Dynamical Symmetries for Nanostructures: Implicit Symmetries in Single-Electron Transport Through Real and Artificial Molecules

Konstantin Kikoin Mikhail Kiselev Yshai Avishai Dynamical Symmetries for Nanostructures Implicit Symmetries in Single-...

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Konstantin Kikoin Mikhail Kiselev Yshai Avishai

Dynamical Symmetries for Nanostructures Implicit Symmetries in Single-Electron Transport Through Real and Artificial Molecules

Ph.D. Konstantin Kikoin School of Physics and Astronomy Tel Aviv University 69978 Tel Aviv Israel [email protected]

Ph.D. Mikhail Kiselev The Abdus Salam Intl. Center for Theoretical Physics Strada Costiera 11 34151 Trieste Italy [email protected]

Ph.D. Yshai Avishai Ben Gurion University 84105 Beer Sheva Israel [email protected]

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, speciﬁcally those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machines or similar means, and storage in data banks. The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. # 2012 Springer-Verlag/Wien SpringerWienNewYork is part of Springer Science+Business Media springer.at Typesetting: SPi Publisher Services, Pondicherry, India Printed on acid-free and chlorine-free bleached paper SPIN: 12590339 Library of Congress Control Number: 2011943752 ISBN 978-3-211-99723-9 e-ISBN 978-3-211-99724-6 DOI 10.1007/978-3-211-99724-6 SpringerWienNewYork

Dedicated to the memory of Yuval Ne’eman and John Hubbard, two great physicists whose ideas are the corner stones of the theories presented in this book.

Preface

The main goal of this monograph is to demonstrate the relevance of dynamical symmetry and its breaking to the rapidly growing ﬁeld of nanophysics in general, and nanoelectronics in particular. It is intended to amalgamate seemingly highly abstract concepts of Group theory with the physics of recently fabricated nanoobjects such as single electron transistors. In all these systems, dynamical symmetries are shown to be intimately related with many-body physics, and in particular, the ubiquitous Kondo effect and other hallmarks of quantum impurity problems. Thereby, we expose yet another facet of the existing deep and profound relations between quantum ﬁeld theory and condensed matter physics. The concept of symmetry in quantum mechanics has had its golden age in the middle of the last century. In that period, the beauty, elegance and efﬁciency of group theoretical physics has been exposed in numerous remarkable revelations, from classiﬁcation of hadron multiplets, isospin in nuclear reactions, the orbital symmetry in Rydberg atoms, point-groups in crystallography, translational symmetry in solid state physics, and so on. At the focus of all these studies stands the symmetry group of the underlying Hamiltonian. Using the powerful formalism of group theory, the energy spectrum of the physical system possessing the pertinent symmetry could be extracted within an elegant and time saving formalism. Exploiting the properties of discrete and inﬁnitesimal rotation and translation operators, general statements about the basic properties of quantum mechanical systems could be formulated in a form of theorems (Wigner theorem, Bloch theorem, Goldstone theorem, Adler principle, etc). The intimate relation between group theory and quantum mechanics is therefore well established and has been exposed in numerous excellent handbooks. A somewhat more subtle aspect featuring group theory and quantum mechanics emerged and was formulated later on, that is, the concept of dynamical symmetry. The notion of dynamical symmetry group is distinct from that of the familiar symmetry group. To understand this distinction in an heuristic way let us recall that all generators of the symmetry group of the Hamiltonian Hˆ encode certain integrals ˆ These operators induce all transformations of the motion, which commute with H. which conserve the symmetry of the Hamiltonian, and may have non-diagonal maˆ On the other trix elements only within a given irreducible representation space of H. vii

viii

Preface

hand, dynamical symmetry of Hˆ is realized by transformations implementing transitions between states belonging to different irreducible representations of the symmetry group. One may then say that the generators of dynamical symmetry group of a quantum mechanical system are in fact the generators of the energy spectrum or some part of it. Special examples of dynamical symmetries in quantum mechanics emerge as hidden symmetries, where additional degeneracy exists due to an implicit symmetry of the interaction. Another example is supersymmetry, where the group algebra includes both commutation and anticommutation relations. The starting point in most of our analysis is a generalized Anderson Hamiltonian which, under certain conditions can be approximated by a generalized spin Hamiltonian encoding a myriad of exchange interactions between localized electrons in nano-objects (such as quantum states in complex quantum dots) and itinerant electrons in the reservoirs made in contact with the localized electrons. These exchange interactions may be due to spin as well as to orbital degrees of freedom. They lead to effective exchange Hamiltonians that display a rich pattern of dynamical symmetries. Mathematically, these symmetries are exposed as the pertinent exchange Hamiltonian includes, in addition to the standard spin operators, new sets of vector operators which form the basis for the representation of irreducible tensor operators entering the effective Hamiltonian. These operators induce transitions between different spin multiplets and generate dynamical symmetry groups (such as SU(n) and SO(n)) that are not exposed within the bare Anderson Hamiltonian. Like in quantum ﬁeld theory, the most dramatic aspects of dynamical symmetry in the present context is not its relation with the spectrum but, rather, the manner in which it is broken. An indispensable tool for manipulating the pertinent mathematics required for identifying the relevant dynamical symmetry groups is the superalgebra of Hubbard operators, upon which we will heavily rely. The role of dynamical symmetries and their manifestations will be reviewed and analyzed in several systems such as complex quantum dots (planar, vertical and self-assembled), molecular complexes adsorbed on metallic surfaces and attached to quantum wires, cold gases conﬁned in magnetic traps. It will be shown how these dynamical symmetries are activated by Coulomb and exchange interactions with itinerant electrons in the macroscopic Fermi or Bose reservoirs (metallic leads and substrates in various nanodevices). We will then develop the concept within numerous physical situations, including the Kondo cotunnelling in various environments. The notion of dynamical symmetry is meaningful also for the systems out of equilibrium, in presence of electromagnetic ﬁeld and stochastic noise and in timedependent problems like Landau –Zener effect. Thus, the main goal of this book is to generalize the principles of dynamical symmetries formulated for the integrable systems to the many-body systems, for which only the low-energy part of the excitation spectrum is known. Tel Aviv - Trieste - Beer Sheva, October 31, 2011

Konstantin Kikoin Mikhail Kiselev Yshai Avishai

Acknowledgements

We acknowledge fruitful discussions with our colleagues Boris Altshuler, Jan von Delft, Peter Fulde, Yuri Galperin, Yuval Gefen, Leonid Glazman, Vladimir Gritsev, David Khmelnitskii, Il’ya Krive, Tetiana Kuzmenko, Stefan Ludwig, Laurens W. Molenkamp, Florina Onufrieva, Michael Pustilnik, Jean Richert, Robert Shekhter, Maarten Wegewijs.

ix

Contents

1

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2

HIDDEN AND DYNAMICAL SYMMETRIES OF ATOMS AND MOLECULES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Rigid Rotator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Hydrogen atom and Runge-Lenz vector . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Dynamical symmetries for spin systems . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Hubbard atom and Fulde molecule . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Three-fold way for Hubbard atom . . . . . . . . . . . . . . . . . . . . . . 2.5 Fock – Darwin atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Dynamical symmetry and supersymmetry . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Manifestations of supersymmetry in atomic models . . . . . . . 2.7 Quasienergy spectrum for periodical time-dependent problems . . . .

5 8 10 15 23 29 31 35 39 45

3

NANOSTRUCTURES AS ARTIFICIAL ATOMS AND MOLECULES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.1 Introductory remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.2 Planar quantum dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.3 Vertical quantum dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.4 Self-assembled quantum dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.5 Complex quantum dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.5.1 Double quