metal to an insulator, and the disorder under discussion involves adding a random component to `ga. Fig. 5.5. Simple rationalization of the CMR e!ect based on a "rst-order transition metal}insulator. In this context CMR can only occur in a narrow window of couplings and densities. Sketched is the ground-state energy vs. a parameter `ga that causes the transition from metal to insulator (coupling or density). The FM phase is shown with and without a magnetic "eld `ha.

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di$cult, and in addition, the recent developments suggest that percolative properties are important in manganites, complicating the theoretical analysis. Nevertheless, the present authors believe that a very simple view of the CMR e!ect could be as follows. It is known that the metallic and insulating phases are separated by "rst-order transitions. Then, when energy is plotted vs. the parameter `ga that transforms one phase into the other (it could be a coupling in the Hamiltonian or the hole density), a level crossing occurs at zero temperature, as sketched in Fig. 5.5. In the vicinity of the transition point, a small magnetic "eld can produce a rapid destabilization of the insulating phase in favor of the metallic phase. This can occur only in a small window of densities and couplings if realistic (small) magnetic "elds are used. At present it is unknown how disorder, and the percolation phenomena it induces, will a!ect these sketchy results. In addition, there are compounds such as Pr Ca MnO that present CMR in a large density window, suggesting that \V V  the simple picture of Fig. 5.5 can be a good starting point, but is incomplete. Thus, quantitative calculations addressing the CMR e!ect are still needed. (e) Does a spin-canted phase ever appear in simple models with competing FM- and AF-phase in the absence of magnetic "elds? Are the regions labeled as spin canted in some experiments truly homogeneous or mixed states? (f) If the prediction of a phase-separated state in the CMR regime of manganites is experimentally fully con"rmed, what are the di!erences between that state and a canonical `spin glassa? Both share complexity and complicated time dependences, but are they in the same class? Stated in more exciting terms, can the phase-separated regime of manganites be considered a `newa state of matter in any respect? (g) Considerable progress has been achieved in understanding the x"0 and 0.5 charge/orbital/spin order states of manganites. But little is known about the ordered states at intermediate densities, both in theory and experiments. Are there stripes in manganites at those intermediate hole densities as recently suggested by experimental and theoretical work? Summarizing, the study of manganites continues challenging our understanding of transition metal oxides. While considerable progress has been achieved in recent years, much work remains to be done. In particular, a full understanding of the famous CMR e!ect is still lacking, although evidence is accumulating that it may be caused by intrinsic tendencies toward inhomogeneities in Mn-oxides and other compounds. Work in this challenging area of research should continue at its present fast pace.

Acknowledgements The authors would like to thank our many collaborators that have helped us in recent years in the development of the phase separation scenario for manganites. Particularly, the key contributions of Seiji Yunoki are here acknowledged. It is remarked that a considerable portion of the subsection entitled `Monte Carlo Simulationsa has been originally prepared by Seiji Yunoki. We also thank C. Buhler, J. Burgy, S. Capponi, A. Feiguin, N. Furukawa, K. Hallberg, J. Hu, H. Koizumi, A. Malvezzi, M. Mayr, D. Poilblanc, J. Riera, Y. Takada, J. A. Verges, for their help in these projects. We are also very grateful to S. L. Cooper, T. Egami, J. P. Hill, J. Lynn, D. Mills, J. Neumeier, A. Pimenov, and P. Schi!er, for their valuable comments on early drafts of the present review.

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E.D. and A.M. are supported by grant NSF-DMR-9814350, the National High Magnetic Field Laboratory (NHMFL), and the Center for Materials Research and Technology (MARTECH). T.H. has been supported by the Ministry of Education, Science, Sports, and Culture (ESSC) of Japan during his stay in the National High Magnetic Field Laboratory, Florida State University. T.H. is also supported by the Grant-in-Aid for Encouragement of Young Scientists under the contact No. 12740230 from the Ministry of ESSC.

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Renormalization group theory in the new millennium edited by Denjoe O'Connor, C.R. Stephens editor: I. Procaccia Contents Editorial J. Zinn-Justin, Precise determination of critical exponents and equation of state by "eld theory methods K. Binder, E. Luijten, Monte Carlo tests of renormalization-group predictions for critical phenomena in Ising models

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M.J. Bowick, A. Travesset, The statistical mechanics of membranes D.I. Kazakov, Supersymmetry in particle physics: the renormalization group viewpoint

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Renormalization group theory in the new millennium 1. Introduction In the last half-century, renormalization group (RG) theory has become a central structure in theoretical physics and beyond, though it is not always clear that di'erent authors mean the same thing when they speak about it. The aim of these reviews is to try and convey some of the power and scope of RG theory and its applications and in the process hopefully convey the underlying unity of the set of ideas involved. The reviews are loosely based on plenary talks given at the conference “RG2000: Renormalization Group Theory at the Turn of the Millennium” held in Taxco, Mexico in January 1999. The chief purpose of the conference was to bring together a group of people who had made signi3cant contributions to RG Theory and its applications, especially those who had contributed to the development of the subject in quantum 3eld theory=particle physics and statistical mechanics=critical phenomena, i.e. the high- and low-energy regimes of RG theory. Interestingly enough, some, such as Michael Fisher and Dimitri Shirkov, had never met before; testament perhaps to the lack of contact between the two principle di'erent streams of RG thought. Although RG theory has had a major impact it has tended to be viewed as a tool rather than as a subject in and of itself. Being presented principally in terms of its applications has therefore meant a lack of contact between practitioners from di'erent 3elds. An important exception to this tendency is the series of RG conferences organized by Dimitri Shirkov and others of the Joint Institute for Nuclear Research, Dubna theoretical physics community. The Taxco conference was in the same spirit. The advent in recent years of conferences on the “exact” RG has also provided an opportunity for practitioners to come together. The only criticism one might have of this latter series is the large emphasis on 3eld theory. This small criticism notwithstanding we hope that there will be continued opportunity to bring together RG theorists from di'erent 3elds. In obtaining contributions for these reviews we did not restrict ourselves to speakers from the conference. A major concern was to avoid producing a typical conference proceedings volume. Hence, the remit given to the contributors was to write as extensively and comprehensively as they saw 3t. Naturally, with such a liberal regime the length of article varies signi3cantly. Our goal was to try and review the state of the art of RG theory given that it could now be considered mature enough to warrant a large scale overview. We believe that we were to some c 2001 Elsevier Science B.V. All rights reserved. 0370-1573/01/$ - see front matter
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extent defeated in our purpose by the very size and range of applicability of the RG. Although we have managed to cover a large gamut we know there are glaring omissions. Nevertheless, we feel it is of great bene3t to have reviews by leading exponents all brought together in the same place even if the range of coverage is suboptimal. A possible remedy to this would be for specialists in areas not adequately covered here to submit articles which would naturally fall into the present series. We particularly wanted to emphasize the idea that although mature enough to warrant a major review, RG theory is young enough, and more signi3cantly, deep enough, that such a review would still only barely scratch its surface. We hope that young researchers will get the feeling that it is still very much an emerging 3eld with a large number of open problems associated with the understanding of RG theory itself and an even larger number associated with its applications. 2. Introduction to the first Volume In this 3rst set of lectures Zinn-Justin reviews progress in the most successful and intensively studied application of RG theory—the calculation of universal quantities in critical phenomena. He reviews the current status of the precise determination of critical exponents, universal amplitudes and the equation of state—showing the present day “gold-standards” by which other calculations may be judged. One may claim that this area is largely understood, however, with recent advances in space-based experiments on liquid helium there has emerged a fresh challenge to theory. In this system the precision of experimentally determined critical exponents now surpasses their theoretical precision. Additionally, as emphasized by Zinn-Justin, the range of universal quantities that has been calculated is quite small. It was the task of Binder and Luijten to review the confrontation between RG theory and “experiment”, or rather numerical simulations. Given the vast applicability of the RG they cleverly chose to focus on one well studied and relatively well understood area—that of ferromagnetic Ising models. One of the most important conclusions of the authors is that even in the most intensively studied model, the 3-D Ising model, important quantities such as the universal equation of state have been neglected relatively speaking and in terms of high precision calculations are rather poorly understood. Additionally, they point to results for dimensions above the upper critical dimension where controversy between theoretical results gleaned from the theory of 3nite size scaling using the RG and numerical simulations arise. Also poorly understood are even the simplest crossover scaling functions such as that between critical behavior and mean 3eld theory for T ¡ Tc . The article by Bowick and Travesset gives an overview of RG results from the emerging 3eld of membrane physics, an area of much current interest that cuts across the boundaries between physics, chemistry, biology and materials science. The authors gather together known RG results, principally using  expansion techniques, about crystalline membranes and compare with both numerical simulations and experiment. Given that the phase diagrams of even the more studied crystalline membrane systems are still not fully understood theoretically the authors emphasize that the application of the RG to such soft condensed matter systems will remain a challenging and fruitful area for the foreseeable future. The current vogue for considering

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membranes as fundamental objects in high-energy physics also makes RG-based studies of membrane Guctuations of great potential interest in this domain also. The article of Kazakov examines the opposite end of the energy spectrum considering how the RG may be used to probe physics beyond the standard model and in particular how the RG may be used to make de3nite predictions at energies currently inaccessible but which may be in the realm of experiment in the near future. Of particular interest here is the extra ingredient of supersymmetry which seems to be necessary to have a self-consistent theory in the ultraviolet. The RG, via an interpolation of the running of the standard model coupling constants, also places stringent consistency checks on possible grand uni3cation scenarios. Acknowledgements We take this opportunity to thank our coorganizers of the Taxco conference, Alberto Robledo, Riccardo Capovilla and Juan Carlos D’Olivo, and the conference secretaries, Trinidad Ramirez and Alejandra GarcKLa. We thank the conference sponsors for their signi3cant 3nancial support: CONACyT, MKexico; NSF, USA; ICTP, Italy; the Depto de FKLsica, Cinvestav, MKexico; Instituto de Ciencias Nucleares, UNAM, MKexico; Instituto de FKLsica, UNAM, MKexico; Fenomec, UNAM, MKexico; Cinvestav, MKexico; DGAPA, UNAM, MKexico and the CoordinaciKon de InvestigaciKon CientKL3ca, UNAM, Mexico. It is fair to say that without this generous support a conference of such caliber could not have taken place. We also take this opportunity to express our gratitude, for their advise and assistance, to the international advisory committee comprised of: A.P. Balachandran, Syracuse University, USA; K. Binder, Mainz, Germany; M.E. Fisher, University of Maryland, USA; N. Goldenfeld, University of Illinois, USA; B.L. Hu, University of Maryland, USA; D. Kazakov, Dubna, Russia; S. Priezzhev, Dubna, Russia; I. Procaccia, Weizmann Institute, Israel; M. Shifman, University of Minasota, USA; D.V. Shirkov, Dubna, Russia; F. Wegner, Heidelberg, Germany; J. Zinn-Justin, Saclay, France. We express our special thanks to Michael Fisher for his cogent advice and organizational help, to Bei Lok Hu for helping organize the US component of the conference and to Itamar Procaccia for organizing an appropriate forum in which to present this overview. Denjoe O’Connor Dept. de Fisica; CINVESTAV, A. Postal 14-740; 07360 Mexico D.F.; Mexico C.R. Stephens Instituto de Ciencias Nucleares; A. Postal 70-543, 04510 Mexico D.F.; Mexico E-mail address: [email protected]

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Precise determination of critical exponents and equation of state by "eld theory methods夽 J. Zinn-Justin CEA-Saclay, Service de Physique The& orique, F-91191 Gif-sur-Yvette, Cedex, France Received June 2000; editor: I. Procaccia Contents 1. Introduction: the e!ective
 "eld theory 1.1. The e!ective quantum "eld theory 1.2. The divergence of the bare coupling constant 2. Critical exponents and series summation 2.1. Borel transformation and conformal mapping 2.2. Exponents 3. The scaling equation of state 3.1. Amplitude ratios 3.2. The
-expansion 4. Parametric representation of the equation of state 4.1. Parametric representation and
-expansion

161 161 161 164 164 165 167 167 167

5. Perturbative expansion at "xed dimension three 5.1. General remarks 5.2. The problem of the low-temperature phase 6. Numerical results 6.1. The small-"eld expansion 6.2. Parametric representation 6.3. Amplitude ratios 7. Concluding remarks Acknowledgements References

170 170 171 173 173 173 175 175 176 176

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Abstract Renormalization group, and in particular its quantum "eld theory implementation has provided us with essential tools for the description of the phase transitions and critical phenomena beyond mean "eld theory. We therefore review the methods, based on renormalized
 quantum "eld theory and renormalization group, which have led to a precise determination of critical exponents of the N-vector model (Le Guillou and Zinn-Justin, Phys. Rev. Lett. 39 (1977) 95; Phys. Rev. B 21 (1980) 3976; Guida and Zinn-Justin, J. Phys. A 31 (1998) 8103; cond-mat/9803240) and of the equation of state of the 3D Ising model (Guida and Zinn-Justin, Nucl. Phys. B 489 [FS] (1997) 626, hep-th/9610223). These results are among the most precise available

夽 Talk given at the Conference `Renormalization Group 2000a, Taxco (Mexico), 11}15 Jan. 1999.  Laboratoire de la Direction des Sciences de la Matie% re du Commissariat a% l'Energie Atomique. E-mail address: [email protected] (J.Z. Zinn-Justin).

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probing "eld theory in a non-perturbative regime. Precise calculations "rst require enough terms of the perturbative expansion. However perturbation series are known to be divergent. The divergence has been characterized by relating it to instanton contributions. The information about large-order behaviour of perturbation series has then allowed to develop e$cient `summationa techniques, based on Borel transformation and conformal mapping (Le Guillou and Zinn-Justin (Eds.), Large Order Behaviour of Perturbation Theory, Current Physics, vol. 7, North-Holland, Amsterdam, 1990). We "rst discuss exponents and describe our recent results (Guida and Zinn-Justin, 1998). Compared to exponents, the determination of the scaling equation of state of the 3D Ising model involves a few additional (non-trivial) technical steps, like the use of the parametric representation, and the order dependent mapping method. From the knowledge of the equation of state a number of ratio of critical amplitudes can also be derived. Finally we emphasize that few physical quantities which are predicted by renormalization group to be universal have been determined precisely, and much work remains to be done. Considering the steady increase in the available computer resources, many new calculations will become feasible. In addition to the in"nite volume quantities, "nite size universal quantities would also be of interest, to provide a more direct contact with numerical simulations. Let us also mention dynamical observables, a largely unexplored territory.  2001 Elsevier Science B.V. All rights reserved. PACS: 05.10.Cc

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1. Introduction: the e4ective 4 5eld theory Second-order phase transitions are continuous phase transitions where the correlation length diverges. Renormalization group (RG) arguments [5], as well as an analysis, near dimension four, of the most IR divergent terms appearing in the expansion around mean-"eld theory [6], indicate that such transitions possess universal properties, i.e. properties independent to a large extent from the details of the microscopic dynamics. Moreover all universal quantities can be calculated from renormalizable quantum "eld theories which are local when the interactions are short range. For an important class of physical systems and models one is led to a
-like euclidean "eld theory with O(N) symmetry. Among those let us mention statistical properties of polymers, liquid}vapour, binary mixtures, super#uid Helium, ferromagnetic transitions, etc. We explain here how critical exponents and other universal quantities have been calculated with the help of renormalization group ideas and their implementation within the quantum "eld theory context [7]. 1.1. The ewective quantum xeld theory Wilson's renormalization group shows that, at least in the neighbourhood of dimension four, the universal properties of statistical models which short-range interactions and O(N) symmetry can generically be described by an ewective euclidean "eld theory, with an action H(
)



H(
)" dBx[
(x)]#r
(x)# g \B(
(x))) ,   r 

(1.1)

where r is a regular function of the temperature ¹. The parameter  has the dimension of a mass and corresponds to the inverse microscopic scale. It also appears as a cut-o! in the Feynman diagrams of the perturbative expansion. For some value r "r(¹ ) the correlation length  diverges (the physical mass m"1/ vanishes).   Near the critical temperature ¹ , in the critical domain ¹!¹ Jr!r ;, 
;B\,    the mass remains small, m;. In this limit a universal behaviour of thermodynamic quantities is expected. The study of the critical domain thus reduces to the study of the large cut-o! behaviour, i.e. to renormalization theory and the corresponding renormalization group. However, in the traditional presentation of quantum "eld theory in the context of particle physics, the dependence of the parameters of the action as a function of the cut-o!  is determined by the condition that renormalized correlation functions should have a "nite cut-o! limit. Here instead the coe$cient g \B/4! of the
 interaction has a dependence on  given a priori. In particular in the  dimensions of interest, d(4, the `barea coupling constant diverges for PR, a re#ection of the IR instability of the Gaussian "xed point, though the "eld theory, being super-renormalizable, requires only a mass renormalization. 1.2. The divergence of the bare coupling constant One solution to the problem of the large coupling constant is provided by Wilson}Fisher's famous
-expansion [8]. One de"nes, at least in perturbation theory, the "eld theory in arbitrary complex space dimension d. Setting d"4!
one then expands both in g and
. The large cut-o! 

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divergences become logarithmic as in four dimensions, and can be removed by standard renormalizations. Renormalization group (RG) equations for correlation functions follow, from which scaling laws can be derived, and which lead to calculations of physical quantities as power series in
. The most convenient RG equations are homogeneous di!erential equations, "rst derived for the critical (massless) theory [9], and then generalized to the whole critical domain rOr [10]. They  correspond to a scheme in which the massless (or critical) theory has a perturbative expansion in g and therefore rely on the double series expansion in g and
. An alternative formalism is based   on the Callan}Symanzik (CS) (inhomogeneous) equations for the massive theory [11]. One introduces correlation functions for a renormalized "eld
"Z\
, expressed in terms of the 0 physical mass m or inverse correlation length \"m and a renormalized coupling constant g. They are implicitly de"ned by the conditions (after factorization of the obvious group indices): I (p; m, g)"m#p#O(p) , 0

(1.2a)

I (p "0; m, g)"m\Bg , 0 G

(1.2b)

where the functions I L are the Fourier transforms of the so-called (1PI) correlation functions L, 0 0 the coe$cients of the expansion of the thermodynamic potential in terms of the local "eld expectation value 
(x). 0 To prove scaling laws from the CS equations it remains necessary to expand in g and
. The form of the CS -function in dimension d is (g)"!(4!d)g#a (d)g#2, a (d)'0 .   For
"4!d small it has a zero gH of order
and one recovers the principle of the
-expansion. However in this framework, because the theory now is massive, the perturbative expansion is IR "nite in any dimension and the CS equations are always satis"ed. Moreover one generally assumes that the results established for the
-expansion remain valid at "nite
. Therefore Parisi [12] has suggested working at "xed dimension d(4, in the massive theory (the massless theory is IR divergent). In contrast with the
-expansion however, at "xed dimensions three or two no small parameter is available. Therefore a precise determination of gH and all other physical quantities depends on the analytic properties of the series, in addition to the number of terms available. A semi-classical analysis, based on instanton calculus, unfortunately indicates that perturbation in quantum "eld theory is always divergent. Therefore to extract any information from perturbation theory a summation method is required. Note "nally that at "xed dimension, at any "nite order, universal quantities are given by expressions which depend on the renormalization scheme, in contrast with the results of the
-expansion. This approach, which is more cumbersome, has eventually been tried for practical reasons: it is easier to calculate Feynman diagrams in dimension three than in generic dimensions, and thus more perturbative orders could be obtained. While this approach is a natural extension of the
-expansion, its interpretation directly at "xed dimension is worth discussing. At "xed dimension d(4, in the critical domain (condition which implies shifting r by r and thus performing a mass renormalization), all terms in the pertur bative expansion have a "nite large cut-o! limit at u "g \B "xed because the theory is  

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super-renormalizable. This means physically that the initial parameters of the "eld theory are "rst tuned to remain arti"cially close to the unstable u "0 Gaussian "xed point. Indeed the true  expansion parameter is a dimensionless quantity and therefore is proportional to g \B/m\B. To  keep this parameter constant when m the physical mass goes to zero for ¹P¹ (rPr ) in the high   temperature phase, one must vary the dimensionless parameter g as  g\BJm/J1/ .  The relevant theory, however, corresponds to the in"nite u limit. One is then confronted with  a serious technical problem: perturbation theory is "nite in the critical domain but one is interested in the in"nite coupling limit, where obviously the perturbative expansion is no longer useful. One thus introduces the renormalized coupling constant g and the "eld renormalization de"ned by the conditions (1.2). Then u "g \B"m\BG(g), (g)"(d!4)G(g)/G (g) .  

(1.3)

When the initial coupling constant u becomes large the new dimensionless coupling g has a "nite  limit provided the -function has an IR stable zero gH: u PR N (gH)"0 and 

, (gH)'0 .

In such a situation the renormalized coupling g is a more suitable expansion parameter than u .  Relations (1.3) then imply that at g "xed, /mPR,  g!gHJ(m/)S .

(1.4)

To the "eld renormalization Z(g) is associated the RG function (g) (g)" (g)Z (g)/Z(g) .

(1.5)

Integration of Eq. (1.5) yields the behaviour of Z(g) near gH Z(g)J(gH!g)ESJ(m/)E ,

(1.6)

where the exponent " (gH) characterizes the "eld anomalous dimension d : 2d "d!2# . ( ( This singular behaviour, consistent with the scaling properties derived from the
-expansion, explains why even in dimension d(4 the introduction of a "eld renormalization is necessary (Table 1). The approach has sometimes be questioned, because it involves this double limit, but the "nal results and their comparison with the other data have shown the consistency of the method. A last remark: in this framework the mass parameter m has still to be related to the temperature. RG equations show that it is singular at ¹ and behaves for t"r!r P0 as   > mJtJJ(¹!¹ )J ,  where is the correlation length exponent.

(1.7)

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Table 1 Critical exponents of the O(N) models from d"3 expansion [2] N

0

1

2

3

gJ H , gH 

  "

1.413$0.006 26.63$0.11 1.1596$0.0020 0.5882$0.0011 0.0284$0.0025 0.3024$0.0008 0.235$0.003 0.812$0.016 0.478$0.010

1.411$0.004 23.64$0.07 1.2396$0.0013 0.6304$0.0013 0.0335$0.0025 0.3258$0.0014 0.109$0.004 0.799$0.011 0.504$0.008

1.403$0.003 21.16$0.05 1.3169$0.0020 0.6703$0.0015 0.0354$0.0025 0.3470$0.0016 !0.011$0.004 0.789$0.011 0.529$0.009

1.390$0.004 19.06$0.05 1.3895$0.0050 0.7073$0.0035 0.0355$0.0025 0.3662$0.0025 !0.122$0.010 0.782$0.0013 0.553$0.012

2. Critical exponents and series summation Critical exponents are the most studied quantities in the theory of phase transitions, because they characterize universality classes and are easier to calculate. They have been extensively used to test RG predictions by comparing them with other results (experiments, high- or low-temperature series expansion, Monte-Carlo simulations) [13]. The "rst precise determination of the exponents of the O(N) symmetric N-vector model has been reported by Le Guillou}Zinn}Justin [1], using the six-loop series for RG functions calculated by Nickel et al. [14]. The summation methods used in [1] were based on a Borel transformation and conformal map. The same ideas have later been applied to the
-expansion [15] when "ve loop series became available [16], and more recently to the equation of state by Guida}Zinn}Justin [3]. With time the method has been re"ned and the e$ciency improved by various tricks but the basic principles have not changed. 2.1. Borel transformation and conformal mapping Let F(g) be any quantity given by a perturbative series F(g)"
F gI . I I A semi-classical analysis of barrier penetration e!ects for negative coupling (instantons) teaches us that in the
 "eld theory for large-order k the coe$cients F behave like  I F J kQ(!a)Ik! . I I The value of a'0 has been determined numerically (while s is known analytically). One then introduces B(b, g), the Borel (rather Borel}Leroy) transform of F(g), which is de"ned by F I gI , B(b, g)"
(b#k#1) I

(2.2)

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165

where b is a free real parameter (b'!1). Formally, i.e. in the sense of series expansion, F(g) can be recovered from



F(g)"



t@e\RB(b, gt) dt .

(2.3)



Using the large-order estimate in (2.2) one veri"es that B(b, z) is analytic at least in a circle of radius 1/a and its singularity closest to the origin is located at z"!1/a. Therefore B(b, z), in contrast with F(z), is determined by its series expansion. However, relation (2.3) can be extended from a relation between formal series to a relation between functions only if B(b, z) is analytic in a neighbourhood of the real positive axis. In the case of the
 "eld theory such a property has been  proven in constructive "eld theory (as well as the property that the function F(g) is indeed given by (2.3)) [17]. Moreover it is necessary to construct an analytic continuation of the function B(b, z) from the circle to the real positive axis. Considerations about more general instanton contributions strongly suggest that the Borel transform actually is analytic in a cut-plane, the cut being on the real negative axis, at the left of !1/a. Therefore an analytic continuation is provided by a conformal map of the cut-plane onto a circle: (1#az!1 . z C u(z)" (1#za#1

(2.4)

The function B[b, z(u)] is then given by a series in powers of u convergent in the cut-plane. The corresponding (hopefully convergent) series expansion for function F(g) takes the form



  F(g)"
; (b) t@e\R[u(gt)]I dt . I  I

(2.5)

The parameter b, as well as a few other parameters introduced in variants, are used to improve the apparent convergence and test the sensitivity of results to their variations. Moreover the value of b has to stay within a reasonable range around the value s predicted by the large-order behaviour. Finally the summation method is expected to be e$cient mainly when the available coe$cients F behave already as predicted by the asymptotic large-order estimate. I 2.2. Exponents The values of critical exponents obtained from "eld theory have remained after about 20 years among the most precise determinations. Only recently have consistent, but signi"cantly more precise, experimental results been reported in low-gravity super#uid experiments [41]. Also the precision of results coming from high temperature expansions [18}29] and various numerical simulations [30}40] on the lattice has kept steadily improving. Recently seven-loop terms have been obtained for 04N43 for two of the three RG functions, related to the
and
 dimensions, by Murray and Nickel [42]. These terms, together with some improvement in the summation methods, have led to the new slightly more precise values of gH and critical exponents displayed in Table 1 (g H "(N#8)gH/48) (Guida}Zinn}Justin [2]). Among the ,G

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exponents , , , , , only two are independent, for example " (2! ), " (1# ), "2!3 ,  but they are calculated independently to test the precision of the summation method. The main improvements concern the exponent which was poorly determined, and the lower value of  for N"0 (polymers). In the framework of the
-expansion, since the series used earlier were a!ected by a small error at order
, the values have also been updated (Table 2). Two kinds of results are reported, free means simple summation as above, bc means that the known values in two dimensions have been incorporated in the summation procedure. It is gratifying that the overall consistency between the two set of values coming from 3D series and
-expansion has improved. Finally O(4) results of interest for simulations of the Higgs phase transition at "nite temperature, obtained from six loop series, have been added (Table 3). Table 2 Critical exponents of the O(N) models from
-expansion [2] N

0

1

2

3

 (free)  (bc)

1.1575$0.0060 1.1571$0.0030

1.2355$0.0050 1.2380$0.0050

1.3110$0.0070 1.317

1.3820$0.0090 1.392

(free)

(bc)

0.5875$0.0025 0.5878$0.0011

0.6290$0.0025 0.6305$0.0025

0.6680$0.0035 0.671

0.7045$0.0055 0.708

(free) (bc)

0.0300$0.0050 0.0315$0.0035

0.0360$0.0050 0.0365$0.0050

0.0380$0.0050 0.0370

0.0375$0.0045 0.0355

(free) (bc)

0.3025$0.0025 0.3032$0.0014

0.3257$0.0025 0.3265$0.0015

0.3465$0.0035

0.3655$0.0035

 

0.828$0.023 0.486$0.016

0.814$0.018 0.512$0.013

0.802$0.018 0.536$0.015

0.794$0.018 0.559$0.017

Table 3 Critical exponents in the O(4) models from d"3 and
-expansion [2]

g H , gH 

  

d"3


: free,bc

1.377$0.005 17.30$0.06 1.456$0.010 0.741$0.006 0.0350$0.0045 0.3830$0.0045 !0.223$0.018 0.774$0.020 0.574$0.020

1.448$0.015, 1.460 0.737$0.008, 0.742 0.036$0.004, 0.033 0.3820$0.0025 !0.211$0.024 0.795$0.030 0.586$0.028

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3. The scaling equation of state Let us "rst recall a few properties of the equation of the state in the critical domain, in the speci"c case N"1 (Ising-like systems), at d(4. The equation of state is the relation between magnetic "eld H, magnetization M"
 (the `barea "eld expectation value) and the temperature which is represented by the parameter t"r!r J¹!¹ . It is related to the free energy per unit volume, in "eld theory language the   generating functional (
) of 1PI correlation functions restricted to constant "elds, i.e the e!ective potential V, V(M)"(M)/vol., by H"RV/RM. In the critical domain the equation of state has Widom's scaling form ("(d#2! )/(d!2# )) H(M, t)"MBf (t/M@)

(3.1)

a form initially conjectured and which renormalization group has justi"ed. One property of the function H(M, t) which plays an essential role in the analysis is Grizth's analyticity: it is regular at t"0 for M'0 "xed, and simultaneously it is regular at M"0 for t'0 "xed. 3.1. Amplitude ratios Universal amplitude ratios are numbers characterizing the behaviour of thermodynamical quantities near ¹ . Several of them commonly considered in the literature can be directly derived  from the scaling equation of state. Let us just give two examples. The singular part of the speci"c heat, i.e. the
 two-point correlation function at zero momentum, behaves like C "A!t\?, tJ¹!¹ P$0 . (3.2) &  The ratio A>/A\ then is universal. The magnetic susceptibility  in zero "eld, i.e. the
two-point function at zero momentum, diverges like "C!t\A, tP$0 .

(3.3)

The ratio C>/C\ is also universal. 3.2. The
-expansion The "rst results concerning the scaling equation of state have been obtained within the framework of the
"4!d expansion. The
-expansion of the scaling equation of state has been determined up to order
 for the general O(N) model [43], and order
 for N"1 [44]. We give here the function f (x"t/M@) of Eq. (3.1) for N"1 up to order
 to display its structure f (x)"1#x#
(x#3)¸#
[  (x#9)¸#  (x#3)¸]#O(
) ,    with ¸"log(x#3).

(3.4)

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Expression (3.4) is not valid for x large, i.e. for small magnetization M. In this regime the magnetic "eld H has a regular expansion in odd powers of M. It is thus convenient to express the equation of state in terms of another scaling variable zJx\@ because zJMt\@ .

(3.5)

The equation of state then takes the form HJt@BF(z) ,

(3.6)

where the relation between exponents " (!1) has been used. Substituting into Eq. (3.4) x"x z\@ (the constant x takes care of the normalization of z) and expanding in
one "nds at   order
 for the function (3.6) F(z)"FI (z)#
FI (z)#
FI (z)#O(
)    with (3.7a) FI "z#z ,   FI "  (!z#¸I (2z#z)) ,   FI "  (!50z#¸I (100z!4z)#¸I (18z#27z)) (3.7b)   and ¸I "log(1#z/2). Within the framework of the formal
-expansion one can easily pass from one expansion to the other. Still a matching problem arises if one wants to use the
-expansion to determine the equation of state for d"3, i.e.
"1. One is thus naturally led to look for a uniform representation valid in both limits. Josephson}Scho"eld parametric representation [45] has this property.

4. Parametric representation of the equation of state In order to implement both Gri$th's analyticity and the scaling relation, one parametrizes the equation of state in terms of two new variables R and , setting M"m R@ ,  t"R(1!) ,

(4.1)

H"h R@Bh() ,  where h , m are two normalization constants. We choose h such that    h()"#O() . In terms of the scaling variables x of Eq. (3.4) or z from Eq. (3.5) this parametrization corresponds to set z"/(1!)@, '0 , x"x \@(1!)\@ ,  where  is some other positive constant.

(4.2) (4.3)

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169

Then the function h() is an odd function of  which from Gri$th's analyticity is regular near "1, which is x small, and near "0 which is x large. It vanishes for " which corresponds to  the coexistence curve H"0, ¹(¹ .  Note that the mapping (4.2) is not invertible for values of  such that z ()"0. The derivative vanishes for "1/((1!2 )+1.69. One has to verify that this value is reasonably larger than  ,  the largest possible value of . Finally it is useful for later purpose to write more explicitly the relation between the function F(z) of Eq. (3.6) and the function h(): h()"\(1!)@BF(z()) .

(4.4)

Expanding both functions zJ> , F(z)"z#z#
F J>  J h()/"1#
h J , J> J one "nds the relations h "! ,   h "(!1)#(2 !)#F  ,     h " (!1)(!2)#  (2 !)(2 !#1)#(4 !)F #F       2

(4.5) (4.6)

(4.7a) (4.7b) (4.7c)

From the parametric representation of the equation of state it is then possible to derive a representation for the singular part of the free energy per unit volume as well as various universal ratios of amplitudes. 4.1. Parametric representation and
-expansion Up to order
 the constant m (or ) can be chosen in such a way that the function h()  reduces to (4.8) h()"(1!)#O(
) .  The minimal model in which h() is approximated by a cubic odd function of  is called the linear parametric model. At order
 the linear parametric model is exact, but at order
 the introduction of a term proportional to  becomes necessary [44,46]. One "nds h()"(1#h #h )#O(
)  

(4.9)

with








 2 , h " ((3)!!) , h "! 1#  27    12 3

(4.10)

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where  is the constant " (1/3)!"1.171953619342 .   The function h() vanishes on the coexistence curve for " : 
 3 #O(
) .  " 1!  2 12






(4.11)

(4.12)

Note that h and thus  are determined only up to order
. It follows   "6(#h )"2(1#
# 
)"3.13$0.13 ,    because h is determined only up to order
.  Remark. In the more general O(N) case, the parametric representation also automatically generates equations of state which satisfy the required regularity properties, and thus leads to uniform approximations. However for N'1 the function h() still has a singularity on the coexistence curve, due to the presence of Goldstone modes in the ordered phase and has therefore a more complicated form. The nature of this singularity can be obtained from the study of the non-linear -model. It is not clear whether a simple polynomial approximation would be useful. For N"1 instead, one expects at most an essential singularity on the coexistence curve, due to barrier penetration, which is much weaker and non-perturbative in the small
- or small g-expansion.

5. Perturbative expansion at 5xed dimension three We now discuss the calculations based on a perturbative expansion at "xed dimension d"3. Five loop series for the renormalized e!ective potential of the
 theory have been "rst reported in  Bagnuls et al. [47], but the printed tables contain some serious misprints. These have been noticed by Halfkann and Dohm, who have published corrected values [48]. These "ve-loop calculations have only been performed for N"1, because they are much more di$cult for NO1 due to the presence of two lengths, the correlation lengths along the applied "eld and transverse to it. 5.1. General remarks The general framework again is the massive theory renormalized at zero momentum. The correlation functions L of the renormalized "eld
"
/(Z are "xed by the normalization 0 0 conditions (1.2). Eventually the renormalized coupling constant g has to be set to its IR "xed point value gH. Conditions (1.2) imply that the free energy F expressed in terms of the `renormalizeda magnetization , i.e. the expectation value of the renormalized "eld "
, has a small 0  expansion of the form (in d dimensions) 1 1 F()"F(0)# m# m\Bg#O() . 4! 2

(5.1)

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171

It is important to remember that the "nite "eld renormalization Z(g) is singular at gH (Eq. (1.6)). It follows that JM/mE. It is convenient to introduce the rescaled variable z z"m\B(gJM/mB\>EJM/t@ ,

(5.2)

(Eq. (1.7)) and set mB F()!F(0)" V(z, g) . g

(5.3)

Taking into account the de"nitions of Section 3, we conclude that the equation of state is related to the derivative F of the reduced e!ective potential V with respect to z RV(z, g) F(z, g)" , Rz

(5.4)

by HJt@BF(z) . Ising symmetry implies that F is an odd function of z 1 F(z, g)"z# z#
F (g)zJ> . J> 6 J

(5.5)

5.2. The problem of the low-temperature phase To determine the equation of state in the whole physical range, or universal ratios of amplitudes, a new problem arises. In this framework it is more di$cult to calculate physical quantities in the ordered phase because the theory is parametrized in terms of the disordered phase correlation length "m\J(¹!¹ )\J which is singular at ¹ (as well as all correlation functions nor  malized as in (1.2)). For example at one-loop order for d"3 the scaling function F(z, g) (Eq. (5.4)) is given by 1 1 F(z, g)"z# z! gz[(1#z/2)!1!z/4] 8 6 1 1 1 "z# z# gz! gz#O(z) , 6 256 2

(5.6)

where the subtractions, due to the mass and coupling normalizations, are determined by conditions (1.2). This expression is adequate for the description of the disordered phase. However when t goes to zero for "xed magnetization, i.e. mP0 at  "xed, then zPR as seen in Eq. (3.5). Thus all terms in the loopwise expansion become singular. In this limit one knows from Eq. (3.1) that the equation of state behaves like H(M, t"0)JMB N F(z)JzB .

(5.7)

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In the framework of the
-expansion the scaling relations (and thus the limiting behaviour (5.7)) are exactly satis"ed order by order. Moreover the change to the variable xJz\@ (more appropriate for the regime tP0) gives an expression for f (x)JF(x\@)x@B that is explicitly regular in x"0 (Gri$th's analyticity): the singular powers of log x induced by the change of variables cancel non-trivially at each order, leaving only regular corrections. The situation changes when one deals with the perturbation theory in d"3 dimensions: scaling is not satis"ed for generic values of g but only at gH. Consequently scaling properties are not satis"ed order by order in g. In particular the change to the Widom function f (x) will introduce singular powers of x that violate Gri$th's analyticity. An analogous problem arises if one "rst sums the series at g"gH before changing to the variable x. In this case the singular contributions (in the form of powers of x) do not cancel, as a result of unavoidable numerical summation errors. Several approaches have been proposed to solve the problem of continuation to the ordered phase. A rather powerful method, motivated by the results obtained within the
-expansion scheme, is based on the parametric representation [3]. 5.2.1. Parametric representation and order-dependent mapping (ODM) The problem that one faces is the following: to reach the ordered region t(0 one must cross the point z"R. However we know from Gri$th's analyticity that F(z)z\B is regular in the variable z\@. This property is naturally satis"ed within the parametric representation. One thus introduces a new variable  and an auxiliary function h() de"ned as in (4.2) and (4.4): the exact function h() is regular near "1 (i.e. z"R) and up to the coexistence curve. The approximate h() that one obtains by summing perturbation theory at "xed dimension, instead is still not regular, because the singular terms generated by the change of parametrization (4.2) at "1 do not cancel exactly due to summation errors. The last step involves a Taylor expansion of the approximate expression of h() around "0 and a truncation of the expansion, to enforce regularity. A question then arises, to which order in  should one expand? Since the coe$cients of the  expansion are in one to one correspondence with the coe$cients of the small z expansion of the function F(z, gH), the maximal power of  in h(), should be equal to the maximal power of z whose coe$cient can be determined with reasonable accuracy. Indeed although the small z expansion of F(z) at each "nite loop order in g contains an in"nite number of terms, the evaluation of the coe$cients of the higher powers of z is increasingly di$cult. The reasons are twofold: (i) The number of terms of the series in g required to get a precise estimate of F increases with J l because the large-order behaviour sets in later. (ii) At any "nite order in g the function F(z) has spurious singularities in the complex z plane (see e.g. Eq. (5.6), z"!2) that dominate the behaviour of the coe$cients F for l large. J In view of these di$culties one has to ensure the fastest possible convergence of the small  expansion. For this purpose one uses the arbitrary parameter  in Eq. (4.2): one determines  by minimizing the last term in the truncated small  expansion, thus increasing the importance of small powers of  which are more precisely calculated. This is nothing but the application to this particular example of the series summation method based on ODM [49].

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6. Numerical results The calculation proceeds in two steps; "rst one determines the "rst coe$cients of the small "eld expansion, then one introduces the parametric representation. 6.1. The small-xeld expansion The determination of the coe$cients F of the small z (small-"eld) expansion of the function J> F(z) relies on exactly the same method as for exponents, i.e. Borel}Leroy transformation and conformal map. In Fig. 1 the behaviour of F in terms of the Borel}Leroy parameter b is displayed.  Increasing #atness of the curves when k increases, i.e. increasing insensitivity to the parameter b, supports the hypothesis that the method indeed converges. Because the asymptotic regime sets in later when l increases, the e$ciency of the summation correspondingly decreases, as expected. Table 4 contains the results of [3] together with other published estimates of the coe$cients of the small z expansion of F(z) coming from hightemperature series [50,52,53], simulations [54] and derivative expansion of the Exact RG [55,56]. 6.2. Parametric representation One then determines by the ODM method the parameter  and the function h(), as explained in Section 5.2. One obtains successive approximations in the form of polynomials of increasing degree for h(). At leading order h() is a polynomial of degree 5, whose coe$cients are given by relations (4.7): h()"[1#h ()#h ()] . (6.1)   For the range of admissible values for F the coe$cient h of  given by Eq. (4.7b) has no real zero   in . It has a minimum instead 1 (!2 ) . " "  12F 

Fig. 1. The summed coe$cient F as a function of the Borel}Leroy parameter b for successive orders k. 

(6.2)

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Table 4 Equation of state


-exp., [3,2]
-exp., [51] d"3, [3,2] HT [50] HT [52] HT [53] MC [54] ERG [56]

gH

F 

F ;10 

F ;10 

23.3 23.4$0.1 23.64$0.07 24.45$0.15 23.72$1.49 23.69$0.10 23.3$0.5 20.72$0.01

0.0177$0.0010 0.01715$0.00009 0.01711$0.00007 0.017974$0.00015 0.0205$0.0052 0.0168$0.0012 0.0227$0.0026 0.01719$0.00004

4.8$0.6 4.9$0.6 4.9$0.5

!3.3$0.3 !5.5$4 !7$5

5.4$0.7

!2.3$1.1

4.9$0.1

!5.2$0.3

Substituting this value of  into expression (6.7) one obtains the "rst approximation for h(). At next order one looks for a minimum  of h (). One "nds a polynomial either of degree 5 in ,   when h has a real zero, or of degree 7 when it has only a minimum. It is not possible to go beyond  h () because already F is too poorly determined.   Note that one here has a simple test of the relevance of the ODM method. Indeed, once h() is determined, assuming the values of the critical exponents  and , one can derive the corresponding function F(z). It has an expansion to all orders in z. As a result one obtains a prediction for the coe$cients F which have not yet been taken into account to determine h(). The relative J> di!erence between the predicted values and the ones directly calculated gives an idea about the accuracy of the ODM method. Indeed from the values F "0.01711, "1.2398, "0.3256,  one obtains F "4.83;10\, F "!3.2;10\, F "1.4;10\ 2 .    We see that the value for F is quite close to the central value one "nds by direct series summation,  while the value for F is within errors. This result gives us con"dence in the method. It also shows  that the value of F obtained by direct summation contains little new information, it provides only  a consistency check. Therefore the simplest representation of the equation of state, consistent with all data, is given by h()"!0.76201(36)#8.04(11);10\

(6.3)

(errors on the last digits in parentheses) that is obtained from "2.8667. This expression of h() has a zero at  "1.154, which corresponds to the coexistence curve. The coe$cient of  in  Eq. (6.3) is smaller than 10\. Note that for the largest value of  which corresponds to  , the   term is still a small correction. Finally the corresponding values for the
-expansion are h "!0.72, h "0.013. These values are reasonably consistent, because a small change in h can    be cancelled to a large extent by a correlated change in . The Widom scaling function f (x), Eq. (3.1), can then easily be obtained numerically from h() and compared with other determinations. The main disagreement with other predictions comes from the region xPR, i.e. from the small magnetization region, where the predictions of the present method should be specially reliable.

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175

6.3. Amplitude ratios Various amplitude ratios can then be derived from h() and the values of the critical exponents determined earlier from longer series. They involve ratios of functions of  at "0 and at  where   is the zero of h() closest to the origin.  Tables 5 and 6 contain a comparison of several amplitude ratios as obtained from RG, lattice calculations (high-temperature series and Monte-Carlo simulations) and experiments on binary mixtures, liquid}vapour, uniaxial magnetic systems. The results from the
-expansion of [57,46] are obtained by direct PadeH summation of each corresponding series, while the results of [3] are obtained by "rst summing h() and then computing ratios, as explained in Section 5.2. The results from the d"3 "xed-dimension expansion of Ref. [47] refer to direct summation up to O(g) while the d"3 values of [2] again rely on the method explained in Ref. [3]. High-temperature results are taken from [50,29] (see also [58,59]). Experimental data are extracted from [60], to which we refer for more results and references. One notes the overall consistency of the results obtained by di!erent methods (see also Ref. [62]).

7. Concluding remarks Within the framework of renormalized quantum "eld theory and renormalization group, the presently available series allow, after proper summation, to determine precisely critical exponents Table 5 Amplitude ratios


-exp., [57,46]
-exp., [3,2] d"3, [47] d"3, [3,2] HT series [29] MC [61] bin. mix. liqu.}vap. magn. syst.

A>/A\

C>/C\

0.524$0.010 0.527$0.037 0.541$0.014 0.537$0.019 0.530$0.003 0.560$0.010 0.56$0.02 0.48}0.53 0.49}0.54

4.9 4.73$0.16 4.77$0.30 4.79$0.10 4.77$0.02 4.75$0.03 4.3$0.3 4.8}5.2 4.9$0.5

R 

R Q

0.0569$0.0035 0.0594$0.001 0.0574$0.0020 0.0564$0.0003

1.67 1.648$0.036 1.7 1.669$0.018 1.662$0.005

0.050$0.015 0.047$0.010

1.75$0.30 1.69$0.14

Table 6 Other amplitude ratios

HT series [50,29] d"3, [3,2]
-expansion, [3,2]

R 

R 

C>/C\  

0.1275$0.0003 0.12584$0.00013 0.127$0.002

6.041$0.011 6.08$0.06 6.07$0.19

!9.1$0.2 !9.1$0.6 !8.6$1.5

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for the N-vector model and the complete scaling equation of state for 3D Ising-like (N"1) systems. In the latter example additional technical tools, beyond Borel summation methods, are required in which the parametric representation plays a central role. From the equation of state new estimates of some amplitude ratios have been deduced which seem reasonably consistent with all other available data. Clearly a similar strategy could be applied to other quantities in a magnetic "eld, in the scaling region. Note also that an extension of the
-expansion of the equation of state for N"1 to order
 or even better
, that does not seem an unrealistic goal, would signi"cantly improve the
-expansion estimates and would therefore be quite useful. Finally the present approach could be extended to systems in the universality class of the (
) "eld theory for NO1,  provided expansions of the renormalized e!ective potential at high enough order can be generated. More generally it must be emphasized that only a small number of universal quantities, as predicted by renormalization group, have yet been calculated. In addition to static in"nite volume quantities, for which much work remains to be done, dynamic properties have not even be touched, "nite-size calculations would be useful for comparison with computer simulations. Considering the increase in computer power more perturbative calculations will become feasible. What has been demonstrated here is that once the series are available, summation methods have been developed which lead to precise determinations.

Acknowledgements The author thanks R. Guida for careful reading of the manuscript.

References [1] [2] [3] [5] [6] [7] [8] [9] [10]

[11] [12] [13]

J.C. Le Guillou, J. Zinn-Justin, Phys. Rev. Lett. 39 (1977) 95; Phys. Rev. B 21 (1980) 3976. R. Guida, J. Zinn-Justin, J. Phys. A 31 (1998) 8103, cond-mat/9803240. R. Guida, J. Zinn-Justin, Nucl. Phys. B 489 [FS] (1997) 626, hep-th/9610223. An early review is K.G. Wilson, J. Kogut, Phys. Rep. C 12 (1974) 75. E. BreH zin, J.C. Le Guillou, J. Zinn-Justin, in: C. Domb, M.S. Green (Eds.), Phase Transitions and Critical Phenomena, Vol. 6, Academic Press, London, 1976. A general reference on the subject is J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, in particular (Chapter 28, 3rd Edition), Clarendon Press, Oxford, 1989 (3rd Edition, 1996). K.G. Wilson, M.E. Fisher, Phys. Rev. Lett. 28 (1972) 240. C. Di Castro, Lett. Nuovo Cimento 5 (1972) 69; G. Mack, in: W. Ruhl, A. Vancura (Eds.), Kaiserslautern 1972, Lecture Notes in Physics, Vol. 17, Springer, Berlin, 1972; B. Schroer, Phys. Rev. B 8 (1973) 4200. E. BreH zin, J.C. Le Guillou, J. Zinn-Justin, Phys. Rev. D 8 (1973) 2418; J. Zinn-Justin, Carge`se Lectures 1973, unpublished, later incorporated in Ref. [6]; C. Di Castro, G. Jona-Lasinio, L. Peliti, Ann. Phys. (NY) 87 (1974) 327. E. BreH zin, J.C. Le Guillou, J. Zinn-Justin, Phys. Rev. D 8 (1973) 434; P.K. Mitter, Phys. Rev. D 7 (1973) 2927. G. Parisi, Carge`se Lectures 1973, published in J. Stat. Phys. 23 (1980) 49. A still partially relevant reference is Phase Transitions, Vol. B72, M. LeH vy, J.C. Le Guillou, J. Zinn-Justin (Eds.), Proceedings of the Carge`se Summer Institute 1980, Plenum, New York, 1982.

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[44] D.J. Wallace, R.P.K. Zia, J. Phys. C 7 (1974) 3480. [45] P. Scho"eld, Phys. Rev. Lett. 23 (1969) 109; P. Scho"eld, J.D. Litster, J.T. Ho, Phys. Rev. Lett. 23 (1969) 1098; B.D. Josephson, J. Phys. C 2 (1969) 1113. [46] J.F. Nicoll, P.C. Albright, Phys. Rev. B 31 (1985) 4576. [47] C. Bagnuls, C. Bervillier, D.I. Meiron, B.G. Nickel, Phys. Rev. B 35 (1987) 3585. [48] F.J. Halfkann, V. Dohm, Z. Phys. B 89 (1992) 79. [49] R. Seznec and J. Zinn-Justin, J. Math. Phys. 20 (1979) 1398; J.C. Le Guillou, J. Zinn-Justin, Ann. Phys. (NY) 147 (1983) 57; R. Guida, K. Konishi, H. Suzuki, Ann. Phys. (NY) 241 (1995) 152; ibidem 249 (1996) 109. [50] S. Zinn, S.-N. Lai, M.E. Fisher, Phys. Rev. E 54 (1996) 1176. [51] A. Pelissetto, E. Vicari, Nucl. Phys. B 519 (1998) 626, cond-mat/9711078; ibidem B 522 (1998) 605, condmat/9801098. [52] T. Reisz, Phys. Lett. B 360 (1995) 77. [53] P. Butera, M. Comi, Phys. Rev. E 55 (1997) 6391. [54] M.M. Tsypin, Phys. Rev. Lett. 73 (1994) 2015. [55] N. Tetradis, C. Wetterich, Nucl. Phys. B 422 (1994) 541. [56] T.R. Morris, Nucl. Phys. B 495 (1997) 477. [57] C. Bervillier, Phys. Rev. B 34 (1986) 8141. [58] A.J. Liu, M.E. Fisher, Physica A 156 (1989) 35. [59] A. Aharony, P.C. Hohenberg, Phys. Rev. B 13 (1976) 3081; Physica B 86}88 (1977) 611. [60] V. Privman, P.C. Hohenberg, A. Aharony, Universal Critical Point Amplitude Relations, in: C. Domb, J.L. Lebowitz (Eds.), Phase Transitions and Critical Phenomena, Vol. 14, Academic Press, New York, 1991. [61] M. Hasenbusch, K. Pinn, J. Phys. A 31 (1998) 6157, cond-mat/9706003; M. Caselle, M. Hasenbusch, J. Phys. A 30 (1997) 4963, hep-lat/9701007; M. Caselle, M. Hasenbusch, Nucl. Phys. Proc. 63 (Suppl.) (1998) 613, hep-lat/9709089. [62] For other articles devoted to the equation of state see for example: J. Rudnick, W. Lay, D. Jasnow, Phys. Rev. E 58 (1998) 2902, cond-mat/9803026; P. Butera, M. Comi, Phys. Rev. B 58 (1998) 11552, hep-lat/9805025; ibidem B 60 (1999) 6749, hep-lat/9903010; A.I. Sokolov, E.V. Orlov, V.A. Ul'kov, S.S. Kashtanov, Phys. Rev. E 60 (1999) 1344, hep-th/9810082; J. Engels, T. Scheideler, Nucl. Phys. B 539 (1999) 557; S. Seide, C. Wetterich, Nucl. Phys. B 562 (1999) 524, cond-mat/9806372; M. Strosser, S.A. Larin, V. Dohm, Nucl. Phys. B 540 (1999) 654, cond-mat/9806103.

Physics Reports 344 (2001) 179 }253

Monte Carlo tests of renormalization-group predictions for critical phenomena in Ising models Kurt Binder*, Erik Luijten Institut fu( r Physik, Johannes-Gutenberg-Universita( t, Staudinger Weg 7, D-55099 Mainz, Germany Received June 2000; editor: I. Procaccia

Contents 1. Introduction 2. Methodological tools 2.1. General aspects 2.2. The Metropolis algorithms and the problem of statistical errors 2.3. Cluster algorithms 2.4. Finite-size scaling 3. Results for the critical behavior of the three-dimensional Ising model with short-range interactions 4. The nearest-neighbor Ising model in d"5 dimensions 4.1. A brief review of the pertinent theory 4.2. Comparison with Monte Carlo results 5. Crossover scaling in Ising systems with large but "nite interaction range in d"2 and d"3 dimensions 5.1. General theory

181 182 182 184 190 192

199 203 203 207

211 211

5.2. Numerical results for d"2 dimensions 5.3. Numerical results in d"3 dimensions and comparison with theoretical predictions 6. Algebraically decaying interactions 6.1. Overview 6.2. Renormalization-group predictions 6.3. Numerical results for the critical exponents 6.4. Finite-size scaling functions 7. The interface localization transition in Ising "lms with competing walls 7.1. A "nite-size scaling study 7.2. Phenomenological mean-"eld theory and Ginzburg criteria 7.3. Monte Carlo test of the theory 8. Summary and outlook Acknowledgements References

214 220 227 227 228 229 231 233 233 235 240 241 246 246

Abstract A critical review is given of status and perspectives of Monte Carlo simulations that address bulk and interfacial phase transitions of ferromagnetic Ising models. First, some basic methodological aspects of these simulations are brie#y summarized (single-spin #ip vs. cluster algorithms, "nite-size scaling concepts), and

* Corresponding author. Tel.: #49-6131-3923348; fax: #49-6131-3925441. E-mail address: [email protected] (K. Binder). 0370-1573/01/$ - see front matter  2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 0 0 ) 0 0 1 2 7 - 7

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then the application of these techniques to the nearest-neighbor Ising model in d"3 and 5 dimensions is described, and a detailed comparison to theoretical predictions is made. In addition, the case of Ising models with a large but "nite range of interaction and the crossover scaling from mean-"eld behavior to the Ising universality class are treated. If one considers instead a long-range interaction described by a power-law decay, new classes of critical behavior depending on the exponent of this power law become accessible, and a stringent test of the
-expansion becomes possible. As a "nal type of crossover from mean-"eld type behavior to two-dimensional Ising behavior, the interface localization}delocalization transition of Ising "lms con"ned between `competinga walls is considered. This problem is still hampered by questions regarding the appropriate coarse-grained model for the #uctuating interface near a wall, which is the starting point for both this problem and the theory of critical wetting.  2001 Elsevier Science B.V. All rights reserved. PACS: 05.10.Cc Keywords: Critical exponents; Finite size scaling; Ising model; Monte Carlo simulation; Renormalization group

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1. Introduction The Ising model [1}3] is one of the workhorses of statistical mechanics, playing a role similar to that of the fruit#y in genetics: techniques such as transfer-matrix methods [2,3] and hightemperature series expansions [4] have initially been formulated for the Ising model and were then generalized and applied to many other problems. Similarly, the Ising model has played a pivotal role in the development of concepts about critical phenomena, from scaling [5,6] and universality [7}9] to the renormalization group [10}22]. In addition, critical phenomena in Ising models have been under study by Monte Carlo (MC) simulations since about thirty years [23}37], and recently these studies have reached an accuracy [36}51] that is competitive with the most accurate renormalization-group estimates [52}55]. As far as the critical properties of the nearest-neighbor ferromagnetic Ising model are concerned, there is also fair agreement with estimates drawn from recent series-expansion analyses [56}59], the Monte Carlo renormalization-group (MCRG) approach [60}63], and the coherent-anomaly method [64]. Despite this impressive progress, there are still many problems left that are less well understood, and hence in the focus of the present article. Monte Carlo simulations of critical phenomena almost always rely on the use of "nite-size scaling theory [24}37,65}97], but certain aspects of this theory still appear to be under discussion even for short-range Ising models [93}97], particularly if the system dimensionality d exceeds the marginal dimension dH (dH"4 here; for d'dH the critical behavior is described by the simple Landau theory [5}22]). There has been a longstanding discrepancy between the simple predictions by BreH zin and Zinn-Justin [74] and Monte Carlo results for the "ve-dimensional Ising model [72,73,98], and although this discrepancy has been resolved now [99}104], there are still controversial issues regarding the form of "nite-size scaling functions above the upper critical dimension [93}96]. The question how to extend "nite-sizescaling analyses to mean-"eld like systems is not a purely academic one, since classical mean-"eld critical behavior can also arise in two- and three-dimensional systems if the interaction is of su$ciently long range [5,14,45}51,87,92,105}108]. If there is an interaction J(r) of in"nite range that decays with distance r like an inverse power law, J(r)Jr\B>N, the marginal dimension for (2 gets lowered to dH"2, and consequently the critical behavior is classical for 0((d/2. However, a very interesting situation also arises if the interaction range R is "nite but large [45,46,48}50,87,92]: then the asymptotic critical behavior for very small temperature distances t"(¹!¹ )/¹ from the critical temperature ¹ is the same as that of the short-range Ising model, but somewhat further away from ¹ (although still in the region where t;1) a crossover to classical critical behavior occurs. While qualitatively the understanding of such a crossover is already provided by the Ginzburg criterion [109], its quantitative description by renormalizationgroup theory has been a longstanding problem [110}124]. Since a related crossover occurs near the unmixing critical point in polymer blends as well (the larger the chain length N of the macromolecules, the more the system behaves mean-"eld like [125}133]), this crossover between Ising-like and mean-"eld critical behavior can be observed experimentally rather directly [134}141]. Various other systems (polymer solutions [142], micellar solutions [143] and ionic systems [144}147] that undergo phase separation) are also candidates for the observation of such crossover phenomena. Therefore, the theoretical understanding of this crossover is relevant to a broad variety of physical systems, and hence the Monte Carlo investigation of these crossover phenomena is one of the main topics of this article.

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A "nal topic that we shall consider here is the statistical mechanics of #uctuating interfaces near walls and the related interface localization}delocalization transitions that occur in Ising "lms with competing boundary "elds of the walls [148}159]. Ising lattices with nearest-neighbor ferromagnetic interactions are expected to exhibit rather complex phase diagrams [150,159]: phase transitions of either second order [148,149,152,154}157] or "rst order [150,158,159] are possible, and in the second-order case one again encounters a problem where crossover between twodimensional Ising criticality and mean-"eld behavior occurs [157]. At the same time, the problem is intimately connected to the problem of critical wetting in the presence of short-range forces [160}177] and the appropriate choice of e!ective interface Hamiltonians [167,178}186], which are the starting point of analytic theories for these phenomena, is still a matter of discussion. The outline of this review is as follows: In Section 2, we brie#y recall the basic methodological aspects of Monte Carlo simulations, as far as they are essential for the reader to easily appreciate their use as a tool for the analysis of critical phenomena, but also for better understanding of the intrinsic limitations of such simulations. In Section 3, we then consider the analysis of critical behavior for the short-range Ising model in d"3 dimensions, while Section 4 is devoted to the case of d"5 dimensions. Section 5 deals with the case of large but "nite interaction ranges, and the problem of crossover scaling between the Ising and mean-"eld universality classes, both in d"2 and in d"3 dimensions. Section 6 summarizes results for Ising models with interactions that decay as a power law. Finally, Section 7 deals with the interface delocalization problem, while Section 8 gives a summary and an outlook to related problems that have not been dealt with in this article.

2. Methodological tools 2.1. General aspects As is well known [29}37], Monte Carlo simulations numerically evaluate canonical thermal averages of some observable A, 1 A " TrA(x) exp[!H(x)/k ¹] , 2 Z

(2.1)

where Z"Trexp[!H(x)/k ¹] is the partition function, k is Boltzmann's constant, ¹ is the absolute temperature and the Hamiltonian in our case is that of the Ising model, where N spins S "$1 are placed on lattice sites labeled by the index i G H (S )"!
J S S !H
S . '  G GH G H G G$H G

(2.2)

Here J are the exchange constants, and we have also included a magnetic "eld H. The points x GH of the (quasiclassical) phase space are the con"gurations that can be taken by the N spins and the trace operation in Eq. (2.1) is a summation over all the 2, states. Monte Carlo simulations replace this exact average by an approximate one, where M states x  are selected by an I

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183

importance-sampling process, + AM "M\
A(x ) . I I

(2.3)

The importance-sampling process consists of the construction of a Markov chain of states (x Px P2Px Px P2), where a suitable choice of the transition probability   I I> =(x Px ) ensures that, for large enough , states x are selected according to the canonical I I> I equilibrium probabilities, P (x )Jexp[!H(x )/k ¹].  I I From this brief description, we can already recognize the main limitations: (i) Only in the limit MPR we can expect to obtain an exact result, while for "nite M a `statistical errora is expected. The estimation of this error is a very nontrivial matter, since } depending on the precise choice of = } subsequently generated states are more or less correlated. In fact, if the Monte Carlo sampling process is interpreted dynamically (associating a (pseudo)time with the label  of subsequent con"gurations, one can interpret the Monte Carlo procedure as the numerical implementation of a master equation describing a kinetic Ising model [187]), one recognizes that the `correlation timea is expected to diverge in the thermodynamic limit at a second-order phase transition (`critical slowing downa [188]). (ii) While the importance sampling method guarantees that, for PR, the states are selected according to P (x ), for choices of  that are not large enough there is still some `memorya of the  I (arbitrary!) initial state with which the Markov chain was started. In practice, one may start with a completely random spin con"guration, or a state where all spins are up, or an (equilibrated) spin con"guration of a previous Monte Carlo run (which, e.g., was carried out at some other temperature ¹). Invoking once more the above dynamic interpretation of Monte Carlo sampling, it is clear that one must `waita until the system has `relaxeda from the initial state toward the correct thermal equilibrium. Also this `nonequilibrium relaxation timea [189}191] is divergent at a second-order phase transition in the thermodynamic limit. Due to "nite-size e!ects, both the critical divergence of the above correlation time and the critical divergence of this nonequilibrium relaxation time will be rounded o! to large but "nite values. However, if one does not omit su$ciently many states generated at the beginning of the Markov chain before the averaging in Eq. (2.3) is started, systematic errors will still be generated. (iii) For the realization of the Markov chain, (pseudo)random numbers are used both for constructing a trial state x from a given state x and for the decision whether or not to accept the I I trial con"guration as a new con"guration (in the Metropolis algorithm this is done if the transition probability = exceeds a random number that is uniformly distributed in the interval from zero to one [29}37]). The Monte Carlo method clearly requires random numbers, however, that are not only uniformly distributed in a given interval, but also uncorrelated, and in practice this absence of correlations is ful"lled only approximately [192}202]. Thus, it is necessary to carefully test the `qualitya of the random numbers for each new application of the Monte Carlo method, and this is again a nontrivial matter, since there is no unique way of testing random-number generators [192}202], and there is no absolute guarantee that a random-number generator that has passed all the standard tests does not yield random numbers that lead, due to some subtle correlations among them, to systematic errors in a particular application. These remarks are not entirely academic } even in studies of the nearest-neighbor Ising model results are documented in the literature

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[203}208] where `bada random numbers have caused systematic errors. Thus for Monte Carlo studies which aim at a high precision it is clearly mandatory to check the results using di!erent high-quality random-number generators and to verify that no systematic discrepancies are found. (iv) Last, but certainly not least, the "nite size of the simulated lattice (typically one chooses a (hyper)cubic lattice of linear dimension ¸ with periodic boundary conditions in all lattice directions) causes a systematic rounding and shifting of the critical singularities one wishes to investigate: singularities of the free energy can only develop in the thermodynamic limit ¸PR. This remark is particularly obvious for the correlation length , which cannot diverge toward in"nity in a "nite simulation box, so that serious "nite-size e!ects must be expected when  has grown to a size comparable to ¸. On the one hand, these "nite-size e!ects constitute a serious limitation, hampering the possibility to extract critical properties from simulations in a direct manner, like it is done in experiments on real systems [209}213]. On the other hand, these "nite-size e!ects o!er a powerful tool, via "nite-size scaling analyses [70,79,80,83,84], for the study of critical phenomena. While the key ideas of "nite-size scaling are quite old [65,66] and their application in the context of simulations is standard [26}37], the optimal use of these concepts remains under discussion [214}223]. In the present article we do not attempt to give a full account of all these problems (i)}(iv), but shall restrict ourselves to a brief discussion of critical slowing down and statistical errors (Section 2.2) and how this problem is eased by the use of cluster algorithms (Section 2.3); in Section 2.4 we then recall those aspects of "nite-size scaling analyses which are most relevant for the simulations described in the later sections. A complete discussion of "nite-size scaling, of course, should also address aspects other than the critical behavior of the Ising model, but this is beyond the scope of our discussion. Other technical aspects, such as histogram extrapolations [224,225], multicanonical Monte Carlo methods [226}230], etc., will not be discussed here either.

2.2. The Metropolis algorithms and the problem of statistical errors In the standard Metropolis algorithm [29}37,231], the Markov chain of states x Px P2Px P2 mentioned above is realized by attempting single-spin #ips, S PS . The   I G H procedure consists of the selection of a lattice site i that is considered for a #ip (one may either choose the sites at random or go through the lattice in a regular fashion) and the calculation of the energy change H in the Hamiltonian (2.2), that would be caused by this #ip. From this energy change one computes the transition probability =, =(xPx )"Min[1, exp(! H/k ¹)] .

(2.4)

If ="1 the spin #ip is always accepted. If =(1, one draws a random number uniformly distributed between zero and unity: if (= the spin is #ipped, while otherwise this trial move is rejected and the old con"guration is counted once more for the averaging, Eq. (2.3). The quantities that are most straightforward to average are the energy per spin, E"H /N, 2 and the magnetization per spin m, or (in the absence of the symmetry-breaking "eld H) its

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absolute value, or higher-order moments

 






, 1 , I
S
S , mI" , k"1, 2,2 . (2.5) G G N 2 2 G G For a discussion of critical behavior, one is also interested in the susceptibility and the speci"c heat C. These are typically calculated from #uctuation relations, which read (again all quantities are normalized per spin to allow for a straightforward thermodynamic limit) 1 m" N

C/k "[H!H]/(Nk ¹) ,

(2.6)

"[m!m]N/(k ¹) ,

(2.7)

"[m!m]N/(k ¹) .

(2.8)

Note that (for H"0) one needs to use the standard expression for [Eq. (2.7)] while for ¹'¹  the expression [Eq. (2.8)] should be used for ¹(¹ [32,36]. The reason for the need of these  two di!erent expressions is the spontaneous symmetry breaking that occurs at ¹ , due to the  appearance of a spontaneous magnetization m : In statistical mechanics, the proper de"nition of  the spontaneous magnetization would be m " lim lim m , (2.9)  2& & , while at all nonzero temperatures one obtains a trivial result if the limits are interchanged, since lim m "0 ∀N . (2.9a) 2& & Thus, for any "nite N and H"0, Eq. (2.7) is equivalent to k ¹ "Nm , which for large 2& but "nite N converges to k ¹ +Nm PN as ¹P0. It follows that as de"ned in Eq. (2.7) is  a monotonically increasing function when ¹ decreases through ¹ which does not show a max imum at any nonzero temperature under equilibrium conditions (if one uses a single-spin #ip algorithm, full equilibrium is actually often not obtained for ¹(¹ , due to the exponential  divergence of the `ergodic timea  with N for ¹(¹ , and a spurious, observation-time-dependent   maximum may occur for as well, as is discussed elsewhere in the literature [32,36]). On the other hand, we have m " lim m (2.10)  2& , and therefore as de"ned in Eq. (2.8) converges to the correct expression for the susceptibility below ¹ . Above ¹ , however, and di!er by a trivial factor in the thermodynamic limit. This is   realized by noting that the probability distribution of m, for su$ciently large N"¸B, is a simple Gaussian [31] p (m)"¸B(2k ¹ *) exp[!m¸B/(2k ¹ *)], ¹'¹ , H"0 , (2.11) *  where the notation * indicates that for large but "nite ¸ there still may be some residual "nite-size e!ect in the susceptibility although in fact we do expect a smooth convergence toward

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the susceptibility  in the thermodynamic limit. From Eq. (2.11), it can easily be shown that [32]

*" *(1!2/) .

(2.12)

For ¹ su$ciently far below ¹ , the distribution function p (m) can, near its peaks (which occur at  * $m as ¸PR), be approximated by a sum of two Gaussians [31]  1 p (m)" ¸B(2k ¹ *)exp[!(m!m* )¸B/(2k ¹ *)] *

 2 # exp[!(m#m* )¸B/(2k ¹ *)] ¹(¹ , H"0 ; (2.13)

  again, the positions $m* of these peaks may di!er from $m by some residual "nite-size

  e!ects, but a smooth convergence of m* toward m is expected as ¸PR. Note that it is (2.8)

  that is appropriate for the widths of the Gaussians in Eq. (2.13). Thus the phase transition of the Ising model in zero "eld shows up via a change of the order-parameter distribution p (m) from a single-peak shape to a double-peak shape. A convenient * measure of this change in behavior is the fourth-order cumulant [31] ; "1!m/[3m] , (2.14) * which converges to zero for ¹'¹ , as one easily derives from Eq. (2.11), while it converges to  ; "2/3 for ¹(¹ . At ¹"¹ , ; converges to a nontrivial universal constant, as will be    * discussed below in the context of "nite-size scaling. We also note that ; appears in the literature * in several variants and under a number of di!erent names. For example, g "!3; is confusing* * ly referred to as the `renormalized coupling constanta [72] } we want to stress here that the use of this nomenclature is to be deprecated! Later this name was correctly used for the related quantity g (¸/ )B where  is the "nite-lattice correlation length [219]. In this article we shall also employ * * * the simpli"ed notation Q "m/m. Other authors use still di!erent normalizations and * names, such as the `Binder parametera. A crucial point of any Monte Carlo simulation is the estimation of the statistical errors of the various observables A that are recorded. Suppose we record M `measurementsa A(X ) that are I perfectly uncorrelated. Then the statistical error A of the estimate AM in Eq. (2.3) can be estimated as [232] A"[A!A]/M ,

(2.15)

while the variance  "A!A can be estimated from  +  +
[A(X )!AM ]/(M!1) . (2.16)  I I Note the denominator M!1 rather than M in Eq. (2.16): this denominator accounts for the fact that AM in Eq. (2.16) is estimated from the same M `measurementsa A(X ) already, and hence there I remain only M!1 rather than M statistically independent `measurementsa of [A(X )!AM ]. In I practice, however, if observables such as the speci"c heat or susceptibilities are sampled from #uctuation relations, this point is sometimes ignored and the same formula Eq. (2.3) is used for the

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estimation of variances as well, +  "
[A(X )!AM ]/M . (2.17)  I I The replacement of M!1 [Eq. (2.16)] in favor of M [Eq. (2.17)] obviously leads to a systematic underestimation of quantities such as C or . While this bias is negligible in the limit of very large M, it does become a problem if the `measurementsa A(X ) are correlated: both the statistical error I and the bias then get strongly enhanced. Let us "rst consider the expectation value of the square of the statistical error, based on M successive observations A ,A(X ) [187]: I I 1 + 
(A !A) ( A)" I M I + 2 + 1 +

(A !A)#
" (2.18) (A  A  !A) . I I I M M I I  I I > Changing the summation index  to  #, Eq. (2.18) can be rewritten as   +  1
I , (2.19) ( A)" [A!A] 1#2
1! M  M I where
I is an autocorrelation function  A A !A . (2.20)
I" I I >I  A!A










 

This result can be interpreted associating a `timea t " t  with the Monte Carlo process, t being I the `time intervala between two successive `measurementsa A and A . It is possible to take I I> t"1/N, i.e., every Monte Carlo move (e.g., every attempted spin #ip in a single-spin-#ip Metropolis algorithm) is included in the calculation, but then it is clear that subsequent states X are highly correlated for large N, since they di!er by at most one out of N spin orientations. I Thus it is more common to choose t"1 (corresponding to a `timea unit of 1 Monte Carlo step per spin), although near ¹ it often is more e$cient to take even less `measurementsa, e.g., t"10.  Thus, we replace
I by
(t) (noting also that in equilibrium there is an invariance with respect to   the choice of the time origin),
(t)"[A(0)A(t)!A]/[A!A] , (2.21)  and, by treating t as a continuous rather than a discrete variable, replace the sum in Eq. (2.19) by an integral, where  "M t is the observation time over which the averaging is extended, + O+ dt t 1 1!
(t) . (2.22) ( A)" [A!A] 1#2  t  M  + De"ning a correlation time as

 




 " 






(t) dt 

 

(2.23)

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and assuming that the observation time  < , one can rewrite Eq. (2.22) as +  1 ( A)" [A!A](1#2 / t) .  M

(2.24)

If  diverges near a critical point and t is kept of order unity, as anticipated above, we have   / t<1 and Eq. (2.24) becomes   ( A)"2  [A!A] .  +

(2.25)

This result shows that the statistical error is independent of the choice of the time interval t and only depends on the number n" /(2 ) of statistically independent `measurementsa: although +  for a given observation time  a choice of a smaller value t results in a correspondingly larger + value of the number of observations, it does not decrease the statistical error. It is only the ratio between the relaxation time  and the observation time  that matters. Thus, the estimation of  + these relaxation times is indispensable for a correct estimation of statistical errors. From these considerations it is already evident how useful it has been to develop algorithms that reduce the critical slowing down near phase transitions or even remove it completely. These correlation times also have a signi"cant e!ect on the problem of biased estimates mentioned above: unbiased estimates of variances are obtained if we take only n measurements A(X ) at time intervals 2 apart, IY  L  +
[A(X )!AM ]/(n!1) .  IY IY

(2.26)

Often n is not very much larger than unity, and then Eq. (2.26) is signi"cantly di!erent from Eq. (2.17). This has been demonstrated by Ferrenberg et al. [233] for the three-dimensional Ising model at ¹ , cf. Fig. 1. One can see that due to the systematic underestimation of both C and (by  a factor 1!1/n for large n) the data approach their asymptotic values always from below with increasing number of measurements M (denoted as N in the "gure). For C the saturation is reached somewhat faster than for , since the correlation time  of the energy is smaller than the # correlation time  for the magnetization. Note also that the time needed to reach the correct K saturation values increases with increasing ¸. This is due to the "nite-size rounding of the relaxation time in the single-spin-#ip kinetic Ising model:  J J¸X ¹"¹ , # K 

(2.27)

with a dynamical critical exponent z+2 [234]. Therefore, there is a strong decrease of the number n of statistically independent `measurementsa with increasing ¸, and it becomes very di$cult to carry out, with single-spin-#ip algorithms, meaningful simulations for values of ¸ distinctly larger than those included in Fig. 1. Another important aspect of statistical errors is the question how these scale with the linear dimension of the simulated systems. If we wish to record the energy or the magnetization, we conclude from Eqs. (2.6)}(2.8) and (2.25) that the squared errors are expected to scale inversely

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Fig. 1. Speci"c heat (a) and susceptibility (b) plotted vs. the number of measurements N in a bin, for t"10 MCS and the simple cubic nearest-neighbor Ising model at ¹ [J/k ¹ "0.221654 was used here]. For each linear lattice dimension   ¸, a total e!ort of at least 3;10 MCS/spin was invested, using a single-spin-#ip algorithm. The `measurementsa of energy and magnetization were grouped into bins containing N entries each, and then C and were calculated using Eq. (2.17). Finally, the results were averaged over all the available bins to reduce the statistical error compared to the systematic error resulting from the bias. From Ferrenberg et al. [233].

with the system volume N"¸B, 2 k ¹C 2 2 k ¹ ( E)" # , ( m)" K [m!m]" K . (2.28)    ¸B ¸B + + + The property that squared errors scale inversely with the volume is called `strong self-averaginga [235] and only holds away from ¹ , whereas at ¹ it is replaced by so-called `weak self-averaginga:   Since, at ¹ , in a "nite system and C scale like powers of ¸ [for d44 we have 

(¹"¹ )J¸AJ, C(¹"¹ )J¸?J, where ,  and  are the standard critical exponents of the   speci"c heat, the susceptibility and the correlation length, respectively, see below], we have 1 1 ( m)J ¸AJ\B, ( E)J ¸?J\B, ¹"¹ .  n n

(2.29)

However, the situation is very di!erent if we consider the errors of quantities that are derived from #uctuation relations, such as and C themselves [Eqs. (2.6)}(2.8)]: Now the quantity A in Eq. (2.25) has to be replaced by ¸B( m) or ¸B( E), respectively, with m"m!m and E"E!E, and hence [235] 1 1 (k ¹  )" ¸B[( m)!( m)], (k ¹C)" ¸B[( E)!( E)] . n n

(2.30)

Using the fact that away from ¹ both m and E are Gaussian distributed for large enough  ¸, ( m)"3( m), ( E)"3( E) can be used to reduce Eq. (2.30) to

 

((k ¹  )"

((k ¹ C)"

 

2 2 ¸B( m)" k ¹ , i.e. (( )/ "(2/n , n n 2 2 ¸B( E)" k ¹C, i.e. ((C)/C"(2/n . n n

(2.31) (2.32)

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Thus we see that the relative error of quantities sampled from #uctuation relations does not decrease at all with increasing ¸, but gets small only when the number n of statistically independent samples becomes large. This property is called `lack of self-averaginga [235]. 2.3. Cluster algorithms From the discussion of statistical errors in the previous section it will be evident that critical slowing down makes it very di$cult to obtain precise estimates for the observables of interest in the critical region, using the Metropolis algorithm and su$ciently large lattices. Only by use of highly e$cient vectorizing algorithms on supercomputers with vector architecture and a massive investment of CPU resources it has been possible to obtain very good results in this way [33,39,233]. Given this situation, the invention of cluster algorithms [225,236}251] that o!er a dramatic reduction of critical slowing down has been crucial for allowing more widespread studies. Here, we "rst discuss the original cluster algorithm for the nearest-neighbor Ising model in zero external "eld. The starting point is the Fortuin}Kasteleyn representation [252] of the partition function, writing K"J/k ¹,






Z"
exp K
S S "
 e)[(1!p)#p G H ] G H 11     6GH7 1G 1G 6GH7

(2.33)

or (2.34) Z"

 e)[(1!p) GH #p G H GH ] , L  1 1 L      1G LGH 6GH7 where the n are auxiliary bond variables that can take the values 0 and 1. Only if the two spins GH S and S are equal, the bond n is `occupieda or `activea (n "1) with probability p, G H GH GH p"1!exp(!2K) , (2.35) while otherwise n "0. In Eq. (2.33) we used the fact that the product S S can only be #1 if both GH G H spins are equal and !1 if they are not, so exp(KS S )"x#y G H is easily solved for x and y. For G H 11 Eq. (2.34) the simple identity  a#b"
(a #b ) (2.36) L L L has been used. Now the Swendsen}Wang algorithm [236] makes use of the joint probability distribution for the spin and bond variables implicit in Eq. (2.34). A `cluster update sweepa [35,251] then consists of alternating updates of the bond variables n for a given spin con"guration GH S  and updates of the spins S for the given bond con"guration. Thus the algorithm consists of the G G following steps: (i) If S OS , set n "0. If S "S , assign values n "1 and n "0 with probability p G H GH G H GH GH [Eq. (2.35)] or 1!p, respectively. (ii) Identify clusters of spins that are connected by `activea bonds (n "1). Clearly, this step is GH relatively time-consuming, but in the context of the bond-percolation problem [253,254] rather e$cient cluster-counting routines have been developed. In any case, this somewhat `technicala issue is out of consideration here.

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Table 1 The critical exponent z of the integrated relaxation time  of various observables (A"energy E, magnetization m, or  susceptibility ) in d"2 and d"3 dimensions ( J¸X). The typical error is of the order of 0.02  Algorithm

d"2

d"3

Observable

Reference

Metropolis Swendsen}Wang

2.16 0.27 0.20 0.26 0.13

2.03 0.50 0.50 0.28 0.14

E and m E

E

[234,256] [238] [238] [238] [238]

Wol!

(iii) Draw a random value $1 independently for each cluster (including clusters containing a single spin only), which then is assigned as the spin value for all spins in this cluster. Note that the clusters de"ned in this way are `stochastica clusters that di!er from clusters that one could de"ne on a purely geometric basis as contours around groups of identical spins [255]. In the present clusters, some bonds between identical spins are deleted with probability p "1!p"exp(!2K): Thus, in the high-temperature limit, KP0, large clusters cannot occur,  unlike the `geometric clustersa where even for KP0 arbitrary large clusters occur with nonzero probability. Obviously the present `stochastica clusters are on average smaller than the `geometric clustersa: Only at ¹P0 (KPR) p vanishes and the present clusters become identical to the  geometric clusters. It turns out that this Swendsen}Wang algorithm spends too much time identifying and handling small clusters, which are present at the critical point, too, although the important physics (such as the diverging correlation length) is embodied in the very large clusters only. For this reason, it is more e$cient to consider a single cluster at a time, instead of making a full decomposition of the whole lattice into clusters. In this `single-cluster algorithma due to Wol! [237}239] one chooses a lattice site at random, constructs only the `stochastica cluster connected to this site, using the same probability p of setting active bonds as discussed above, and then #ips all the spins of this cluster. If c denotes the average cluster size, a `sweepa through the lattice is de"ned by the number ¸B/c of such single-cluster steps, while in the Metropolis single-spin-#ip algorithm a `sweepa is de"ned when (on average) every of the ¸B lattice sites has been considered once for a spin-#ip. With these de"nitions of `sweepsa, a sensible comparison of the correlation times of various quantities for the di!erent algorithms is possible (Table 1) [35,256]. An important generalization of cluster algorithms is the extension to long-range interactions due to Luijten and BloK te [250]. It has the remarkable property that the e$ciency is independent of the number of interactions per spin, and can be applied to any O(n) and Potts model without frustration. Furthermore, it is applicable to any interaction pro"le and dimensionality. Obviously, this implies an enormous improvement compared to conventional single-spin-#ip and cluster algorithms alike, with a typical speed-up of several orders of magnitude. The essential ingredient of this method is that one does not consider every single interacting neighbor of a spin for inclusion into the cluster being built, but instead calculates which spin is the next one to be considered for inclusion. For the technical details, the reader is referred to the original literature [250], while also Ref. [257] contains a pedagogical introduction.

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2.4. Finite-size scaling As already mentioned in the introduction, the literature on "nite-size scaling is rich and in some aspects still controversial [24}37,65}97], so that we cannot give an exhaustive review here; instead we focus on those aspects which are most pertinent to the Monte Carlo studies that are reviewed in the following sections. The key idea (at least for systems that obey the hyperscaling relation [12}22] between critical exponents, d"#2) is that the linear dimension ¸ `scales with the correlation length a, Jt\J, t"¹/¹ !1. For the probability distribution of observing a magnetization m in a cubic  box of size ¸ (with periodic boundary conditions) this idea implies [31] P (m, t)"@JP(¸/, m@J)"¸@JPI (¸/, m¸@J) ¸PR, PR, ¸/ "xed . (2.37) * Here the function P (m, t), a function of three variables ¸, m and t, is, via the scaling assumption, * reduced to a `scaling functiona (P or PI , respectively) that depends on two arguments only (¸/ and m@J or m¸@J, respectively). For the sake of a transparent formulation, we have replaced the variable t by the related variable \J in Eq. (2.37). We note from the outset that "nite-size scaling holds in the limit where both ¸PR and tP0 (and consequently mP0), as is the case for all scaling descriptions near critical points. At "xed ¸ and/or "xed , corrections to the asymptotic critical behavior embodied in Eq. (2.37) must be considered, as is discussed below. The scale factors @J or ¸@J in Eq. (2.37) trivially follow from the condition that P (m, t) is * normalized to unity:



> P (m, t) dm"1 ∀t, ∀¸ . (2.38) * \ In the limit considered, P (m, t) is nonzero only near m"0 and hence the integration limits $1 * can be replaced by $R with negligible error. Naturally, the scaling forms with P and PI in Eq. (2.37) are fully equivalent, since with the arguments X"¸/, >"m@J we can form new variables X"¸/, Z">X@J"m¸@J. The moments mI considered in Eq. (2.5) can be found straightforwardly from Eq. (2.37) as



mI"

>

mIP (m, t)dm"¸\I@JM I !(¸/)"¸\I@JM (¸Jt) , (2.39) * I I \ where M I !(X) is the resulting scaling function, with the symbol $ referring to the sign of t. It is I more convenient to work with the single function M which may have both positive and negative I arguments and in fact is analytic near t"0, while M I ! would be singular near XP0. Eqs. (2.7), I (2.8) and (2.14) then yield (using m,0 in zero external "eld) m"¸\@JM (¸Jt) , 

"(¸B/k ¹)¸\@JM (¸Jt) , 

"(¸B/k ¹)¸\@JM (¸Jt)![M (¸Jt)]  

(2.40) (2.41) (2.42)

and ; "1!M (¸Jt)/3[M (¸Jt)],;I (¸Jt) . (2.43) *   Analogously, the quantity Q "m/m can be written as Q "QI (¸Jt) in this limit. * *

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From Eqs. (2.40)}(2.42) we recover simple power laws in the limit ¸/PR or ¸JtPR, requiring that powers of ¸ must cancel to obtain a sensible thermodynamic limit. Hence (taking X ,¸Jt), M (X P!R)J(!X )I@, mJ\@JJ(!t)@, t(0 , (2.44) I M (X PR)J(X )\IA, k ¹ Jk ¹ J¸B\@J\AJtA, t'0 . (2.45) I We recognize from this result that hyperscaling is built into the description (2.37), since the power of ¸ in Eq. (2.45) only cancels if d"2# [31]. Just like Eqs. (2.40)}(2.42) imply simple power laws in t and hence in  if ¸PR, one "nds simple power laws in ¸ in the limit PR, i.e., when considering the behavior right at ¹ :M I >(0)"M I \(0) are simple constants, and hence  I I (2.46) m  J¸\@J , ;  ";I (0) . *2 2 While the power laws for m and (or , respectively) involve nonuniversal prefactors (the critical amplitudes), both in the t dependence [Eqs. (2.44) and (2.45)] and in the ¸ dependence [Eq. (2.46)], these amplitudes cancel out in the ratio ;I (0) [or QI (0)], which therefore is a universal constant that has proven very useful for the identi"cation of the corresponding universality class. For completeness, we mention that the "nite-size scaling description can be extended to the energy per spin E and the speci"c heat C [Eq. (2.6)], E"E #¸\\?JEI (¸Jt) , A C/k "¸?JCI (¸Jt) .

(2.47) (2.48)

Imposing once more that powers of ¸ must cancel in the thermodynamic limit, one "nds EI (X P$R)JX \?, CI (X P$R)JX \?, CJt\? ,

(2.49)

while for t"0 we have the power laws in ¸ (we assume '0 here) E!E J¸\\?J, CJ¸?J . (2.50) A Clearly, we have only considered the leading critical behavior in this description, and neither nonanalytic corrections to scaling nor analytic background terms (which are, e.g., expected for quantities such as the speci"c heat C) have been taken into account yet. It is also of interest to estimate the correlation length itself from the simulations. One can either sample the spin pair correlation function G(r)"S(r )S(r #r) (2.51) G G and study its asymptotic decay for large r in order to obtain the true correlation range [G(r)Jexp(!r/) for r<; note that  de"ned in this way in general depends on the direction of the lattice [6]], or one can obtain the second-moment correlation length. This quantity is de"ned from the behavior of the structure factor S(k) at small k, "[S(0)/S(k)!1]/k ,

(2.52)

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Fig. 2. Plot of !3;"m/m!3 (left) and g "3(¸/); (right) vs. K"J/k ¹ for the nearest-neighbor Ising 0 model on the simple cubic lattice. The data were obtained from histogram reweighting of data taken at K "0.221655;  thus the values of ; and g shown as di!erent points are not based on independent runs, but are highly correlated, and 0 the plotted error bars are just the statistical error observed at K . Three choices of ¸ are shown, data being obtained  with the Swendsen}Wang cluster algorithm at the Thinking Machine CM-5 computers, based on about 500 000 measurements per size. The estimates for K are taken from the crossing points of data for ¸"128 and ¸"256. From  Gupta and Tamayo [62].

where 1 S(k)" ¸B




S(r ) exp(ik ) r ) G G G

 

.

(2.53)

We now brie#y discuss how these "nite-size scaling relations can be utilized to investigate critical phenomena. The "rst task is to locate the critical temperature ¹ . A method that does not need any  a priori knowledge of critical exponents is based on the observation [31] that at ¹ the cumulant  ; should take a universal value ;I (0), Eq. (2.46). On the other hand, we know that for ¹(¹ , *  ; converges to ; "2/3 for ¸PR, while for ¹'¹ , ; P0 in this limit, and *   * d; /d¹  J;I (0)¸J [cf. Eq. (2.43)]. These results imply that ; (¹) for di!erent ¸ should be * * 2 a family of curves which all merge at ; "2/3 at low temperatures, splay out and intersect in * a unique intersection point ;I (0) at ¹ , and then spread out again, since the slope at this  intersection point scales as ¸J. If one is satis"ed with a modest accuracy, this is indeed a very useful and simple method, and it therefore has been of widespread use for a variety of systems [32}37]. However, if a very high precision in the location of ¹ is desired, deviations from this intersection property will be  found, due to the residual e!ect of corrections to "nite-size scaling. These corrections to the leading "nite-size scaling behavior are only small if su$ciently large lattice sizes are available. As an example, Fig. 2 presents a plot of !3; and the renormalized coupling constant g "(¸/)B[3!m/m] vs. K,J/k ¹ for the nearest-neighbor Ising model [62]. Three 0 rather large values of ¸ are included: ¸"64, ¸"128 and ¸"256. Due to the very "ne resolution

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of both abscissa and ordinate scale, one clearly "nds three distinct intersection points in the range from K"0.221652 to K"0.221656, and there is a corresponding uncertainty in !3; (the intersection is in the range between !1.39 and !1.41, corresponding to QI (0)"0.625$0.004). The relative inaccuracy of K in this determination is only 10\ or perhaps even less [since other  criteria can be derived from the same simulation, the authors suggest K "0.221655 (1) [62]].  Now an alternative way to "nd ¹ is to locate the positions K (¸) where quantities such as C and  

have a peak, and to extrapolate these peak positions toward ¸PR(we recall that itself does not have a peak, but is monotonically increasing toward ¸B/k ¹ as ¹P0). Actually, in practice the peak of C is not well suited, since the speci"c-heat exponent  is rather small and the position t where the function CI (X ) takes its maximum is strongly a!ected both by singular corrections to

 scaling and by regular background terms, unless ¸ is exceedingly large. Only when the singular contribution written in Eq. (2.48) is actually dominant, we can conclude that the maximum of C occurs at a value X ! "¸Jt! where the function CI (X ) has its maximum



 t! "X ! ¸\J . (2.54)



 We emphasize that there is no reason to expect that the maximum of C and the maximum of [Eq. (2.42)], with

"(¸AJ/k ¹ )  (¸Jt) , (2.55)  will coincide: Actually one "nds a maximum at a di!erent value X QY of the scaling variable, and

 hence tQY "X Q ¸\J . (2.55)



 Similar behavior occurs for other quantities which exhibit a maximum, such as the temperature derivative of moments, dm/d¹, dm/d¹, etc. These temperature derivatives (or corresponding derivatives dm/dK,2) can be obtained from suitable #uctuation relations dA/dK"(AE!AE)¸B .

(2.56)

Histogram reweighting [224,225] is a convenient technique to record these correlation functions AE!AE for a range of values around a suitably chosen value K and to "nd their  maxima. The recipe is then to extrapolate all the resulting `pseudocriticala couplings K(¸) vs.  ¸\J and to attempt to locate a unique critical point K "K(R), cf. Fig. 3. Here, we have   included data from both Gupta and Tamayo [62] (using the Swendsen}Wang algorithm, for sizes ¸"64, 128, 256) and Ferrenberg and Landau [39] (using the Metropolis algorithm for sizes in the range 244¸496) to show that reasonably consistent results are obtained. To take into account the leading singular correction to scaling in the extrapolation shown in Fig. 3, one should use a formula [39] K(¸)"K #a¸\J[1#b¸\S#2] , (2.57)   involving a (universal) correction-to-scaling exponent  and a (nonuniversal) amplitude factor b. The data in Fig. 3 do not give a clear indication whether or not such corrections to scaling are important in this range of system sizes, although the #uctuation in the cumulant intersection points suggests that such corrections are still present.

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Fig. 3. (a) Estimate of K by extrapolation of data for K(¸) for di!erent observables A, as shown in the "gure, plotting   K(¸) vs. ¸\J assuming "0.625. The data are for K "0.221655. The resulting best estimate for K is    K "0.221654 (2). From Gupta and Tamayo [62]. (b) Same as in Fig. 3(a), but using data for much smaller sizes  (244¸496) on more expanded scales. A rather good linear "t is obtained, using "0.629, and the resulting estimate for K then is K "0.2216595 (26), which is nowadays believed to be somewhat too high (cf. Table 2). From Ferrenberg   and Landau [39].

In principle, corrections to scaling could be safely ignored if still much larger sizes could be simulated reliably. While calculations of Ising models have been performed up to lattice sizes of 4800 [258] (as well as short runs for 5888 [259] and [in d"2] 496640 [259] and [in d"5] 112 [258] spins), these calculations could not produce well-equilibrated con"gurations at ¹ , thus  prohibiting a meaningful "nite-size analysis. Also it may well be doubted that an appreciable statistical accuracy can be reached for these systems. An alternative strategy is not to strive for as large lattices as possible but on the contrary work with relatively small lattices (or, at best, lattices of intermediate size) and to obtain results of very high statistical quality, so that the problem of various corrections to the leading asymptotic behavior can be addressed in a serious way [39}51,214]. This will be the strategy emphasized in the following sections. For this purpose, it is helpful not to proceed simply on an entirely phenomenological level [as done in Eqs. (2.37)}(2.57)], but to take some guidance from the renormalization-group description of "nite-size scaling [67,83,260]. Denoting the normalized free-energy density by f"F/(k ¹¸B), where the free energy F is related to the partition function Z as F"!k ¹ ln Z, we obtain the following behavior under a renormalization transformation with a scaling factor b (see, e.g., Ref. [42]): (2.58) f (t, h, u, ¸\)"b\Bf (bWR t, bWF h,bWG u , b/¸)#g(t, h,u ) i"1, 2,2 . G G   Here t and h are the temperature-like and magnetic-"eld-like scaling "elds, which are `relevanta in the renormalization-group sense, while u  is the set of all other scaling "elds, which are G `irrelevant variablesa [14}22]. The pertinent exponents are denoted as y , y (which are both R F positive) and y  ((0) respectively. The function g(t, h,u ) is the regular (analytic) part resulting G G from the scale transformation, and f the singular part. The key aspect of the renormalization  group theory of "nite-size scaling is that the inverse linear dimension ¸\ simply scales with b; no

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other scaling power appears here. For our purposes, it will be su$cient to keep one irrelevant variable (the one with the largest exponent y ,!), which we shall simply denote as u. Note  also the relation with the standard critical exponents [14}22] y "1/, y "d!/ . R F By di!erentiating k times with respect to h, and setting b"¸ and h"0, one obtains

(2.59)

(2.60) f I(t, 0, u, ¸\)"¸IWF \Bf I (¸WR t, 0, ¸\Su, 1)#gI(t, 0, u) .   We now "rst concentrate on the moments m and m, which follow from di!erentiations of the free energy with respect to the physical magnetic "eld H, m"!¸\B(Rf/RH) , m"!¸\B(Rf/RH) #3¸\B(Rf/RH) . (2.61) & & & The Ising spin up}spin down symmetry ensures that the scaling "eld h is an odd function of H. However, it does not need to be simply proportional to it, but can contain higher odd powers of H as well. Similarly, the scaling "eld t has a power-series expansion in K !K [we write   J /k ¹"K for the nearest-neighbor coupling here to emphasize that one may allow for   further-neighbor interactions as well [42]], where both even and odd powers can occur. Thus











Rh  Rf Rh  Rh Rf , #4f  "f  "f  RH RH RH RH RH

Rh . RH

(2.62)

Both the analytic term in Eq. (2.60) and the second term on the right-hand side of the above expression for Rf/RH give rise to corrections to scaling. We now use Eqs. (2.59)}(2.62) to derive a systematic expansion of Q "m/m in powers of t and u (or K !K and u, respectively). *   After some algebra, the result is, for the limit where t is small and ¸ large but "nite [42], Q (K )"Q#a (K !K )¸J#a (K !K )¸J#a (K !K )¸J#2 *           #b ¸\S#b ¸\B>@J#2 , (2.63)   where Q is a universal constant, whereas the expansion coe$cients a , a , a ,2 and b , b ,2 are      nonuniversal. The powers of the geometric factor Rh/RH have canceled in the "rst term on the right-hand side of Eq. (2.63) and the last term results from the analytic background term in Eq. (2.60). The corrections resulting from Eq. (2.62) are decaying even more rapidly with ¸ and have been omitted. The magnetic susceptibility "¸Bm/k ¹ becomes [42] k ¹ "g(t)#¸WF \Bf  (¸WR t, 0, ¸\Su, 1) ,   which yields upon expansion in t and u

(2.64)

k ¹ "c #c (K !K )#2     #¸AJ[a #a (K !K )¸J#a (K !K )¸J#2#b ¸\S#2] ,         (2.65) where the a , b and c are nonuniversal coe$cients, di!erent from those used in (2.63). Similar G G G expansions can be derived from Eq. (2.60) for k"0 to obtain both the energy and the speci"c heat

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[42]. Extending the treatment to include local "elds H , Hr that couple to spins at positions 0 and  r, the spin}spin correlation function G(r) [Eq. (2.51)] can be included in this treatment as well, and also expansions for the temperature derivatives of and Q can be derived in the same way [42]. The strategy is then to obtain very precise data for a broad range of sizes (34¸440 was studied by BloK te et al. [42] for the short-range Ising model in d"3 dimensions) and all these quantities. Each quantity is "tted independently, using K , the critical exponents and the nonuniversal constants as  "t parameters. In order to avoid ambiguities with these "tting procedures, it is advisable to proceed in steps, using the values of the critical exponents from "eld-theoretic renormalization [20] as initial guesses to obtain a good "rst estimate of K . This result is then used as input in an analysis  where one tries to obtain the critical exponents from the "t, and thus the "tting procedures can be iterated. In a "nal stage, one can also keep a subset of the exponents "xed to "t the remaining ones, etc., and apply many consistency checks. The outcome of such an analysis will be described in the next section. As emphasized above, we have assumed the validity of hyperscaling throughout, and thus hyperscaling is implicit in the expansions Eqs. (2.63) and (2.65) on which the analysis of the next section is based. However, we shall discuss the necessary modi"cation of this treatment in Section 4, where we consider Ising models in more than four dimensions, where hyperscaling is violated. Also the extension of "nite-size scaling to properly describe the crossover from one universality class to another one will be described only in the context where this is needed to understand the crossover between the Ising universality class and the mean-"eld universality class (Section 5). Even for the short-range Ising model in d"3 dimensions, our discussion of the use of "nite-size scaling has been far from exhaustive: we have not discussed deviations from "nite-size scaling predicted to occur in the limit ¸PR at "xed (R [97]. If one considers the relative di!erence between a quantity in the thermodynamic limit (such as the susceptibility ) and its counterpart in the "nite system, "¸Bm /k ¹, * * ,( ! )/ "(, ¸) , *

(2.66)

it is argued that, in the limit ¸PR at "xed ,  is not in accord with "nite-size scaling [which would imply that (, ¸) only depends on /¸]. However, since the scale for this deviation from "nite-size scaling is very small [of the order of exp(!¸/) which is negligible in the considered limit], this problem is presumably not a practical limitation on "nite-size scaling analyses of critical phenomena. We also do not address the approach of numerically computing "nite-size scaling functions [216}218,220], where one attempts to construct ratios such as / as function of ¸/ ,  being * * * derived from Eq. (2.52). It has been suggested that this type of "nite-size scaling works already for small values of ¸ and that no a priori information on the critical behavior of the system is required. This procedure would test "nite-size scaling itself by means of the resulting data collapse. The extrapolated results for , , etc. can be "tted straightforwardly to power laws in order to extract the critical exponents. Although the results of this approach so far look quite encouraging [220], we feel there is a need to worry about the e!ects of the various corrections to scaling, that are possibly masked in this approach, leading to systematic deviations in the estimates of critical properties. Thus, even for the short-range Ising model in d"3 the "nite-size scaling analysis of critical phenomena is still an active topic of research.

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Table 2 The critical coupling of the nearest-neighbor simple cubic Ising lattice K  Reference and year Method

0.221655 (5) [56] 1983

0.221620 (6) 0.221663 (9) 0.2216544 (3) [58] 1989 [261] 1997 [262] 1998 High-temperature series extrapolation

K  Reference and year Method

0.221654 (6) [60] 184

0.221652 (6) 0.221652 (3) 0.221655 (1) [264] 1989 [61] 1992 [62] 1996 Monte Carlo renormalization group

K  Reference and year Method

0.2216595 (26) [39] 1991

0.2216544 (10) 0.221657 (3) 0.221648 (4) [38] 1991 [265] 1991 [40] 1993 Monte Carlo (MC) and "nite-size scaling

0.2216576 (22) [41] 1994

K  Reference and year Method

0.2216546 (10) [42] 1995

0.2216544 (6) 0.221655 (15) 0.221655 (1) [43] 1996 [220] 1996 [62] 1996 Monte Carlo (MC) and "nite-size scaling (FSS)

0.2216546 (10) [266] 1998

0.221660 (4) [263] 1998

3. Results for the critical behavior of the three-dimensional Ising model with short-range interactions In this section we summarize the results that were obtained in the last decade for various critical properties of the Ising model. Table 2 quotes estimates for the critical coupling K of the  nearest-neighbor model (some of the exponent estimates have also been obtained for models including next-nearest and 3rd nearest-neighbor interactions, in an attempt to reduce corrections to scaling [42,63]; these results are not included here). It now appears well established that the critical point occurs at K "0.221655 (5) (3.1)  which was already suggested from high-temperature series extrapolations as early as 1983 [56]. However, among the attempts to narrow down the error bar of this estimate, the method of high-temperature series extrapolation [58,261}263] seems to be rather inconclusive, as is shown by a comparison of the more recent estimates obtained by this technique. Thus, if we ignore this method and rely exclusively on MCRG [60}62,264] and MC [38}43,265,266] estimates, we see that it is rather certain that K "0.221655 (2) .  This re"ned estimate still barely excludes the Rosengren conjecture [267]

(3.2)

tanh(K )"((5!2) cos(/8), i.e., K "0.221658632 .   On the other hand, there are strong theoretical arguments against Eq. (3.3) [268].

(3.3)

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Fig. 4. Log}log plot of derivatives dQ/dK of several quantities Q, namely Q"!3;, Q"lnm and Q"lnm,

 vs. ¸ (Q is here not to be confused with the amplitude ratio it generally indicates). Corresponding estimates for  are quoted in the "gure (left). Log}log plot of m at K"K "0.221655 vs. ¸ (curve labeled (right), curve labeled refers  to m!m). From Gupta and Tamayo [62].

Having established the location of the critical point with a satisfactory accuracy, one can proceed to estimate the critical exponents. Often, "ts are carried out where K is kept "xed. Di!erent  choices for K create di!erent biases for the resulting exponent estimates; unfortunately the quoted  uncertainties do not always re#ect the e!ect of this bias. As indicated in Section 2.3, di!erent strategies can be applied with respect to the treatment of corrections to scaling: one, naive, strategy is to choose in a "nite-size scaling analysis the range of linear dimensions ¸ su$ciently large and simply "t the Monte Carlo data at ¹ to the power laws  in Eq. (2.46) that yield / and /. Corresponding data for temperature derivatives at ¹  Jd lnm/dK Jd lnm/dK J¸J, ¸PR , (3.4) d;/dK ) ) )) are "tted to obtain 1/. To avoid the bias from the choice of K one can also use the corresponding  derivative at the value K"K where the derivative is maximal (Fig. 4). The second strategy is to K also include smaller values of ¸ in the analysis and not just rely on the leading power laws [Eqs. (2.46) and (3.4)] but rather include corrections to "nite-size scaling in the "t, applying expansions such as Eqs. (2.63) and (2.65). Table 3 summarizes the corresponding results, again comparing the outcome of di!erent methods. We have put the results stemming from "eld-theoretic renormalization-group methods [53,55] at the top of the table: over the last twenty years, these results have not been questioned, and even in the light of all the additional work summarized in this table, there is no good reason to question the accuracy of the estimates obtained by Zinn-Justin et al. [53,55]. In contrast, the estimates obtained both from extrapolation of high-temperature series [56,57,261}263,269}271] and from Monte Carlo-based methods [39,42,60}63,220,222,223, 264,266,272] are more varying, although they now both appear to have reached a level that is comparable in accuracy. In some cases the table includes entries where the error has not been given(!), and in some cases extremely small error estimates are quoted } these estimates seem overly optimistic to us, however: We feel that both high-temperature series extrapolations and

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Table 3 Estimates for the exponents y "1/, y "3!/ and  R F Method

Ref.

Year

y R

y F



Field-theoretic renormalization group

[53] [55]

1980 1998

1.587 (4) 1.586 (3)

2.485 (2) 2.483 (2)

0.79 (3) 0.799 (11)

High-temperature series extrapolation

[333] [334] [56] [269] [270] [271] [57] [261] [262] [263]

1981 1981 1983 1985 1987 1990 1991 1997 1998 1998

1.585 (8) 1.586 (4) 1.585 (10) 1.5823 (25) 1.582 (7) 1.587 (4) 1.587 1.577 (5) 1.5835 (20) 1.587 (12)

2.482 (5) 2.482 (3) 2.482 (6) 2.4806 (15) 2.480 (6) 2.4821 (4) 2.4823 2.481 (5) 2.4808 (20) 2.483 (6)

CAM

[64]

1995

1.586 (4)

2.482 (4)

MCRG

[60] [264] [61] [62] [63]

1984 1989 1992 1996 1996

1.590 1.590 1.602 1.600 1.585

2.485 (3) 2.4865 (25) 2.4870 (15) 2.488 (3) 2.481 (1)

[39] [41] [42] [62] [220] [44] [266] [223] [222] [272]

1991 1994 1995 1996 1996 1997 1999 1999 1999 1999

1.590 (2) 1.590 (2) 1.587 (2) 1.585 (7) 1.558 1.585 (3) 1.5865 (14) 1.5883 (5) 1.5878 (13) 1.5888 (12)

MC FSS

(10) (8) (5) (3) (3)

2.4914 (28) 2.482 (7) 2.4815 (15) 2.487 (3) 2.465 2.4815 2.4821 2.4817 2.4821

(4) (2) (4) (5)

0.90 (11) 0.85 (8) 0.83 (5) 0.84

0.825 (25) 0.7

0.82 (6)

0.82 0.845 (10) 0.87 (4)

Monte Carlo methods at this point have reached an accuracy that is competitive with the accuracy of "eld-theoretic renormalization-group results for the Ising model, but it would be premature to claim that the accuracy already is much better. Thus, in our opinion it is rather certain that the exponent estimates that follow from Monte Carlo analysis can be summarized as y "1.588 (2), y "2.482 (2), "0.83 (4) , (3.5) R F well consistent with the estimates of [53,55]. It remains to be seen whether the high accuracy claimed in very recent work [222,223,266,272] can be maintained. We also note that some rather recent estimates [220] fall well outside the range of Eq. (3.5), but these estimates were obtained from "ts of extrapolated bulk data over a fairly extended range in the reduced temperature t, and no corrections to scaling terms were allowed for. Thus, these estimates (which deliberately did not quote any estimates for the error) really should be discarded, and we have included them in

202

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Table 3 just for the sake of completeness of the historical record. Some other estimates included in Table 3 were not taken from simulations of the Ising model but rather from the
 model on the simple cubic lattice [222,223]; at this point we rely on the hypothesis that the Ising model (with discrete spins S "$1) and the
 model (with continuous degrees of freedom !R(
(R) G G do belong to the same universality class. Some of the studies mentioned above have attempted to extract more than two exponents (y and R y or  and , respectively) from the "nite-size scaling analysis (e.g., one can obtain / and / F independently, or one can try to estimate (1!)/ from the critical part of the energy, etc.). By means of such analyses, both the hyperscaling relation (/#2/"d) and the thermodynamic scaling relation (#2"2!) can be tested. Within the quoted accuracy of the various studies, the scaling relations always were found to be ful"lled rather convincingly. For details of these studies we refer the interested reader to the original publications quoted in Table 3. Other critical properties of the Ising model have also been estimated occasionally, such as the "nite-size scaling invariant Q"m/m at K"K . While early estimates [39] only reached  a relatively modest accuracy, Q+0.63 (1), BloK te et al. [42] obtained the rather precise estimate Q"0.6233 (4) and this result was even more re"ned by BloK te et al. [266], Q"0.62358 (15), while Ballesteros et al. [272] obtained Q"0.6238 (4) and Hasenbusch et al. [222] found Q"0.62393 (13). The related renormalized coupling constant g, de"ned by 0 ¸ B [3!m/m] (3.6) g" lim lim 0  )) * has also received considerable attention in the literature [219,221,262,273}275]. Estimates from "eld-theoretic calculations [275] yielded g "23.73 (2), while series expansions gave 0 g "23.69 (10) [273] or g "23.55 (15) [274]. While originally it was concluded [219] that 0 0 Monte Carlo results strongly disagree with these estimates, Ballesteros et al. [221] have presented new data that are compatible with these estimates, though not really accurate! The problem is complicated, because g (¸, K) approaches its limiting behavior nonuniformly, i.e., g di!ers from 0 0 the quantity g obtained by taking the limits in Eq. (3.6) in the reverse order: g "5.25 (3) [221]. 0 0 There is also a considerable interest in precisely estimating the (nonuniversal) critical amplitudes of various quantities, in order to "nd results for the universal critical amplitude ratios [213]. While such critical amplitude ratios have been estimated both by "eld-theoretic renormalization-group methods [55] and by high-temperature series expansions [58,213], recent Monte Carlo estimates of these quantities are comparably scarce [276}278]. We de"ne the critical amplitudes of the speci"c heat C, the correlation length , the susceptibility , the order parameter m, the singular part of the free energy f and the surface tension  as follows [213]:   C"(A!/)t\?, m"B(!t)@, "!t\A, "!t\J , (3.7)  f "f !t\?, " (!t)\?\J . (3.8)     Two-scale factor universality [279] implies that all of the following combinations of these amplitudes are universal




A>/A\, >/\, R "A>>/B, >/\, R>"(A>)B> , !   K  (>)Bf >, (\)Bf \, f >/f \, r ">(>)B\ "[4 (\)B\]\ .       NK    

(3.9)

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203

Table 4 Selected critical amplitude values for the d"3 Ising model Quantity


-expansion

Field-theoretic RG in d"3

A>/A\

 

 

0.524$0.010 [280] 0.527$0.037 [55]

>/\ >/\  

4.9 [280] 4.73$0.16 [55]

1.91 [281]

0.541$0.014 [280]

High-T/Low-T series

Monte Carlo

0.523$0.009 [58]

0.550$0.012 [276]

4.77$0.30 [280] 4.79$0.10 [55]

4.95$0.15 [58]

2.013$0.028 [282]

1.96$0.01 [58]

   

0.537$0.019 [55] 4.75$0.03 [277] 5.18$0.33 [283] 2.06$0.01 [283] 1.95$0.02 [277]

3\/[(\)B] 

14.4$0.2 [282]

>/[(>)B] 

14.8$1.0 [58]

3.02$0.08 [280]

3.09$0.08 [58]

0.2700$0.0007

0.2659$0.0007 [58]

17.1$1.9 [283] 14.3$0.1 [277] 3.36$0.23 [283] 3.05$0.05 [277]

R> K f>(>)   R NK 

0.27 [280]

0.0355$0.0015 [277] 0.2 [284] 1.5 [284]

0.39$0.03 [285]

0.36$0.01 [286] 0.86 [288]

0.34$0.02 [278] 1.2$0.3 [287]



0.88$0.04 [278]

Of course, not all of these ratios are independent: e.g., f >/f \ and A>/A\ are obviously related as   the speci"c heat is the second temperature derivative of the free energy and from >/\ and   R one can easily obtain  (which should not be confused with the leading irrelevant exponent). NK Nevertheless, we want to mention these combinations, because they are commonly used in the literature. Many more universal amplitude ratios come into play if one considers the dependence of various quantities on the magnetic "eld at t"0 [213]. However, we are not aware that this problem has found much attention from simulations recently } some more work in this direction would be desirable! Table 4 summarizes some of the predictions on the ratios de"ned in Eq. (3.9) that can be found in the literature [55,58,276}288]. The accuracy of early estimates for these quantities was often overestimated, and even the most recent results for these ratios are much less accurate than these for the corresponding exponents. Also here, more work on these quantities would clearly be desirable.

4. The nearest-neighbor Ising model in dⴝ5 dimensions 4.1. A brief review of the pertinent theory One of the basic results of renormalization-group theory [10}22] is that, for systems with short-range interactions, nonclassical critical exponents only occur for d(4 dimensions, while for

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d'4 the `classicala exponents of simple Landau theory apply. The case d"4 itself is a borderline case, where logarithmic corrections to the classical power laws are present. A test of this case with Monte Carlo simulations is fairly involved [289] and will remain out of consideration here. However, the situation in d"5 should be much simpler and Monte Carlo simulations for this case should provide a good testing ground to check whether our current theoretical understanding of systems above their upper critical dimensionality is in fact complete. While obviously no laboratory experiment can be carried out in d"5 spatial dimensions, the study of such high (but integer) [290] dimensions is in fact straightforward by means of simulational techniques. We start by reviewing the adaption of the theory of "nite-size scaling to this case [72}74]. Using the fact that for d'dH"4 we have the Landau values for the exponents y , y and the (leading) R F correction to scaling exponent , y "2, y "1#d/2, "d!4 , (4.1) R F the singular part of the free-energy density f of a "nite system with linear dimensions ¸ in * a external "eld h is written as [cf. Eq. (2.58) for b"¸] [71]






¸  (4.2) , h¸>B, u¸\B . f (t, h, ¸)"¸\B fI t * >  Here we have used the mean-"eld result ">t\ (for t'0) and scaled the length ¸ with the  length > to make the "rst argument of fI dimensionless [and to remind the reader of the "nite-size  scaling principle that this term simply can be interpreted as (¸/)]. Now an important issue is that although (!)"4!d for d'4 is negative and hence u¸\BP0 for ¸PR, one nevertheless may not omit the last term, because u is a `dangerous irrelevant variablea [291,292]. This statement means that the scaling function fI (x, x , y) is singular in the limit yP0 and cannot be simply replaced by f (x, x , 0). Also the variable y is normalized such that it is dimensionless, and this is borne out by our notation by writing uJlB\, where l is   another (microscopic) length, so that y"(¸/l )\B. For brevity we will henceforth simply write   instead of >.   Taking suitable derivatives of Eq. (4.2) with respect to the "eld h we can write for the order parameter m, the (high-temperature) susceptibility and the ratio Q (in zero "eld) m"¸\B\P t(¸/ ), (¸/l )\B , K   k ¹ "!(Rf /Rh) "¸Bm"¸P t(¸/ ), (¸/l )\B * 2 Q  

(4.3) (4.4)

and Q"m/m"P t(¸/ ), (¸/l )\B , (4.5) /   where P , P and P are the (universal [293]) "nite-size scaling functions of the quantities m, K Q /

and Q. The question how these functions behave in the limit yP0 was "rst addressed in [72],

 Preliminary data have also been obtained for the case d"6.

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where it was assumed that the dangerous irrelevant variable y enters in the form of multiplicative singular powers of y, e.g., (4.6) f (x, x , y)+yN fI (xyN , x yN ) as yP0 , * * with p , p and p suitable exponents [72]. This assumption was in the "rst place motivated by the    fact that this is the mechanism that operates for the scaling in the bulk for d'4 [292]. In addition, Eq. (4.6) is supported by various phenomenological arguments. In particular, it was argued [72,73] that standard thermodynamic #uctuation theory requires, for ¹(¹ and su$ciently large ¸, that  the distribution function P (m) of the magnetization per spin for m near the spontaneous magnetiz* ation $m can be written as a sum of two Gaussians [cf. Eq. (2.13)]. Using m "B(!t) and  

"\(!t)\ the arguments of the exponential functions in Eq. (2.13) have the form 1 ! [(m/B)(!t)\G1](¸/l )B , 2 2

(4.7)

where the `thermodynamic lengtha l is de"ned as 2 l "[m\ ]B"(B\\)B(!t)\B . (4.8) 2  Taking moments of P (m) it was then concluded that the scaling functions P , P and P in * K Q / Eqs. (4.3)}(4.5) should reduce to scaling functions of a single variable (¸/l )B"t¸B\l\B"x/(y , 2   m"¸\BPI (t¸B\l\B) , K  

"¸BPI (t¸B\l\B) Q  

(4.9) (4.10) (4.11)

and Q"PI (t¸B\l\B) . (4.12) /   Scale factors for the magnetization and the susceptibility have been absorbed in P (or PI ) and K K P (or PI ), respectively, while in ratios like Q (and hence in P and PI ) such scale factors cancel out Q Q / / and fully universal scaling functions remain. While these arguments did not yield explicit expressions for the scaling functions, BreH zin and Zinn-Justin [74] did propose such an explicit form, suggesting that for d'dH"4 one could split the argument of the Boltzmann factor into a contribution due to the uniform magnetization (the `zero modea) and contributions of nonuniform magnetization #uctuations, which can be treated perturbatively. Thus









HS  (m/m !1) G "exp !  ¸B##uctuation contributions . (4.13) exp ! k ¹ 8k ¹ /m  The #uctuation contributions were argued to yield only a shift of ¹ ,  ¹ (¸)/¹ (R)!1J¸\B , (4.14)   which is of higher order than the rounding of the singularities [scaling like ¸\B, cf. Eq. (4.10)}(4.12)] and hence negligible in the limit of large ¸, compared to the "nite-size e!ects

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included in Eqs. (4.10)}(4.12). If the #uctuation contributions in Eq. (4.13) are ignored completely, this so-called `zero-mode theorya yields an explicit result for the distribution function of the magnetization, P (m)J¸B expB/8 . (4.15) * 2 From this result it is straightforward to obtain PI , PI and PI . In particular, one derives the K Q / universal constant at ¹ [74]  Q  "PI (0)"8/(1/4)+0.456947 . (4.16) 2 / This treatment has recently been criticized by Chen and Dohm [93}95] who presented detailed arguments that for d'dH the standard treatment of the
 "eld theory in continuous space yields a misleading description of the "nite-size behavior, being di!erent from the "nite-size behavior of a
 model on a lattice, which is believed to be equivalent to an Ising Model [10}22]. Chen and Dohm emphasized that therefore the justi"cation given for the zero-mode theory is invalid and stated that the moment ratio mentioned above does not have the universal properties predicted previously [94]. Their analysis is based on the
-model on a (hyper)cubic lattice,





J r (4.17) H"

#u (
) #
GH (
!
) , G  G H 2 G 2 GH G where r , u and the J are constants, and
is an n-component vector. Chen and Dohm have   GH G considered both the limit nPR, which can be treated exactly, and the case n"1 (which corresponds to the Ising model); the latter, however, is only treated to one-loop order in a perturbation expansion. In terms of the reduced moments



 (>)" K







1 d

K exp ! >
!
 2







1 d
exp ! >
!
 , 2

(4.18)   the scaling functions P (x, y) and P (x, y) are written Q /  (>(x, y)) 1  P (x, y)" , (4.19) Q J (y#36I (r )y   P (x, y)"[ (>(x, y))]/ (>(x, y)) , (4.20) /   with J "
J (r !r )/d¸B and  GH GH G H (4.21) >(x, y)"[x!12yI (r )!144 (x/(y)I (r )y]/[(y(1#36I (r )y)] .     Here further abbreviations r ,x#12 (x/(y)(y and    B > B (4.22) I (x)"(2)\K dyyK\ exp(!xy/4) !
e\WN #1 K y  N\ were introduced. Eqs. (4.18)}(4.22) are the "rst results for the scaling functions P and P that contain the Q / dependence on both variables x"t¸\ and y"(¸/l )\B separately and explicitly. However,   at the same time these results do con"rm the validity of the `zero-modea results, such as







 

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207

Eqs. (4.10)}(4.12) and (4.16), in the limit of large enough ¸ (when yP0). This is obvious from Eq. (4.21), since for yP0 we "nd >(x, y)Px/(y, P (x, y)"(J (y)\ (x/(y), P "[ (x/(y)]/ (x/(y) , Q   /  

(4.23)

fully compatible with Eqs. (4.9)}(4.12) and the results proposed by BreH zin and Zinn-Justin [74], including Eq. (4.16). 4.2. Comparison with Monte Carlo results While already in early Monte Carlo work it was claimed from very small linear dimensions, ¸"3 to 7 [72,73], that the data agree with the scaling structure of Eqs. (4.10)}(4.12), there seemed to occur a discrepancy with Eq. (4.16) and this discrepancy was con"rmed in later work using somewhat larger sizes (¸415) [98]. In contrast, it was then shown in Ref. [99] for a completely di!erent class of models, which are however expected to be described by the same renormalization-group equations as high-dimensional Ising models, that Eq. (4.16) holds to a very high accuracy, for a wide range of distances to the upper critical dimension. It was also demonstrated how the "ndings of Ref. [98] could be traced back to the neglect of certain corrections to scaling and an insu$cient statistical accuracy. Nevertheless, several additional attempt were undertaken to clarify matters by more extensive simulations of the "ve-dimensional Ising model itself [100}104] and it is fair to say that on the simulational side there no longer exists any discrepancy regarding the value of the parameter Q at criticality. On the analytical side, however, matters appear to be less clear, in particular regarding the shape of the "nite-size scaling functions [93}95]. Here we shall describe the approach taken in the most recent analysis [104] where a comparison with both the `zero-modea results [74] and the alternative expressions, Eqs. (4.18)}(4.22), was made. Fig. 5 shows that if one studies the variation of Q over a wide range (0.3:Q:1) and uses a correspondingly wide range of the scaling variable t¸, !44t¸4#4, the data suggest that already for these relatively limited system sizes the scaling prediction, Eq. (4.12) is a reasonable approximation, although there are still slight deviations. The accuracy of the currently available data is such that statistical scatter and systematic deviations can be clearly distinguished. This allows a closer look at the behavior of the amplitude ratio in the neighborhood of ¹ , revealing  how the deviations found in Refs. [72,73] and also in Ref. [98] could come about. For this, we refer to Fig. 6, where only data for Q near t¸"0 are included. If corrections to Eq. (4.12) were negligible, we would expect in this plot the three curves for the three values of ¸ to intersect at t¸"0 at the critical value of Q (4.16). On the other hand, if corrections to scaling are nonnegligible, one would have expected that there is no longer an unique intersection point at all, but a di!erent intersection for each pair of curves, i.e., a behavior qualitatively analogous to Fig. 2. However, what happens instead is that the three curves apparently still have a intersection point, but at a wrong value: this intersection occurs not at t"0, but at a negative value of t¸, and the value of Q is here correspondingly higher than in Eq. (4.16). Interestingly, the theory of Chen and Dohm [95] qualitatively predicts precisely such a behavior with a spurious `intersection pointa, although it is clearly not in quantitative agreement with the Monte Carlo data either.

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Fig. 5. Plot of Q"m/m for the nearest-neighbor Ising model in d"5 dimensions vs. the scaling variable t¸ [cf. Eq. (4.12)], including both Monte Carlo (MC) data for ¸"4 [72], ¸"8, 12 [103] and the results of Chen and Dohm (CD) [95], Eqs. (4.18)}(4.22). Note that the result of the zero-mode theory [74] results from setting >"x/(y in Eqs. (4.18)}(4.22). The critical temperature was estimated as J/k ¹ "0.1139155 (2), and the constants  , l were estimated    as  "0.549 (2) and l "0.603 (13), see the text. From Luijten et al. [104].  

Fig. 6. Magni"ed plot of Q vs. t¸ near t¸"0, to demonstrate the occurrence of spurious cumulant intersections. Broken curves are again the predictions of Chen and Dohm (CD) [95], Eqs. (4.18)}(4.22), with the parameters as quoted in the caption of Fig. 5, while the symbols indicate the Monte Carlo data. From Luijten et al. [104].

For the comparison between the theory, Eqs. (4.18)}(4.22), and the simulation it is essential to determine the parameters ¹ ,  and l correctly. Using data for in the range 54¸422,    a "nite-size scaling expansion similar to Eq. (2.65) was used [104],

"¸B(c #c tK ¸WHR #c tK ¸WHR #q ¸\B#q ¸\B) ,      where tK is a variable that includes a "nite-size shift of ¹ ,  tK "t#¸\B ,

(4.24) (4.25)

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209

Fig. 7. Plot of (a) ¸\ and (b) Q vs. ¸ at J/k ¹ "0.1139150. The dashed curve in (a) is the "t according to Eq. (4.24),  while the dotted curve is the result of Chen and Dohm [95], Eq. (4.19). In (b) the horizontal straight line highlights the asymptotic value, Eq. (4.16), while the broken curve is a "t of Q according to Eq. (5.28), and the dotted line represents again the result of Chen and Dohm [95], Eq. (4.20). From Luijten et al. [104].

and the parameters c , c , c , q , q and  are adjustable constants. In a "rst step of the      "tting procedure, yH was also treated as an adjustable constant, which yielded yH"2.53 (4) and R R J/k ¹ "0.1139152 (4). Obviously this result does not indicate any serious problem with the  prediction [cf. Eq. (4.9)] yH"y #/2"2#(d!4)/2"5/2. In the second step of the "tting R R procedure, this exponent was also "xed at its theoretical value, yielding then, in particular, J/k ¹ "0.1139155 (2) and c "1.91 (2). Using now the asymptotic result following from   Eq. (4.19), namely (J "2J/k ¹)  ¸B (3/4) ¸ (0)  " ,

"¸BP (0, 0)" Q (l J (1/4) J (y   

(4.26)

one recognizes that the constant c is directly related to the length l , for which one "nds (in units   of the lattice spacing) l "0.603 (13) [104]. 

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The procedure of estimating l is reviewed here in such detail since it has been criticized by Chen  and Dohm [95] in a note added to their paper. While we consider this criticism as unfounded, we leave it to the reader to form his own opinion on this apparently controversial issue. Also the parameter  can be extracted from the Monte Carlo data, since in the limit yP0, at  "xed small t, Eqs. (4.4), (4.19) imply x W 

t"   (>(x, y)) P  /J . (4.27)    J (y#36I (r )y   Alternatively, in Ref. [104] it was found convenient to use the result from high-temperature series [294] for this purpose, "A/(1!v/v ) where v"tanh(J/k ¹) and A"1.311 (9). Rewriting this in A terms of t yields "1.322t\ and Eq. (4.27) gives  "0.549 (2).  Having "xed all constants of the theories [Eqs. (4.9)}(4.12) or (4.18)}(4.22), respectively], we show in Fig. 7 the approach of both and Q at ¹ to their limiting behaviors. For Q an expansion  analogous to Eq. (4.24) was used [103] Q(¸, t)"PI (0)#c tK ¸B#c tK ¸B#q ¸\B#q ¸\B . (4.28) /     In addition, analyses have been performed where Q (R, 0) was not "xed at its theoretical value PI (0), Eq. (4.16), but also in this case the results were nicely compatible with Eq. (4.28) [289]. The / data show that even for ¸"22 the data are still far from their asymptotic values, due to strong corrections to the leading "nite-size scaling behavior Q(¸, 0)"PI (0) and ¸\"c . The theory /  of Chen and Dohm [95], which is claimed to describe exactly the leading corrections (of order ¸\) to the asymptotic behavior, is not useful in this regime. In particular, for ¸\ it predicts a monotonic decrease toward the asymptotic value, while the Monte Carlo data reach a shallow maximum "rst and then a decrease in a much less pronounced way (note that, by construction, both curves in Fig. 7b converge to the same constant c !).  Thus the present state of a!airs concerning this model is somewhat disappointing: although one knows all the critical exponents exactly (including those of the correction terms), and even "nite-size scaling functions are believed to be known exactly both in the limit yP0 (where the `zero-modea results [74] hold) and also beyond it, where nonuniform terms in Eq. (4.13) were computed via perturbation theory in "rst-order loop expansion for the
 model [Eq. (4.17)], there is no explicit understanding of the behavior found in the accessible range of ¸ (Fig. 7). The simulation data are compatible with the approach to `zero-modea results [74] for ¸PR, but the approach is surprisingly slow, and for the accessible range of ¸ other corrections than those derived by Chen and Dohm [95] are dominant. Luijten et al. [104] have speculated that this discrepancy could be due to the need of including second-order terms in the loop expansion, or that corrections might be present due to the fact that an Ising model only asymptotically agrees with a
 model [Eq. (4.17)]. Thus the e!ects of statistical #uctuations on critical behavior are not even completely clear when ultimately the critical behavior is mean-"eld like.

 After submission of the present review, Chen and Dome [335] attempted a reanalysis allowing for an additional amplitude factor A"0.678 (the present treatment means A,1). Thus "tting an additional parameter the discrepancy between their theory and the Monte Carlo results can be reduced.

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211

Table 5 The cuto! distance R of the interaction function, the corresponding coordination number q, and the e!ective range of K interaction R for the equivalent-neighbor Ising model in d"2 and d"3 dimensions d"2

d"3

q

R K

R

q

R K

4 8 12 20 24 36 60 100 160 224 316 436

1 2 4 6 8 10 18 32 50 72 100 140

1 3/2 7/3 17/5 25/6 6 148/15 81/5 517/20 1007/28 4003/79 7594/109

6 18 26 32 56 80 92 122 146 170 178 202 250

1 2 3 4 5 6 8 9 10 11 12 13 14

R 1 5/3 27/13 39/16 99/28 171/40 219/46 354/61 474/73 606/85 654/89 810/101 1146/125

5. Crossover scaling in Ising systems with large but 5nite interaction range in dⴝ2 and dⴝ3 dimensions 5.1. General theory In this section we consider the Hamiltonian [45}51,92] H/k ¹"!

K(r !r )S S !h
S , S "$1 G H G H  G G G HG G with an interaction K(r) de"ned as

(5.1)

K(r),cR\B r"r !r 4R ; K(r)"0 r'R . (5.2) G H K K Here c is a constant, R a cuto! distance and the range R of the interaction K(r) is de"ned in terms K of its second moment, as usual,
(r !r )K(r !r ) 1 H G H "
r !r  . R" H$G G (5.3) G H
K(r !r ) q H$G G H H$G Here q is the coordination number of this `equivalent-neighbor Ising modela and
indicates that the summation is restricted to r4R . For large R we have a simple proportionality between K K R and R , R"R /2 (d"2) or 3R /5 (d"3), whereas for small R there are lattice e!ects. In K K K K Table 5 we have listed the choices of R that have been studied [49,92]. K For RPR this model crosses over to the trivial mean-"eld model of a ferromagnet, in which every spin interacts equally with every other spin, and then the simple Weiss molecular "eld theory becomes exact [9]. However, when R is large but "nite one expects that mean-"eld theory describes

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Fig. 8. Qualitative picture of the renormalization trajectory describing the crossover from the Gaussian "xed point H"(r "0, u"0) to the Ising "xed point H"(rH, uH). From Luijten et al. [92].   

the behavior of the model quite well, except in a very narrow neighborhood of the critical point: There ultimatively mean-"eld theory must break down, and a crossover from mean-"eld critical behavior (critical exponents "0 ["nite jump in the speci"c heat], "1/2, "1, "1/2, "0 [8,9,15,16]) to the critical behavior of the `Ising universality classa [13}21] occurs. The analysis of this crossover by means of Monte Carlo simulations is in the focus of the present section. To analyze this crossover it is instructive to consider the associated Ginzburg}Landau "eld theory in continuous space [92]

 



1 2

dr



dr





c 1
(r)
(r ) ! v
(r)!u
(r)#h
(r) ,   RB 2

(5.4) 4 r\rY0K where
(r) is the single-component order-parameter "eld, v is a temperature-like variable, and u is  a constant. After Fourier transformation and suitable rescaling of
, this can be rewritten as (remember that N"¸B is the total number of lattice sites) H(
)/k ¹"!



1 r u h H M ( )/k ¹"
k#  k  k #


k k k  k k k ! I \ \ \ \ 2k R 4RN k k k R 







N k 0 . 2  (5.5)

See Ref. [92] for a detailed derivation of the relation between the new parameters u, h and the old ones (u , h ). The variable r is proportional to the relative deviation of the temperature from its    critical-point value in mean-"eld theory. We are interested in identifying the crossover scaling variable associated with the crossover from the Gaussian "xed point (u"0, r "0) to the nontrivial "xed point (Fig. 8). Because of the trivial 

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213

character of the Gaussian "xed point and the fact that the crossover scaling description should hold all the way from the Ising "xed point to the Gaussian "xed point, one can infer the length scale l exactly,  l "R\B . (5.6)  This is done by considering a renormalization transformation by a length scale l, k"kl, N "Nl\B,  "l\ . IY I Note that Eq. (5.7) was constructed such that the Hamiltonian is left invariant,

 

(5.7)



u r 1 l\B


 k   k   k   k  k  k  H M ( )/k ¹"
k #  l k  k # Y \ Y 4RN \ \ \ IY R 2 k k k    IY h N ! l>B k 0 . (5.8) Y R 2 For d(4, the  term grows and the system moves away from the Gaussian "xed point H toward  the Ising "xed point. The crossover to Ising-like critical behavior occurs when the coe$cient of the  term is of the same order as the k term, which is unity: This happens when l"l as given in  Eq. (5.6). By comparing the coe$cient of the  term to that of the r  term, one derives  a criterion that states for which temperatures the critical behavior will be Ising-like and for which temperatures it will be classical: actually this is nothing but the well-known Ginzburg criterion [109]! One expects the Gaussian "xed point to dominate the renormalization #ow if, irrespective of l, the  coe$cient is small compared to the temperature coe$cient. Thus, one requires the scaled combination uR\l\B/(r R\l)\B to be small, or, equivalently (see also [87])  r\BRB/u<1 . (5.9)  Since the total free energy is conserved along the renormalization trajectory, we can conclude that the singular part of the free-energy density resulting from Eqs. (5.4), (5.5) and (5.8) must satisfy the scaling relation fI











u h r u h r   l, l\B, l>B . "l\BfI  R R  R, R, R R

(5.10)

We see that a "nite and nonzero value for the second argument of fI is retained exactly when l takes  the value of the crossover scale l . Thus, we conclude from Eq. (5.10) that the singular part of the  free energy scales with R as follows: fI "R\B\BfK (r RB\B, u , hRB\B) . (5.11)    In Eq. (5.10) we have anticipated that a natural choice of coordinates (Fig. 8) is to measure r and  u as distances from the Ising "xed point, unlike in the original Hamiltonian, where r and u are   distances from the Gaussian "xed point. Eq. (5.11) describes how the temperature distance r from  criticality and the magnetic "eld h scale with the range of interaction R: Note that here the crossover exponent is known exactly, unlike for other cases of crossover, such as the crossover between the Ising and Heisenberg universality classes in isotropic magnets with varying uniaxial anisotropy [295]. Obviously, the same result for the crossover exponent follows from

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simple-minded arguments using the Ginzburg criterion [87,109]. The location of the nontrivial "xed point H (Fig. 8), the associated critical exponents y , y ,  and the explicit form of the scaling R F functions fI or fK , respectively, cannot be obtained exactly. The calculation of these scaling functions   (as well as of corresponding scaling functions of free-energy derivatives, e.g. of the susceptibility) remains a nontrivial task for both renormalization-group approaches [110}124] and Monte Carlo calculations [45}50,87], as will be described below. If we carry out a rescaling transformation by a factor b in the neighborhood of the Ising "xed point, the nontrivial exponents y , y and  must R F show up in the transformation as follows: (5.12) fK "b\BR\B\BfK (tRB\BbWR , u b\S, hRB\BbWF ) ,   where for simplicity we have replaced the variable r by the reduced temperature distance t from  the true critical point, suppressing the prefactor in the relation r Jt. From Eq. (5.12), one can  derive scaling relations for critical amplitude prefactors of the magnetization, the susceptibility, the speci"c heat, etc. [87,92], in powers of R. In addition, we can generalize Eq. (5.12) immediately to the case of "nite-size scaling, by including l/¸ in Eq. (5.10) or l b/¸"bR\B/¸ in Eq. (5.12) as  an additional scaling variable. The "nite-size scaling behavior is then found by choosing b such that l b/¸"1, i.e.,  b"¸R\\B . (5.13) The critical behavior of the magnetization m and the susceptibility are then obtained in terms of the "rst ( fK ) and second ( fK ) derivative of the scaling function fK for the free energy as follows [92]:    m"¸WF \BRB\WF \BfK t¸WR R\WR \B\B, u ¸\SRS\B, h¸WF RB\WF \B , (5.14) 

"¸WF \BRB\WF \BfK t¸WR R\WR \B\B, u ¸\SRS\B, h¸WF RB\WF \B . (5.15)  The "nite-size scaling right at ¹ (t"0, h"0) can then be written as  m"¸WF \BRB\WF \Bm (¸R\\B)"¸\Bm\ (¸R\\B) , (5.16)

"¸WF \BRB\WF \B (¸R\\B)"¸B \ (¸R\\B) ,

(5.17)

Q"Q[ (¸R\\B) ,

(5.18)

and where m , m\ , , \ and Q[ are suitable scaling functions. Note that these functions have been de"ned such as to bring out a simple limiting behavior in the respective limits, namely m (PR)"const,  (PR)"const, while m\ (P0)"const, \ (P0)"const. In the opposite limit the functions then must be simple power laws, which are easily extracted from Eqs. (5.16) and (5.17). On the other hand, the function Q[ smoothly interpolates from the constant PI (0) [Eq. (4.16)] / that is reached for P0 to the constant Q[ (PR) which is the respective universal constant Q [Eq. (2.63)] for the corresponding universality class of the short-range Ising model at the respective dimensionality d(4. 5.2. Numerical results for d"2 dimensions The "rst task is again the accurate numerical determination of the critical temperature. Note that the straightforward cumulant intersection method (Fig. 2) is not expected to work here, due to the

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Fig. 9. The amplitude ratio Q (K ) at the critical point of the two-dimensional Ising model with interactions up to *  R "(140 as a function of linear system dimension ¸ (discrete points). For large ¸, Q (K ) approaches the Ising limit K *  Q"0.856216 [296], dotted line. For decreasing ¸, Q (K ) approaches the mean-"eld limit PI (0) [Eq. (4.16)], until the *  / linear dimension becomes smaller than the range R and strong "nite-size e!ects come into play. To illustrate that the K system indeed behaves mean-"eld like for small sizes, Q was also plotted for "nite systems in which all spins interact equally strongly (dashed curve). The points are seen to approach this curve for small ¸. From Luijten et al. [92].

presence of the argument ¸R\B\"¸/R in Eq. (5.18). As illustrated in Fig. 9, there is indeed a slow crossover that is spread out over several decades in ¸. Obviously, it is very di$cult to cover the full crossover with a single choice of R. In order to reproduce the mean-"eld limit described in Eq. (4.16), one needs to choose R relatively large; at the same time, however, ¸ must not become smaller than R, because otherwise every spin interacts with the same strength with every other spin and the character of "nite-size e!ects is di!erent in this limit: For such a "nite mean-"eld system containing N spins, one easily derives Q"PI (0)#0.214002/(N #O(1/N) [250] (dotted curved / in Fig. 9). Fig. 9 shows that even for R "(140 one does not yet fully reach the mean-"eld result K Q"PI (0)+0.456947. On the other hand, in order to reach the Ising limit, values of ¸/R+10 / are required (Fig. 10). The asymptotic value Q"Q[ (R) for the Ising limit is known with very high precision, Q"0.856216 (1) [296], and this number is used as an input for the analysis. Luijten et al. [46,92] used linear sizes up to ¸"500 for R 410, and for larger ranges R system sizes up K K to ¸"700 or even ¸"800 (R "100, 140). For each run 10 Wol! clusters were generated after K equilibration of the system, sampling the various thermodynamic quantities after every tenth Wol! cluster. Q (K) was then "tted to Eq. (2.63), using the exponents appropriate to the d"2 * Ising case, "1,

"1/8, "2 .

(5.19)

The resulting estimates for the critical coupling K are plotted vs. R\ in Fig. 11. Motivated by  renormalization-group arguments [92], these data are "tted by a power law with a logarithmic correction, qK "1#R\(a#b ln R)#cR\ , 

(5.20)

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Fig. 10. Finite-size crossover scaling curves plotted in d"2 dimensions at ¹"¹ (R) vs. ¸/R for (a) Q,  (b) m¸/C[m], and (c) ¸\/C[ ]. In order to remove some corrections to scaling, corrections C[m]"1!R\(b #b ln R) and C[ ]"1#R\(q #q ln R)#q R\, with b , b , q , q and q adjustable           constants, have been "tted to the data. (Note that the functional form of these "nite-range corrections can be justi"ed by renormalization-group arguments). Di!erent symbols stand for the di!erent choices of R , as indicated in the "gure. K Mean-"eld and Ising asymptotes are included in parts b and c. From Luijten et al. [46].

where a"!0.267 (6), b"1.14 (6) and c"!0.27 (3). This relation (the curve drawn in Fig. 11) is useful for providing estimates for very large R (i.e., for 5004R 410000), where direct K determinations of K from "nite-size scaling would no longer be feasible.  Fig. 10 has already provided evidence that the crossover from mean-"eld behavior to shortrange Ising behavior occurs in a smooth fashion and is spread out over several decades in the crossover scaling variable, which is ¸/R for d"2 [cf. Eqs. (5.16)}(5.18)]. Also the thermal crossover spans a comparatively wide regime but now the crossover scaling variable is tR [cf. Eq. (5.12)]. Putting b"1 in Eq. (5.12) one directly obtains m"t@RB@\B\Bm (tRB\B)"R\B\Bm\ (tRB\B) R R

(5.21)

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Fig. 11. Plot of 1/(zK ) vs. R\, where z denotes the number of equivalent neighbors, for the d"2 equivalent-neighbor  Ising model. The dotted line denotes the extrapolation to the mean-"eld limit. The inset shows 1/(zK ) over the full range  of R\ between the Ising and the mean-"eld limit. From Luijten et al. [92].

and

"t\ARB\A\B  (tRB\B)"RB\B \ (tRB\B) , (5.22) R R where we have attached the subscript t to the crossover scaling functions in order to distinguish them from those for the "nite-size scaling crossover, Eqs. (5.16) and (5.17). Fig. 12 shows a corresponding plot for m and Fig. 13 for (at temperatures ¹(¹ ). One sees that the `raw dataa for  m and only show a rough collapse and that there are various systematic deviations from a perfect match to a master curve. For example, for very small values of tR the data start to deviate from the slope of the Ising asymptote and approximately cross over, for temperatures closer to ¹ , to a constant value. This crossover to a horizontal slope occurs at an ¸-dependent location  and is due to residual "nite-size e!ects: Eqs. (5.21) and (5.22) only follow from Eqs. (5.16) and (5.17) in the limit ¸PRat "xed tRB\B. The data which are a!ected by such "nite-size e!ects were all omitted in part (b) of Figs. 12 and 13. However, even in the remaining data one "nds that, in the regime where mR is already proportional to (!tR) for small tR, there is a systematic o!set in the prefactor: One obtains a set of parallel straight lines rather than a collapse on a single line. It turns out that this e!ect results from a `"nite-rangea correction C[m] to the critical amplitude. The same correction was already involved in the "nite-size crossover scaling plot, Fig. 10. The presence of such corrections actually is no surprise at all } the treatment presented in the previous section applies for RPR, tP0 and tRB\B "nite or RPR, ¸PR and ¸R\\B "nite, whereas we have included rather small values of R in Figs. 10}13. If one would only include data lying in the appropriate scaling limit into the analysis, there would be no need for the present corrections: However, in order to reach, for large R, the scaling limit of the Ising universality class, one would have to simulate huge systems very close to ¹ (R). Presently, such simulations are not feasible, and  therefore Luijten et al. [45}51,92] decided to include relatively small values of R in the analysis as well and to apply appropriate corrections. The functional form of this range-dependent corrections can be justi"ed by renormalization-group arguments.

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Fig. 12. Log}log plot of (a) mR and (b) mR/C[m] vs. tR, for the d"2 equivalent-neighbor Ising model, where the reduced temperature t is de"ned as t"[¹!¹ (R)]/¹ (R) and the symbols denote various choices of R as indicated. In   K (a) no additional correction terms have been used, while in (b) the factor C[m] has been divided out (see text), data points in the "nite-size regime have been omitted, and data for R 572 have been corrected for saturation e!ects. The dotted K straight lines show the asymptotic power laws in the Ising and mean-"eld regime, respectively. From Luijten et al. [46].

Fig. 13. Log}log plot for the susceptibility ,[m!m]¸B/(k ¹), normalized by a factor RC[ ], see text, vs. the thermal crossover variable tR. Various choices of R are included as indicated. In part (b) data points in the "nite-size K regime, which are included in (a), have been omitted, and data for R 572 were corrected for saturation e!ects. Dotted K straight lines show the asymptotic power laws in the Ising and mean-"eld regime, respectively. From Luijten et al. [46].

A further systematic deviation from a data collapse for m is encountered for large values of tR: There the curves systematically bend away from the mean-"eld asymptote toward smaller values. This results from a saturation of the order parameter at low temperatures, as can be understood already within mean-"eld theory: the magnetization must fall below the power law m"(3(!t) as ¹P0 since then mP1 rather than mP(3. In fact, from the self-consistent molecular-"eld equation m"tanh(m¹ /¹) one can derive [46]  2 12 (!tR)! (!tR) . (5.23) mR+(3(!tR) 1! 5R 175R





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219

Using the term in square brackets to correct the numerical data for large tR, one obtains the almost perfect data collapse for the magnetization shown in Fig. 12b. Similar e!ects occur in the susceptibility, Fig. 13: For small tR, the curves bend away from the Ising asymptote toward ¸-dependent plateau values because of "nite-size e!ects and for large tR the curves decrease more strongly than expected from the mean-"eld asymptote, which is again a saturation e!ect, since "(1!m)/(t#m) in the molecular "eld approximation, and the numerator of this expression vanishes as mP1. In fact, it is straightforward to derive the expansion [46]



9 36 36 13 428 t! t! t

+$(¹(¹ )+(!2t)\ 1# t!  5 175 175 67 375



(5.24)

and using again the term in square brackets to correct the data for for these saturation e!ects, one obtains the perfect collapse on a master curve shown in part b of Fig. 13. A very remarkable feature occurs in the central part of the crossover, for !14tR4!10\, where the scaled data fall below the mean-"eld asymptote before reaching the Ising asymptote. This means that in the intermediate regime of the crossover scaling one "nds an e!ective exponent  (1! These e!ective exponents are traditionally de"ned as logarithmic derivatives [297],   ,d lnm/d ln t,  "!d ln /d ln t"!td ln /dt . (5.25)   While  varies monotonically from the mean-"eld value  "1/2 for large tR to the Ising  +$ value "1/8 for small tR [46], the variation of  is monotonic above ¹ but nonmonotonic   below ¹ (Fig. 14). The physical reason why > varies monotonically while \ shows this    `underswinga is unclear, and there are presently no theoretical predictions for the crossover scaling functions shown in Figs. 13b and 14a! However, for ¹'¹ ,  has been found by Pelissetto et al.   [124] by means of a systematic perturbation around the mean-"eld limit, and the agreement with Fig. 14b is almost perfect, cf. Fig. 14c. One question that has been left unanswered by the treatment of Section 5.1 is the universality of crossover scaling functions such as fK [Eq. (5.12)] or the functions fK , fK , m , , m ,  [Eqs.    R R (5.14)}(5.17), (5.21), (5.22)] derived from it: Is there a single variable R that controls such crossover phenomena, or are there additional parameters on which these crossover scaling functions might depend? There is no complete consensus on this problem in the literature [110}124]. In the absence of clear theoretical guidance, and because of suggestions that the presence of an additional length scale might in#uence the nature of the crossover [121], it has been speculated that the shape of the interaction function K(r) might play a role [50]. In order to check this conjecture, a function K(r) was constructed that did not only involve a single length scale R, but rather two di!erent scales R , R , by choosing K(r)"K for 0(r(R and K(r)"K for R (r(R . In order to create        a strong asymmetry between the two domains, a large strength ratio was chosen, K /K "16.   Now it is possible to choose very di!erent combinations R , R that yield the same e!ective range   R, de"ned from Eq. (5.3) in the usual way. For example, choosing R "93, R "140 yields   R"48.8 while the very di!erent choice R "4, R "140 yields the quite similar value   R"49.99. Indeed, one "nds that both choices yield appreciably di!erent results for K , although  R is almost the same. However, the crossover scaling function for both choices is precisely identical! Thus, the idea that the crossover scaling functions might be nonuniversal because they depend on

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Fig. 14. E!ective susceptibility exponents (a) \ and (b) > for ¹(¹ and ¹'¹ , respectively, plotted vs. tR. The     Ising value ("7/4) and the mean-"eld (MF) limit ("1) are indicated. The symbols indicate di!erent values of R. From Luijten et al. [45]. (c) E!ective exponent  for the d"2 Ising model for ¹'¹ , as a function of tR, comparing results   of Luijten et al. [46] with results from a systematic perturbation expansion around the mean-"eld limit. Pluses, crosses, squares and diamonds correspond to data for R "10, 72, 140 and 1000, respectively. From Pelissetto et al. [124]. K

the detailed shape of K(r) can be refuted } although this "nding clearly cannot rule out that other parameters that induce nonuniversal variations of the scaling functions might exist. 5.3. Numerical results in d"3 dimensions and comparison with theoretical predictions The analysis of Luijten and Binder [48,49] in the case d"3 closely followed the procedures that already had been applied earlier in the case of d"2, and thus we keep the description of the results rather brief. The "nite-size crossover scaling variable is now ¸/R\B"¸/R [cf. Eqs. (5.16)}(5.18)]. Thus one has to reach the regime where ¸/R<1 and at the same time ful"ll the requirement R<1 in order to span the full crossover region, and these simulations are therefore computationally quite demanding. In fact, simulations have been carried out for linear dimensions up to ¸"200, i.e., 8 million lattice sites, attempting to nevertheless obtain Q (K ) with a precision *  of one part in a thousand.

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Fig. 15. Finite-size crossover scaling curves for d"3 dimensions at ¹"¹ (R) vs. ¸/R for (a) Q, (b) m¸/C[m] and  (c) ¸\/C[ ]. Systems with up to q"52 514 interacting neighbors have been included. From Luijten [49].

Again Q (K) was "tted to Eq. (2.63), using now the value Q"0.6233 (4) as an input (see * Section 3), as well as the exponents 1/"1.587 (2) and "0.82(6), as obtained from a similar data analysis for the nearest-neighbor model. The resulting values for K (R) are compatible with an  expression expected from renormalization-group arguments, c c c ln R c #2 qK "1#  #  # >   R R R

(5.26)

In comparison with the corresponding result for d"2, Eq. (5.20), the logarithmic correction appears in rather high order only, and hence the resulting coe$cient c (as well as the coe$cients of  the neighboring orders c and c ) are di$cult to determine. Only c "0.498 (2) is known to a high    precision [49]. Fig. 15 shows the analog of the plots in Fig. 10 for the "nite-size crossover scaling. One observes that the crossover spans at least four decades in the variable ¸/R. Correction terms of the form C[m]"1#aR\, C[ ]"1#bR\#cR\ were applied to correct for residual "nite-range

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Fig. 16. Log}log plot of the scaled susceptibility, G, vs. the scaled temperature distance from criticality, t/G, for the three-dimensional equivalent-neighbor model and temperatures ¹'¹ . Di!erent symbols show di!erent interaction  ranges R, as indicated in the key of the "gure. The `Ginzburg numbera G is de"ned in Eq. (5.28). The crossover functions due to Belyakov and Kiselev (BK) [119] and Bagnuls and Bervillier (BB) [112] are also included, but are indistinguishable from each other (and from the numerical data) on the scale of this graph. The asymptotic power laws in the mean-"eld limit ( "1/t) and in the Ising limit ( ">t\A, with "1.237 and >"1.1025, cf. text) are included as well (dotted straight lines). From Luijten and Binder [48].

e!ects. Coordination numbers from q"18 to q"52 514 were included in Fig. 15. The straight lines marked `Isinga in parts (b) and (c) have the theoretical slopes y !9/4"0.2315 and F 2y !9/2, respectively. The amplitudes of the horizontal mean-"eld asymptotes in these plots are F known exactly. For RPR one has m¸"12

¸\"(12

(1/2) +0.909891 , (1/4)

(3/4) +1.170829 , (1/4)

(5.27)

which agrees well with the data. Fig. 16 presents the thermal crossover for the susceptibility for ¹'¹ . In this "gure, the  reduced susceptibility  was de"ned as (¹ (R)/¹)¸m, so that in the mean-"eld limit one simply  has "t\, while for the nearest-neighbor Ising model ">t\A where "1.237 [42] and >"1.1025 [58] have been taken. From Eq. (5.22) we notice that the crossover scaling variable is tR in d"3 dimensions. When discussing the universality of the crossover scaling description, a suitable amplitude factor for this crossover scaling variable must be taken into account. In order to make contact with experimental analyses of crossover phenomena, we introduce the notion of the `Ginzburg numbera G [19,109]. Writing the phenomenological Ginzburg criterion in terms of the mean-"eld power laws for the order parameter (m"B (!t)), the susceptibility +$ ( "> t\) and the correlation length ("> t\), mean-"eld theory is valid for t ) +$ [v /(> )]"G R\ . (5.28) G"  4 (B )  +$ +$




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Fig. 17. (a) E!ective exponent > of the susceptibility of the three-dimensional equivalent-neighbor Ising model above  ¹ plotted vs. the logarithm of the crossover scaling variable t/G, along with three theoretical calculations, obtained from  the work of Seglar and Fisher (SF) [110], Bagnuls and Bervillier (BB) [112] and Belyakov and Kiselev (BK) [119], as described in the text. (b) Same as (a) but for the exponent \ for ¹(¹ . `Appa denotes the approximation discussed in   the text. From Luijten and Binder [48].

Here v is the unit of volume (v "1 in the Ising model, if length is measured in units of the lattice   spacing). For the gas}liquid transition or for unmixing transitions in mixtures of small molecules, v is the volume per molecule and in the case of polymer mixtures v is interpreted as the volume   per monomer [132}141]. Since in the mean-"eld theory of an equivalent-neighbor Ising model > JR and the other amplitudes occurring in Eq. (5.28) do not depend on R, the simple power +$ law G"G R\ is obtained. In many experimental studies one actually does not attempt to extract  all the amplitudes in Eq. (5.28) from experimental data, but often simply takes G as an adjustable constant [141]. In principle, the crossover scaling description of Eqs. (5.21) and (5.22) is only expected to hold in the limit tP0, RPR, with tRB\B"tR "nite. However, Fig. 16 shows that it is even possible to approximately represent data for all R down to R"1 by a single master curve. This is possible because the amplitude G +0.1027 in Eq. (5.28) was chosen such that the scaling function  of Ref. [119] precisely reproduces the amplitude >"1.1025 for R"1 [48]. Also a scale parameter in the theory [112] included in Fig. 16 was adjusted such that the prefactors of the two asymptotic power laws are reproduced. One should not be misled by the apparently perfect agreement between these theories and the simulation results, however: Since the ordinate scale spans 10 decades, systematic deviations in the crossover regime cannot be detected here. Therefore it is again important to also consider the variation of the e!ective exponents > and \ , as   was done already in the case d"2 (Fig. 17). While in the direct representation of the susceptibility data (Fig. 16) the crossover seems to be rather sharp, the plot of the e!ective exponent shows that also in d"3 dimensions the crossover is actually spread out over many decades in the scaling variable t/G. Several theoretical descriptions have been included in Fig. 17a. The "rst one is based on an extrapolation of a "rst-order
-expansion [110], which we write as > "1#(!1)/1#exp[ ln(ct/G)] ,  

(5.29)

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where the constant c is an ad hoc "t parameter introduced in [48] to "t the initial rise of > with  decreasing log (t/G). The second curve (labeled as BB) results from a renormalization treatment  [112] and also involves a single adjustable parameter for the abscissa scale. Given the great success of this technique in accurately predicting the Ising critical exponents (see Table 3, Section 3), one expects that this theory provides the most accurate description in the limit where GP0 but t/G remains "nite. The third description is another phenomenological generalization of "rst-order
-expansions [119], which is now very popular with experimentalists (e.g., it has been used to analyze crossover scaling phenomena in polymer mixtures [140,141]), and hence we describe it here in some detail. The susceptibility is written as the solution of the following implicit equation t/G"[1#( G)FA]A\F( G)\#[1#( G)FA]\AF ,

(5.30)

where +2.333 is a universal constant, and "+0.508 (25) [43] is the critical exponent of the leading correction to scaling. As mentioned above, G "0.1027 must be chosen in order to  describe the Ising asymptote in Fig. 16, while the theoretical value [119] for G in our model would  be G "27/+0.27718. We shall return to this problem below. Here we only emphasize that all  three theoretical formulas as well as the Monte Carlo data imply that in the symmetric phase (¹'¹ ) the variation of the e!ective exponent with t/G is monotonic, in contrast to a conjecture  of Fisher [115] who suggested that a nonmonotonic variation of > might be a property of the  universal scaling function. However, the data in the phase of broken symmetry (¹(¹ , Fig. 17b)  show that here \ stays close to the mean-"eld value  "1 over a much more extended region of  +$ log (!t/G), namely from large values of log (!t/G) down to about log (!t/G)+1, followed    by a rather sharp rise of \ , similar to the situation for ¹(¹ in d"2 (Fig. 14a). In fact, a very   small underswing (\ ( "1) cannot be ruled out for the data near log (!t/G)"1. This  +$  would then be a precursor of the pronounced underswing found in d"2 (Fig. 14a). Later systematic expansions around mean-"eld theory [124] have in fact predicted such a slight underswing (Fig. 18). Unfortunately, the theoretical predictions for \ are rather scarce: The "eld-theoretical calcu lations [298] have only been formulated for relatively small values of t/G and hence do not cover the entire crossover region. Motivated by the success of Eq. (5.29) for ¹'¹ , Luijten and Binder  [48] used a similar, but now purely phenomenological, expression for ¹(¹ ,  \ "1#(!1)/[1#exp(ln c t/G)] , (5.31)  where c is another adjustable constant. This approximation happens to "t the numerical results fairly well (Fig. 17b), although it does not yield the above-mentioned `underswinga predicted by Pelissetto et al. [124]. We now turn to the interpretation of the systematic deviations between the numerical data for > (Fig. 17a) and all the theories, occurring near the Ising limit: Does this systematic deviation  mean that the theories fail to predict the universal crossover scaling limit (tP0, GP0, t/G "nite) for small t/G (small tR)? Actually we believe that this is not the case, but that the discrepancies are caused by the inclusion of too small values of R (such as models with interactions between only nearest (R"1) or nearest and next-nearest (R"1.3) neighbors, for instance). In fact, in Figs. 10, 12 and 15 it was already emphasized that corrections due to the "nite range R are required in order to obtain valid data in the crossover scaling limit. Of course, the quantitatively accurate estimation of

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Fig. 18. E!ective susceptibility exponent as a function of tR for the high and low-temperature phase of the three-dimensional Ising model. The curves are the result of a systematic perturbative expansion around the mean-"eld limit. From Pelissetto et al. [124].

such correction terms is a delicate matter, and hence residual systematic errors must be anticipated for too small R. This conclusion is also corroborated by the systematic perturbation expansions around the mean-"eld limit (which is exact for RPR at "xed tO0) [124]. This work shows that correction terms to the crossover scaling limit appear in various quantities and scale to leading order like R\ ln R for d"2 and R\ for d"3, respectively. Hence, these corrections have the same form as the deviations from the mean-"eld result for the critical temperature [cf. Eq. (5.20) and (5.26), respectively], as found from renormalization-group arguments [49,92]. Further evidence that the Monte Carlo data for small t/G in Fig. 17a do not re#ect the behavior of the universal crossover scaling limit comes from analysis of these data [299] in terms of a recent phenomenological two-parameter description of crossover scaling [120,121] that has proven very useful to account for various experimental data of #uids and binary #uid mixtures [142,145,146,300,301]. This description was derived from a renormalization-group matching for the free-energy density, and although it is based on a "rst-order
"4!d expansion, using the correct exponents of the d"3 Ising model instead of their values in "rst order in
makes this description a #exible and useful interpolation scheme. It contains two crossover parameters, a parameter u related to the coe$cient u of the quartic term in Eq. (5.5) and a parameter  which is essentially a large-momentum cuto!, both of which are system dependent. Then there exist nonuniversal scale factors c , c in the inverse square correlation length \ and the inverse R M susceptibility \, which are both written in terms of a crossover function >(u , ): \"c t>J\F , R

(5.32)

1!(1!u )>"u [1#()]>JF ,

(5.33)

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¹ uH (1!u )>

\"cc  t>A\F 1# 2[()\#1] /# !(2!1)/ M R¹ 2 1!(1!u )>

\

.

(5.34) Here uH"0.472 is the universal coupling constant at the RG "xed point [302]. In the approximation of an in"nite cuto! PR, which physically corresponds to neglecting the discrete structure of matter, u P0 and the two crossover parameters u ,  in Eqs. (5.32)}(5.34) collapse into a single one, u , which is related to the Ginzburg number G by G"g (u )/c where g +0.028 is  R  a universal constant [120]. In this limit Eqs. (5.32)}(5.34) reduce to Eq. (5.30). This single-parameter crossover is universal, and has been calculated by powerful "eld-theoretic methods by Bagnuls and Bervillier [112], as noted above. Pelissetto et al. [124] have investigated the numerical accuracy of Eqs. (5.32)}(5.34) [299]. The single-parameter scaling (u "0 reduces to the curve labeled BK in Fig. 17a) does not "t the data for small R , and one rather needs u "1.22 to "t the data for R "1. K K As an interpolation, we have used u "u R\ with u "1.22, and " for a three-dimensional   Ising lattice. Since the e!ective exponent > (u , , t) is a nontrivial function of three parameters  [121] the di!erent choices of R (or R) corresponding to di!erent choices of u do not lead to K a single function > (tR) but to a whole family of functions (see Fig. 19). The choice of u "1.22   implies a variation of the Ginzburg number G"G>R\ with G>+0.24, qualitatively consistent   with Eq. (5.28). However, a similar "t of \ implies that for ¹(¹ G"G\/G>"2.58, which     appears at variance with the theoretical result G\/G>"3.125 [303].   While it is gratifying that the same crossover scaling model [Eqs. (5.32)}(5.34)] can describe both the equivalent-neighbor Ising model and various experimental systems (although we note that this model is essentially of a phenomenological nature!), it is clear that some problems regarding the quantitative accuracy of both the model [Eqs. (5.32)}(5.34)] and the Monte Carlo results [48] still need to be resolved. Of course, the problem that the simulations cannot easily reach the crossover scaling limit (tP0, GP0, t/G "xed) for small t/G is also shared by many experiments, where typically G cannot be varied at all. Thus, it is of interest to compare the Monte Carlo results directly to the experiments. This has recently been performed for both Xe and He [304], and for small t/G very good agreement between experiment and the Ising model simulations has been found. Using data for the compressibility above the gas}liquid transition and for the coexistence curve below it, the Ginzburg parameter G (or G , respectively) was used as a single adjustable > \ parameter, resulting in G (Xe)"0.07 (2), G (He)"0.070 (8), G (Xe)"0.018 (2), G (He)" \ \ > > 0.0025 (10). The compressibility data for He strongly deviate from the crossover scaling function for t/G>910, unlike the Xe data. This distinction was tentatively attributed to quantum e!ects [304]. Further problems with the theoretical interpretation of experimental crossover scaling data occur for polymer mixtures [140,141], where one does not "nd the theoretically predicted [125}131] behavior GJN\ where N is the degree of polymerization of the polymers. In this case, the discrepancy is attributed to a pressure dependence of the e!ective interaction parameter not taken into account in the theories. This problem also deserves further investigation. In any case, it is clear that quantitative studies of crossover phenomena from the Ising universality class to mean-"eld behavior by means of either simulations or experiments have only started a few years ago, and much more work remains to be done.

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Fig. 19. The e!ective susceptibility exponent > for the three-dimensional variable-range Ising model and ¹'¹ . The   symbols indicate numerical simulation data for various choices of R , as indicated (for clarity, the error bars have been K omitted; they are all of the order of 0.004). The solid curves were calculated from Eqs. (5.32)}(5.34), using u "u R\ with  u "1.22. The dotted curve corresponds to the alternative choice u "1.22, while the dash-dotted curve refers to the  crossover scaling limit. Note that "tting the data for the di!erent choices of R separately yields di!erent cuts of a twoK dimensional surface and thus one does not "nd a unique solid curve but a whole family of curves, which merge on the curve for u "0 for large enough R . From Anisimov et al. [299]. K

6. Algebraically decaying interactions 6.1. Overview In addition to the dimensionality, the range of the interactions is one of the few parameters that in#uence the universal critical properties of Ising spin models. A natural extension of the "niteranged interactions considered hitherto are interactions that decay as a power of the distance between the spins. Following "rst calculations for the spherical model by Joyce [305] and a number of rigorous results for the one-dimensional Ising model [306}308] in the 1960s, this case was considered within the framework of the renormalization-group (RG) theory in the seminal work by Fisher et al. [105]. As one of their central results emerged an explicit dependence of the upper critical dimension on the decay power of the interactions. In the following decades, a limited amount of numerical work was performed, almost exclusively restricted to the case d"1 and mainly concerned with the calculation of the (nonuniversal) critical temperature as a function of the power-law decay (cf. the references cited in Ref. [47]). Also on the analytical side progress focused on the one-dimensional case, in particular on the pivotal case with inverse-square interactions [309,310]. Obviously, the study of these systems is greatly complicated by the long-ranged nature of the interactions. As far as numerical approaches are concerned, this leads to prohibitively large computational requirements, restricting practical calculations to very small systems. Fortunately,

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for the case of arbitrary ferromagnetic interactions this bottleneck could be resolved by the construction of a novel Monte Carlo algorithm which has an e$ciency that is independent of the number of interactions per spin [250]. Indeed, this algorithm has allowed a major step forward in the numerical treatment of several outstanding problems in the "eld of long-range interactions. For a detailed description of the algorithm the reader is referred to Refs. [250,251]. 6.2. Renormalization-group predictions Let us "rst give a brief summary of the RG predictions as obtained in Ref. [105]. We employ here the standard notation J(r)"r\B>N for the spin}spin interactions. The corresponding Landau} Ginzburg}Wilson (LGW) Hamiltonian in momentum space then takes its standard
 form, except that the k term resulting from the Fourier transform of the (
) term (representing the short-range interactions) is replaced by a term kN (we ignore here additional logarithmic factors appearing for integer ). It follows then that the upper critical dimension is given by d "2. For  smaller values of  (more slowly decaying interactions), the critical behavior is mean-"eld-like and the critical exponents take their standard classical values. An exception to this are the correlationfunction exponent "2! and the correlation-length exponent "1/. For 'd/2, the exponents become continuous functions of  and can be calculated by means of an
expansion, where
"2!d [105]. There are strong indications, however, that for the exponent all correction terms vanish identically, such that also in the nonclassical regime is given by 2!. This places us in the particular situation that we can study the critical properties of systems for which (a) one of the two independent exponents is (presumably) known exactly and in addition (b) the RG predictions can be veri"ed by numerical means for arbitrarily small
, since
is a continuous parameter even if the dimensionality has to take integer values. So we can speak of a truly ideal testing ground here! At some border-line value of , the critical behavior changes to the standard short-range universality class. Originally, it was concluded from the RG calculations that this occurred at "2, where the kN term in the LGW Hamiltonian coincides with the short-range k term. This conclusion entails two remarkable implications. First, it would imply a jump discontinuity in the exponent as a function of , since lim

"0 and lim

" O0, Nt Ns  where is the exponent belonging to the corresponding short-range system. Secondly, the  one-dimensional case would not comply with this classi"cation, as rigorous results have shown that a phase transition is absent here already for '1 [306]. Subsequent work by Sak [106] appeared to have resolved this point, with a smooth crossover at "2! , but later studies  [311}313] have casted some new doubts upon this issue. Another area where RG predictions can be tested by means of numerical methods is the perturbative calculation of "nite-size scaling functions. For systems with short-range interactions, it was shown by BreH zin and Zinn-Justin [74] that an RG calculation of such functions is indeed possible below the upper critical dimension, although the resulting expansion in powers of
"4!d is a singular one. More speci"cally, the amplitude ratio Q,m/m, where m is the order parameter or magnetization density, is given by a Taylor series in (
. A veri"cation of this rather striking result has not been possible to date, as the calculation has been carried out only to second order in (
, i.e., corrections are of O(
), and the numerical results are obviously restricted to
"1, 2, 3. It is thus interesting to note that very recently the calculation of Ref. [74] has been

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generalized to periodic systems with an n-component order parameter and algebraically decaying interactions [314,315]. It could be shown that, upon replacement of the original expansion parameter (4!d) by (2!d), the singular nature of the expansion is preserved, i.e., Q is again given by a power series in (
. Explicit expressions for the coe$cients of the "rst two correction terms have been obtained, enabling a comparison to numerical results for small
. At criticality, the amplitude ratio is given by













() 1 () () 1 () 4()  (6x# 13  !2 6x#O(x) , #  1# 4  ! Q" () () 2 () () 16 ()     

(6.1)

with



1 n#2 x "(
 2 (3(n#8)





() I (2, , 0)#O(
) . N 

(6.2)

The integral I (d, d/2, 0), which has been indicated only symbolically here, can be evaluated  numerically, yielding !2.9207092, !3.9002642 and !4.8227192, for d"1, 2, 3, respectively [314]. 6.3. Numerical results for the critical exponents Since there exist very few systematic, precise comparisons between RG predictions for longrange systems and corresponding numerical results beyond those obtained by means of the Monte Carlo method introduced in Ref. [250], we will in this section exclusively concentrate on the latter. The numerical results can be divided into two regimes: (i) 0(4d/2, where (up to logarithmic corrections) classical critical behavior is expected and (ii) 'd/2, where one expects nonclassical behavior. Regime (i) has been investigated in Ref. [47], where accurate results have been presented for systems containing up to 300 000 spins, with d"1, 2 and 3. The "nite-size analysis concentrated on the magnetization density, the magnetic susceptibility and Q. From the former two quantities, the renormalization exponents yH and yH were determined, where the asterisk indicates that these R F exponents are modi"ed due to the so-called dangerous-irrelevant-variable mechanism. Within the numerical accuracy, the exponents agreed, over the entire  regime and for all three lattice dimensionalities, with the predicted values yH"d/2 and yH"3d/4. Even at the upper critical R F dimension, these values could be con"rmed to within 1}2 parts in a thousand after imposing the predicted logarithmic factors in the analysis. Obviously, these systems also provide an excellent way to study the behavior of Q above the upper critical dimension. Indeed, the so-called zero-mode value predicted in Ref. [74] could be con"rmed with considerable accuracy over a wide range of values for  and d [99], thus lending strong support to the expectation that the cumulant takes this value over the entire classical regime. Finally, Ref. [47] also demonstrated that essentially two types of behavior can be distinguished for the spin}spin correlation function g(r) in "nite systems: at short distances (r/¸;1) it decays according to the predicted value for , i.e., like r\B\N, to be compared to a decay r\B>N for the spin}spin interactions. If one, however, considers g(r) at r"¸/2, one essentially studies the k"0 mode and g(r)J¸\B for all 0((d/2.

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Fig. 20. Thermal exponent for the one-dimensional Ising model with algebraically decaying interactions, as a function of the decay power. From Ref. [289]. Both the agreement with the second-order
expansion and the approach of the Kosterlitz}Thouless transition at "1 can be clearly observed.

The "ndings for regime (ii), which are at least as interesting, are unfortunately almost exclusively available in Ref. [289, Chapter 5]. Here, we concentrate on the cases d"1 and d"2. Fig. 20 shows the Monte Carlo results for the thermal exponent as a function of , together with
expansion of Ref. [105]. Up to "0.75 (
"0.5) the agreement is most satisfactory, where it is stressed that the RG curve contains only terms up to second order in
, without the application of any series resummation. For higher values of , the numerical values exhibit a rapid decay toward zero, in agreement with the presence of a Kosterlitz}Thouless transition at "1 [316}318] and the consequential absence of an algebraic temperature dependence of the correlation length. We have also included an expansion for y around "1 [318] in Fig. 20 and the numerical data indeed R appear to approach this curve for su$ciently large . An interesting implication of the displayed behavior of the thermal exponent has been pointed out in Ref. [319]: since y ' only for R  (:0.65, the speci"c heat will consequentially only diverge for this part of the nonclassical  regime and display a cusp singularity for larger values of . (In the classical regime, the speci"c heat also exhibits a cusp singularity, but of a di!erent nature, cf. Ref. [319].) The corresponding graph for the magnetic exponent (as obtained from a "nite-size scaling analysis of the magnetic susceptibility) is displayed in Fig. 21. As can be seen, the Monte Carlo data follow the predicted dependence y "(d#)/2 (corresponding to "2!) very closely: for "0.95, the relative deviation lies F below one part in a thousand. The analysis of the two-dimensional system is more involved, due to the crossover to short-range critical behavior and the unknown nature of the corrections to scaling. Both at the upper critical dimension ("1) and in the short-range regime, the thermal exponent is equal to unity. In the intermediate long-range regime, the
expansion suggests (like for d"1) an initial increase of y as a function of , which is consistent with the trend exhibited by the numerical R results, although these mostly lie within one standard deviation from unity. The behavior of the magnetic exponent o!ers the opportunity to determine the location of the transition between the intermediate long-range critical regime and the short-range critical regime. Postponing a detailed

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Fig. 21. Magnetic exponent for the one-dimensional Ising model with algebraically decaying interactions, as a function of the decay power. From Ref. [289]. The linear behavior of the numerical data in the nonclassical regime (41 lends  strong support to the conjectured exactness of the zeroth-order RG result.

discussion to a later work [320], we only mention here that the numerical results strongly suggest a crossover at "2! , i.e., at " for d"2.   6.4. Finite-size scaling functions The Monte Carlo simulations in regime (ii) also permit an accurate determination of the amplitude ratio Q as a function of . In order to provide a reference frame, Fig. 22 shows known results for Q in periodic linear, square and cubic Ising systems, along with the "nite-size scaling function calculated in Ref. [74]. Clearly, it is impossible to draw any conclusion on the singular nature of the
expansion from this graph. Figs. 23 and 24 then show Q as a function of  for d"1 and d"2, respectively, where also the corresponding values of
have been indicated along the upper horizontal axis. While a very close approach of the upper critical dimension is hampered by the corresponding strong corrections to scaling, the numerical results clearly deeply penetrate into the regime where the convergence of the
expansion does not have to be doubted. In the same "gures, the second-order (
expansion of Eqs. (1) and (2) has been included. The agreement is obviously extremely poor: the numerical results rather fall strikingly well onto a straight line, with a weak deviation upon increasing
. Let us point out two important indications for the consistency of the numerical data: for d"1 the order-parameter jump implied by the Kosterlitz}Thouless transition at "1 leads to Q"1, in full agreement with the trend exhibited in Fig. 23. For d"2, Q+0.856 in the short-range regime, in good agreement with the observed value at the transition point "1.75. The source of this discrepancy has not been identi"ed yet. While a slow convergence of the
expansion appears as a natural suggestion, it is pointed out that marked deviations already occur for values of
as low as 0.2 and that the relative size of the coe$cients appearing in

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Fig. 22. The amplitude ratio Q"m/m at criticality for periodic, short-range Ising models in d"1, 2, 3, along with the singular "nite-size scaling function obtained by means of the
expansion. For the numerical results, the symbol size by far exceeds the uncertainty. Fig. 23. The critical amplitude ratio Q(¹ ) for the one-dimensional Ising model with algebraically decaying interactions  together with its second-order (
expansion, as a function of the decay power. The corresponding values for
are indicated along the upper horizontal axis. From Ref. [314].

Fig. 24. The analog of Fig. 23 for the two-dimensional long-range Ising model. From Ref. [314].

Eq. (1) does not hint at unusually strong higher-order corrections. In addition, it is not obvious how such correction terms would largely compensate the singular contribution of the "rst-order term. In any case, an explicit calculation of the correction terms to O(
) and O(
), respectively, seems highly desirable in order to shed some light onto this unexplained discrepancy.

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7. The interface localization transition in Ising 5lms with competing walls 7.1. A xnite-size scaling study In the previous sections, ferromagnetic Ising systems in various dimensionalities have been considered, and di!erent choices for the interaction range were treated, but bulk homogeneous systems were studied exclusively. Although also in these cases there are some open questions, we feel that the basic aspects of critical phenomena in these systems are well understood. The situation is rather di!erent, however, when we consider inhomogeneous systems con"ned by walls. Many new phenomena can arise because surface magnetic "elds may act at the walls, exchange constants near walls may have values di!erent from those in the bulk, etc. Here we are not at all aiming at an exhaustive review of such phenomena, but only treat one case which provides a good example of the wealth of new physics that is involved. This case concerns a nearest-neighbor Ising ferromagnet on the simple cubic lattice in an ¸;¸;D geometry, where periodic boundary conditions act in the x and y direction, while one chooses two free ¸;¸ surfaces in the z-direction, where surface "elds H , act in the layer n"1 and H "!H in the layer n"D [148}159],  "  (7.1)
S . H"!J
S S !H
S !H G G " G H  GZ  6GH7 GZ  L" L In the limit ¸PR, for any "nite D the transition from the disordered phase for temperatures ¹ above the bulk critical temperature ¹ to the ordered phase at ¹(¹ is a gradual, rounded 

transition without any singular behavior, although the system is already in"nite in two spatial directions. This smearing of the transition happens because for ¹:¹ the state of the system is 
characterized by the presence of an interface which runs parallel to the wall and has an average position in the center of the "lm (Fig. 25). For ¹<¹ due to the action of the surface "elds there is 
already a region of thickness  (the correlation length in the bulk) adjacent to the left wall where
the local magnetization m(z) is positive, and similarly in the region D! we have m(z)(0. Near
¹ , when D becomes comparable to 2 , these two regions start to interact, and a pro"le m(z) 

develops, which decays smoothly from m(z"0) near the bulk magnetization m toward m(z"D)
near !m . Although m(z)"0 only for z"D/2, in the center of the "lm, and we have (for ¹:¹

and large D) a coexistence of two oppositely oriented domains with magnetizations $m
separated by an interface, the thin "lm as a whole still has zero magnetization, since "m(z)dz"0  when H "!H . "  However, at some lower temperature ¹ (D), which does not converge to ¹ when DPR,  
a phase transition does occur where the interface is either bound to the left wall (then m (0)  or the right wall (then m '0), cf. Fig. 25. The spontaneous symmetry breaking that occurs  here involves an one-component order parameter, and if a continuous transition occurs it should fall in the universality class of the two-dimensional Ising model. If one studies this transition with a "nite-size scaling analysis of Monte Carlo data, applying the most straightforward version of the techniques described in Section 2.4, where one attempts to "rst locate ¹ by looking for cumulant  crossings (analogous to Fig. 2), one experiences a bad surprise (Fig. 26): Even on a very coarse scale of inverse temperature, there is a distinct spread of cumulant crossings, and thus the extrapolation of these crossing points toward ¸PRcan yield only a rather crude estimate of J/k ¹ in a plot vs. 

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Fig. 25. Schematic description of the interface localization}delocalization transition of an Ising model con"ned between two walls a distance D apart, where at one wall (left side) a positive "eld H acts, while on the other wall (right side)  a negative "eld H acts. For ¹(¹ (D) the interface is bound either to the left or the right wall, and then the average "  magnetization m of the "lm is nonzero. For ¹ (D)(¹(¹ , however, the interface #uctuates in the center of the   
"lm, and thus m "0, although there is still a nonzero bulk magnetization $m in an in"nite system, as well as 
  locally in the "lm away from the interface. For ¹'¹ , however, the "lm is disordered (apart from the regions near the 
walls, where due to the response of the system to the surface "elds there occurs a local nonzero magnetization). The description of the interface in terms of a coordinate z"l(
) with
being the x, y coordinates in the plane formed by the left wall is also indicated. From Binder et al. [157].

1/¸ (Fig. 27). The alternative method (shown in the same "gure) of extrapolating the locations K (¸) of the maxima of speci"c heat (C ), susceptibility ( ), etc. is not at all better. It is also





 very disturbing that the ordinates of these cumulant crossings (; ) are far o! the theoretical  value (;H"0.6107 in the d"2 Ising model [296], although they possibly converge to the correct value as ¸PR(at least for D"6 and D"8 this seems plausible). Problems are also encountered for the critical exponents: testing for the law (K (¸)), J¸AJ"¸  (Fig. 28) we "nd



 rough agreement with the predicted power law for D"6, while systematically smaller e!ective exponents occur for D"8 and D"12; note that this analysis is not a!ected by the uncertainty in "nding the correct K (D). And the thermal "nite-size scaling, as predicted in Eq. (2.40),  m¸@J"M (¸Jt) [m¸"M (¸t) in our case] works only roughly for D"6 and if we   restrict ourselves to ¸5128 (Fig. 29). In fact, despite considerable e!ort (for ¸"256, D"6 we have in total 393 216 spins in the system, and the relaxation time at criticality is expected to be of the order J¸ +1.6;10 Monte Carlo steps/site) a data collapse is obtained (Fig. 29) which only has a quality comparable to that of ¸;¸ square lattices with ¸"20 to 40, which would be a student's exercise for a small personal computer.

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Fig. 26. Cumulant ; plotted vs. J/k ¹, for the case H /J"0.55 and "lm thickness D"6 (a) and D"12 (b), and *  several choices of ¸, as indicated. Arrows in (a) indicate the locations of cumulant crossings of two neighboring lattice sizes ¸, ¸ , while the arrow in part (b) indicates the location of the transition temperature of the wetting transition ¹ at  the surface of a semi-in"nite system. From Binder et al. [156].

7.2. Phenomenological mean-xeld theory and Ginzburg criteria The data shown in Figs. 26}29 already point to the fact that the critical region is very narrow due to crossover phenomena. The theoretical interpretation of these crossover phenomena can in fact be provided by the concept of the e!ective interface Hamiltonian H (l) [160}162,164}167], 

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Fig. 27. Cumulant crossing values ; vs. ¸\ for D"6, 8 and 12 (upper part). The arrow shows the value of the  two-dimensional Ising universality class, and the curves are only guides to the eye. The lower part shows, for D"6 only, the extrapolations of inverse temperatures K (¸) of cumulant crossings as well as the inverse temperatures K (¸) 

 where susceptibility ( ) and speci"c heat (C ) have their maxima. The arrow with the error bar marks the "nal



 estimate of J/k ¹(D"6), while straight lines indicate possible extrapolations. From Binder et al. [156].

a concept which is a very popular starting point for the description of wetting phenomena, surface-induced disordering, surface melting, and related phenomena [167]. In its simplest version, H (l) is written in terms of a single collective coordinate l(
), the local distance of the interface  from the left wall at position
in the xy plane (Fig. 25),

 



 H (l)" d
(l)#l(
) .  2

(7.2)

Here a factor of (k ¹)\ is absorbed in the Hamiltonian throughout, and we have assumed that the interfacial sti!ness  is a constant (rather than considering a dependence of  on the distance l of the interface from the wall, (l) [171,172,179,180]). For short-range forces due to the walls, such as

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Fig. 28. Log}log plot of ( J) "J (K (¸)) vs. ¸ for D"6, 8 and 12. Straight line shows the theoretically predicted



 slope /"1.75, while broken curves are guides to the eye only. From Binder et al. [156].

the surface "elds H , H [Eq. (7.1)], the e!ective potential is phenomenologically assumed as  " follows [152]: (l)"!
a [e\Gl#e\G"\l]#b [e\Gl#e\G"\l]!h(l!D/2), h,2m H , 

,(¹ !¹)/¹ . (7.3)   Here a , b are constants,
is the normalized distance from a second-order wetting transition   which would occur at the temperature ¹ in a semi-in"nite system, and \ is a transverse length  scale associated with the interface (if one derives Eq. (7.3) approximately from a Ginzburg}Landau theory for the magnetization pro"le m(z) across the "lm, \" is obtained [152,171,172]). The
last term on the right-hand side of Eq. (7.3) accounts for the e!ect of a bulk magnetic "eld H, which is included here for convenience. Apart from the questionable approximations that yield the explicit form (7.3) of the interface potential (l), Eq. (7.2) has also been questioned on more general grounds, suggesting that one needs (at least) a second collective coordinate l (
), which is related to #uctuations of the local  magnetization m(z) very close to the wall [181}184]. It is argued that the coupling between l(
) and

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Fig. 29. Log}log plot of m¸ vs. ¸t for D"6. Three choices of ¸ are included. Data for ¹(¹ (D) are shown by  open symbols, data for ¹'¹ (D) by full symbols. The straight lines through the data were "tted using the correct  exponents for order parameter ("1/8) and susceptibility ("7/4) [recall that for ¹'¹ (D) we have [31]  mJ(mJ(k ¹ /¸) in d"2 dimensions, and hence ¸mJ(t¸)\]. From Binder et al. [157].

l (
) has important e!ects, and when one approximately integrates out l (
), one e!ectively   obtains Eqs. (7.2) and (7.3), but \" is no longer true and one has instead [181}184]
\" (1#/2) ,


(7.4)

where "[4]\ is the universal constant entering the theory of critical wetting
[160}162,164}167]. At this point, we only aim at a description of the interface localization transition in terms of the most simple mean-"eld theory. In this spirit, #uctuations of the interface position, included in Eq. (7.2) via the (l) term, are neglected altogether. Hence (l) is treated as an e!ective free energy function, which simply needs to be minimized in order to "nd the average position of the interface l in thermal equilibrium. For zero "eld h"0 and ¹'¹ all terms in Eq. (7.3) are positive, and   hence (l) assumes its minimum for l "D/2, i.e., we are in the `delocalized phasea. For ¹(¹ ,   however,
'0 in Eq. (7.3), so that the terms within the "rst pair of square brackets compete with those within the second pair and a solution l OD/2 is expected, i.e., a nonzero order parameter.  For h"0 the equilibrium condition hence yields



R(l) "
a \Gl!e\G"\l]!2b \Gl!e\G"\l] 0"   Rl l l 

(7.5)

and setting l "D/2# one obtains an equation for the order parameter ,  cosh()"
a exp (D/2)/4b .  

(7.6)

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For X";1, the expansion cosh x+1#x/2 yields 4b exp(D/4) (!t) , !t,
!  exp(!D/2) . (7.7) X"$ a (2b /a    Here t,[¹ (D)!¹]/¹ is nothing but the reduced distance from the transition temperature in   the thin "lm. It is seen that ¹ (D) di!ers from ¹ only by terms exponentially small in D/2 [152].   The order parameter m (i.e., the magnetization) is then given by this mean-"eld (MF) theory  as follows: m+$ "2
/D"$(2a /b m exp(D/4)(!t)/(D) . (7.8)  K  
Thus one sees that the order parameter satis"es the standard mean-"eld power law, m+$ "B (!t), but the amplitude is not B "(3 as in the molecular-"eld approxima +$ +$ tion of a ferromagnet, but scales exponentially with the "lm thickness: B Jexp(D/4)/(D) . (7.9) +$ De"ning the response function for nonzero h as ,(Rl /Rh) and noting that Eq. (7.3) also  2" implies "1/(R(l)/Rl)l l , we obtain from Eq. (7.3) for t'0  R "2 exp(!D/2)[!a
#4b exp(!D/2)] , (7.10)   Rl l





where on the right-hand side l "D/2 has already been inserted. From Eq. (7.7) we see that  R/Rll vanishes at t"0, as expected. Thus

"exp(D/2)t\/(2a ) , (7.11)  and noting that "Rm /RH"4m /D one gets 


"> t\, > "(2m/a ) exp(D/2)/(D) . (7.12) +$ +$ +$
 Finally, the correlation length  for #uctuations of the order parameter is computed. In the , framework of Eqs. (7.2) and (7.3), these #uctuations arise from #uctuations of the local interface position around its mean value, l(
)"l(
)!l . From a quadratic expansion of Eq. (7.2) around  l"l it follows that the problem is formally analogous to the treatment of correlations in the  standard Ginzburg}Landau theory. Thus the correlation function G (
!
)" l(
) l(
) has , the standard Ornstein}Zernike form, with a correlation length  related to the coe$cient /2 , of the (l) term in Eq. (7.2) and the second derivative R/Rl l in the standard way, \"(R/Rl l )/, and hence ,  "> t\, > "(/2a ) exp(D/4)/ . (7.13) ,+$ +$ +$  The most interesting aspect of this mean-"eld theory is the unusual behavior of the critical amplitudes, which has the consequence that the regime where two-dimensional Ising critical behavior occurs is unusually narrow and hence mean-"eld theory is unusually good. This is seen when we work out the Ginzburg criterion, which states that mean-"eld theory is good if G;t;1

(7.14)

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where the Ginzburg number G for d-dimensional critical behavior is (suppressing prefactors of order unity) [321]





v \B  . G"(> )\BB\\B +$ +$ (> )B +$ Here the dimensionality of the problem is d"2, and so we have instead of Eq. (5.28)

(7.15)

G"> B\[v /(> )]Jexp(!D/2) . (7.16) +$ +$  +$ As a result, we predict that there is a crossover scaling similar to that described in Section 5.1, except that the parameter RB\BJG\ has to be replaced using Eq. (7.16). As an example, m "D\ exp(D/4)tM I [t exp(D/2)]  is the analog of Eq. (5.21) and the cumulant ratio at t"0 is

(7.17)

; ";[ [exp(D/2)/¸] , * analogous to Eq. (5.18).

(7.18)

7.3. Monte Carlo test of the theory First, the prediction that the critical amplitudes in the mean-"eld regime vary exponentially with "lm thickness D is considered. This is most easily studied for the susceptibility at temperatures ¹'¹ (D), Eq. (7.12). Noting that ¹ (D) di!ers from ¹ only by small terms proportional to    exp(!D/2), cf. Eq. (7.7), one can check Eq. (7.12) by simply studying (or related quantities referring to the layer magnetization m in the nth layer, such as "(Rm /RH) or " L L L 2 LL (Rm /RH ) , the response function to a "eld acting in the nth layer) at "xed temperature as function L L2 of D (Fig. 30). The theory outlined above implies J

J

Jexp(D/2), and such a behavior L LL seems indeed more or less compatible with the data. There is clearly some curvature present in these plots, and it would have been desirable to try to estimate these susceptibilities not only for the systematically investigated range 64D420, but for still larger thicknesses as well. This task would be very di$cult, however, because then also the lateral dimension ¸ must be substantially larger than the choice ¸"128 used in Fig. 30, and also much larger sampling times are necessary to avoid the bias shown in Fig. 1b. Thus the points for D"28 included in Fig. 30 may already be subject to rather large systematic errors. Nevertheless one can try to estimate  by considering ln(

)/D as function of D (Fig. 31). Rather LL clear evidence is obtained that the identi"cation "\ proposed by Parry and Evans [152] is
not correct; however, the idea of Parry and Boulter [181}184] that  e!ectively gets renormalized [Eq. (7.4)] is better compatible with the data (Fig. 32). This choice of  [from Eq. (7.4)] then also needs to be used in the crossover scaling relations Eqs. (7.17) and (7.18). The analysis shows (Fig. 33) that the surprising behavior of the cumulant ; (Figs. 26 and 27) and the maximum value * of the susceptibility (Fig. 27) can indeed be interpreted in terms of the predicted crossover

 scaling e!ects. However, unlike the data in Section 5 } which have bene"ted from a very e$cient novel Monte Carlo algorithm } here only data of a much poorer statistical quality are available, and systematic errors (e.g., due to the smallness of D) are clearly still present. In addition, only a small part of the crossover regime could be explored [the mean-"eld limit ;H"0.2704, cf.

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241

Fig. 30. Semilog plot of (a) the total susceptibility "Rm /RH and (b) the maximum value of the (symmetrized)   layer susceptibility vs. "lm thickness. Di!erent symbols show various inverse temperatures J/k ¹, as indicated in the L "gure. Straight lines indicate the predicted exponential variation with thickness, while the broken curve is only a guide to the eye. Linear dimension ¸"128 was used throughout. From Binder et al. [156].

Eq. (4.16), is beyond the scale of Fig. 33]. While the qualitative understanding of the interface localization transition in thin Ising "lms as provided by the simulations reviewed here and the pertinent theory is certainly encouraging, a quantitative understanding of the critical behavior of this transition and the associated crossover phenomena remains a challenge for the future. Note that the theoretical understanding of this transition is also rather rudimentary } it is just in the stage of mean-"eld theory and phenomenological Ginzburg criteria, while a renormalization approach only exists for critical wetting: for the interface localization transition between competing walls it remains to be developed!

8. Summary and outlook In this review we have summarized results on the static critical behavior of ferromagnetic Ising models as obtained from Monte Carlo simulations by various groups, and have contrasted them to pertinent theoretical predictions. Our emphasis has been on bulk properties; the critical behavior of

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Fig. 31. Plot of the logarithm of the maximum value of the symmetrized layer susceptibility [( ) "( # )/2, LL  LL LYLY where n "D!n#1], divided by the "lm thickness, as a function of D. Di!erent symbols show various temperatures, as indicated in the key. Curves are guides to the eye only, to illustrate that ln(

) /D actually approaches plateau values LL  for each temperature, which can be interpreted as (¹)/2 according to the theory [Eq. (7.12)]. However, the identi"cation (¹)/2"(2 )\ does not hold, as shown by the corresponding broken horizontal straight lines, that were calculated
from the series expansions of Liu and Fisher [58]. From Binder et al. [156].

the interfacial tension has only been mentioned in connection with pertinent universal amplitude ratios, and the e!ects of free surfaces have not been analyzed here at all. Both the critical behavior of free surfaces and the problem of critical wetting with short-range forces have been outside the scope of this review, although these topics still involve many fascinating and partially unsolved questions. We have also emphasized the approach of "nite-size scaling analysis of Monte Carlo data calculated in the (grand)-canonical ensemble as the appropriate technique to study critical phenomena by means of simulations. Other approaches exist in the literature (e.g., the analysis of critical phenomena in the micro-canonical ensemble, or the use of the initial behavior of critical relaxation to extract static critical exponents as well, etc.), but have been deliberately left out here. Despite the interest of having several di!erent approaches, we felt that these `o!-mainstreama approaches are not as thoroughly explored and have not yielded as accurate results as the methods described here. From the simulation data reviewed here, it is clear that with respect to bulk properties of Ising models with short-range interactions in various dimensionalities, accurate numerical data can be obtained by means of the single-cluster Wol! algorithm, allowing the extraction of estimates for the critical temperature, critical exponents and critical amplitudes with an accuracy that is already

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243

Fig. 32. Plot of the inverse length scale /2 vs. J/k ¹. Here full dots represent estimates of (2 )\,  being the true

correlation range in a lattice direction, obtained from the leading term of the PadeH approximant to the low-temperature series analysis of Liu and Fisher (LF) [58]. Open squares are corresponding Monte Carlo estimates of Hasenbusch and Pinn (HP) [278]. Open circles are the direct estimates of /2 extracted by Binder, Landau and Ferrenberg (BLF) [156], as shown in Fig. 31. The dash-dotted curve shows the suggestion of Parry and coworkers [181}184] that /2"[ (2#)]\, Eq. (7.4), using +0.86 [173] in the temperature region of interest. Arrows (with error bars) at the
abscissa show the location of ¹ (D) for D"12, 8 and 6, respectively. From Binder et al. [157]. 

better than the analysis of systematic high- and low-temperature series expansions. Universal properties (critical exponents, universal amplitude ratios) can be obtained by "eld-theoretic renormalization-group methods with a competitive accuracy: this method, however, does not yield the nonuniversal characteristics of the critical behavior at the same time. Furthermore, some properties that "eld-theoretic renormalization may yield in principle, like the crossover scaling function for the universal crossover limit (RPR, tP0, tRB\B "nite), are so far only available for ¹'¹ , but not for ¹(¹ . In contrast, Monte Carlo simulations yield such results for ¹'¹    and for ¹(¹ equally well. Thus, utilizing a novel extension of the cluster algorithm to Ising  models with interaction of arbitrary long range, the crossover in the critical behavior from the Ising universality class in d"2 and d"3 dimensions to mean-"eld behavior, with increasing range R of the interaction, has been thoroughly investigated, and we have reviewed these recent studies here in detail. Results such as the nonmonotonic variation of the e!ective exponent  of the susceptibility  below ¹ so far could not be obtained with any other method than Monte Carlo simulation.  For the nearest-neighbor Ising model on the simple cubic lattice, there now exist numerous high-precision Monte Carlo studies, and the value of ¹ as well as the critical exponents are known 

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Fig. 33. Plot of the scaled susceptibility maximum (a) and of the cumulant (b) [Eq. (7.18)] vs. the crossover scaling

 variable exp( D/2)/¸, using Eq. (7.4) and all values of ¸ that were available. Curves are guides to the eye only. The arrow A on the ordinate in (b) shows ;H. From Binder et al. [157].

very accurately. However, much less e!ort has been devoted to critical amplitude ratios, and a matter which has been particularly neglected is the equation of state as a function of magnetic "eld near the critical point. Thus, even for the d"3 nearest-neighbor Ising model still interesting and important studies need to be made! Another problem which we have emphasized in this article are short-range Ising models at high dimensionality (such as d"5) and Ising models with a long-range interaction, described by a power-law decay. Both cases are convenient testing grounds for our understanding of the theory of phase transitions. In the former problem, the critical behavior is mean-"eld like: not only the critical exponents are known, but it is also possible to systematically compute all quantities of

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interest by means of perturbation expansions. However, it is somewhat disturbing that there still occur deviations between the theoretical description of "nite-size behavior and the simulational results and our main hope is that the theoretical description will be extended to included higher-order corrections. The case of long-range interactions is particularly rewarding, since here the distance to the marginal dimension dH where mean-"eld behavior becomes valid is a parameter that can be varied continuously. Also for this problem it appears that some properties are incompletely understood, such as the variation of the invariant Q(K ) with d!dH in the non mean-"eld regime. As a "nal case study reviewed here, we have taken the interface localization transition in Ising "lms con"ned between competing walls. We have argued that the theoretical understanding of this problem remains on a qualitative rather than a quantitative level: already the starting point of the theory } the e!ective interface Hamiltonian } involves fundamental questions, which hamper other problems (such as critical wetting) as well and theories beyond the mean-"eld level remain to be developed. Unfortunately, also the quality of the available Monte Carlo data is much lower than for the other problems described in the present article } available studies are based on the use of simple Metropolis algorithms and hence severely su!er from the problem of critical slowing down. In conclusion we draw attention to extensions that were not dealt with here at all. One such extension is to allow for antiferromagnetic interactions to more distant neighbors competing with the nearest-neighbor exchange: this yields the possibility of di!erent types of ordering beyond the Ising universality class, and nontrivial phase diagrams occur [321]. A particularly fascinating topic is the ANNNI (axial next-nearest-neighbor Ising model) [322], where the competing antiferromagnetic exchange occurs in one lattice direction only. Beyond a certain critical strength of this exchange, a transition to a modulated phase rather than to ferromagnetic order occurs. The critical properties of the multicritical point that separates these di!erent types of order (the `Lifshitz pointa [322]) are known only very roughly. Another fascinating extension of the Ising ferromagnet is to re-interpret it as a lattice gas model and assign charges to the particles, adding an electric "eld acting on these charges in order to maintain an electrical current through the system. Beyond a critical interaction strength, this model undergoes a phase separation into a lattice gas of high density coexisting with a low-density lattice gas, analogous to the transition of the standard Ising-lattice gas model, but in a di!erent universality class. The critical exponents of this model have been controversial for a long time, and only recent work applying an anisotropic extension of "nite-size scaling seems to settle the issue [323]. Particularly interesting problems occur when we generalize the Ising model by introducing quenched disorder, such as random bonds or random "elds [324]. The problem becomes much harder now, because quantities such as the susceptibility have to be obtained by a double average, k ¹ "¸B[m ] , where 2 can still be obtained by the Monte Carlo methods as 2  2 described in the present article, but an additional average [2] has to be carried out (via a simple  random sampling) over the distribution over the random bonds or random "elds, respectively. The simplest case is that in which one has only random-bond disorder with bonds that still are all ferromagnetic (i.e., a random mixture of stronger and weaker bonds). This problem has been studied extensively by Monte Carlo simulations [325] and high-temperature series extrapolations [326], but many problems remain. Much harder, however, is the problem of the random-"eld Ising model (RFIM) [327] or the Ising spin glass [328}331]. In the RFIM, it is still controversial whether one has a second-order transition from the paramagnetic to the ferromagnetic phase in d"3, or

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a weak "rst-order transition, or whether one has a glass-like phase intervening between the paraand ferromagnetic phase. Ising spin glasses, on the other hand, where one has exchange constants of random sign $J, have already been studied since about twenty-"ve years by Monte Carlo methods [328]: still it seems controversial whether in d"3 the transition occurs at ¹ +1.1J [329] or  ¹ +1.3J [330], and the nature of the ordered phase is also a point of debate [330}332]. Thus,  much still remains to be done in the "eld of simulation studies of Ising models in the next decades!

Acknowledgements Some of the original research reviewed here has been carried out together with H.W.J. BloK te (Sections 3}6), some with D.P. Landau, R. Evans and A.M. Ferrenberg (Section 7). It is a great pleasure to thank all these colleagues for a particularly pleasant and fruitful collaboration. In addition, we thank them and numerous other colleagues for helpful discussions, pertinent preprints and reprints, etc., which all helped to improve the present manuscript. One of us (E.L.) acknowledges support from the Max-Planck-Institut fuK r Polymerforschung through a Fellowship.

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Physics Reports 344 (2001) 255}308

The statistical mechanics of membranes Mark J. Bowick*, Alex Travesset Physics Department, Syracuse University, Syracuse, NY 13244-1130, USA Received June 2000; editor: I. Procaccia Contents 1. 2. 3. 4.

Introduction Physical examples of membranes The Renormalization Group Crystalline membranes 4.1 Phantom 4.2. Self-avoiding 5. Anisotropic membranes 5.1. Phantom 5.2. The tubular phase 6. Defects in membranes: The crystalline-#uid transition and #uid membranes 6.1. Topological defects 6.2. The hexatic membrane 7. The #uid phase

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8. Conclusions Acknowledgements Appendix A. Useful integrals in dimensional regularization Appendix B. Some practical identities for RG quantities Appendix C. Discretized model for tethered membranes Appendix D. The crumpling transition Appendix E. The #at phase Appendix F. The self-avoiding phase Appendix G. The mean "eld solution of the anisotropic case References

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Abstract The #uctuations of two-dimensional extended objects (membranes) is a rich and exciting "eld with many solid results and a wide range of open issues. We review the distinct universality classes of membranes, determined by the local order, and the associated phase diagrams. After a discussion of several physical examples of membranes we turn to the physics of crystalline (or polymerized) membranes in which the individual monomers are rigidly bound. We discuss the phase diagram with particular attention to the dependence on the degree of self-avoidance and anisotropy. In each case we review and discuss analytic, numerical and experimental predictions of critical exponents and other key observables. Particular emphasis is given to the results obtained from the renormalization group
-expansion. The resulting renormalization group #ows and "xed points are illustrated graphically. The full technical details necessary to perform actual

* Corresponding author. Tel.: #1-315-443-5979; fax: #1-315-443-9103. E-mail addresses: [email protected] (M.J. Bowick), [email protected] (A. Travesset). 0370-1573/01/$ - see front matter  2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 0 0 ) 0 0 1 2 8 - 9

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calculations are presented in the Appendices. We then turn to a discussion of the role of topological defects whose liberation leads to the hexatic and yuid universality classes. We "nish with conclusions and a discussion of promising open directions for the future.  2001 Elsevier Science B.V. All rights reserved. PACS: 11.10.Kk; 64.60.Ak; 64.60.Cn; 64.60.Fr; 82.70.Kj; 82.70.Uv Keywords: Membrane; Random surfaces; Fluctuating geometries; Surfaces; Statistical mechanics; Interfaces; Polymerized membranes; Tethered membranes; Crumpling

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1. Introduction The statistical mechanics of one-dimensional structures (polymers) is fascinating and has proved to be fruitful from the fundamental and applied points of view [1,2]. The key reasons for this success lie in the notion of universality and the relative simplicity of one-dimensional geometry. Many features of the long-wavelength behavior of polymers are independent of the detailed physical and chemical nature of the monomers that constitute the polymer building blocks and their bonding into macromolecules. These microscopic details simply wash out in the thermodynamic limit of large systems and allow predictions of critical exponents that should apply to a wide class of microscopically distinct polymeric systems. Polymers are also su$ciently simple that considerable analytic and numerical progress has been possible. Their statistical mechanics is essentially that of ensembles of various classes of random walks in some d-dimensional bulk or embedding space. A natural extension of these systems is to intrinsic two-dimensional structures which we may generically call membranes. The statistical mechanics of these random surfaces is far more complex than that of polymers because two-dimensional geometry is far richer than the very restricted geometry of lines. Even planar two-dimensional } monolayers } are complex, as evidenced by the KTNHY [3}5] theory of defect-mediated melting of monolayers with two distinct continuous phase transitions separating an intermediate hexatic phase, characterized by quasi-long-range bond orientational order, from both a low-temperature crystalline phase and a high-temperature yuid phase. But full-#edged membranes are subject also to shape #uctuations and their macroscopic behavior is determined by a subtle interplay between their particular microscopic order and the entropy of shape and elastic deformations. For membranes, unlike polymers, distinct types of microscopic order (crystalline, hexatic, #uid) will lead to distinct long-wavelength behavior and consequently a rich set of universality classes. Flexible membranes are an important member of the enormous class of soft condensed matter systems [6}9], those which respond easily to external forces. Their physical properties are to a considerable extent dominated by the entropy of thermal #uctuations. In this review we will describe some of the presently understood behavior of crystalline ("xed-connectivity), hexatic and #uid membranes, including the relevance of self-avoidance, intrinsic anisotropy and topological defects. Emphasis will be given to the role of the renormalization group in elucidating the critical behavior of membranes. The polymer pastures may be lovely but a dazzling world awaits those who wander into the membrane meadows. The outline of the review is the following. In Section 2 we describe a variety of important physical examples of membranes, with representatives from the key universality classes. In Section 3 we introduce basic notions from the renormalization group and some formalism that we will use in the rest of the review. In Section 4 we review the phase structure of crystalline membranes for both phantom and self-avoiding membranes, including a thorough discussion of the "xed-point structure, RG #ows and critical exponents of each global phase. In Section 5 we turn to the same issues for intrinsically anisotropic membranes, with the new feature of the tubular phase. In Section 6 we address the consequences of allowing for membrane defects, leading to a discussion of the hexatic membrane universality class. We end with a brief discussion of #uid membranes in Section 7 and conclusions.

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2. Physical examples of membranes There are many concrete realizations of membranes in nature, which greatly enhances the signi"cance of their study. Crystalline membranes, sometimes termed tethered or polymerized membranes, are the natural generalization of linear polymer chains to intrinsically two-dimensional structures. They possess in-plane elastic moduli as well as bending rigidity and are characterized by broken translational invariance in the plane and "xed connectivity resulting from relatively strong bonding. Geometrically speaking they have a preferred two-dimensional metric. Let us look at some of the examples. One can polymerize suitable chiral oligomeric precursors to form molecular sheets [10]. This approach is based directly on the idea of creating an intrinsically two-dimensional polymer. Alternatively one can permanently cross-link #uid-like Langmuir}Blodgett "lms or amphiphilic bilayers by adding certain functional groups to the hydrocarbon tails and/or the polar heads [11,12] as shown schematically in Fig. 1. The cytoskeletons of cell membranes are beautiful and naturally occurring crystalline membranes that are essential to cell membrane stability and functionality. The simplest and most thoroughly studied example is the cytoskeleton of mammalian erythrocytes (red blood cells). The human body has roughly 5;10 red blood cells. The red blood cell cytoskeleton is a "shnet-like network of triangular plaquettes formed primarily by the proteins spectrin and actin. The links of the mesh are spectrin tetramers (of length approximately 200 nm) and the nodes are short actin "laments (of length 37 nm and typically 13 actin monomers long) [13,14], as seen in Figs. 2 and 3.

Fig. 1. The polymerization of #uid-like membrane to a crystalline membrane. Fig. 2. An electron micrograph of a region of the erythrocyte cytoskeleton. The skeleton is negatively stained (magni"cation 365 000) and has been arti"cially spread to a surface area nine to ten times as great as in the native membrane [15].

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Fig. 3. An extended view of the crystalline spectrin/actin network which forms the cytoskeleton of the red blood cell membrane.

There are roughly 70 000 triangular plaquettes in the mesh altogether and the cytoskeleton as a whole is bound by ankyrin and other proteins to the cytoplasmic side of the #uid phospholipid bilayer which constitutes the other key component of the red blood cell membrane. There are also inorganic realizations of crystalline membranes. Graphitic oxide (GO) membranes are micron size sheets of solid carbon with thicknesses on the order of 10 As , formed by exfoliating carbon with a strong oxidizing agent. Their structure in an aqueous suspension has been examined by several groups [16}18]. Metal dichalcogenides such as MoS have also been observed  to form rag-like sheets [19]. Finally similar structures occur in the large sheet molecules, shown in Fig. 4, believed to be an ingredient in glassy B O .   In contrast to crystalline membranes, #uid membranes are characterized by vanishing shear modulus and dynamical connectivity. They exhibit signi"cant shape #uctuations controlled by an e!ective bending rigidity parameter. A rich source of physical realizations of #uid membranes is found in amphiphilic systems [20}22]. Amphiphiles are molecules with a two-fold character } one part is hydrophobic and

 We thank David Nelson for providing us with this image from the laboratory of Dan Branton.

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Fig. 4. The sheet molecule B O .   Fig. 5. Schematic of experimental procedure to make a black membrane.

another part hydrophilic. The classic examples are lipid molecules, such as phospholipids, which have polar or ionic head groups (the hydrophilic component) and hydrocarbon tails (the hydrophobic component). Such systems are observed to self-assemble into a bewildering array of ordered structures, such as monolayers, planar (see Fig. 5) and spherical bilayers (vesicles or liposomes) (see Fig. 6) as well as lamellar, hexagonal and bicontinuous phases [23]. In each case the basic ingredients are thin and highly #exible surfaces of amphiphiles. The lipid bilayer of cell membranes may itself be viewed as a #uid membrane with considerable disorder in the form of membrane proteins (both peripheral and integral) and with, generally, an attached crystalline cytoskeleton, such as the spectrin/actin mesh discussed above. A complete understanding of these biological membranes will require a thorough understanding of each of its components (#uid and crystalline) followed by the challenging problem of the coupled system with thermal #uctuations, self-avoidance, potential anisotropy and disorder. The full system is currently beyond the scope of analytic and numerical methods but there has been considerable progress in the last "fteen years. Related examples of #uid membranes arise when the surface tension between two normally immiscible substances, such as oil and water, is signi"cantly lowered by the surface action of amphiphiles (surfactants), which preferentially orient with their polar heads in water and their hydrocarbon tails in oil. For some range of amphiphile concentration both phases can span the system, leading to a bicontinuous complex #uid known as a microemulsion. The oil}water interface of a microemulsion is a rather unruly #uid surface with strong thermal #uctuations [24] (see Fig. 7). The structures formed by membrane/polymer complexes are of considerable current theoretical, experimental and medical interest. To be speci"c it has recently been found that mixtures of cationic liposomes (positively charged vesicles) and linear DNA chains spontaneously self-assemble into a coupled two-dimensional smectic phase of DNA chains embedded between lamellar lipid bilayers [25,26]. For the appropriate regime of lipid concentration the same system can also

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Fig. 6. The structure of a liposome with its pure lipid spherical bilayer. Fig. 7. The structure of a microemulsion formed by the addition of surfactant to an oil}water mixture.

Fig. 8. Metal-coated #uid microcylinders (tubules) formed by chiral lipids.

form an inverted hexagonal phase with the DNA encapsulated by cylindrical columns of liposomes [27] (see Fig. 9). In both these structures the liposomes may act as non-viral carriers (vectors) for DNA with many potentially important applications in gene therapy [28]. Liposomes themselves have long been studied and utilized in the pharmaceutical industry as drug carriers [29]. On the materials science side the self-assembling ability of membranes is being exploited to fabricate microstructures for advanced material development. One beautiful example is the use of chirallipid-based #uid microcylinders (tubules) as a template for metallization. The resultant hollow metal needles may be half a micron in diameter and as much as a millimeter in length [30,31], as illustrated in Fig. 8. They have potential applications as, for example, cathodes for vacuum "eld emission and microvials for controlled release [30].

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Fig. 9. The lamellar and inverted hexagonal DNA-membrane complexes from the work of [27].

3. The Renormalization group The Renormalization Group (RG) has provided an extremely general framework that has uni"ed whole areas of physics and chemistry [32]. It is beyond the scope of this review to discuss the RG formalism in detail but there is an ample literature to which we refer the reader (see the articles in this issue). It is the goal of this review to apply the RG framework to the statistical mechanics of membranes, and for this reason we brie#y emphasize and review some well-known aspects of the RG and its related
-expansion. The RG formalism elegantly shows that the large distance properties (or equivalently low p-limit) of di!erent models are actually governed by the properties of the corresponding Fixed Point (FP). In this way one can compute observables in a variety of models, such as a molecular dynamics simulation or a continuum Landau phenomenological approach, and obtain the same long wavelength result. The main idea is to encode the e!ects of the short-distance degrees of freedom in rede"ned couplings. A practical way to implement such a program is the Renormalization Group Transformation (RGT), which provides an explicit prescription for integrating out all the high p-modes of the theory. One obtains the large-distance universal term of any model by applying a very large (R to be rigorous) number of RGTs.

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The previous approach is very general and simple but presents the technical problem of the proliferation in the number of operators generated along the RG #ow. There are established techniques to control this expansion, one of the most successful being the
-expansion. The
-expansion may also performed via a "eld theoretical approach using Feynman diagrams and dimensional regularization within a minimal subtraction scheme, which we brie#y discuss below. Whereas it is true that this technique is rather abstract and intuitively not very close to the physics of the model, we "nd it computationally much simpler. Generally we describe a particular model by several "elds 
, ,2 and we construct the Landau free energy by including all terms compatible with the symmetries and introducing new couplings (u, v,2) for each term. The Landau free energy may be considered in arbitrary dimension d, and then, one usually "nds a Gaussian FP (quadratic in the "elds) which is infrared stable above a critical dimension (d ). Below d there are one or several couplings that de"ne 3 3 relevant directions. One then computes all physical quantities as a function of
,d !d, that is, as 3 perturbations of the Gaussian theory. In the "eld theory approach, we introduce a renormalization constant for each "eld (Z , Z ,2) ( Q and a renormalization constant (Z , Z ,2) for each relevant direction below d . If the model has S T 3 symmetries, there are some relations among observables (Ward identities) and some of these renormalization constants may be related. This not only reduces their number but also has the added bonus of providing cross-checks in practical calculations. Within dimensional regularization, the in"nities of the Feynman diagrams appear as poles in
, which encode the short-distance details of the model. If we use these new constants (Z's) to absorb the poles in
, thereby producing a complete set of "nite Green's functions, we have succeeded in carrying out the RG program of including the appropriate short-distance information in rede"ned couplings and "elds. This particular prescription of absorbing only the poles in
in the Z's is called the Minimal Subtraction Scheme (MS), and it considerably simpli"es practical calculations. As a concrete example, we consider the theory of a single scalar "eld
with two independent coupling constants. The one-particle irreducible Green's function has the form






1 , ,(k ; u , v , M)"Z,, k ; u, v; G 0 G 0 0 (


(1)

where the function on the left depends on a new parameter M, which is unavoidably introduced in eliminating the poles in
. The associated correlator also depends on rede"ned couplings u and v . 0 0 The RHS depends on the poles in
, but its only dependence on M arises through Z . This ( observation allows one to write M






R R R N d (Z\,,)" M #(u ) #(v ) !  ,"0 , 0 0 Ru 0 Rv RM 2 ( 0 dM ( 0 0

where u "M\CF(Z , Z ,2Z )u, v "M\CF(Z , Z ,2Z )v , 0 ( Q S 0 ( Q T Ru Rv  (u , v )" M 0 ,  (u , v )" M 0 , S 0 0 T 0 0 RM RM ST ST R ln Z ( .  " M ( RM ST
















(2)

(3)

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The -functions control the running of the coupling by M

du dv 0 " (u , v ), M 0 " (u , v ) . S 0 0 T 0 0 dM dM

(4)

The existence of a FP, at which couplings cease to #ow, requires (uH, vH)"0 for all -functions of 0 0 the model. Those are the most important aspects of the RG we wanted to review. In Appendix B we derive more appropriate expressions of the RG-functions for practical convenience. For a detailed exposition of the
-expansion within the "eld theory framework we refer to the excellent book by Amit [33].

4. Crystalline membranes A crystalline membrane is a two-dimensional "sh-net structure with bonds (links) that never break } the connectivity of the monomers (nodes) is "xed. It is useful to keep the discussion general and consider D-dimensional objects embedded in d-dimensional space. These are described by a d-dimensional vector ro (x), with x the D-dimensional internal coordinates, as illustrated in Fig. 10. The case (d"3, D"2) corresponds to the physical crystalline membrane. To construct the Landau free energy of the model, one must recall that the free energy must be invariant under global translations, so the order parameter is given by derivatives of the embedding ro , that is to "Rro /Ru , with "1,2, D. This latter condition, together with the invariance under ? ? rotations (both in internal and bulk space), give a Landau free energy [34}36]

  

F(ro )" d"x #



1 t (Rro )# (R ro )#u(R ro R ro )#v(R ro R?ro ) ? ? @ ? 2 2 ?

b d"x d"y B(ro (x)!ro (y)) , 2

Fig. 10. Representation of a membrane.

(5)

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Fig. 11. Examples of (a) crumpled phase, (b) crumpling transition (crumpled phase) and (c) a #at phase. Results correspond to a numerical simulation of the phantom case [37] and gives a very intuitive physical picture of the di!erent phases.

where higher-order terms may be shown to be irrelevant at long wavelength, as discussed later. The physics in Eq. (5) depends on "ve parameters, E , bending rigidity: This is the coupling to the extrinsic curvature (the square of the Gaussian mean curvature). Since reparametrization invariance is broken for crystalline membranes, this term may be replaced by its long-wavelength limit. For large and positive bending rigidities #atter surfaces are favored. E t, u, v, elastic constants: These coe$cients encode the microscopic elastic properties of the membrane. In a #at phase, they may be related to the LameH coe$cients of Landau elastic theory (see Section 4.1.3). E b, excluded volume or self-avoiding coupling: This is the coupling that imposes an energy penalty for the membrane to self-intersect. The case b"0, i.e. no self-avoidance, corresponds to a phantom model. We generally expand ro (x) as ro (x)"(x#u(x), h(x)) ,

(6)

with u the D-dimensional phonon in-plane modes, and h the d!D out-of-plane #uctuations. If "0 the model is in a rotationally invariant crumpled phase, where the typical surfaces have fractal dimension, and there is no real distinction between the in-plane phonons and out-of plane modes. For a pictorial view, see cases (a) and (b) in Fig. 11.

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If O0 the membrane is #at up to small #uctuations and the full rotational symmetry is spontaneously broken. The "elds h are the analog of the Goldstone bosons and they have di!erent naive scaling properties than u. See Fig. 11 for a visualization of a typical con"guration in the #at phase. We will begin by studying the phantom case "rst. This simpli"ed model may even be relevant to physical systems since one can envision membranes that self-intersect (at least over some time scale). One can also view the model as a fascinating toy model for understanding the more physical self-avoiding case to be discussed later. Combined analytical and numerical studies have yielded a thorough understanding of the phase diagram of phantom crystalline membranes. 4.1. Phantom The Phantom case corresponds to setting b"0 in the free energy Eq. (5):

 

F(ro )" d"x



1 t (Rro )# (R ro )#u(R ro R ro )#v(R ro R?ro ) . ? @ ? 2 ? 2 ?

(7)

The mean "eld e!ective potential, using the decomposition of Eq. (6), becomes






(8)

t50 , t : t(0 . ! 4(u#vD)

(9)

<()"D

t #(u#vD) , 2

with solutions



"

0:

There is, consequently, a #at phase for t(0 and a crumpled phase for t'0, separated by a crumpling transition at t"0 (see Fig. 12). The actual phase diagram agrees qualitatively with the phase diagram of the model shown schematically in Fig. 13. The crumpled phase is described by a line of equivalent FPs (GFP). There is a general hyper-surface, whose projection onto the }t plane corresponds to a one-dimensional curve (CTH), which corresponds to the crumpling transition. Within the CTH there is an infrared stable FP (CTFP) which describes the large distance properties of the crumpling transition. Finally, for large enough values of and negative values of t, the system is in a #at phase described by the corresponding infra-red stable FP (FLFP). Although the precise phase diagram turns out to be slightly more complicated than the one depicted in Fig. 13, the additional subtleties do not modify the general picture. The evidence for the phase diagram depicted in Fig. 13 comes from combining the results of a variety of analytical and numerical calculations. We present in detail the results obtained from the
-expansion since they have wide applicability and allow a systematic calculation of  The FLFP is actually a line of equivalent "xed points.

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Fig. 12. Mean "eld solution for crystalline membranes.

Fig. 13. Schematic plot of the phase diagram for phantom membranes. GFP are the equivalent FPs describing the crumpled phase. The crumpling transition is described by the Crumpling transition critical line (CTH), which contains the Crumpling Transition FP (CTFP). The Flat phase is described by the (FLFP).

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the -function and the critical exponents. We also describe brie#y results obtained with other approaches. 4.1.1. The crumpled phase In the crumpled phase, the free energy Eq. (7) for D52 simpli"es to



t F(ro )" d"x(R ro )#Irrelevant terms , ? 2

(10)

since the model is completely equivalent to a linear sigma model in D42 dimensions having O(d) symmetry, and therefore all derivative operators in ro are irrelevant by power counting. The parameter t labels equivalent Gaussian FPs, as depicted in Fig. 13. In RG language, it de"nes a completely marginal direction. This is true provided the condition t'0 is satis"ed. The large distance properties of this phase are described by simple Gaussian FPs and therefore the connected Green's function may be calculated exactly with result



G(x)&

x\" DO2 , logx

D"2 .

(11)

The associated critical exponents may also be computed exactly. The Hausdor! dimension d , or & equivalently the size exponent "D/d , is given (for the membrane case D"2) by & d "R ( "0)PR &log ¸ . (12) & % The square of the radius of gyration R scales logarithmically with the membrane size ¸. This % result is in complete agreement with numerical simulations of tethered membranes in the crumpled phase where the logarithmic behavior of the radius of gyration is accurately checked [37}46]. Reviews may be found in [47,48]. 4.1.2. The crumpling transition The free energy is now given by



F(ro )" d"x[(Rro )#u(R ro R ro )#v( (R ro R?ro )] ,  ? ? @ ?

(13)

where the dependence on may be included in the couplings u and v( . With the leading term having two derivatives, the directions de"ned by the couplings u and v( are relevant by naive power counting for D44. This shows that the model is amenable to an
-expansion with
"4!D. For practical purposes, it is more convenient to consider the coupling v"v( #u/D. We provide the detailed derivation of the corresponding  functions in Appendix D. The result is 1 (d/3#65/12)u #6u v #4/3v  ,  (u , v )"!
u # 0 0 0 0 S 0 0 0 8
 1 21/16u #21/2u v #(4d#5)v  .  (u , v )"!
v # 0 0 0 0 T 0 0 0 8


(14)

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269

Rather surprisingly, this set of  functions does not possess a FP, except for d'219. This result would suggest that the crumpling transition is "rst order for d"3. Other estimates, however, give results which are consistent with the crumpling transition being continuous. These are E Limit of large elastic constants [49]: The crumpling transition is approached from the #at phase, in the limit of in"nite elastic constants. The model is H

,*



" d" (ro ) , 2

(15)

with the further constraint R ro R ro " . Remarkably, the -function may be computed within ? @ ?@ a large d expansion, yielding a continuous crumpling transition with size exponent at the transition (for D"2) 2d 1 d " P "1! . & d!1 d

(16)

E SCSA approximation [50]: The Schwinger}Dyson equations for the model given by Eq. (13) are truncated to include up to four point vertices. The result for the Hausdor! dimension and size exponent is d "2.73P "0.732 . &

(17)

E MCRG calculation [51]: The crumpling transition is studied using MCRG (Monte Carlo Renormalization Group) techniques. Again, the transition is found to be continuous with exponents d "2.64(5)P "0.85(9) . &

(18)

Each of these three independent estimates give a continuous crumpling transition with a size exponent in the range &0.7$0.15. It would be interesting to understand how the
-expansion must be performed in order to reconcile it with these results. Further evidence for the crumpling transition being continuous is provided by numerical simulations [37}46] where the analysis of observables like the speci"c heat (see Fig. 14) or the radius of gyration radius give textbook continuous phase transitions, although the precise value of the exponents at the transition are di$cult to pin down. Since this model has also been explored numerically with di!erent discretizations on several lattices, there is clear evidence for universality of the crumpling transition [45], again consistent with a continuous transition. In Appendix C we present more details of suitable discretizations of the energy for numerical simulations of membranes. 4.1.3. The yat phase In a #at membrane (see Fig. 15), we consider the strain tensor u "R u #R u #R hR h . ?@ ? @ @ ? ? @

(19)

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Fig. 14. Plot of the speci"c heat observable [37]. The growth of the speci"c heat peak with system size indicates a continuous transition. Fig. 15. Coordinates for #uctuations in the #at phase.

The free energy Eq. (7) becomes

 

F(u, h)" d"x



(  (R R h)#u u?@# (u? ) , ?@ 2 ? @ 2 ?

(20)

where we have dropped irrelevant terms. One recognizes the standard Landau free energy of elasticity theory, with LameH coe$cients  and , plus an extrinsic curvature term, with bending rigidity ( . These couplings are related to the original ones in Eq. (7) by "u\", "2v\", ( " \" and t"!4(#(D/2))"\. The large distance properties of the #at phase for crystalline membranes are completely described by the free energy Eq. (20). Since the bending rigidity may be scaled out at the crumpling transition, the free energy becomes a function of /  and / . The -function for the couplings u, v at "1 may be calculated within an
-expansion, which we describe in detail in Appendix E. Let us recall that the dependence on may be trivially restored at any stage. The result is






 d  ( ,  )"!
 # 0  #20A , I 0 0 0 8
 3




(21)



1 d   #2(d #10A)  #2d  ,  ( ,  )"!
 #  0 0  0 H 0 0 0 8
 3 0 where d "d!D, and A"( # )/(2 # ). These  functions show four "xed points whose  0 0 0 0 actual values are shown in Table 1.

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271

Table 1 The FPs and critical exponents of the #at phase FP

H 0

H 0



 S

FP1 FP2

0 0

0 0

0 0

FP3

12
20#d  12
24#d 

0 2
/d  !6
20#d  !4
24#d 


2#d /10 
2#d /12 


1#20/d 
1#24/d 

FLFP

Fig. 16. Phase diagram for the phantom #at phase. There are three infra-red unstable FPs, labelled by FLP1, FLP2 and FLP3, but the physics of the #at phase is governed by the infra-red stable FP (FLFP).

As apparent from Fig. 16, the phase diagram of the #at phase turns out to be slightly more involved than the one shown in Fig. 13, as there are three FPs in addition to the FLFP already introduced. These additional FPs are infra-red unstable, however, and can only be reached for very speci"c values of the LameH coe$cients. So for any practical situation we can regard the FLFP as the only existing FP in the #at phase. 4.1.3.1. The properties of the yat phase. The #at phase is a very important phase as will be apparent once we study the full model, including self-avoidance. For that reason we turn now to a more detailed study of its most important properties. Fig. 11(c) gives an intuitive visualization of a crystalline membrane in the #at phase. The membrane is essentially a #at two-dimensional object up to #uctuations in the perpendicular direction. The rotational symmetry of the model is spontaneously broken, being reduced from O(d)

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to O(d!D);O(D). The remnant rotational symmetry is realized in Eq. (20) as h (x)Ph (x)#AG?x , (22) G G ? u (x)Pu !AG?h ! GH(AG?A@Hx ) , ? ? G  @ where AG? is a D;(d!D) matrix. This relation is very important as it provides Ward identities which simplify enormously the renormalization of the theory. Let us "rst study the critical exponents of the model. There are two key correlators, involving the in-plane and the out-of-plane phonon modes. Using the RG equations, it is easy to realize that at any given FP, the low-p limit of the model is given by  (po )&po >ES , SS  (po ),po  (po )&po \E , FF

(23)

where the last equation de"nes the anomalous elasticity (po ) as a function of momenta po . These two exponents are not independent, since they satisfy the scaling relation [52]  "4!D!2 , (24) S which follows from the Ward identities associated with the remnant rotational symmetry (Eq. (22)). Another important exponent is the roughness exponent , which measures the #uctuations transverse to the #at directions. It can be expressed as "(4!D!)/2. The long wavelength properties of the #at phase are described by the FLFP (see Fig. 16). Since the FLFP occurs at non-zero renormalized values of the LameH coe$cients, the associated critical exponents discussed earlier are clearly non-Gaussian. Within an
-expansion, the values for the critical exponents are given in Table 1. There are alternative estimates available from di!erent methods. These are E Numerical simulation: In [37] a large-scale simulation of the model was performed using very large meshes. The results obtained for the critical exponents are very reliable, namely  "0.50(1) "0.750(5) "0.64(2) . (25) S For a review of numerical results see [47,48]. E SCSA approximation: This consists of suitably truncating the Schwinger}Dyson equations to include up to four-point correlation functions [50]. The result for general d is 4 , (d)" d #(16!2d #d)    which for d"3 gives  "0.358 "0.821 "0.59 . S E Large d expansion: The result is [49] 2 " P(3)"2/3 . d

(26)

(27)

(28)

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We regard the results of the numerical simulation as our most accurate estimates, since we can estimate the errors. The results obtained from the SCSA, which are the best analytical estimate, are in acceptable agreement with simulations. Finally there are two experimental measurements of critical exponents for the #at phase of crystalline membranes. The static structure factor of the red blood cell cytoskeleton (see Section 1) has been measured by small-angle X-ray and light scattering, yielding a roughness exponent of "0.65(10) [13]. Freeze-fracture electron microscopy and static light scattering of the conformations of graphitic oxide sheets (Section 1) revealed #at sheets with a fractal dimension d "2.15(6). & Both these values are in good agreement with the best analytic and numerical predictions, but the errors are still too large to discriminate between di!erent analytic calculations. The Poisson ratio of a crystalline membrane (measuring the transverse elongation due to a longitudinal stress [53]) is universal and within the SCSA approximation, which we regard as the more accurate analytical estimate, is given by 1 P (2)"!1/3 .

(D)"! D#1

(29)

This result has been accurately checked in numerical simulations [54]. Rather remarkably, it turns out to be negative. Materials having a negative Poisson ratio are called auxetic. This highlights potential applications of crystalline membranes to materials science since auxetic materials have a wide variety of potential applications as gaskets, seals, etc. Finally another critical regime of a #at membrane is achieved by subjecting the membrane to external tension [55,56]. This allows a low-temperature phase in which the membrane has a domain structure, with distinct domains corresponding to #at phases with di!erent bulk orientations. This describes, physically, a buckled membrane whose equilibrium shape is no longer planar. 4.2. Self-avoiding Self-avoidance is a necessary interaction in any realistic description of a crystalline membrane. It is introduced in the form of a delta-function repulsion in the full model Eq. (5). We have already analyzed the phantom case and explored in detail the distinct phases. The question before us now is the e!ect of self-avoidance on each of these phases. The "rst phase we analyze is the #at phase. Since self-intersections are unlikely in this phase, it is intuitively clear that self-avoidance should be irrelevant. This may also be seen if one neglects the e!ects of the in-plane phonons. In the self-avoiding term for the #at phase we have





b b d"x d"y B(ro (x)!ro (y))" d"x d"y "((x!y)#u(x)!u(y)) B\"(h(x)!h(y)) 2 2



b & d"x d"y "((x!y)) B\"(h(x)!h(y))"0 , 2

(30)

as the trivial contribution where the membrane equals itself is eliminated by regularization. The previous argument receives additional support from numerical simulations in the #at phase, where it is found that self-intersections are extremely rare in the typical con"gurations appearing in those simulations [79]. It seems clear that self-avoidance is most likely an irrelevant operator, in the RG

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sense, of the FLFP. Nevertheless, it would be very interesting if one could provide a more rigorous analytical proof for this statement. A rough argument can be made as follows. Shortly we will see that the Flory approximation for self-avoiding membranes predicts a fractal dimension d "2.5. & For bulk dimension d exceeding 2.5 therefore we expect self-avoidance to be irrelevant. A rigorous proof of this sort remains rather elusive, as it involves the incorporation of both self-avoidance and non-linear elasticity, and this remains a di$cult open problem. The addition of self-avoidance in the crumpled phase consists of adding the self-avoiding interaction to the free energy of Eq. (10)





b 1 F(ro )" d"x(R ro (x))# d"x d"y B(ro (x)!ro (y)) , ? 2 2

(31)

which becomes the natural generalization of the Edwards' model for polymers to D-dimensional objects. Standard power counting shows that the GFP of the crumpled phase is infra-red unstable to the self-avoiding perturbation for
(D, d),2D!d

2!D '0 , 2

(32)

which implies that self-avoidance is a relevant perturbation for D"2-objects at any embedding dimension d. The previous remarks make it apparent that it is possible to perform an
-expansion of the model [57}60]. In Appendix F, we present the calculation of the -function at lowest order in
using the MOPE (Multi-local-operator-product-expansion) formalism [61,62]. The MOPE formalism has the advantage that it is more easily generalizable to higher orders in
, and enables concrete proofs showing that the expansion may be carried out to all orders. At lowest order, the result for the -function is




 



(2!D)\>B 2
" >B (D/2!D) d (2!D) b 0 #  (b )"!
b # @ 0 0 (D/2) (2D/2!D) 2 2D 2 (4
)B ,!
b #a b . (33) 0  0 The infra-red stable FP is given at lowest order in
by bH"
/a , which clearly shows that the GFP 0  of the crumpled phase is infra-red unstable in the presence of self-avoidance. The preceding results are shown in Fig. 17 and may be summarized as E The #at phase of self-avoiding crystalline membranes is exactly the same as the #at phase of phantom crystalline tethered membranes. E The crumpled phase of crystalline membranes is destabilized by the presence of any amount of self-avoidance. The next issue to elucidate is whether this new SAFP describes a crumpled self-avoiding phase or a #at one and to give a more quantitative description of the critical exponents describing the universality class. Supposing that the SAFP is, in fact, #at we must understand its relation to the FLFP describing the physics of the #at phase and the putative phase transitions between these two. 4.2.1. The nature of the SAFP Let us study in more detail the model described in Eq. (31). The key issue is whether this model still admits a crumpled phase, and if so to determine the associated size exponent. On general

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Fig. 17. The addition of self-avoidance at the crumpled and #at phases.

Fig. 18. The size exponent as a function of d. There is a critical dimension d below which the crumpled phase disappears. 

grounds we expect that there is a critical dimension d , below which there is no crumpled phase  (see Fig. 18). An estimate for the critical dimension may be obtained from a Flory approximation in which one minimizes the free energy obtained by replacing both the elastic and self-avoiding terms with the radius of gyration raised to the power of the appropriate scaling dimensions. Within the Flory treatment a D-dimensional membrane is in a crumpled phase, with a size exponent given by "(D#2)/(d#2) .

(34)

From this it follows that d "D (see Fig. 18). The Flory approximation, though very accurate for  polymers (D"1), remains an uncontrolled approximation. In contrast, the
-expansion provides a systematic determination of the critical exponents. For the case of membranes, however, some extrapolation is required, as the upper critical dimension is in"nite. This was done in [63], where it is shown that reconsidering the
-expansion as a double expansion in
and D, critical quantities may be extrapolated for D"2-dimensional objects. At lowest order in
, the membrane is in a crumpled phase. The enormous task of calculating the next correction (
) was successfully carried out in [64], employing more elaborate extrapolation methods than those in [63]. Within this calculation, the d"3 membrane is still in a crumpled phase, but with a size exponent now closer to 1. It cannot be ruled out that the
-expansion, successfully carried out to all orders could give a #at phase "1. In fact, the authors in [64,65] present some arguments in favor of a scenario of this type, with a critical dimension d &4. 

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Other approaches have been developed with di!erent results. A Gaussian approximation was developed in [66,67]. The size exponent of a self-avoiding membrane within this approach is "4/d ,

(35)

and since one has '1 for d44, one may conclude that the membrane is #at for d4d "4. Since  we cannot determine the accuracy of the Gaussian approximation this estimate must be viewed largely as interesting speculation. Slightly more elaborate arguments of this type [8] yield an estimated critical dimension d "3.  4.2.1.1. Numerical simulations. We have seen that numerical simulations provide good support for analytic results in the case of phantom membranes. When self-avoidance is included, numerical simulations become invaluable, since analytic results are harder to come by. It is for this reason that we discuss them in greater detail than in previous sections. A possible discretization of membranes with excluded volume e!ects consists of a network of N particles arranged in a triangular array. Nearest neighbors interact with a potential



0 for ro (b , < (ro )" ,, R for ro 'b ,

(36)

although some authors prefer a smoothened version, with the same general features. The quantity b is of the order of a few lattice spacings. This is a lattice version of the elastic term in Eq. (31). The discretization of the self-avoidance is introduced as a repulsive hard sphere potential, now acting between any two atoms in the membrane, instead of only nearest neighbors. A hard sphere repulsive potential is, for example,



R for ro ( , < (ro )" #VA 0 for ro ' ,

(37)

where is the range of the potential, and (b. Again, some smoothened versions, continuous at ro " , have also been considered. This model may be pictured as springs, de"ned by the nearest-neighbor potential Eq. (36), with excluded volume e!ects enforced by balls of radius (Eq. (37)). This model represents a lattice discretization of Eq. (31). Early simulations of this type of model [38,39] provided a "rst estimate of the size exponent at d"3 fully compatible with the Flory estimate Eq. (34). The lattices examined were not very large, however, and subsequent simulations with larger volumes [68,69] found that the d"3 membrane is actually #at. This result is even more remarkable if one recalls that there is no explicit bending rigidity. The #at phase was a very surprising result, in some con#ict with the insight provided from the analytical estimates discussed in the previous subsection. An explanation for it came from the observation [70] that excluded volume e!ects induce bending rigidity, as depicted in Fig. 19. The reason is that the excluded volume e!ects generate a non-zero expectation value for the bending rigidity, since the normals can be parallel, but not anti-parallel (see 19). This induced bending rigidity was estimated and found to be big enough to drive the self-avoiding membrane well within

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Fig. 19. Visualization of bending rigidity generated by a hard sphere potential. Normals no and no cannot be   anti-parallel.

the #at phase of the phantom one. This means that this particular discretization of the model renders any potential SAFP inaccessible and the physics is described by the FLFP. In [71] the structure function of the self-avoiding model is numerically computed and found to compare well with the analytical structure function for the #at phase of phantom crystalline membranes, including comparable roughness exponents. The natural question then to ask is whether it is possible to reduce the bending rigidity su$ciently to produce a crumpled self-avoiding phase. Subsequent studies addressed this issue in various ways. The most natural way is obviously to reduce the range of the potential su$ciently that the induced bending rigidity is within the crumpled phase. This is the approach followed in [72]. The #at phase was found to persist to very small values of , with eventual signs of a crumpled phase. This crumpled phase may essentially be due to the elimination at self-avoidance at su$ciently small . A more comprehensive study, in which the same limit is performed this time with an excluded volume potential which is a function of the internal distance along the lattice [73], concluded that for large membranes, inclusion of excluded volume e!ects, no matter how small, leads to #atness. A di!erent approach to weakening the #at phase, bond dilution [74], found that the #at phase persists until the percolation critical point. In conclusion the bulk of accumulated evidence indicates that #atness is an intrinsic consequence of self-avoidance. If this is indeed correct the SAFP coincides with FLFP and this feature is an inherent consequence of self-avoidance, rather than an artifact of discretization. Given the di$culties of "nding a crumpled phase with a repulsive potential, simulations for larger values of the embedding space dimension d have also been performed [75,76]. These simulations show clear evidence that the membrane remains #at for d"3 and 4 and undergoes a crumpling transition for d55, implying d '4.  An alternative approach to incorporating excluded volume e!ects corresponds to discretize a surface with a triangular lattice and imposing the self-avoidance constraint by preventing the triangular plaquettes from inter-penetrating. This model has the advantage that is extremely #exible, since there is no restriction on the bending angle of adjacent plaquettes (triangles) and therefore no induced bending rigidity (see Appendix C). The "rst simulations of the plaquette model [77] found a size exponent in agreement with the Flory estimate Eq. (34). A subsequent simulation [78] disproved this result, and found a size exponent "0.87, higher than the Flory estimate, but below one. More recent results simulating larger lattices with more sophisticated algorithms agree completely with the results obtained from the ball and spring models [79].

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Further insight into the lack of a crumpled phase for self-avoiding crystalline membranes is found in the study of folding [80}86]. This corresponds to the limit of in"nite elastic constants studied by David and Guitter with the further approximation that the space of bending angles is discretized. One quickly discovers that the re#ection symmetries of the allowed folding vertices forbid local folding (crumpling) of surfaces. There is therefore essentially no entropy for crumpling. There is, however, local unfolding and the resulting statistical mechanical models are non-trivial. The lack of local folding is the discrete equivalent of the long-range curvature}curvature interactions that stabilize the #at phase. The dual e!ect of the integrity of the surface (time-independent connectivity) and self-avoidance is so powerful that crumpling seems to be impossible in low embedding dimensions. 4.2.2. Attractive potentials Self-avoidance, as introduced in Eq. (37) is a totally repulsive force among monomers. There is the interesting possibility of allowing for attractive potentials also. This was pioneered in [70] as a way to escape to the induced bending rigidity argument (see Fig. 19), since an attractive potential would correspond to a negligible (or rather a negative) bending rigidity. Remarkably, in [70], a compact (more crumpled) self-avoiding phase was found, with fractal dimension close to 3. This was further studied in [87], where it was found that with an attractive Van der Waals potential, the crystalline membrane underwent a sequence of folding transitions leading to a crumpled phase. In [88] similar results were found, but instead of a sequence of folding transitions a crumpled phase was found with an additional compact (more crumpled phase) at even lower temperatures. Subsequent work gave some support to this scenario [89]. On the analytical side, the nature of the -point for membranes and its relevance to the issue of attractive interactions has been addressed in [90]. We think that the study of a tether with an attractive potential remains an open question begging for new insights. A thorough understanding of the nature of the compact phases produced by attractive interactions would be of great value. 4.2.3. The properties of the SAFP The enormous e!orts dedicated to study the SAFP have not resulted in a complete clari"cation of the overall scenario since the existing analytical tools do not provide a clear picture. Numerical results clearly provide the best insight. For the physically relevant case d"3, the most plausible situation is that there is no crumpled phase and that the #at phase is identical to the #at phase of the phantom model. For example, the roughness exponents  from numerical simulations of 1 self-avoidance at d"3 using ball-and-spring models [75] and impenetrable plaquette models [79] and the roughness exponent at the FLFP, Eq. (25), compare extremely well  "0.64(4), "0.64(2) . (38) 1 So the numerical evidence allows us to conjecture that the SAFP is exactly the same as the FLFP, and that the crumpled self-avoiding phase is absent in the presence of purely repulsive potentials (see Fig. 20). This conjecture is strengthened by the "nding that the Poisson ratio of SA membranes is the same as that of #at phantom membranes [123]. This identi"cation of "xed points enhances

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Fig. 20. The conjectured phase diagram for self-avoiding crystalline membranes in d"3. With any degree of selfavoidance the #ows are to the #at phase "xed point of the phantom model (FL).

the signi"cance of the FLFP treated earlier. It would be very helpful if analytical tools were developed to further substantiate this statement.

5. Anisotropic membranes An anisotropic membrane is a crystalline membrane having the property that the elastic or the bending rigidity properties in one distinguished direction are di!erent from those in the D!1 remaining directions. As for the isotropic case we keep the discussion general and describe the membrane by a d-dimensional ro (x , y), where now the D dimensional coordinates are split into , D!1 x coordinates and the orthogonal distinguished direction y. , The construction of the Landau free energy follows the same steps as in the isotropic case. Imposing translational invariance, O(d) rotations in the embedding space and O(D!1) rotations in internal space, the equivalent of Eq. (5) is now



1 F(ro (x))" d"\x dy[ (R ro )# (Rro ) , , , W W 2 # Rro ) R ro #t (R,ro )#t (R ro ) ,W W , , ? W W u u # ,, (R,ro ) R,ro )# WW (R ro ) R ro ) @ W 2 W 2 ?

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v #u (R,ro ) R ro )# ,, (R,ro ) R,ro ) ,W ? W ? 2 ? #v (R,ro )(R ro )] ,W ? W

 

b # d"x d"x B(ro (x)!ro (x)) . 2

(39)

This model has eleven parameters, representing distinct physical interactions: E , , bending rigidity: the anisotropic versions of the isotropic bending rigidity splits into , W ,W three distinct terms. E t , t , u , u , v , v elastic constants: there are six quantities describing the microscopic elastic , W ,, WW ,, ,W properties of the anisotropic membrane. E b, self-avoidance coupling: This particular term is identical to its isotropic counterpart. Following the same steps as in the isotropic case, we split ro (x)"( x #u (x),  y#u (x), h(x)) , , , , W W

(40)

with u being the D!1-dimensional phonon in-plane modes, u the in-plane phonon mode in the , W distinguished direction y and h the d!D out-of-plane #uctuations. If  " "0, the membrane is , W in a crumpled phase and if both  O0 and  O0 the membrane is in a #at phase very similar to , W the isotropic case (how similar will be discussed shortly). There is, however, the possibility that  "0 and  O0 or  O0 and  "0. This describes a completely new phase, in which the , W , W membrane is crumpled in some internal directions but #at in the remaining ones. A phase of this type is called a tubular phase and does not appear when studying isotropic membranes. In Fig. 21

Fig. 21. Examples of (a) the tubular phase, (b) the crumpled phase and (c) the #at phase of an anisotropic phantom crystalline membrane taken from the simulations of [91].

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281

we show an intuitive visualization of a tubular phase along with the corresponding #at and crumpled phases of anisotropic membranes. We will start by studying the phantom case. We show, using both analytical and numerical arguments, that the phase diagram contains a crumpled, tubular and #at phase. The crumpled and #at phases are equivalent to the isotropic ones, so anisotropy turns out to be an irrelevant interaction in those phases. The new physics is contained in the tubular phase, which we describe in detail, both with and without self-avoidance. 5.1. Phantom 5.1.1. The phase diagram We "rst describe the mean "eld theory phase diagram and then the e!ect of #uctuations. There are two situations depending on the particular values of the function , which depends on the elastic constants u , v , u and v . Since the derivation is rather technical, we refer to Appendix ,, ,W WW ,, G for the details. E Case A ('0): the mean "eld solution displays all possible phases. When t '0 and t '0 the W , model is in a crumpled phase. Lowering the temperature, one of the t couplings becomes negative, and we reach a tubular phase (either N or y-tubule). A further reduction of the temperature eventually leads to a #at phase. E Case B ((0): in this case the #at phase disappears from the mean "eld solution. Lowering further the temperature leads to a continuous transition from the crumpled phase to a tubular phase. Tubular phases are the low temperature stable phases in this regime. This mean "eld result is summarized in Fig. 22. Beyond mean "eld theory, the Ginsburg criterion applied to this model tells us that the phase diagram should be stable for physical membranes D"2 at any embedding dimension d, so the mean "eld scenario should give the right qualitative picture for the full model. Numerical simulations have spectacularly con"rmed this result. We have already shown in Fig. 21 the results from the numerical simulation in [91], where it was shown that changing the temperature generates a sequence of transitions crumpled-to-tubular and tubular-to-#at, in total agreement with Case A) in the mean "eld result illustrated in Fig. 22.

Fig. 22. The phase diagram for anisotropic phantom membranes.

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We now turn to a more detailed study of both the crumpled and #at anisotropic phases. Since we have already studied crumpled and #at phases we just outline how those are modi"ed when anisotropy is introduced. 5.1.2. The crumpled anisotropic phase In this phase t '0 and t '0, and the free energy Eq. (39) reduces for D52 to W ,



1 F(ro (x))" d"\x dy[t (R,ro )#t (R ro )]#Irrelevant . , , ? W W 2

(41)

By rede"ning the y direction as y"(t /t )y this reduces to Eq. (10), with t,t . We have proved , W , that anisotropy is totally irrelevant in this particular phase. 5.1.3. The yat phase This phase becomes equivalent to the isotropic case as well. Intuitively, this may be obtained from the fact that if the membrane is #at, the intrinsic anisotropies are only apparent at short distances, and therefore by analyzing the RG #ow at larger and larger distances the membrane should become isotropic. This argument may be made slightly more precise [92]. 5.2. The tubular phase We now turn to the study of the novel tubular phase, both in the phantom case and with self-avoidance. Since the physically relevant case for membranes is D"2 the y-tubular and N-tubular phase are the same. So we concentrate on the properties of the y-tubular phase. The key critical exponents characterizing the tubular phase are the size (or Flory) exponent , giving the scaling of the tubular diameter R with the extended (¸ ) and transverse (¸ ) sizes of the  W , membrane, and the roughness exponent  associated with the growth of height #uctuations h PKQ (see Fig. 23): R (¸ , ¸ )J¸J S (¸ /¸X ) ,  , W , 0 W , h

PKQ

(¸ , ¸ )J¸D S (¸ /¸X ) . , W W F W ,

(42)

Here S and S are scaling functions [93,94] and z is the anisotropy exponent. 0 F

Fig. 23. A schematic illustration of a tubular con"guration indicating the radius of gyration R and the height  #uctuations h . PKQ

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283

Fig. 24. The phase diagram for self-avoiding anisotropic membranes with the Gaussian "xed point (GFP), the tubular phase "xed point (TPFP) and the self-avoidance "xed point (SAFP). Fig. 25. The phase diagram for self-avoiding anisotropic membranes with the Gaussian "xed point (GFP), the tubular phase "xed point (TPFP), the self-avoidance "xed point (SAFP) and the bending rigidity "xed point (BRFP).

The general free energy described in Eq. (39) may be simpli"ed considerably in a y-tubular phase. The analysis required is involved and we refer the interested reader to [95,96]. We just quote the "nal result. It is



1 F(u, ho )" d"\x dy[ (Rho )#t(R ho ) , W ? 2 #g (R u#R ho R ho )#g (R u#(R ho ))] , ? ? W W W  W



b # dy d"\x d"\x B\(ho (x , y)!ho (x , y)) . , , , , 2

(43)

Comparing with Eq. (39), this free energy does represent a simpli"cation as the number of couplings has been reduced from eleven to "ve. Furthermore, the coupling g is irrelevant by standard power , counting. The most natural assumption is to set it to zero. In that case the phase diagram one obtains is shown in Fig. 24. Without self-avoidance b"0, the Gaussian Fixed Point (GFP) is unstable and the infra-red stable FP is the tubular phase FP (TPFP). Any amount of selfavoidance, however, leads to a new FP, the Self-avoiding Tubular FP (SAFP), which describes the large distance properties of self-avoiding tubules. We just mention, though, that other authors advocate a di!erent scenario [94]. For su$ciently small embedding dimensions d, including the physical d"3 case, these authors suggest the existence of a new bending rigidity renormalized FP (BRFP), which is the infra-red FP describing the actual properties of self-avoiding tubules (see Fig. 25). Here we follow the arguments presented in [96] and consider the model de"ned by Eq. (43) with the g -term as the model describing the large distance properties of tubules. One can prove then ,

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Table 2 Di!erent extrapolations for the size exponent at di!erent embedding dimensions d [96] d



$

4

$ 

8 7 6 5 4 3

0.333(5) 0.374(8) 0.42(1) 0.47(1) 0.54(2) 0.62(2)

0.34(1) 0.39(2) 0.44(2) 0.51(3) 0.60(4) 0.71(6)

0.34(1) 0.39(2) 0.44(4) 0.51(5) 0.60(6) 0.70(9)

0.333 0.375 0.429 0.500 0.600 0.750

The "rst column gives the corrections to the mean "eld result, the second corrections to the Flory estimate and the third corresponds to corrections to the Gaussian approximation. The last column quotes the Flory estimate for comparison.

that there are some general scaling relations among the critical exponents. All three exponents may be expressed in terms of a single exponent 3 1!D , " # 2z 2

"z .

(44)

Remarkably, the phantom case as described by Eq. (43) can be solved exactly. The result for the size exponent is 1 5!2D , (2)" (D)" .  .  4 4

(45)

with the remaining exponents following from the scaling relations Eq. (44). The self-avoiding case may be treated with techniques similar to those in isotropic case. The size exponent may be estimated within a Flory approach. The result is D#1 . " $  d#1

(46)

The Flory estimate is an uncontrolled approximation. Fortunately, a
-expansion, adapting the MOPE technique described for the self-avoiding isotropic case to the case of tubules, is also possible [95,96]. The -functions are computed and provide evidence for the phase diagram shown in Fig. 24. Using rather involved extrapolation techniques, it is possible to obtain estimates for the size exponent, which are shown in Table 2. The rest of the exponents may be computed from the scaling relations.

6. Defects in membranes: the crystalline-6uid transition and 6uid membranes A #at crystal melts into a liquid when the temperature is increased. This transition may be driven by the sequential liberation of defects, as predicted by the KTNHY theory. The KTNHY theory is

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Fig. 26. Two stage melting according to KTNHY theory.

schematically shown in Fig. 26. With increasing temperature, a crystal melts "rst to an intermediate hexatic phase via a continuous transition, and "nally goes to a conventional isotropic #uid phase via another continuous transition. We will not review here either the KTNHY theory or the experimental evidence in its favor [97]. We just want to emphasize here that the KTNHY theory is in general agreement with existing experiments, although there are two main points worth keeping in mind when studying the more di$cult case of #uctuating geometries. (1) The experimental evidence for the existence of the hexatic phase is not completely settled in those transitions which are continuous. (2) Some 2D crystals (like Xenon absorbed on graphite) melt to a #uid phase via a "rst-order transition without any intermediate hexatic phase. The straight-forward translation of the previous results to the tethered membrane would suggest a similar scenario. There would then be a crystalline to hexatic transition and a hexatic to #uid transition, as schematically depicted in Fig. 26. Although the previous scenario is plausible, there are no solid experimental or theoretical results that establish it. From the theoretical point of view, for example, an important open problem is how to generalize the RG equations of the KTNHY theory to the case of #uctuating geometry. The situation looks even more uncertain experimentally, especially considering the elusive nature of the hexatic phase even in the case of #at monolayers. In this review we will assume the general validity of the KTNHY scenario and we describe models of hexatic membranes, as well as #uid membranes. The study of the KTNHY theory in #uctuating geometries is a fascinating and challenging problem that deserves considerable e!ort. In this context, let us mention recent calculations of defects on frozen topographies [98], which show that even in the more simpli"ed case when the geometry is frozen, defects proliferate in an attempt to screen out Gaussian curvature, even at zero temperature, and organize themselves in rather surprising and unexpected structures. These results hint at a rich set of possibilities for the more general case of #uctuating geometries. 6.1. Topological defects A crystal may have di!erent distortions from its ground state. Thermal #uctuations are the simplest. Thermal #uctuations are small displacements from the ground state, and therefore one may bring back the system to its original positions by local moves without a!ecting the rest of the lattice. There are more subtle lattice distortions though, where the lattice cannot be taken to its ground state by local moves. These are the topological defects. There are di!erent possible topological defects that may occur on a lattice, but we just need to consider dislocations and disclinations. Let us review the most salient features. E Dislocation: Represents the breaking of the translational holonomy. A path that would naturally close in a perfect lattice fails to close by a vector bo , the Burgers vector, as illustrated in Fig. 27. In

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Fig. 27. Example of a dislocation showing the breaking of the translational holonomy measured by the Burgers vector bo .

Fig. 28. Example of a minus disclination (left "gure) and a plus disclination (right "gure). The orientational holonomy is broken by $
/3 respectively.

a #at monolayer, the energy is




R K bo  E"  ln , a 8


(47)

where K is the Young modulus. It diverges logarithmically with system size.  E Disclination: Represents the breaking of the rotational holonomy. The bond angle around the point defect is a multiple of the natural bond angle in the ground state (
/3 in a triangular lattice), as illustrated in Fig. 28 for a # and ! disclination. The energy for a disclination in a #at monolayer is given by K s E"  R . 32
Note the quadratic divergence of the energy with system size R.

(48)

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Inspection of Fig. 27 shows that a dislocation may be regarded as a tightly bound #,! disclination pair. 6.1.1. Topological defects in yuctuating geometries The problem of understanding topological defects when the geometry is allowed to #uctuate was addressed in [99] (see [100] for a review). The important new feature is that the energy of a disclination defect may be lowered considerably if the membrane buckles out-of-the-plane. That is, the membrane trades elastic energy for bending rigidity. The energy for a buckled free disclination is given by E"f ( , K , q ) ln(R/a) ,  G

(49)

where f is some complicated function that may be evaluated numerically for given values of the parameters. It depends explicitly on q , which implies that the energies for positive and negative G disclinations may be di!erent, unlike the situation in #at space. The extraordinary reduction in energy from R to ln R is possible because the buckled membrane creates positive Gaussian curvature for the plus disclination and negative curvature for the negative disclination. This is a very important physical feature of defects on curved surfaces. The defects attempt to screen out like-sign curvature, and analogously, like-sign defects may force the surface to create like-sign curvature in order to minimize the energy. The reduction in energy for a dislocation defect is even more remarkable, since the energy of a dislocation becomes a constant, independent of the system size, provided the system is larger than a critical radius R . Again, by allowing the possibility of out-of-plane buckling, a spectacular  reduction in energy is achieved (from R to ln R). The study of other topological defects, e.g. vacancies, interstitials, and grain boundaries, may be carried out along the same lines. Since we are not going to make use of it, we refer the reader to the excellent review in [100]. 6.1.2. Melting and the hexatic phase The celebrated Kosterlitz}Thouless argument shows that defects will necessarily drive a 2D crystal to melt. The entropy of a dislocation grows logarithmically with the system size, so for su$ciently high temperature, entropy will dominate over the dislocation energy (Eq. (47)) and the crystal will necessarily melt. If the same Kosterlitz}Thouless is applied now to a tethered membrane, the entropy is still growing logarithmically with the system size, while the energy becomes independent of the system size, as explained in the previous subsection, so any "nite temperature drive the crystal to melt, and the low temperature phase of a tethered membrane will necessarily be a #uid phase, either hexatic if the KTNHY melting can be applied, or a conventional #uid if a "rst-order transition takes place, or even some other more perverse possibility. This problem has also been investigated in numerical simulations [101], and this provides some concrete evidence in favor of the KTNHY scenario, although the issue is far from being settled. It is apparent from these arguments that a hexatic membrane is a very interesting and possibly experimentally relevant membrane to understand.

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6.2. The hexatic membrane The hexatic membrane is a #uid membrane that, in contrast to a conventional #uid, preserves the orientational order of the original lattice (six-fold (hexatic) for a triangular lattice). The mathematical description of a #uid membrane is very di!erent from those with crystalline order. Since the description cannot depend on internal degrees of freedom, the free energy must be invariant under reparametrizations of the internal coordinates (that is, should depend only on geometrical quantities, or in more mathematical terminology, must be di!eomorphism invariant). The corresponding free energy was proposed by Helfrich [102] and it is given by





H & " (g# dx(gHo  , 2 ¹

(50)

where  is the bare string tension, the bending rigidity, g is the determinant of the metric of the surface g (x)"R ro (x)R ro (x) IJ I J

(51)

and Ho is the mean Gaussian curvature of the surface. For a good description of the di!erential geometry relevant to the study of membranes we refer to [103]. A hexatic membrane has an additional degree of freedom, the bond angle, which is introduced as a "eld on the surface . The hexatic free energy [104] is obtained from adjoining to the #uid case of Eq. (50), the additional energy of the bond angle



H K  "  dx (ggIJ(R # !*)(R # !*) , I   I J   J ¹ 2

(52)

where K is called the hexatic sti!ness, and  is the connection two form of the metric, which may  I be related to the Gaussian curvature of the surface by 1
IJR  K(x)" I J (g

(53)

and  is similarly related to the topological defect density   1 s(x)"
IJR  , I  J (g


1 ,  q (x, x ) . s(x)" G G 3 (g G

(54)

From general theorems on di!erential geometry one has the relation



, (gs(x)"4
P  q "2 , G G

(55)

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where  is the Euler characteristic of the surface. Therefore the total free energy for the hexatic membrane is given by





H/¹" (g# dx(gHo  2



K #  dx(ggIJ(R # !*) (R # !*) . I   I J   J 2

(56)

The partition function is therefore

Z()"  ,> \,\ Q y,> >,\ N !N ! ,> ,\ > \ ,> ,\ ; D[ro ]D[]  dx> (g  dx\ (ge\HPo xFx2 , (57) I J I J where y is the fugacity of the disclination density. The partition function includes a discrete sum over allowed topological defects, those satisfying the topological constraint Eq. (55), and a path integral over embeddings ro and bond angles . The previous model remains quite intractable since the sum over defects interaction is very di$cult to deal with. Fortunately, the limit of very low fugacity yP0 is analytically tractable as was shown in the beautiful paper [106]. The RG functions can be computed within a combined large d and large bending rigidity expansion. The  functions in that limit is given by





1 (!D##O(1/K )) , ()"   4
K  (K )"0 , 

(58)

where "1/ . The physics of hexatic membranes in the limit of very low fugacity is very rich and show a line of "xed points parametrized by the hexatic sti!ness K . The normal}normal correla tion function, for example, reads [106] no (r)no (0)&x\E ,

(59)

with "(2/3
)d(d!2)k ¹/K . The FPs of Eq. (58) describe a new crinkled phase, more rigid than  a crumpled phase but more crumpled than a #at one. The Hausdor! dimension at the crinkled phase is given by [106] d(d!2) k ¹ . d "2# & K 3


(60)

From the RG point of view, the properties of these crinkled phases are really interesting, since they involve a line of xxed points which are inequivalent in the sense that the associated critical exponents depend continuously on K , a situation reminiscent of the X>-model. In [105] the 

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Fig. 29. Phase diagram according to [105]. The hexatic membrane interpolates between a crystalline membrane and a #uid one. For small rigidity and large hexatic sti!ness the RG #ows towards a #uid crumpled phase. For small hexatic sti!ness and large it #ows towards a tethered like phase, whereas for both large and K the #ow is to the crinkled  phases described by the non-trivial FP of Eq. (58).

phase diagram is discussed, and the authors propose the scenario depicted in Fig. 29. How these scenarios are modi"ed when the fugacity is considered is not well established and we refer the reader to the original papers [107,108,105]. The shape #uctuations of hexatic vesicles, for large defect core energies, have also been investigated [109,110].

7. The 6uid phase The study of #uid membranes is a broad subject, currently under intense experimental and theoretical work. The Hamiltonian is given by







( H/¹" (g# dx (gHo # (gK . 2 2

(61)

This corresponds to the Helfrich hamiltonian together with a term that allows for topology changing interactions. For "xed topology it is well-known [111}114] that the one-loop beta function for the inverse-bending rigidity has a "xed point only at "0, which corresponds to the bending rigidity being irrelevant at large length scales. The RG #ow of the bending rigidity is given by 3¹ ln(l/a) , (l)" !  4


(62)

where a is a microscopic cuto! length. The #uid membrane is therefore crumpled, for arbitrary microscopic bending rigidity , at length scales beyond a persistence length which grows  exponentially with . For a #uid membrane out-of-plane #uctuations cost no elastic energy (the 

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membrane #ows internally to accommodate the deformation) and the bending rigidity is therefore softened by thermal undulations at all length scales, rather than sti!ening at long length scales as in the crystalline membrane. So far we have assumed an in"nite membrane, which is not always a realistic assumption. A thickness may be taken into account via a spontaneous extrinsic curvature Ho . The model  described by Eq. (61) gets replaced then by







( H/¹" (g# dx (g(Ho !Ho )# (gK .  2 2

(63)

Further e!ects of a "nite membrane size for spherical topology have been discussed in [115}117]. The phase diagram of #uid membranes when topology change is allowed is fascinating and not completely understood. A complete description of these phases goes beyond the scope of this review } we refer the reader to [47,48,21,24,124,125] and references therein.

8. Conclusions In this review we have described the distinct universality classes of membranes with particular emphasis on crystalline membranes. In each case we discussed and summarized the key models describing the interactions of the relevant large distance degrees of freedom (at the micron scale). The body of the review emphasizes qualitative and descriptive aspects of the physics with technical details presented in extensive appendices. We hope that the concreteness of these calculations gives a complete picture of how to extract relevant physical information from these membrane models. We have also shown that the phase diagram of the phantom crystalline membrane class is theoretically very well understood both by analytical and numerical treatments. To complete the picture, it would be extremely valuable to "nd experimental realizations for this particular system. An exciting possibility is a system of cross-linked DNA chains together with restriction enzymes that catalyze cutting and rejoining [118]. The di$cult chemistry involved in these experiments is not yet under control, but we hope that these technical problems will be overcome in the near future. There are several experimental realizations of self-avoiding polymerized membranes discussed in the text. The experimental results compare very well with the theoretical estimates from numerical simulations. As a future theoretical challenge, analytical tools need to be sharpened since they fail to provide a clear and uni"ed picture of the phase diagram. On the experimental side, there are promising experimental realizations of tethered membranes which will allow more precise results than those presently available. Among them there is the possibility of very well controlled synthesis of DNA networks to form physical realizations of tethered membranes. The case of anisotropic polymerized membranes has also been described in some detail. The phase diagram contains a new tubular phase which may be realized in nature [126]. There is some controversy about the precise phase diagram of the model, but de"nite predictions for the critical exponents and other quantities exist. Anisotropic membranes are also experimentally relevant. They may be created in the laboratory by polymerizing a #uid membrane in the presence of an external electric "eld.

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Probably the most challenging problem, both theoretically and experimental, is a complete study of the role of defects in polymerized membranes. There are a large number of unanswered questions, which include the existence of hexatic phases, the properties of defects on curved surfaces and its relevance to the possible existence of more complex phases. This problem is now under intense experimental investigation. In this context, let us mention very recent experiments on Langmuir "lms in a presumed hexatic phase [119]. The coalescence of air bubbles with the "lm exhibit several puzzling features which are strongly related to the curvature of the bubble. Crystalline membranes also provide important insight into the #uid case, since any crystalline membrane eventually becomes #uid at high temperature. The physics of #uid membranes is a complex and fascinating subject in itself which goes beyond the scope of this review. We highlighted some relevant experimental realizations and gave a quick overview of the existing theoretical models. Due to its relevance in many physical and biological systems and its potential applications in material science, the experimental and theoretical understanding of #uid membranes is, and will continue to be, one of the most active areas in soft condensed matter physics. We have not been able in this review, simply for lack of time, to address the important topic of the role of disorder. We hope to cover this in a separate article. We hope that this review will be useful for physicists trying to get a thorough understanding of the fascinating "eld of membranes. We think it is a subject with signi"cant prospects for new and exciting developments. Note: The interested reader may also "nd additional material in a forthcoming review by Wiese [120], of which we have seen only the table of contents.

Acknowledgements This work was supported by the US Department of Energy under contract No. DE-FG0285ER40237. We would like to thank Dan Branton and Cyrus Sa"nya for providing us images from their laboratories and Paula Herrera-SikloH dy for assistance with the "gures. MJB would like to thank Riccardo Capovilla, Chris Stephens, Denjoe O'Connor and the other organizers of RG2000 for the opportunity to attend a wonderful meeting in Taxco, Mexico.

Appendix A. Useful integrals in dimensional regularization In performing the
-expansion, we will be considering integrals of the form



q  2q L ? ? . I  2 L (a, b)(po )" d"q( ?  ? (po #qo )?qo @

(A.1)

These integrals may be computed exactly for general D, a, b and "1,2, N. The result will be published elsewhere. We will content ourselves by quoting what we need, the poles in
, for the

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integrals that appear in the diagrammatic calculations. We just quote the results 1 1 I   (2, 2)"! p p , ?? 8
p ? ?


(A.2)

1 1 I    (2, 2)" p p p  , ??? ? ? ?
8
p

(A.3)






1 p 1 I     (2, 2)"! p p p p ! ( #     #     ) , ???? ?? ?? ?? ??
8
p ? ? ? ? 24 ? ? ? ?






p 1 1 (p    #p    #p    )!p  p  p  . I    (2, 1)"! ? ? ? ? ? ? ? ? ? ? ? ? ???
8
p 6

(A.4)

(A.5)

Appendix B. Some practical identities for RG quantities The beta functions de"ned in Eq. (3) may be re-expressed as




R ln u  (u , v ) S 0 0 "!
Ru0 R ln v  (u , v ) T 0 0 Ru 0








R ln u Rv \ 1 0 . R ln v 1 Rv 0




(B.1)

The previous expression may be further simpli"ed noticing




R ln u Ru 0 A" R ln v Ru 0 so that




R ln u 1 Rv 0 " u0 R ln v 0 Rv 0





A\" 1#




"

u 0 0

u 0 0

0

0 v 0

v

0

1 v 0

#D ,


D


!

0

u 0 0

\ u0 0 0 v 0

0 v 0


D

u 0 0

(B.2)

 0 v

0



#2 ,

(B.3)

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where the last result follows from Taylor expansion. These formulas easily allow to compute the corresponding -functions. If

 

 

1 u"MC u # (a u #a u v #a v ) , 0
 0  0 0  0 1 v"MC v # (a u #a u v #a v ) , 0
 0  0 0  0

(B.4)

from Eqs. (B.3) and (B.1) we easily derive the leading two orders in the couplings  (u , v )"!
u #a u #a u v #a v , S 0 0 0  0  0 0  0  (u , v )"!
v #a u #a u v #a v . T 0 0 0  0  0 0  0

(B.5)

The formula for  in Eq. (3) may also be given a more practical expression. It is given by




" 



R R # ln Z . S Ru T Rv ( 0 0

(B.6)

Those are the formulas we need in the calculations we present in this review.

Appendix C. Discretized model for tethered membranes In this appendix we present appropriate discretized models for numerical simulation of tethered , with membranes. The surface is discretized by a triangular lattice de"ned by its vertices ro  ?2 a corresponding discretized version of the Landau elastic term Eq. (20) given by [99]  F "  (ro !ro !1) , ? @ Q 2 6?@7

(C.1)

where a, b are nearest-neighbor vertices. If we write ro "x #u with x de"ning the vertices of ? ? ? ? a perfectly regular triangular lattice and u the small perturbations around it, one gets ro !ro "1#u x?x@#2 , @ ?@ ?

(C.2)

with u being a discretized strain tensor and we have neglected higher-order terms. Plugging the ?@ previous expression into Eq. (C.1) and passing from the discrete to the continuum language, we obtain



(3 F"  dx(2u #u ) Q ?@ ?? 8 which is the elastic part of the free energy Eq. (20) with ""((3/4).

(C.3)

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The bending rigidity term is written in the continuum as





S " dx (gKI K?@" du (gg?@ no  no , CVR ?@ I ? @

(C.4)

where no is the normal to the surface and  is the covariant derivative (see [121] for a detailed description of these geometrical quantities). We discretize the normals from the previous equation by



du (gg?@ no  no P  (no !no )"2  (1!no no ) . ? @ @ ? ? @ 6?@7 6?@7

(C.5)

The two terms, Eqs. (C.1) and (C.5), provide a suitable discretized model for tethered membranes. However, in actual simulations, the even more simpli"ed discretization F"  (ro !ro )#  (1!no no ) ? @ G H 6?@7 6GH7

(C.6)

is preferred since it is simpler and describes the same universality class (see [37] for a discussion). Anisotropy may be introduced in this model by ascribing distinct bending rigidities to bending across links in di!erent intrinsic directions [91]. Self-avoidance can be introduced in this model by imposing that the triangles that de"ne the discretized surface cannot self-intersect. There are other possible discretizations of self-avoidance that we discuss in Section 4.2. In order to numerically simulate the model Eq. (C.6) di!erent algorithms have been used. A detailed comparison of the performance of each algorithm may be found in [122].

Appendix D. The crumpling transition The free energy is given by Eq. (13)

 

F(ro )" d"x








1 

(Rro )#u R ro R ro ! ?@ (R o ) #v(R ro R?ro ) , ? ? @ 2 ? D ?P

(D.1)

where the dependence on is trivially scaled out. The Feynman rules for the model are given in Fig. 30. We need three renormalized constants, namely Z, Z and Z in order to renormalize the theory. S T We de"ne the renormalized quantities by ro "Z\ro , u "M\CZZ\u, v "M\CZZ\v . 0 S 0 T

(D.2)

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Fig. 30. Feynman rules for the model at the crumpling transition.

Fig. 31. Diagrams to consider at one loop.

Then Eq. (D.1) becomes

 

F(ro )" d"x








Z 

(Rro )#MCZ u R ro R ro ! ?@ (R ro ) #MCZ v(R ro R?ro ) . T ? 0 0 S ? 0 @ 0 2 ? 0 D ? 0

(D.3)

In order to compute the renormalized couplings, one must compute all relevant diagrams at one loop. Those are depicted in Fig. 31. Within dimensional regularization, diagrams (1a) and (1b) are zero, which in turns imply that the renormalized constant is Z"1 at leading order in
, similarly as in linear models. Using the integrals in dimensional regularization (see Appendix A) diagram (2a) gives the result










u u d 1

G G H H (po ) po po ) po #po ) po po ) po )# v! po ) po po ) po .     24     48     8



(D.4)

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And diagrams (2b) and (2c) may be computed at once, since the result of (2c) is just (2b) after interchanging po po , so the total result (2a)#(2b) is   1 1

G G H H 16









#

 

7 v 61 u# uv# ( po ) po po ) po #po ) po po ) po )         12 6 96



v uv u # # po ) po po ) po . 6 12 96    

(D.5)

And the result for (2d) and (2e) is just identical, so the total result (2d)#(2e) is 1 1

G G H H 16








u 1 # uv (po ) po po ) po #po ) po po ) po )         24 6








13 u # v# uv! po ) po po ) po . 6 48    

(D.6)

Adding up all these contributions taking into account the di!erent combinatorial factors (4 the "rst contribution, 8 the last two ones) and recalling Z"1, we get

 






1 1 u"MC u # 0
8




 

d 65 4 # u #6u v # v 0 0 3 P 3 12 0

1 1 21 21 u # u v #(4d#5)v v"MC v # 0
8
 16 0 0 2 0 0

,

.

(D.7)

The resultant -functions are then readily obtained by applying Eq. (B.5).

Appendix E. The 6at phase The free energy is given in Eq. (20), and it is given by

 

F(u, h)" d"x



 ( (R R h)#u u?@# (u? ) . ?@ 2 ? 2 ? @

(E.1)

The Feynman rules are shown in Fig. 32, it is apparent that the in-plane phonons couple di!erent from the out-of-plane, which play the role of Goldstone bosons. We apply standard "eld theory techniques to obtain the RG-quantities. Using the Ward identities, the theory can be renormalized using only three renormalization constants Z, Z and Z , I H corresponding to the wave function and the two coupling renormalizations. Renormalized quantities read h "Z\h, u"Z\u , 0  "M\CZZ\,  "M\CZZ\ . 0 I 0 H

(E.2)

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Fig. 32. Feynman rules for the model in the #at phase.

Fig. 33. Diagrams to consider at one loop.

Then Eq. (E.1) becomes



F(u, h)" d"x[Z(R R h )#2MCZ  u u?@#MCZ (u? )] . ? @ 0 I 0 0?@ 0 H 0?

(E.3)

We now compute the renormalized quantities from the leading divergences appearing in the Feynman diagrams. The diagrams to consider are given in Fig. 33. These can be computed using the integrals given in Appendix A. The result of diagram (1a) is given by








p p p p 1 d   ! ? @ #3(#2#2) ? @ p . ?@ p p
6


(E.4)

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Diagrams (2b) and (2c) are identically zero, so (2a) is the only additional diagram to be computed. The result is 1 GH (#) 10(p) !
8
 2#

(E.5)

from Eq. (E.5) and the de"nitions in Eq. (E.2) 1 10  ( # ) 0 0 0 . Z"1!
8
 2 # 0 0

(E.6)

Using the previous result in diagrams (1a) whose result is in Eq. (E.4) we obtain 1 d   , Z "1# I
24
 0 1 d  ( #6  #6 )/ Z "1# H 0 0 0 0
24
 0

(E.7)

and we deduce the renormalized couplings

 





 

1 10 ( # ) d 0 0 #   , "MC  # 0
4
 2 # 24
 0 0 0



1 10 ( # ) d 0 0   #  ( #6  #6 ) "MC  # 0
4
 2 # 0 0 24
 0 0 0 0 0 0

(E.8)

from which the -functions trivially follow with the aid of Eq. (B.5).

Appendix F. The self-avoiding phase The model has been introduced in Eq. (31) and is given by





b 1 F(ro )" d"x(R ro (x))# d"x d"y B(ro (x)!ro (y)) . ? 2 2

(F.1)

We follow the usual strategy of de"ning the renormalized quantities by ro "Zro , 0 b"MCZ ZBb , @ 0

(F.2)

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and the renormalized free energy by





b Z F(ro )" d"x(R ro (x))#MCZ 0 d"x d"y B(ro (x)!ro (y)) . ? 0 @2 0 0 2

(F.3)

The -function being non-local adds some technical di$culties to the calculation of the renormalized constants Z and Z . There are di!erent approaches available but we will follow the @ Multilocal-Operator-Product-Expansion (MOPE), which we will just explain in a very simpli"ed version. A rigorous description of the method may be found in the literature. The idea is to expand the -function term in Eq. (F.3)








L  b e\$Po "e\8 "x.? Po 0 x ;  !MCZ 0 d"x d"y B(ro (x)!ro (y)) , @2 0 0 L

(F.4)

with this trick, the delta-function term may be treated as expectation values of a Gaussian free theory. This observation alone allows to isolate the poles in
. We write the identity e Io Po x \Po x ": e Io Po x \Po x : eI%V \V  ,

(F.5)

where G(x) is the two point correlator x\" , !G(x)"!ro (x)ro (0)" (2!D)S "

(F.6)

with S being the volume of the D-dimensional sphere. The symbol :: stands for normal ordering. " A normal ordered operator is non-singular at short distances, so it may be Taylor expanded e Io Po x \Po x "(1#i(x !x )?(ko R ro )   ? ! (x !x )?(x !x )@(ko R ro )(ko R ro )#2)eI%V \V  .    ? @  

(F.7)

To isolate the poles in
we do not need to consider higher-order terms as it will become clear. If we now integrate over ko , we get 1 1

B(ro (x )!ro (x ))"   (4
)B(!G(x !x ))B   !

(x !x )?(x !x )@ 1     R ro (x)R ro (x)#2 ? 4 (4
)B(!G(x !x ))B> @  

,C(x !x )1#C?@(x !x )R ro (x)R ro (x)#2 ,     @ ?

(F.8)

where we omit higher-dimensional operators in ro , which are irrelevant by power counting, so since the theory is renormalizable they cannot have simple poles in
. Additionally, we have de"ned

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Fig. 34. Diagrammatic expansion to isolate the poles in
within the MOPE formalism at lowest non-trivial order. Solid lines represent -function terms and dashed lines indicate that points inside are taken arbitrarily close. Higher-orders contributions arise in the same way.

x"(x #x )/2. One recognizes in Eq. (F.8) Wilson's operator product expansion, applied to the   non-local delta-function operator. Following the same technique of splitting the operator into a normal ordered part and a singular part at short distances, it just takes a little more e!ort to derive the OPE for the product of two delta functions, the result is

B(ro (x )!ro (y )) B(ro (x )!ro (y ))"C(x !x , y !y ) B(ro (x)!ro (y))#2        

(F.9)

with 1 . C(x !x , y !y )"     (4
)B(!G(x !x )!G(y !y ))B     The terms omitted are again higher dimensional by power counting and so do not contribute at this order. The OPE Eqs. (F.8) and Eq. (F.9) is all we need to compute the renormalization constants at lowest order in
, but the calculation may be pursued to higher orders in
. In order to do that, one must identify where poles in
arise. In the previous example poles in
appear whenever the internal coordinates (x and x in Eq. (F.8), x , x , y and y in Eq. (F.9)) are pairwise       made to coincide. This is diagrammatically shown in Fig. 34. It is possible to show, that higher poles appear in the same way, if more -product terms are considered. Let us consider the "rst delta-function term corresponding to n"1 in the sum Eq. (F.4). Using Eq. (F.8) we have



b MC ! P Z d"x d"y B(ro (x)!ro (y)) @ 0 0 2

 

b MC "! P Z d"x d"y(C(x!y)#C?@(x!y)R ro (x)R ro (x)#2 ? 0 @ @ 0 2



b MC

C?@ "! P Z d"x R ro (x) R?ro (x) d"y ?@ #2 . 0 @ ? 0 2 D

(F.10)

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The "rst term just provides a renormalization of the identity operator, which we can neglect. From



 + y






C?@ 1 M\C 2
" >B d"y ?@ (y)"! (4
)\B(2!D)>B , D 4D
(D/2)

(F.11)

and we can absorb the pole by Z if we de"ne






2
" >B b (4
)\B . (2!D)>B Z"1# 0 4D (D/2)


(F.12)

From the short distance behavior in the sum Eq. (F.4) corresponding to n"2 we get

 

bMC ! P d"x d"y d"x d"y B(ro (x )!ro (y )) B(ro (x )!ro (y )), 0  0  0  0      8



bMC ! P d"x d"y B(ro (x)!ro (y)) d"z d"w C(z, w) , 0 0 8

(F.13)

where, in order to isolate the pole we can perform the following tricks:



d"z d"w C(z, w)







   

  

2
"  +\ +\ z"\w"\ dz dw " (4
)\BSB(2!D)B " (D/2) (z\"#w\")B   2
"  M\C   x"\"y"\" " (4
)\BSB(2!D)B dx dy " (D/2) (2!D) (x#y)\B   2
"  M\C " (4
)\BSB(2!D)B " (D/2) (2!D)

dx dy

V>WX

x"\"y"\" (x#y)\B

2
"  M\C (D/2!D) 1 " (4
)\BSB(2!D)B " (D/2) (2!D) (2D/2!D)


(F.14)

since changing the boundary of integration from a square to a circle does not a!ect the residue of the pole. We "nally have






(D/2!D) 2
" >B b 1 (4
)\B , Z "1# 0 (2!D)\>B @ (2D/2!D) (D/2)
2

(F.15)

and the -function follows from the de"nitions Eq. (F.2) together with Eq. (F.12) and Eq. (B.5).

M.J. Bowick, A. Travesset / Physics Reports 344 (2001) 255}308

303

Appendix G. The mean 5eld solution of the anisotropic case The free energy has been introduced in Eq. (39). Let us "rst show the constraints on the couplings so that the free energy is bounded from below. E u '0: This follows trivially. WW E u ,v #u /(D!1)'0: De"ne AG "R rG(x) then from Eq. (39) we get ,, ,, ,, ? ? v u u ,, #v ,, Tr(AA2)# ,, (Tr AA2)5 2(Tr AA2) ,, 2 D!1 2






u " ,, (Tr AA2) , 2 which implies u '0. ,, E v '!(u u ): de"ning bo "R ro (x), it is derived from ,W ,, WW W u u ,, (Tr(A2A))# WW (b)#v b2b Tr AA2'0 . ,W 2 2 Introducing the variables u v # ,, v ,, D!1 ,W , b"(t , t ) A" , W v u ,W WW and w"((D!1) , ), the mean "eld e!ective potential may be written as , W <(w)"¸"\¸ [wb#wAw] . W   , In this form, it is easy to "nd the four minima of Eq. (G.4), which are






(G.1)

(G.2)

(G.3)

(G.4)

1. Crumpled phase:  "0, , < "0 .

 "0, W 2. Flat phase: u t !v t , W,  "! WW , , (D!1)

(G.5)

<



¸"\¸ W [u t#u t !2v t t ] . "! , ,, W WW , ,W , W 4

!v t #u t ,W , ,, W , " W  3. N-Tubule: !t ,,  " ¸"\¸ , u W < "! , ,,

 4 "0, W

t , . u ,,

(G.6)

(G.7)

304

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4. y-Tubule:  "0, ¸"\¸ , W < "! , t

 4 "! W , W u WW

t W . u WW

(G.8)

The regions in which each of the four minima prevails depend on the sign of . E '0: Let us see under which conditions the #at phase may occur. We must satisfy the equations u t (v t , WW , ,W W

u t (v t . ,, W ,W ,W

(G.9)

If v '0 these inequalities can only be satis"ed if both t and t have the same sign. If they are ,W , W positive, the equations (G.9) imply t (0 or t (0, which by the assumption '0 cannot , W be satis"ed. The #at phase exists for t (0 and t (0 and satisfying Eq. (G.9). If t '0 and W , W t '0 then the #at phase or the tubular cannot exist (see Eqs. (G.7) and (G.8)) so those are the , conditions for the crumpled phase. Any other case is a tubular phase, either N-tubule or y-tubule, depending on which of the inequalities Eq. (G.9) is not satis"ed. If v (0 it is easily ,W checked from Eq. (G.6) that the #at phase exists as well and the same analysis apply. E (0: From inequality (G.2) we have v '0. The inequalities are now ,W u t ( WW t , W v , ,

v t ' ,W . W u t ,,Y ,W

(G.10)

Now, in order to have a solution for both inequalities we must have u /v 'v /u which WW , ,W ,, requires '0. This proves that the #at phase cannot exist. There is then a crumpled phase for t '0 and t '0 and tubular phase when either one of these two conditions are not satis"ed. W ,

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Physics Reports 344 (2001) 309}353

Supersymmetry in particle physics: the renormalization group viewpoint夽 D.I. Kazakov Bogoliubov Laboratory of Theoretical Physics (BLTP), Joint Institute of Nuclear Physics (JINR), 141 980 Dubna and ITEP, Moscow Region, Russia Received June 2000; editor: I. Procaccia

Contents 1. Introduction 1.1. The Standard Model 1.2. RG #ow in the Standard Model 2. Supersymmetry 2.1. Motivations of SUSY 2.2. Global SUSY: algebra and representations 2.3. SUSY Lagrangians 3. Minimal Supersymmetric Standard Model 3.1. The "eld content 3.2. Lagrangian of the MSSM 3.3. Soft SUSY breaking 3.4. Masses 3.5. RG #ow in the MSSM

311 311 314 317 317 317 320 321 322 323 324 325 328

4. Renormalization of softly broken SUSY theories 5. RG #ow for the soft terms 5.1. Radiative electroweak symmetry breaking 6. Infrared quasi-"xed points 6.1. Low tan
regime 6.2. Large tan
regime 7. Higgs boson mass prediction in the SM and MSSM 7.1. The Higgs boson mass in the SM 7.2. The Higgs boson mass in the MSSM 8. Conclusion Acknowledgements References

330 332 333 334 334 338 344 344 345 351 351 351

Abstract An attempt is made to present modern hopes to "nd manifestation of supersymmetry, a new symmetry that relates bosons and fermions, in particle physics from the point of view of renormalization group #ow.

夽 Review talk given at the Conference `Renormalization Group at the Turn of the Millenniuma, Taxco, Mexico, January 1999. E-mail address: [email protected] (D.I. Kazakov).

0370-1573/01/$ - see front matter  2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 0 0 ) 0 0 1 2 9 - 0

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The Standard Model of particle interactions is brie#y reviewed, the main notions of supersymmetry are introduced. In more detail the RG #ow in the Minimal Supersymmetric Standard Model is considered. Predictions of particle spectrum are obtained based on the RG "xed points.  2001 Elsevier Science B.V. All rights reserved. PACS: 11.10.Gh; 11.10.Hi; 11.30.Pb Keywords: Renormalization group; Supersymmetry; Fixed point; Minimal supersymmetric Standard Model

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311

1. Introduction 1.1. The Standard Model The Standard Model (SM) of fundamental interactions describes strong, weak and electromagnetic interactions of elementary particles [1]. It is based on a gauge principle, according to which all the forces of Nature are mediated by an exchange of the gauge "elds of a corresponding local symmetry group. The symmetry group of the SM is S; (3)S; (2); (1) ,     

(1.1)

whereas the "eld content is the following: Gauge sector: Spin"1. The gauge bosons are spin 1 vector particles belonging to the adjoint representation of group (1.1). Their quantum numbers with respect to S;(3)S;(2);(1) are: G? : (8, 1, 0) S; (3) g , I A Q   intermediate weak bosons =G : (1, 3, 0) S; (2) g , I *   abelian boson B : (1, 1, 0) ; (1) g , I 7   where the coupling constants are usually denoted by g , g and g, respectively. Q Fermion sector: Spin"1/2. The matter "elds are fermions belonging to the fundamental representation of the gauge group. These are believed to be quarks and leptons at least of three generations. The SM is left}right asymmetric. Left- and right-handed fermions have di!erent quantum numbers. Quarks: gluons







;G uG cG tG , , ,2(3, 2, 1/3) QG " ? " ?* DG * dG * sG * bG *   ? ;G "u , c , t ,2(3H, 1, 4/3) ?0 G0 G0 G0   DG "d , s , b ,2(3H, 1,!2/3) ?0 G0 G0 G0   Leptons:






   ¸ " C , I , O ,2(1, 2,!1) ?* e *  *  *   E "e ,  ,  ,2(1, 1,!2) ?0 0 0 0   i"1, 2, 3 } colour, "1, 2, 3,2 } generation. Higgs sector: Spin"0. In the minimal version of the SM there is one doublet of Higgs scalar "elds




H"

H

H\

(1, 2,!1) ,  

(1.2)

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which is introduced in order to give masses to quarks, leptons and intermediate weak bosons via spontaneous breaking of electroweak symmetry. In Quantum Field Theory framework the SM is described by the following Lagrangian: L"L #L #L ,   7  &  L "!G? G? !=G =G !B B    IJ IJ  IJ IJ  IJ IJ #i¸M ID ¸ #iQM ID Q #iEM ID E ? I ? ? I ? ? I ? # i;M ID ; #iDM ID D #(D H)R(D H) , ? I ? ? I ? I I where

(1.3)

(1.4)

G? "R G? !R G? #g f ?@AG@ GA , IJ I J J I Q I J =G "R =G !R =G #g GHI=H =I , IJ I J J I I J B "R B !R B , IJ I J J I g g D ¸ " R !i G=G #i B ¸ , I ? I I 2 2 I ?






D E "(R #igB )E , I ? I I ? g g g D Q " R !i G=G !i B !i Q ?G? Q , I ? I I I ? 2 6 I 2






 



2 g D ; " R !i gB !i Q ?G? ; , I I I ? I ? 3 2 1 g D D " R #i gB !i Q ?G? D , I I ? I ? 3 I 2 "y* ¸M E H#y" QM D H#y3 QM ; HI #h.c. , 7  ?@ ? @ ?@ ? @ ?@ ? @ where HI "i HR . 

L "!<"mHRH! (HRH) . &  2 L

(1.5)

(1.6)

Here y are the Yukawa and is the Higgs coupling constants, both dimensionless, and m is the only dimensional mass parameter. The Lagrangian of the SM contains the following set of free parameters: E E E E E

three gauge couplings g , g, g; Q three Yukawa matrices y* , y" , y3 ; ?@ ?@ ?@ Higgs coupling constant ; Higgs mass parameter m; number of matter "elds (generations).  We use the usual for particle physics units c" "1.

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313

All the particles obtain their masses due to spontaneous breaking of S; (2) symmetry group  via a non-zero vacuum expectation value (v.e.v.) of the Higgs "eld




H"

v

0

, v"m/( .

(1.7)

As a result the gauge group of the SM is spontaneously broken down to S; (3)S; (2); (1)NS; (3); (1) . A * 7 A #+ The physical weak intermediate bosons are the linear combinations of the gauge ones = Gi= I , Z "!sin  B #cos  = =!" I I I 5 I 5 I (2

(1.8)

with masses 1 gv, m "m /cos  , tan  "g/g , m " 8 5 5 5 5 (2

(1.9)

while the photon "eld  "cos  B #sin  = I 5 I 5 I

(1.10)

remains massless. The matter "elds acquire masses proportional to the corresponding Yukawa couplings: MS "yS v, MB "yB v, MJ "yJ v, m "(2m . ?@ ?@ ?@ ?@ ?@ ?@ &

(1.11)

Explicit mass terms in the Lagrangian are forbidden because they are not S; (2) symmetrical and  would destroy the renormalizability of the Standard Model. The SM has been constructed as a result of numerous e!orts both theoretical and experimental. At present the SM is extraordinarily successful, the achieved accuracy of its predictions corresponds to the experimental data within 5% [2]. All the particles except for the Higgs boson have been discovered experimentally. However the SM has its natural drawbacks and unsolved problems. Among them are: E large number of free parameters, E formal uni"cation of strong and electroweak interactions, E the Higgs boson has not yet been observed and it is not clear whether it is fundamental or composite, E the problem of CP-violation is not well understood including CP-violation in strong interaction, E #avour mixing and the number of generations are arbitrary, E the origin of the mass spectrum is unclear. The answer to these problems lies beyond the SM.

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1.2. RG yow in the Standard Model The question is: how to go beyond the SM? Apparently, we are talking about new particles, new structures, new interactions and new symmetries. The answer is not obvious. We describe below one of the options and assume that at high energies (small distances) fundamental interactions possess wider symmetries, in particular, new kind of symmetry, symmetry between bosons and fermions, called sypersymmetry. This is a wide subject by itself, however, we would like to look at it from the point of view of renormalization group. Let us try to go along the road o!ered by the renormalization group #ow. Let us take the Lagrangian of the SM and see what happens with its parameters (the couplings) when the energy scale increases. As it follows from Eq. (1.1) one has three gauge couplings corresponding to S;(3) ;S;(2) ;;(1) gauge groups, g , g and g, respectively. In what follows, A * 7 Q it will be more useful to consider the set g , g , g "(3/5g, g, g . Besides, there are three Q    Yukawa couplings y , y and y , which are 3;3 matrices in the generation space and one Higgs 3 " * coupling . To simplify the picture we consider only the third generation Yukawa couplings replacing the Yukawa matrices by their (3, 3) elements. Then the RG equations in the leading one-loop order are [3]: da G "b a, t"log(Q/) , G G dt






(1.12)

 

d> 9 17 9 R "!> 8a # a # a ! > , R    dt 4 20 2 R d> 9 1 3 @ "!> 8a # a # a ! > , @  4  4  2 R dt

(1.13)



9 9 d> O "!> a # a !3> , O 4  4  R dt d 9 9 27 9 9 " a # a a # a ! a ! a #6 > !6>#6  , R R dt 8  20   200  2  10 

(1.14)

where to simplify the formulas we have used the notation  g a, G, G G 4
16


(i"1, 2, 3),

(yI ) > ,  I 16


(k"t, b, ) .

For the SM the coe$cients b are G b 0 4/3 1/10  b " b " !22/3 #N 4/3 #N 1/6 . $ &  G  b !11 4/3 0 







(1.15)

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315

Here N is the number of generations of matter multiplets and N is the number of Higgs $ &  doublets. We use N "3 and N "1 for the minimal SM. $ &  The initial conditions for the couplings can be taken at some scale which is experimentally favourable. Thus, for the gauge couplings one has precise measurement at the Z-boson mass scale obtained at LEP accelerator [4]:  (M )"0.017,  (M )"0.034,  (M )"0.118$0.005 . (1.16)  8  8  8 As for the Yukawa couplings, they are related to the running quark masses by Eq. (1.11), where v is the vacuum expectation value of the Higgs "eld. It can be calculated, for instance, from the Z-boson mass according to Eq. (1.9) and is equal to v"174.1 GeV. Thus, knowing the quark masses one can "nd the values of the Yukawa couplings. One should, however, distinguish between the running and the pole quark masses which are determined experimentally. Having all this in mind and solving Eqs. (1.12) and (1.13) one has the following qualitative picture (see Fig. 1). The behaviour of the Higgs quartic coupling strongly depends on initial conditions which are unknown in this case. We return to this subject later. The qualitative picture presented in Fig. 1 contains an obvious uni"cation pattern. The three gauge couplings are seen to unify at energy of the order of 10}10 GeV and so do the Yukawa couplings > and > . What does it mean? The usual answer is given in the framework of the Grand @ O Uni"cation hypothesis [5]: three-gauge interactions are the three branches of a single-gauge interaction described by a simple gauge group with a single coupling. All quarks and leptons belong to some representation of this group. This explains the equality of gauge and (some) Yukawa couplings at the uni"cation scale. The GUT hypothesis has many far reaching consequences, however, one can see that the uni"cation scale is very high. This is not only di$cult to check experimentally, but creates a big problem, called the hierarchy problem. The point is that in a theory with two so very di!erent scales: M &10 GeV and 5 M &10 GeV, it is very di$cult both to achieve this hierarchy of 10 in a natural way and to %32 preserve it against the radiative corrections. Indeed, due to modern point of view, the mass scales in the SM and in GUT are de"ned by vacuum expectation values of the scalar "elds, called the Higgs "elds. Non-zero v.e.v.'s of these "elds lead to spontaneous breaking of the corresponding gauge symmetry and provide masses to all the particles. So, we have at least two scalar particles with the masses of the order of 10 and 10 GeV. However, these masses obtain the radiative corrections proportional to the masses of the interacting particles. Due to inevitable interaction between the light and heavy "elds the radiative corrections to the light Higgs mass are proportional to the heavy one m&gM , where g is some coupling. Assuming m&10 GeV, M&10 GeV, g&0.1, one gets the radiative correction which is 10 times bigger than the mass itself. This correction obviously spoils the hierarchy unless it is cancelled. A cancellation with a precision &10\ needs a very accurate "ne tuning of the coupling constants. Solution to the "ne-tuning problem has been found in the framework of a revolutionary hypothesis: the existence of a new type of symmetry, the symmetry between bosons and fermions, called supersymmetry.

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Fig. 1. Evolution of the gauge (left) and Yukawa couplings (right) in the Standard Model. The width of the curves corresponds to the experimental error.

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317

2. Supersymmetry 2.1. Motivations of SUSY Supersymmetry or fermion}boson symmetry has not yet been observed in Nature. This is a purely theoretical invention [6]. Its validity in particle physics follows from common belief in uni"cation. The general idea is a uni"cation of all forces of Nature. It de"nes the strategy: increasing uni"cation towards smaller distances up to l &10\ cm including quantum gravity. However, .J the graviton has spin 2, while the other gauge bosons (photon, gluons, = and Z weak bosons) have spin 1. Uni"cation of spins 2 and 1 gauge forces within unique algebra is forbidden due to the no-go theorems for any symmetry but SUSY. If Q is a generator of SUSY algebra, then Qboson"fermion and Qfermion"boson . Hence starting with the graviton spin 2 state and acting by SUSY generators we get the following chain of states: spin 2Pspin 3/2Pspin 1Pspin 1/2Pspin 0 . Thus, a partial uni"cation of matter (fermions) with forces (bosons) naturally arises out of an attempt to unify gravity with the other interactions. The uniqueness of SUSY is due to a strict mathematical statement that algebra of SUSY is the only graded (i.e. containing anticommutators as well as commutators) Lie algebra possible within relativistic "eld theory [7]. The other motivation of SUSY is the solution of the hierarchy problem mentioned above. At the moment supersymmetry is the only known way to achieve the cancellation of quadratic terms in radiative corrections (also known as the cancellation of the quadratic divergences). Moreover, SUSY automatically cancels quadratic corrections in all orders of perturbation theory [8]. 2.2. Global SUSY: algebra and representations As can be easily seen, supersymmetry transformations di!er from ordinary global transformations as far as they convert bosons into fermions and vice versa. Indeed if we symbolically write SUSY transformation as B" ) f , where B and f are boson and fermion "elds, respectively, and is an in"nitesimal transformation parameter, then from the usual (anti)commutation relations for (fermions) bosons  f, f "0, [B, B]"0 , we immediately "nd  , "0 .

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This means that all the generators of SUSY must be fermionic, i.e. they must change the spin by a half-odd amount and change the statistics. Combined with the usual PoincareH and internal symmetry algebra the Super-PoincareH Lie algebra contains additional generators [6]: [QG , P ]"[QM G , P ]"0 , ? I ? I [QG , M ]"( )@QG , [QM G , M ]"!QM GQ ( )@ Q , ? IJ  IJ ? @ ? IJ  @ IJ ? [QG , B ]"(b )G QH , [QM G , B ]"!QM H (b )G , ? P PH ? ? P ? PH QG , QM HQ "2GH(I) Q P , ? @ ?@ I QG , QH "2 ZGH, Z "aP b , ZGH"Z> , ? @ ?@ GH GH P GH QM G , QM HQ "!2  Q ZGH, [Z , anything]"0 , ? @ ?@ GH , ,
,
Q "1, 2, i, j"1, 2,2, N .

(2.1)

Here P and M are four-momentum and angular momentum operators respectively, B are I IJ P internal symmetry generators, QG and QM G are spinorial SUSY generators and Z are the so-called GH central charges. , ,
,
Q are spinorial indices. In the simplest case one has one spinor generator Q (and the conjugated one QM  ) that corresponds to an ordinary or N"1 sypersymmetry. When ? ? N'1 one has an extended sypersymmetry. An elegant formulation of supersymmetry transformations and invariants can be achieved in the framework of superspace [9]. Superspace di!ers from the ordinary Euclidean (Minkowski) space by addition of two new coordinates,  and M  , which are grassmannian, i.e. anticommuting, ? ? variables  ,  "0, M  , M Q "0, "0, ? @ ? @ ? Thus, we go from space to superspace

M  "0, ,
, ,
Q "1, 2 . ?

Space N Superspace . x ,  , M  I ? ? A SUSY group element can be constructed in superspace in the same way as an ordinary translation in the usual space x

I

G(x, , M )"e \VI.I >F/>FM /M  . It leads to the supertranslation in superspace, x Px #i  !i  M , I I I I P# , M PM #  ,

(2.2)

where and  are grassmannian transformation parameters. Taking them to be local or space}time dependent one gets local translation. And the theory that is invariant under local translations is

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319

general relativity. Thus, local supersymmetry is just the theory of gravity or supergravity [18]. This way following the gauge principle one gets a uni"ed theory of all four interactions known as SUGRA theory. To de"ne the "elds on a superspace consider representations of the Super-PoincareH group (2.1) [10]. The simplest one is a scalar super"eld F(x, , M ) which is SUSY invariant. Its Taylor expansion in  and M has only several terms due to the nilpotent character of grassmannian parameters. However, this super"eld is a reducible representation of SUSY. To get an irreducible one, we de"ne a chiral super"eld which obeys the equation R DM F"0, where DM "! !iIR . I RM

(2.3)

Its Taylor expansion looks like (y"x#i )
(y, )"A(y)#(2(y)#F(y) "A(x)#iIM R A(x)#M M 䊐A(x) I  i R (x)IM #F(x) . #(2(x)! I (2

(2.4)

The coe$cients are ordinary functions of x being the usual "elds. They are called the components of a super"eld. In Eq. (2.4) one has 2 bosonic (complex scalar "eld A) and 2 fermionic (Weyl spinor "eld ) degrees of freedom. The component "elds A and  are called the superpartners. The "eld F is an auxiliary "eld, it has the `wronga dimension and has no physical meaning. It is needed to close algebra (2.1). One can get rid of the auxiliary "elds with the help of equations of motion. Thus, a super"eld contains an equal number of bosonic and fermionic degrees of freedom. Under SUSY transformation they convert one into another. The product of chiral super"elds
,
, etc., is also a chiral super"eld, while the product of chiral and antichiral ones
>
is a general super"eld. To construct the gauge invariant interactions, we will need a real vector super"eld <"<>. It is not chiral but rather a general super"eld. Its expansion over  and M looks like <(x, , M )"C(x)#i(x)!iM (x) i i # [M(x)#iN(x)]! M M [M(x)!iN(x)] 2 2





i ! IM v (x)#iM M (x)# IR (x) I I 2









i 1 1 ! iM M  # IR (x) # M M D(x)# 䊐C(x) . I 2 2 2

(2.5)

The physical degrees of freedom corresponding to a real vector super"eld are the vector gauge "eld v and its superpartner the Majorana four-component spinor "eld made of two Weyl spinors and I

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M . All other components are unphysical and can be eliminated. Thus, we again have an equal number of bosonic and fermionic degrees of freedom. One can choose a gauge (Wess}Zumino gauge) where C""M"N"0, leaving us with the physical degrees of freedom except for the auxiliary "eld D. In this gauge <"!IM v (x)#iM M (x)!iM M  (x)#M M D(x) , I  <"!M M v (x)vI(x) , I  <"0 , etc. One can de"ne also a "eld strength tensor (as analog of F in gauge theories) IJ = "!DM e\4D e4, = M  "!De\4DM e4 , ?  ? ?  ? which is needed to construct gauge invariant Lagrangians.

(2.6)

(2.7)

2.3. SUSY Lagrangians In the super"eld notation SUSY invariant Lagrangians are the polynomials of super"elds. Having in mind that for component "elds we should have the ordinary terms, the general SUSY invariant Lagrangian has the form [8}10]





L" d dM
>
# d[
#m

#y


]#h.c. , G G G G  GH G H  GHI G H I

(2.8)

where the "rst part is a kinetic term and the second one is a superpotential W. Here instead of taking the proper components we use an integration over the superspace according to the rules of grassmannian integration [11]



d "0, ?



 d " . ? @ ?@

Performing this integration we get in components L"iR M I #AH䊐A #FHF I G G G G G G #[ F #m (A F !  )#y (A A F !  A )#h.c.] GHI G H I G H I G G GH G H  G H or solving the constraints

(2.9)

L"iR M I #AH䊐A !m   !mH M M I G G G G  GH G H  GH G H !y   A !yH M M AH!<(A , A ) , (2.10) GHI G H I GHI G H I G H where <"FHF . Note that because of the renormalizability constraint <4A the superpotential I I should be limited by W4
 as in Eq. (2.8).

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The gauge "eld part of a Lagrangian is





1 1 M ? = M  L" d =?= # dM = ? 4 ? 4 " D!F FIJ!i ID M . (2.11)   IJ I To obtain a gauge-invariant interaction with matter chiral super"elds, consider their gauge transformation
Pe\ E
,
>P
>e E>,

) ,

where  is a gauge parameter (chiral super"eld). It is clear now how to construct both SUSY and gauge-invariant interaction which is equivalent to transition from the usual to covariant derivatives





d dM
>
N d dM
>eE4
. G G G G

Thus, the form of the Lagrangian is practically "xed by symmetry requirements. The only freedom is the "eld content, the value of the gauge coupling g, the Yukawa couplings y and the GHI masses. This allows one to construct a SUSY generalization of the SM.

3. Minimal Supersymmetric Standard Model As has been already mentioned, in SUSY theories the number of bosonic degrees of freedom equals that of fermionic. In the SM one has 28 bosonic and 90 fermionic degrees of freedom. So the SM is in great deal non-supersymmetric. Trying to add some new particles to supersymmetrize the SM, one should take into account the following observations: 1. there are no fermions with quantum numbers of the gauge bosons; 2. Higgs "elds have a non-zero v.e.v.s, hence they cannot be superpartners of quarks and leptons since this would induce a spontaneous violation of baryon and lepton numbers; 3. one needs at least two complex chiral Higgs multiplets to give masses to Up and Down quarks. The latter is due to the form of a superpotential and chirality of matter super"elds. Indeed, the superpotential should be invariant under S;(3);S;(2);;(1) gauge group. If one looks at the Yukawa interaction in the Standard Model, Eq. (1.5), one "nds that it is indeed ;(1) invariant since the sum of hypercharges in each vertex equal zero. In the last term this is achieved by taking the conjugated Higgs doublet HI "i HR instead of H. However, in SUSY H is a chiral super"eld and  hence a superpotential, which is constructed out of chiral "elds, can contain only H but not HI , which is an antichiral super"eld. Another reason for the second Higgs doublet is related to chiral anomalies. It is known that chiral anomalies spoil the gauge invariance and, hence, the renormalizability of the theory. They are cancelled in the SM between quarks and leptons in each generation. However, if one introduces

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a chiral Higgs super"eld, it contains higgsinos, which are chiral fermions, and contain anomalies. To cancel them one has to add the second Higgs doublet with the opposite hypercharge. Therefore, the Higgs sector in SUSY models is inevitably enlarged, it contains an even number of doublets. Conclusion: In SUSY models supersymmetry associates known bosons with new fermions and known fermions with new bosons. 3.1. The xeld content Consider the particle content of the Minimal Supersymmetric Standard Model [12]. According to the previous discussion in the minimal version we double the number of particles (introducing a superpartner to each particle) and add another Higgs doublet (with its superpartner). The particle content of the MSSM is given in Table 1 [13]. In Table 1, a"1, 2,2, 8 and k"1, 2, 3 are S;(3) and S;(2) indices, respectively, and i"1, 2, 3 is the generation index. Hereafter tilde denotes a superpartner of an ordinary particle. Thus, the characteristic feature of any supersymmetric generalization of the SM is the presence of superpartners. If supersymmetry is exact, superpartners of ordinary particles should have the same masses and have to be observed. The absence of them at modern energies is believed to be explained by the fact that their masses are very heavy, that means that supersymmetry should be broken. Hence, if the energy of accelerators is high enough, the superpartners will be created. Table 1 Particle content of the MSSM Super"eld

Bosons

Fermions

S; (3) A

S; (2) *

Gauge G?
Gluon g? Weak =I (=!, Z) Hypercharge B()

Gluino g ? Wino, zino w I (w !, z ) Bino bI ()

8 1 1

0 3 1

0 0 0

1

2

!1

1

1

2

3

2

1/3

3H

1

!4/3

3H

1

2/3

1

2

!1

1

2

1

; (1) 7

Matter ¸ G E G Q G ; G D G Higgs H  H 



¸I "(, e ) G * EI "e G 0 QI "(u , dI ) G * Squarks ;I "u G 0 DI "dI G 0 Sleptons



Higgses



H



H





¸ "(, e) G * E "e G 0 Q "(u, d) G * Quarks ; "uA G 0 D "dA G 0

Leptons



Higgsinos



HI  HI 

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The presence of an extra Higgs doublet in SUSY model is a novel feature of the theory. In the MSSM one has two doublets with the quantum numbers (1, 2,!1) and (1, 2, 1), respectively [14]:











S #iP H>   H v #  H> S    H " " , H " " v #  #iP (2 ,   H\ H  (2   H\ 







(3.1)

where v are the vacuum expectation values of the neutral components. G Hence, one has 8"4#4"5#3 degrees of freedom. As in the case of the SM, 3 degrees of freedom can be gauged away, and one is left with 5 physical states compared to 1 state in the SM. Thus, in the MSSM, as actually in any two Higgs doublet model, one has "ve physical Higgs bosons: two CP-even neutral, one CP-odd neutral and two charged. We consider the mass eigenstates below. 3.2. Lagrangian of the MSSM The Lagrangian of the MSSM consists of two parts; the "rst part is SUSY generalization of the Standard Model, while the second one represents the SUSY breaking as mentioned above: L"L #L , 1317    where L

1317

L

% 

and

"L

% 

1 "  4 13 13 3

7 



(3.3)

#L 7 

#  +  L

(3.2)








d  ¹r =?= # dM ¹r = M ? = M  ? ?



d  dM
ReE 4K  >E 4K  >E 4K 
, G G

" d(W #W )#h.c. . 0 ,0

(3.4) (3.5)

The index R in a superpotential refers to the so-called R-parity which adjusts a `#a charge to all the ordinary particles and a `!a charge to their superpartners [15]. The "rst part of W is R-symmetric = " (y3 QH ;A HG #y" QH DA HG #y* ¸H EA HG #HG HH ) , (3.6) 0 GH ?@ ? @  ?@ ? @  ?@ ? @    where i, j"1, 2, 3 are S;(2) and a, b"1, 2, 3 are the generation indices; colour indices are suppressed. This part of the Lagrangian almost exactly repeats that of the SM except that the "elds are now the super"elds rather than the ordinary "elds of the SM. The only di!erence is the last term which describes the Higgs mixing. It is absent in the SM since we have only one Higgs "eld there. The second part is R-non-symmetric = " ( * ¸G ¸H EA # *Y ¸G QH DA # ¸G HH )# ;A DA DA . ,0 GH ?@B ? @ B ?@B ? @ B ? ?  ?@B ? @ B

(3.7)

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These terms are absent in the SM. The reason is very simple: one cannot replace the super"elds in Eq. (3.7) by the ordinary "elds like in Eq. (3.6) because of the Lorentz invariance. These terms have the other property, they violate either lepton (the "rst line in Eq. (3.7)) or baryon number (the second line). Since both e!ects are not observed in Nature, these terms must be suppressed or be excluded. In the minimal version of the MSSM these terms are not included, they are forbidden by R-parity conservation [16]. 3.3. Soft SUSY breaking To introduce supersymmetry breaking as required by the absence of superpartners at modern energies, one has to be careful not to spoil the cancellation of divergencies which allows to solve the hierarchy problem. This is achieved by spontaneous breaking of SUSY in the same way as spontaneous electroweak symmetry breaking. One introduces the "eld whose vacuum expectation value breaks supersymmetry. However, due to a special character of SUSY, this should be a super"eld, whose auxiliary F or D components acquire non-zero v.e.v.'s. This leads to appearance of the so-called soft breaking terms. In the simplest version the soft supersymmetry breaking terms are [17]: !L

  




1 "m   # m  I I ? ?  G 2  ? G



#A[y3 QI ;I A H #y" QI DI A H #y* ¸I EI A H ]#B[H H ]#h.c. , (3.8) ?@ ? @  ?@ ? @  ?@ ? @    where we have suppressed S;(2) indices. Here  are all scalar "elds, I are the gaugino "elds, G ? QI , ;I , DI and ¸I , EI are the squark and slepton "elds, respectively, and H are the S;(2) doublet  Higgs "elds. These terms are obtained via supergravity mechanism and are usually introduced at the GUT scale. We have assumed in Eq. (3.8) the so-called universality of the soft terms, namely, we put all the spin 0 particle masses to be equal to the universal value m , all the spin 1/2 particle (gaugino)  masses to be equal to m and all the cubic and quadratic terms, proportional to A and B, to repeat  the structure of the Yukawa superpotential (3.6). This is an additional requirement motivated by the supergravity mechanism of SUSY breaking as mentioned earlier [18]. Universality is not a necessary requirement and one may consider non-universal soft terms as well. However, it will not change the qualitative picture presented below, so for simplicity, in what follows, we consider the universal boundary conditions. It should be noted that supergravity-induced universality of the soft terms is more likely to be valid at the Planck scale, rather than at the GUT one. This is because a natural scale for gravity is M , while M is the scale for the gauge interactions. However, due to a small di!erence .  %32 between these two scales, it is usually ignored in the "rst approximation resulting in minor uncertainties in the low-energy predictions [19]. The soft terms explicitly break supersymmetry. As will be shown later they lead to the mass spectrum of superpartners di!erent from that of the ordinary particles. Remind that the masses of quarks and leptons remain zero until S;(2) invariance is spontaneously broken.

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3.4. Masses With given values of m , m , , > , > , > , A, and B at the GUT scale, one can solve the   R @ O corresponding RG equations thus linking the values at the GUT and electroweak scales. Substituting these parameters into the mass matrices one can predict the mass spectrum of superpartners [12,20]. 3.4.1. Gaugino}higgsino mass terms The mass matrix for the gauginos, the superpartners of the gauge bosons, and for higgsinos, the superpartners of the Higgs bosons, is non-diagonal, thus leading to their mixing. The mass terms look like "!M M !M!(M MA#h.c.) ,   ? ?  %  }&   where , a"1, 2,2, 8, are the Majorana gluino "elds and ? BI 

(3.9)

L





"

= I 




, "

= I >

(3.10) HI > HI   HI   are, respectively, the Majorana neutralino and Dirac chargino "elds. The neutralino mass matrix is M  0

0



!M cos
sin M sin
sin 8 5 8 5 M cos
cos !M sin
cos 8 5 8 5 . (3.11) 0 !

M  !M cos
sin M cos
cos 8 5 8 5 M sin
sin !M sin
cos ! 0 8 5 8 5 The physical neutralino masses M  G are obtained as eigenvalues of this matrix after diagonaliQ zation. For charginos one has M"






(2M sin
 5 .  (2M cos
5 This matrix has two chargino eigenstates ! with mass eigenvalues  M "[ M ##2M  5   G((M !)#4M cos 2
#4M (M ##2M  sin 2
)] .  5 5   MA"

M

(3.12)

(3.13)

3.4.2. Squark and slepton masses The non-negligible Yukawa couplings cause a mixing between the electroweak eigenstates and the mass eigenstates of the third generation particles. The mixing matrices for the m , m  and R @ m  are O m  m (A ! cot
) R* R R , (3.14) m (A ! cot
) m  R R R0






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m (A ! tan
) @ @ , m  @0




m (A ! tan
) O O m  O0

m  @* m (A ! tan
) @ @

m  O* m (A ! tan
) O O



(3.15)



(3.16)

with m  "m  #m#(4M !M ) cos 2
, R* / R  5 8 m  "m  #m!(M !M ) cos 2
, R0 3 R  5 8 m  "m  #m!(2M #M ) cos 2
, 5 8 @* / @  m  "m  #m#(M !M ) cos 2
, @0 " @  5 8 m  "m  #m!(2M !M ) cos 2
, O* * O  5 8 m  "m  #m#(M !M ) cos 2
O0 # O 5 8 and the mass eigenstates are the eigenvalues of these mass matrices. 3.4.3. The Higgs potential As has been already mentioned, the Higgs potential in MSSM is totally de"ned by superpotential (and the soft terms). Due to the structure of W the Higgs self-interaction is given by the D-terms, while the F-terms contribute only to the mass matrix. The tree level potential is < (H , H )"m H #m H !m (H H #h.c.)           g g#g (H !H )# H>H  , #   2   8

(3.17)

where m "m  #, m "m  #. At the GUT scale m "m "m # , m "!B .  &  &       Notice, that the Higgs self-interaction coupling in Eq. (3.17) is "xed and is de"ned by the gauge interactions as opposite to the SM. Potential (3.17), in accordance with supersymmetry, is positively de"nite and stable. It has no non-trivial minimum di!erent from zero. Indeed, let us write the minimization condition for potential (3.17) 1 < g#g "m v !m v # (v !v )v "0 ,        2 H 4  1 < g#g "m v !m v # (v !v )v "0 ,        2 H 4  where we have introduced the notation v

H ,v "v cos
, H ,v "v sin
, v"v #v , tan
,  .       v 

(3.18) (3.19)

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Solution of Eqs. (3.18) and (3.19) can be expressed in terms of v and sin
: 4(m !m tan
)   v" , (g#g)(tan
!1)

2m  sin 2
" . (3.20) m #m   One can easily see from Eq. (3.20) that if m "m "m # , v happens to be negative, i.e. the     minimum does not exist. In fact, real positive solutions to Eqs. (3.18) and (3.19) exist only if the following conditions are satis"ed [13]: m #m '2m , m m (m , (3.21)       which is not the case at the GUT scale. This means that spontaneous breaking of the S;(2) gauge invariance, which is needed in the SM to give masses for all the particles, does not take place in the MSSM. This strong statement is valid, however, only at the GUT scale. Indeed, going down with energy the parameters of potential (3.17) are renormalized. They become the `runninga parameters with the energy scale dependence given by the RG equations. The running of the parameters leads to a remarkable phenomenon known as a radiative spontaneous symmetry breaking which we discuss below. Provided conditions (3.21) are satis"ed the mass matrices at the tree level are CP-odd components P and P :   tan
1 R< " m , (3.22) M"  RP RP G G 1 cot
G H & T CP-even neutral components S and S :   tan
!1 cot
!1 R< M" " m # M cos
sin
. (3.23)  8 RS RS G G !1 cot
!1 tan
G H & T Charged components H\ and H>:



























tan
1 R< " (m #M cos
sin
) . M "  5 RH>RH\ G G 1 cot
G H & T Diagonalizing the mass matrices one gets the mass eigenstates [13]: G"!cos
P #sin
P , Goldstone boson PZ ,    A"sin
P #cos
P , Neutral CP"!1 Higgs ,   G>"!cos
(H\)H#sin
H>, Goldstone boson P=> ,   H>"sin
(H\)H#cos
H>, Charged Higgs ,   h"!sin S #cos S , SM Higgs boson CP"1 ,   H"cos S #sin S , Extra heavy Higgs boson ,   where the mixing angle  is given by




tan 2"!tan 2


m #M  8 m !M  8



.

(3.24)

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The physical Higgs bosons acquire the following masses [12]: CP-odd neutral Higgs A: m "m #m ,    Charge Higgses H!:

m ! "m #M , &  5

(3.25)

CP-even neutral Higgses H, h: m "[m #M $((m #M )!4m M cos 2
] , &F   8  8  8

(3.26)

where as usual g g#g M " v, M " v . 5 8 2 2 This leads to the once celebrated SUSY mass relations: m ! 5M , & 5 m 4m 4M , F  & m 4M cos 2
4M , F 8 8 m#m "m #M . F &  8 Thus, the lightest neutral Higgs boson happens to be lighter than Z boson, that clearly distinguishes it from the SM one. Though we do not know the mass of the Higgs boson in the SM, there are several indirect constraints leading to the lower boundary of m1+5135 GeV [31]. After F including the radiative corrections the mass of the lightest Higgs boson in the MSSM, m , increases. F We consider it in more detail below. 3.5. RG yow in the MSSM If one compares the RG #ow in the SM and the MSSM, one "nds additional contributions from superpartners to the RG equations. Consider the gauge couplings. In the SM the RG #ow is given by Eqs. (1.12). We have mentioned already in Section 1 that it o!ers the uni"cation pattern supporting the GUT hypothesis. However, if one looks at the curves more attentively, one "nds that the situation is not that good. Indeed, let us consider the solution to the RG equations in more detail. The result is demonstrated in the left part of Fig. 2, which shows the evolution of an inverse of the couplings as function of a logarithm of energy [21]. In this presentation the evolution becomes a straight line in "rst order. The second-order corrections are small and do not cause any visible deviation from a straight line. Fig. 2 clearly demonstrates that within the SM the coupling constants uni"cation at a single point is impossible. It is excluded by more than 8 standard deviations [21}23]. This result means that the uni"cation can only be obtained if new physics enters between the electroweak and the Planck scales.

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Fig. 2. Evolution of the inverse of the three coupling constants in the Standard Model (left) and in the supersymmetric extension of the SM (MSSM) (right). Only in the latter case uni"cation is obtained. The SUSY particles are assumed to contribute only above the e!ective SUSY scale M of about 1 TeV, which causes the change in slope in the evolution 1317 of couplings. The thickness of the lines represents the error in the coupling constants [21].

In the MSSM the slopes of the RG evolution curves are modi"ed. The coe$cients b in Eq. (1.12) G now are





 b



0

2

3/10

b " b " !6 #N (3.27) 2 #N 1/2 , G $ &   b !9 2 0  where use N "3 and N "2, which corresponds to the MSSM. $ &  It turns out that within the SUSY model perfect uni"cation can be obtained if the SUSY masses are of the order of 1 TeV. This is shown on the right part of Fig. 2; the SUSY particles are assumed to contribute e!ectively to the running of the coupling constants only for energies above the typical SUSY mass scale, which causes the change in the slope of the lines near 1 TeV. From the "t requiring uni"cation one "nds for the breakpoint M and the uni"cation point M [21]: 1317 %32 M "10 ! !  GeV , 1317 M "10 ! !  GeV , %32 \ "(26.3$1.9$1.0) . (3.28) %32 The "rst error originates from the uncertainty in the coupling constant, while the second one is due to the uncertainty in the mass splittings between the SUSY particles. For SUSY models, the dimensional reduction DR scheme is used [24]. This uni"cation of the gauge couplings was considered as the "rst `evidencea for supersymmetry, especially since M was found in the range preferred by the "ne-tuning arguments. 1317

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It should be noted, that the uni"cation of the three curves at a single point is not that trivial as it may seem from the existence of three free parameters (M ,M and  ). The reason is 1317 %32 %32 simple: when introducing new particles one in#uences all three curves simultaneously, thus giving rise to strong correlations between the slopes of the three lines. For example, adding new generations and/or new Higgs doublets never yield uni"cation.

4. Renormalization of softly broken SUSY theories To "nd the RG #ow for the soft terms one has to know how they are renormalized. Remarkably that the renormalizations in softly broken SUSY theories follow a simple pattern which is completely de"ned by an unbroken theory [25}28]. The main idea is that a softly broken supersymmetric gauge theory can be considered as a rigid SUSY theory imbedded into external space}time-independent super"eld, so that all couplings and masses become external super"elds. The crucial statement is that the singular part of e!ective action depends on external super"eld, but not on its derivatives, so that one can calculate it when the external "eld is a constant, i.e. in a rigid theory [29]. This approach to a softly broken sypersymmetric theory allows one to use remarkable mathematical properties of N"1 SUSY theories such as non-renormalization theorems, cancellation of quadratic divergences, etc. The renormalization procedure in a softly broken SUSY gauge theory can be performed in the following way [27]: One takes renormalization constants of a rigid theory, calculated in some massless scheme, substitutes instead of the rigid couplings (gauge and Yukawa) their modi"ed expressions, which depend on a grassmannian variable, and expand over this variable. This gives renormalization constants for the soft terms. Di!erentiating them with respect to a scale one can "nd corresponding renormalization group equations. In fact, as it has been shown in [30] this procedure works at all stages. One can make the abovementioned substitution on the level of the renormalization constants, RG equations, solutions to these equations, approximate solutions, "xed points, "niteness conditions, etc. Expanding then over a grassmannian variable one obtains corresponding expressions for the soft terms. We demonstrate now how this procedure works in the MSSM. Using notation introduced above the modi"ed couplings in the MSSM are (", "M ) (4.1) a "a (1#M #M M #(M M M # G )) , G G G G G G ? >I "> (1!A !AM #(A AM # )) , (4.2) I I I I I I I where M are the gaugino masses, A are the trilinear scalar couplings,  are the certain G I I combinations of soft squark and slepton masses entering the Yukawa vertex and  G are the SUSY ? ghost soft terms  "m  #m  #m ,  "m  #m  #m , R / 3 & @ / " &  "m  #m  #m ,  G "M#m  G G EF O * # & ? and m  is the soft scalar ghost mass, which is eliminated by solving the RG equation. In one-loop EF order m  "0. EF

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To get now the RG equations for the soft terms one just has to take the corresponding RG equations for the rigid couplings and perform the grassmannian expansion. The one-loop RG equations for the MSSM couplings are [34]: da G "b a , G G dt








16 13 d> 3 "!> a #3a # a !6> !> , * 3   15  3 " dt



d> 16 7 " "!> a #3a # a !> !6> !> , " 3   15  3 " * dt



9 d> * "!> 3a # a !3> !4> . *  " * 5  dt

(4.3)

Performing the grassmannian expansion one "nds: dM G "b a M , G G G dt 16 13 dA 3 " a M #3a M # a M #6> A #> A ,     3 3 " " 3 15   dt 16 7 dA " " a M #3a M # a M #6> A #> A #> A ,   15   " " 3 3 * * 3   dt 9 dA * "3a M # a M #3> A #4> A ,   5   " " * * dt dB 3 "3a M # a M #3> A #3> A #> A .   3 3 " " * * dt 5  




dm  / "! dt



16 1 a M #3a M # a M !> (m  #m  #m  #A )   15   3 / 3 & 3 3  



!> (m  #m  #m  #A ) , " " / " &







 

 

dm  3 "! dt

16 16 a M # a M !2> (m  #m  #m  #A ) , 3 / 3 & 3 3   15  

dm  " "! dt

16 4 a M # a M !2> (m  #m  #m  #A ) , " / " & " 3   15  





1 dm  * "! 3 a M # a M !> (m  #m  #m #A ) ,   * * # & * 5   dt dm  # "! dt





12 a M !2> (m  #m  #m  #A ) , * * # & * 5  

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d 1 "! 3 a # a !(3> #3> #> ) ,  3 " * dt 5 



1 dm  & "! 3 a M # a M !3> (m  #m  #m #A )   " / " & " dt 5  



!> (m  #m  #m  #A ) , * * # & *








1 dm  & "! 3 a M # a M !3> (m  #m  #m #A ) ,   5   3 / 3 & 3 dt

(4.4)

where we have already substituted the solution m  "0 in the one-loop order. (Note that to get the EF RG equation for the individual squark and slepton masses one needs to know the anomalous dimensions for the corresponding "elds.)

5. RG 6ow for the soft terms Having all the RG equations, one can "nd now the RG #ow for the soft terms. To see what happens at lower scales one has to run the RG equations for the mass parameters from GUT to the EW scale. Let us take some initial values of the soft masses at the GUT scale in the interval between 10 and 10 GeV consistent with SUSY scale suggested by uni"cation of the gauge couplings (3.28). This leads to the following RG #ow of the soft terms shown in Fig. 3 [20] (note that we perform the running of soft parameters in the opposite direction, from GUT to EW scale).

Fig. 3. An example of evolution of sparticle masses and soft supersymmetry breaking parameters m "m  # and  & m "m  #.  &

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One should mention the following general features common to any choice of initial conditions: (i) The gaugino masses follow the running of the gauge couplings and split at low energies. The gluino mass is running faster than the others and is usually the heaviest due to the strong interaction. (ii) The squark and slepton masses also split at low energies, the stops (and sbottoms) being the lightest due to relatively big Yukawa couplings of the third generation. (iii) The Higgs masses (or at least one of them) are running down very quickly and may even become negative. To calculate the masses one has also to take into account the mixing between various states (see Eqs. (3.11), (3.12), (3.14)}(3.16)). 5.1. Radiative electroweak symmetry breaking The running of the Higgs masses leads to the phenomenon known as a radiative electroweak symmetry breaking. By this we mean the following: at the GUT energy scale both the Higgs mass parameters m and m are positive and the Higgs potential has no non-trivial minima. However,   when running down to the EW scale due to the radiative corrections they may change sign so that the potential develops a non-trivial minimum. At this minimum the electroweak symmetry happens to be spontaneously broken. Thus, contrary to the SM where one has to choose the negative sign of the Higgs mass squared `by handa, in the MSSM the e!ect of spontaneous symmetry breaking is triggered by the radiative corrections. Indeed, one can see in Fig. 3 that m (or both m and m ) decreases when going down from the    GUT scale to the M scale and can even become negative. This is the e!ect of the large top (and 8 bottom) Yukawa couplings in the RG equations. As a result, at some value of Q conditions (3.21) are satis"ed, so that the non-trivial minimum appears. This triggers spontaneous breaking of the S;(2) gauge invariance. The vacuum expectations of the Higgs "elds acquire non-zero values and provide masses to the quarks, leptons, S;(2) gauge bosons, and additional masses to their superpartners. This way one obtains also the explanation of why the two scales are so much di!erent. Due to the logarithmic running of the parameters one needs a long `running timea to get m to be negative  when starting from a positive value of the order of M scale &10}10 GeV. 1317 6. Infrared quasi-5xed points Examining the RG equations for the Yukawa couplings one "nds that they possess the infrared "xed points. This is a very typical behaviour for RG equations. In this section we give a short description of the infrared quasi-"xed points (IRQFP) [33] in the MSSM. They play an important role in predictions of the mass spectrum. As in the previous section we consider the RG #ow in the direction from GUT to EW scale, the running parameter being t"log M /Q. This corresponds to the opposite sign in RG equations %32 (4.3) and (4.4).

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Fig. 4. The infrared quasi-"xed points for  "> /a (a) and  "A /M (b). The x-axis shows the values of  . R R   R  

6.1. Low tan
regime Consider "rst the low tan
regime. In this case, the only important Yukawa coupling is the top-quark one, all the others can be put equal to zero and the RG equations can be solved analytically [34] a > E (t)   R a (t)" , > (t)" , (6.1) G R 1#a b t 1#6> F (t)  G  R where 13 16 E (t)" (1#b a t)ARG @G , c " , 3, , R G  RG 3 15 G R F (t)" E (t) dt . R R  In the IR regime solution (6.1) possesses a quasi-"xed point. Indeed taking the limit > "  > (0)PR one can drop 1 in the denominator of Eq. (6.1) and obtain the IRQFP R E (t) , (6.2) >(t)N>$." R R 6F (t) R which is independent of the initial condition [33,35]. Though perturbation theory is not valid for > '1, it does not prevent us from using the "xed R point (6.2) since it attracts any solution with > 'a or, numerically, for > '0.1/4
. Thus, for    a wide range of initial values > is driven to the IR quasi-"xed point given by Eq. (6.2) which R








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335

numerically corresponds to y (M )+1.125. It is useful to introduce the ratio  ,> /a since the R 8 R R  strong coupling is the leading one in the IR regime. At the "xed point  (M )+0.84 and is R 8 approached in the IR regime when Q decreases. The behaviour of (t) is shown in Fig. 4 [36]. To get the solutions for the soft terms it is enough to perform the substitution aPa and >P>I and expand over  and . Expanding the gauge coupling in (6.1) up to  one has (hereafter we assume M "m ) G  m  . (6.3) M (t)" G 1#b a t G  Performing the same expansion for the Yukawa coupling one "nds [30]






t dE 6> A R!   !m (tE !F ) . (6.4) A (t)"  E dt R R R 1#6> F 1#6> F R  R  R To get the solution for the  term one has to make expansion over  and . This leads to








d t dE (A #m 6> (tE !F )) 6> dE  !A R !   R R #m   #  t R ,  (t)"   dt E dt R (1#6> F ) 1#6> F dt 1#6> F R  R  R  R (6.5) With analytic solutions (6.4), (6.5) one can analyse asymptotics and, in particular, "nd the infrared quasi-"xed points which correspond to > PR  t dE tE !F R! R R , A$."!m (6.6) R  E dt F R R tE !F  d t dE t dE R R # R ! R . $."m (6.7) R  F dt E dt F dt R R R The FP solutions (6.6), (6.7) can be directly obtained from a "xed point for the rigid Yukawa coupling (6.2) by grassmannian expansion. This explains, in particular, why "xed point solutions for the soft couplings exist if they exist for the rigid ones and with the same stability properties [37]. The behaviour of  "A /M as a function of a for a "xed ratio > /a "5 is shown in Fig. 4b.  R     One can observe the strong attraction to the IR stable quasi-"xed point  +!0.62 [36].  One can also write down solutions for the individual masses. This can be obtained using the grassmannian expansion of solutions for the corresponding super"eld propagators. For the "rst two generations one has























m 16 1 f #3f # f , m  * "m #    15  / 3  2

 

m 16 16 m  0 "m #  f # f , 3  3  15  2 m 16 4 f # f , m  0 "m #   " 3  15  2



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m 3 m  "m #  3f # f , &   2 5  m 3 m * "m #  3f # f ,   15  * 2 m 12 f m 0 "m #   # 5  2

,

where






1 1 f " 1! . G b (1#b a t) G G  The third generation masses get the contribution from the top Yukawa coupling [38]: m 0 "m  0 , @ " m * "m  * #/6 , / @ m 0 "m  0 #/3 , R 3 m * "m  * #/6 , / R m  "m  #/2 , & & where  is related to  (6.5) by R d t dE R . " ! !m R   dt E dt R There is no obvious infrared attractive "xed point for m  . However, one can take the linear & combination m "m  #2m  which together with m  shows the IR "xed point behaviour in & & > & the limit > PR.  m$. 1 m m$. & +!0.12  #3.4 . > +!0.73, (6.8) 2 m M M    In Eq. (6.8) one has only weak dependence on the ratio m /m . One can "nd the IR quasi-"xed   point m  /M +!0.40 which corresponds to m /m "0. As for the combination m , the    > & dependence on initial conditions disappears completely, as it follows from (6.8). The situation is illustrated in Fig. 5 [36]. Consider now the squark masses. In the limit > PR these solutions are driven to the IRQFPs R 1 m m $. m $.  #5.8 . / +0.12 3 +0.48, (6.9) 2 m M M    As it follows from Eq. (6.9), the solution for m  /M becomes independent of the initial conditions 3  m /m and A /m , when the top-quark Yukawa coupling is initially large enough. As a result,     the solutions of RGEs are driven to the "xed point (6.9) for a wide range of m /m (Fig. 6b). As for   m , the dependence on initial conditions does not completely disappear; however, it is rather weak / like in the case of m  and approaches the value m  /M +0.69 (Fig. 6a) [36]. /  &
















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Fig. 5. The infrared quasi-"xed points for m  /M (a) and (m  #2m  )/M (b). &  & & 

Fig. 6. The infrared quasi-"xed points for m  /M (a) and m  /M (b). /  3 

The bilinear SUSY breaking parameter B does not exhibit a "xed-point behaviour in the limit >





B B 1 A  !  !0.8 . +0.35 M m 2m   

(6.10)

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Table 2 Comparison of one- and two-loop IRQFPs

One-loop level Two-loop level

A /M R 

m  /M 3 

m /M > 

m  /M & 

m  /M / 

!0.62 !0.59

0.48 0.48

!0.73 !0.72

!0.40 !0.46

0.69 0.75

It is clear that neither B /m nor A /m may be neglected. As a consequence, no "xed point     behaviour for the ratio B/M is observed.  We have considered the one-loop RGEs. It is interesting to see, however, how our results are modi"ed when two-loop RGEs are used. For comparison we present the two-loop IRQFP values [39] together with our one-loop results in Table 2. As one can see from this table, the di!erence between the one- and two-loop results is negligible for A /M , m  /M and m /M . As for m  /M and m  /M , the two-loop corrections to the "xed  /  R  3  >  & points are about two times as small as deviations from them. As it was mentioned above, such corrections have a negligible impact on our main results. 6.2. Large tan
regime We now give a short description of the infrared behaviour of the RGEs in the MSSM for the large tan
regime. While with a single Yukawa coupling the analytical solution to the one-loop RG equations has been known for long, for increasing number of Yukawa couplings it has been obtained quite recently [40] in the form that allows iterative representation. One can write down the one-loop RG equations (4.3) as da G "b a , G G dt

(6.11)






d> I "!>  c a ! a > , I IG G IJ J dt G J where

(6.12)

b "33/5, 1,!3 , G c "13/15, 3, 16/3 , c "7/15, 3, 16/3 , c "9/5, 3, 0 , RG @G OG a "6, 1, 0 , a "1, 6, 1 , a "0, 3, 4 . RJ @J OJ Then the general solution to Eqs. (6.11) and (6.12) can be written in the form [40] a  a" , G 1#b a t G  >u I I , > " I 1#a >R u II I  I

(6.13) (6.14)

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where the functions u obey the integral system of equations I E R u" , R (1#6>R u ) @  @

E @ u " , @ (1#6>R u )(1#4>R u ) R  R O  O

E O u" O (1#6>R u ) @  @

(6.15)

and the functions E are given by I  E "  (1#b a t)AIG @G . I G  G

(6.16)

Let us stress that Eqs. (6.13) and (6.14) give the exact solution to Eqs. (6.11) and (6.12), while the u 's in Eqs. (6.15), although solved formally in terms of the E 's and >'s as continued integrated I I I fractions, should in practice be solved iteratively. Let us now perform the substitution (4.1), (4.2) in (6.13)}(6.15) and expand over  and . Then the linear term in  will give us the solution for M and A and the  terms the ones for  . The G I I resulting exact solutions look similar to those for the rigid couplings (6.13)}(6.15) [41] M G , M" G 1#b at G G

(6.17)

A/>#a u e II I I , A "!e # I I I I 1/>#a u I II I

(6.18)

(A)/>!/>#a u  I I I II I I ,  " #A#2e A ! I I I I I I 1/>#a u I II I

(6.19)

where the new functions e and  have been introduced which obey the iteration equations I I 1 dEI Au !u e R# @ @ @ @ , e" R E d 1/>#6u R @ @ Au !u e Au !u e 1 dEI R R# O O O O , @# R R e " @ E d 1/>#6u 1/>#4u R R O O @ 1 dEI Au !u e O #3 @ @ @ @ , e" O E d 1/>#6u O @ @




  

Au !u e  1 dEI 1 dEI Au !u e R #2 R @ @ @ @ #7 @ @ @ @ " R E d d d 1/> #6u 1/>#6u E @ @ @ @ R R








 


! (#(A)) u !2A u e # u  @ @ @ @ @ @ @ @

1 #6 u , @ > @

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Au !u e Au !u e R R R R# O O O O 1/>#6u 1/>#4u R R O O  Au !u e  O O #5 O O 1/>#4u O O Au !u e O O O O 1/>#4u O O 1 ! (#(A)) u !2A u e # u  #6 u R R R R R R R R R > R 1 ! (#(A)) u !2A u e # u  #4 u , O O O O O O O O O > O Au !u e  1 dEI Au !u e 1 dEI O #6 O @ @ @ @ # 27 @ @ @ @ " O E d d 1/>#6u E d 1/>#6u @ @ @ @ O O 1 #6 u . (6.20) ! 3 (#(A)) u !2A u e # u  @ @ @ @ @ @ @ @ @ > @ Here the variations of EI should be taken at ""0. When solving Eqs. (6.15) and (6.20) in the I nth iteration one has to substitute in the r.h.s. the (n!1)th iterative solution for all the corresponding functions. The solutions for the individual soft masses are linearly expressed through 's [38] 1 dEI 1 dEI @ #2 @  " @ E d d E d @ @ Au !u e R R #7 R R 1/>#6u R R Au !u e R R #2 R R 1/>#6u R R

m   "m #m   /



  
   
 
   
 

128f #87f !11f 17( !)#20( !)!5( !)   # R R @ @ O R , 122 122

m   "m #m 3  

42( !)!8( !)#2( !) 144f !108f #144/5f R R @ @ O R ,   # 122 122

m   "m #m   "

112f !84f #112/5f !8( !)#48( !)!12( !)   # R R @ @ O R , 122 122

m  "m #m &  

!9( !)#54( !)#17( !) !240f !3f !57/5f R R @ @ O R ,   # 122 122

m  "m #m   &

!272f #21f !89/5f 63( !)!12( !)#3( !)   # R R @ @ O R , 122 122

m  "m #m   *

80f #123f !103/5f 3( !)!18( !)#35( !)   # R R @ @ O R , 122 122

m  "m #m #  

160f !120f #32f 6( !)!36( !)#70( !)   # R R @ @ O R . 122 122

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Solutions (6.13)}(6.15), (6.17)}(6.20) have a nice property since they contain explicit dependence on initial conditions and one can trace this dependence in the "nal results. This is of special importance for the non-universal case since one can see which of the parameters is essential and which is washed out during the evolution. In particular, the solution for the Yukawa couplings exhibit the "xed point behaviour when the initial values are large enough. More precisely, in the regime >, >, >PR with "xed "nite ratios >/>"r , >/>"r , it is legitimate to drop 1 in R @ O R @  @ O  the denominators of Eqs. (6.14) and (6.15) in which case the exact Yukawa solutions go to the IRQFP de"ned by [41] u$. >$." I I a u$. II I

(6.21)

E E E @ O R , u$." , u$." u$." @ O R (u$.)(u$.) (u$.) (u$.) R O @ @

(6.22)

with

extending the IRQFP (6.2) to three Yukawa couplings. What is worth stressing here is that both the dependence on the initial condition for each Yukawa as well as the e!ect of Yukawa nonuni"cation, (r , r ), have completely dropped out of the runnings.   This in turn leads to the IRQFPs for the soft terms. Disappearance of > in the FP solution I naturally leads to the disappearance of A and  in the soft term "xed points. I I Below we present the result of numerical analysis. We begin with Yukawa couplings and assume the equality of the Yukawa couplings of the third generation at the GUT scale: > (M )" R %32 > (M )"> (M ). @ %32 O %32 In Figs. 4a}c the numerical solutions of the RGEs are shown for a wide range of initial values of  (M )" (M )" (M ) from the interval 0.2, 5, where  "> /a . One can clearly R %32 @ %32 O %32 G G  see the IRQFP-type behaviour when the parameter  at the GUT scale is big enough [42]. G We have found the following values of the Yukawa couplings y at the M scale: G 8 y 3 0.787, 1.048, y 3 0.758, 0.98, y 3 0.375, 0.619 . R @ O Comparing y and y one can see that the ratio belongs to a very narrow interval R @ y /y 3 1.039, 1.069. R @ Now, we proceed with the discussion of RGEs for trilinear scalar couplings, A , i"(t, b, ). The G results are shown in Figs. 4d and e for the following quantities  G "A /M , i"(t, b) for di!erent G   initial values at the GUT scale and for  (M )"5. One can see the strong attraction to the "xed G %32 points [42]. The question of stability of these IRQFPs becomes important for further consideration. Analysing their stability under the change of the initial conditions for  (M ) one "nds remarkable G %32 stability, which allows to use them as "xed parameters at the M scale. In Fig. 7f a particular 8 example of stability of IRQFP for A is shown. As a result one has the following IRQFP values for R the parameters  G :   R +!0.619,  @ +!0.658,  O +0.090 .   

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Fig. 7. The infrared quasi "xed points for  "> /a , i"t, b,  (a, b, c),  G "A /M , i"t, b (d, e) and  R with G G   G    R (a )"1 for di!erent initial values of  (a ) (f).   R 

The last step in the investigation of the RGEs is the calculation of the soft mass parameters. As one can see from Figs. 8a and b there exist IRQFPs [42] m  /M +!0.306, m  /M +!0.339 . &  &  The numbers correspond to the initial condition m /m "0. Later the initial values for the ratio   m /m belonging to the following interval m /m 3 0, 2 are considered.     In Figs. 8c}e the infrared behaviour of the soft SUSY breaking squark masses is shown. One can immediately see that all masses have IRQFPs which are used in the next section to "nd the mass spectrum. For further analysis only the squark masses are important. As for sleptons they also have an attractive infrared behaviour but it does not in#uence the mass spectrum of the Higgs bosons in which we are interested in below and we do not show them explicitly.

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Fig. 8. The infrared quasi-"xed points for mass parameters.

Numerical values of the ratios are the following [42]: m /M +0.58, m /M +0.52, m /M +0.53 , /  3  "  obtained for m /m "0. One again "nds a very weak dependence on initial values of the Yukawa   couplings. The behaviour of the bilinear SUSY breaking parameter B is the same as in low tan
case. The ratio B/M does not exhibit the infrared quasi-"xed point behaviour.  Thus, one can see that solutions of RGEs for all MSSM SUSY breaking parameters (the only exception is the parameter B) are driven to the infrared attractive "xed points if the Yukawa couplings at the GUT scale are large enough. Our analysis is constrained by the one-loop RG equations. The di!erence between one- and two-loop IRQFPs is similar to the low tan
case (see Table 1) and is less than 10%. At the same time the deviations from the IRQFPs obtained by one-loop RGEs are also of the same order which de"nes the accuracy of our predictions. The only place where it really matters is the prediction of the lightest Higgs boson mass where all the proper corrections are taken into account.

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7. Higgs boson mass prediction in the SM and MSSM 7.1. The Higgs boson mass in the SM The last unobserved particle from the Standard Model is the Higgs boson. Its discovery would allow one to complete the SM paradigm and con"rm the mechanism of spontaneous symmetry breaking. On the contrary, the absence of the Higgs boson would awake doubts about the whole picture and would require new concepts. Experimental limits on the Higgs boson mass come from a direct search at LEP II and Tevatron and from indirect "ts of electroweak precision data, "rst of all from the radiative corrections to the W and top quark masses. A combined "t of modern experimental data gives [43] m "78> GeV , (7.1) F \ which at the 95% con"dence level leads to the upper bound of 260 GeV (see Fig. 9). At the same time, recent direct searches at LEP II for the c.m. energy of 189 GeV give the lower limit of almost 95 GeV [43]. From theoretical point of view low Higgs mass could be a hint for physics beyond the SM, in particular for the supersymmetric extension of the SM. Within the Standard Model the value of the Higgs mass m is not predicted. However, one can F get the bounds on the Higgs mass [31,32]. They follow from the behaviour of the quartic coupling which obeys the RG equation (1.14). Since the quartic coupling grows with rising energy inde"nitely, an upper bound on m follows F from the requirement that the theory be valid up to the scale M or up to a given cut-o! scale .   below M [31]. The scale  could be identi"ed with the scale at which a Landau pole .  develops. The upper bound on m depends mildly on the top-quark mass through the impact of the F top-quark Yukawa coupling on the running of the quartic coupling . On the other hand, the requirement of vacuum stability in the SM (positivity of ) imposes a lower bound on the Higgs boson mass, which crucially depends on the top-quark mass as well as on the cut-o!  [31,32]. Again, the dependence of this lower bound on m is due to the e!ect of the R top-quark Yukawa coupling on the quartic coupling in Eq. (1.14), which drives to negative values at large scales, thus destabilizing the standard electroweak vacuum. From the point of view of LEP and Tevatron physics, the upper bound on the SM Higgs boson mass does not pose any relevant restriction. The lower bound on m , instead, is particularly F important in view of search for the Higgs boson at LEP II and Tevatron. For m &174 GeV and R  (M )"0.118 the results at "10 GeV or at "1 TeV can be given by the approximate Q 8 formulae [32]:

 

m '72#0.9[m !174]!1.0 F R

 

 (M )!0.118 Q 8 , "10 GeV , 0.006

m '135#2.1[m !174]!4.5 F R

 (M )!0.118 Q 8 , "1 TeV , 0.006

where the masses are in units of GeV.  The last run of LEP II at 200 GeV c.m. energy has increased this bound up to 103 GeV [44].

(7.2) (7.3)

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345

Fig. 9. The  distribution as function of the Higgs mass from the SM "t to the electroweak precision observables and the top mass. The shaded area is excluded by the direct searches. Fig. 10. Strong interaction and stability bounds on the SM Higgs boson mass.  denotes the energy scale up to which the SM is valid.

Fig. 10 [45] shows the perturbativity and stability bounds on the Higgs boson mass of the SM for di!erent values of the cut-o!  at which new physics is expected. We see from Fig. 10 and Eqs. (7.2), (7.3) that indeed for m &174 GeV the discovery of a Higgs particle at LEP II would R imply that the Standard Model breaks down at a scale  well below M or M , smaller for %32 .  lighter Higgs. Actually, if the SM is valid up to &M or M , for m &174 GeV only %32 .  R a small range of values is allowed: 134(m (&200 GeV. For m "174 GeV and m (100 GeV F R F (i.e. in the LEP II range) new physics should appear below the scale &a few to 100 TeV. The dependence on the top-quark mass however is noticeable. A lower value, m K170 GeV, would R relax the previous requirement to &10 TeV, while a heavier value m K180 GeV would R demand new physics at an energy scale as low as 10 TeV. 7.2. The Higgs boson mass in the MSSM It has been already mentioned that in the MSSM the mass of the lightest Higgs boson is predicted to be less than the Z-boson mass. This is, however, the tree level result and the masses acquire the radiative corrections. With account of the radiative corrections the e!ective Higgs bosons potential is < "< #< , &  

(7.4)

where < is given by Eq. (3.17) and in one-loop order 






m 3 1 (!1)(I (2J #1)c m log I ! . < " I I I   Q 2 64
 I

(7.5)

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Here the sum is taken over all the particles in the loop, J is the spin and m is the "eldI I dependent mass of a particle at the scale Q. These radiative corrections vanish when supersymmetry is not broken and are positive in softly broken case. The leading contribution comes from (s)top loops














m  3 m  3 m 3 3 m  log R ! #m  log R ! !2m log R ! < " R R   32
 R Q Q 2 2 Q 2



.

(7.6)

Contributions from the other particles are much smaller [46}48]. These corrections lead to the following modi"cation of the tree level relation for the lightest Higgs mass m  m  3gm R log R R . (7.7) m+M cos 2
# F 8 m 16
M R 5 One "nds that the one-loop correction is positive and increases the mass value. Two-loop corrections have the opposite e!ect but are smaller and result in slightly lower value of the Higgs mass. To "nd out numerical values of these corrections one has to determine the masses of all superpartners. This means that one has to know the initial conditions for the soft parameters. Fortunately, due to the IRQFP solutions the dependence on initial conditions may disappear at low energies. This allows one to reduce the number of unknown parameters and make predictions. Due to extreme importance of the Higgs mass predictions for experimental searches this problem has been the subject of intense investigation. There is a considerable amount of papers devoted to this topic (see e.g. [47}49,52,57,60]). And though the initial assumptions and the strategy are di!erent, the general conclusions are very similar. In what follows, we accept the strategy advocated in Refs. [36,42]: As input parameters one takes the known values of the top-quark, bottom-quark and -lepton masses, the experimental values of the gauge couplings, and the mass of the Z-boson [1]. To reduce the arbitrariness of the soft terms we use the "xed-point values for the Yukawa couplings and SUSY breaking parameters. The value of tan
is determined from the relations between the running quark masses and the Higgs v.e.v.'s in the MSSM m "y v sin
, (7.8) R R m "y v cos
, (7.9) @ @ m "y v cos
. (7.10) O O The Higgs mixing parameter  is de"ned from the minimization conditions for the Higgs potential and requirement of radiative EWSB. As an output we determine the mass spectrum of superpartners and of the Higgs bosons as functions of the only free parameter, namely m , which is directly related to the gluino mass M .   Varying this parameter within the experimentally allowed range, one gets all the masses as functions of this parameter (see Table 3). For low tan
the value of sin
is determined from Eq. (7.8), while for high tan
it is more convenient to use the relation tan
"(m /m )y /y , since the ratio y /y is almost a constant in the R @ @ R R @ range of possible values of y and y . R @

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Table 3 The values of input parameters and the output of "xed point analysis Input

Output

m (pole)"173.8$5.2 GeV R m (pole)"4.94$0.15 GeV @ m (pole)"1.7771$0.0005 GeV O  (M )"0.118$0.005  8  (M )"0.034  8  (M )"0.017  8 M "91.187$0.007 GeV 8

> ,> ,> R @ O A ,A ,A R @ O tan
,  m  , m  , m  / 3 " m  , m  * # m  , m  & & m , m , m , m! F  & &

N

For the evaluation of tan
one "rst needs to determine the running top and bottom-quark masses. One can "nd them using the well-known relations to the pole masses (see e.g. [50}52]), including both QCD and SUSY corrections. For the top-quark one has m  R . (7.11) m (m )" R R 1#(m /m ) #(m /m ) R R /!" R R 1317 Then, the following procedure is used to evaluate the running top mass. First, only the QCD correction is taken into account and m (m ) is found in the "rst approximation. This allows one to R R determine both the stop masses and the stop mixing angle. Next, having at hand the stop and gluino masses, one takes into account the stop/gluino corrections. For the bottom quark the situation is more complicated because the mass of the bottom quark m is essentially smaller than the scale M and so one has to take into account the running of this @ 8 mass from the scale m to the scale M . The procedure is the following [51,53,54]: one starts with @ 8 the bottom-quark pole mass, m "4.94$0.15 [55] and "nds the SM bottom-quark mass at the @ scale m using the two-loop QCD corrections @ m  @ . (7.12) m (m )1+" @ @ 1#(m /m ) @ @ /!" Then, evolving this mass to the scale M and using a numerical solution of the two-loop SM RGEs 8 [51,54] with  (M )"0.12 one obtains m (M ) "2.91 GeV. Using this value one can calculate  8 @ 8 1+ the sbottom masses and then return back to take into account the SUSY corrections from massive SUSY particles m (M )1+ @ 8 m (M )" . (7.13) @ 8 1#(m /m ) @ @ 1317 When calculating the stop and sbottom masses one needs to know the Higgs mixing parameter . For determination of this parameter one uses the relation between the Z-boson mass and the low-energy values of m  and m  which comes from the minimization of the Higgs potential: & & M m  # !(m  # ) tan
 &  8 #" & , (7.14) tan
!1 2

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where  and  are the radiative corrections [47]. Large contributions to these functions come   from stops and sbottoms. This equation allows one to obtain the absolute value of , the sign of  remains a free parameter. Whence the quark running masses and the  parameter are found, one can determine the corresponding values of tan
with the help of Eqs. (7.8) and (7.9). This gives in low and high tan
cases, respectively [36,42,56]: tan
"1.47$0.15$0.05 for '0 , tan
"1.56$0.15$0.05 for (0 , tan
"69.3$0.6$0.3

for '0 ,

tan
"38.1$0.9$0.4

for (0 .

The deviations from the central value are connected with the experimental uncertainties of the top-quark mass,  (M ) and uncertainty due to the "xed point values of y (M ) and y (M ).  8 R 8 @ 8 Having all the relevant parameters at hand it is possible to estimate the masses of the Higgs bosons. With the "xed point behaviour the only dependence left is on m or the gluino mass M .   It is restricted only experimentally: M '144 GeV [43] for arbitrary values of the squarks masses.  Let us start with low tan
case. The masses of CP-odd, charged and CP-even heavy Higgses increase almost linearly with M . The main restriction comes from the experimental limit on the  lightest Higgs boson mass. It excludes (0 case and for '0 requires the heavy gluino mass M 5750 GeV. Subsequently, one obtains [36]  m '844 GeV, m ! '846 GeV, m '848 GeV, for '0 ,  & & i.e. these particles are too heavy to be detected in the nearest experiments. For high tan
already the requirement of positivity of m excludes the region with small M . In   the most promising region M '1 TeV (m '300 GeV) for the both cases '0 and (0 the   masses of CP-odd, charged and CP-even heavy Higgses are also too heavy to be detected in the near future [42] m '1100 GeV for '0, m '570 GeV for (0 ,   m ! '1105 GeV for '0, m ! '575 GeV for (0 , & & m '1100 GeV for '0, m '570 GeV for (0 . & & The situation is di!erent for the lightest Higgs boson h, which is much lighter. Radiative corrections in this case are crucial and increase the value of the Higgs mass substantially [48,47]. They have been calculated up to the second loop order [58,60]. As can be seen from Eq. (7.7) the one-loop correction is positive increasing the tree level value almost up to 100% and the second one is negative thus decreasing it a little bit. We use in our analysis the leading two-loop contributions evaluated in Ref. [48]. As has been already mentioned these corrections depend on the masses of squarks and the other superpartners for which we substitute the values obtained above from the IRQFPs.

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349

Fig. 11. (A) The dependence of the mass of the lightest Higgs boson h on M "(m  m  ) (shaded area) for '0, low 1317 R R tan
. The dashed line corresponds to the minimum value of m "90 GeV allowed by experiment. (B), (C) The mass of the F lightest Higgs boson h as function of M for di!erent signs of , large tan
. The curves (a,b) correspond to the upper 1317 limit of the Yukawa couplings and to m /m "0 (a) or to m /m "2 (b). The curves (c, d) correspond to the lower     limit of the Yukawa couplings and to m /m "0 (c) or to m /m "2 (d). Possible values of the mass of the lightest     Higgs boson are inside the areas marked by these lines.

The results depend on the sign of parameter . It is not "xed since the requirement of EWSB de"nes only the value of . However, for low tan
negative values of  lead to a very small Higgs mass which is already excluded by modern experimental data, so further on we consider only the positive values of . Fot high tan
both signs of  are allowed. Consider "rst the low tan
regime. At the upper part of Fig. 11 it is shown the value of m for F '0 as a function of the geometrical mean of stop masses } this parameter is often identi"ed with

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a supersymmetry breaking scale M . One can see that the value of m quickly saturates close to 1317 F &100 GeV. For M of the order of 1 TeV the value of the lightest Higgs mass is [36] 1317 m "(94.3#1.6#0.6$5$0.4) GeV for M "1 TeV , (7.15) F 1317 where the "rst uncertainty comes from the deviations from the IRQFPs for the mass parameters, the second one is related to that of the top-quark Yukawa coupling, the third re#ects the uncertainty of the top-quark mass of 5 GeV, and the last one comes from that of the strong coupling. One can see that the main source of uncertainty is the experimental error in the top-quark mass. As for the uncertainties connected with the "xed points, they give much smaller errors of the order of 1 GeV. The obtained result (7.15) is very close to the upper boundary, m "97 GeV, obtained in F Refs. [52,57]. Note, however, that the uncertainties mentioned above as well as the upper bound [52,57] are valid for the universal boundary conditions. Loosing these requirement leads to increase of the upper bound values of the Higgs mass in case of low tan
up to &102 GeV [59]. For the high tan
case the lightest Higgs is slightly heavier, but the di!erence is crucial for LEP II. The mass of the lightest Higgs boson as a function of M is shown in the lower part of Fig. 11. 1317 One has the following values of m at a typical scale M "1 TeV (M +1.3 TeV) [42]: F 1317  m "128.2!0.4!7.1$5 GeV for '0 , F m "120.6!0.1!3.8$5 GeV for (0 . F The "rst uncertainty is connected with the deviations from the IRQFPs for mass parameters, the second one with the Yukawa coupling IRQFPs, and the third one is due to the experimental uncertainty in the top-quark mass. One can immediately see that the deviations from the IRQFPs for mass parameters are negligible and only in#uence the steep fall of the function on the left, which is related to the restriction on the CP-odd Higgs boson mass m . In contrast with the low tan
 case, where the dependence on the deviations from Yukawa "xed points was about 1 GeV, in the present case it is much stronger. The experimental uncertainty in the strong coupling constant  is Q not included because it is negligible compared to those of the top-quark mass and the Yukawa couplings and is not essential here contrary to the low tan
case. One can see that for large tan
the masses of the lightest Higgs boson are typically around 120 GeV that is too heavy for observation at LEP II. Note, however, that the uncertainties increase if one considers the non-universal boundary conditions which decreases the lower boundary for the Higgs mass. At the same time recent experimental data has practically excluded the low tan
scenario and in the next year the situation will be completely clari"ed. Thus, one can see that in the IRQFP approach all the Higgs bosons except for the lightest one are found to be too heavy to be accessible in the nearest experiments. This conclusion essentially coincides with the results of more sophisticated analyses. The lightest neutral Higgs boson, on the contrary is always light. The situation may improve a bit if one considers more complicated models with enlarged Higgs structure [61]. However, it does not change the generic feature of SUSY theories, the presence of light Higgs boson.  Lep II is now increasing its energy and may possibly reach the lower bound of high tan
scenario predictions.

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In the near future with the operation of the Tevatron and LHC proton}antiproton accelerators with the c.m. energy of 2 and 15 TeV, respectively, the energy range from 100 to 1000 GeV will be scanned and supersymmetry will be either discovered or the minimal version will be abandoned in particle physics. The Higgs boson might be the "rst target in search of SUSY.

8. Conclusion We have attempted to show how following the pattern of RG #ow one can explore physics lying beyond the Standard Model of fundamental interactions. The methods are essentially based on renormalization group technique in the framework of quantum "eld theory. The running of parameters is the key ingredient in attempts to go beyond the range of energies accessible to modern accelerators, to "nd manifestation of new physics. In the absence of solid experimental facts the RG #ow happens to be the only source of de"nite predictions. The peculiarity of the moment is that these predictions are the subject of experimental tests today, and in the near future, so one may hope to check the ideas described above at the turn of the Millennium.

Acknowledgements The author is grateful to S. Codoban, A. Gladyshev, M. Jurc\ is\ in, V. Velizhanin, K. TerMartirossian, R. Nevzorov, W. de Boer and G. Moultaka for useful discussions and to S. Codoban for help in preparing the manuscript. Special thanks to the organizers of the conference D. O'Connor and C. Stephens. Financial support from RFBR grants C 99-02-16650 and C 9615-96030 is kindly acknowledged.

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CONTENTS VOLUME 344 E. Dagotto, T. Hotta, A. Moreo. Colossal magnetoresistant materials: the key role of phase separation

1

Editorial

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J. Zinn-Justin. Precise determination of critical exponents and equation of state by "eld theory methods

159

K. Binder, E. Luijten. Monte Carlo tests of renormalization-group predictions for critical phenomena in Ising models

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M.J. Bowick, A. Travesset. The statistical mechanics of membranes

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D.I. Kazakov. Supersymmetry in particle physics: the renormalization group viewpoint

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Contents volume 344

355

Forthcoming issues

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PII S0370-1573(01)00027-8

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FORTHCOMING ISSUES J. von Delft, D.C. Ralph. Spectroscopy of discrete energy levels in ultrasmall metallic grains F. Ehlotzky. Atomic phenomena in bichromatic laser "elds M. Chaichian, W.F. Chen, C. Montonen. New superconformal "eld theories in four dimensions and N"1 duality J. Manjavidze, A. Sissakian. Very high multiplicity hadron processes R. Merkel. Force spectroscopy on single passive biomolecules and single biomolecular bonds M. Krawczyk, A. Zembrzuski, M. Staszel. Survey of present data on photon structure functions and resolved processes K. Michielsen, H. De Raedt. Integral-geometry morphological image analysis E. Nielsen, D.V. Fedorov, A.S. Jensen, E. Garrido. The three-body problem with short-range interactions C. Song, Dense nuclear matter: Landau Fermi-liquid theory and chiral Lagrangian with scaling D.R. Grasso, H.R. Rubinstein. Magnetic "elds in the early Universe E.L. Nagaev. Colossal-magnetoresistance materials: manganites and conventional ferromagnetic semiconductors D. Atwood, S. Bar-Shalom, G. Eilam, A. Soni. CP violation in top physics V.K.B. Kota. Embedded random matrix ensembles for complexity and chaos in "nite interacting particle systems V.M. Loktev, R.M. Quick, S. Sharapov. Phase #uctuations and pseudogap phenomena T. Nakayama, K. Yakubo. The forced oscillator method: eigenvalue analysis and computing linear response functions S.O. Demokritov, B. Hillebrands, A.N. Slavin. Brillouin light scattering studies of con"ned spin waves: linear and nonlinear con"nement I.M. Dremin, J.W. Gray. Hadron multiplicities C.O. Wilke, C. Ronnewinkel, T. Martinez. Dynamic "tness landscapes in molecular evolution V. Narayanamurti, M. Kozhevnikov. BEEM imaging and spectroscopy of buried structures in semiconductors J. Richert, P. Wagner. Microscopic model approaches to fragmentation of nuclei and phase transitions in nuclear matter

PII S0370-1573(01)00028-X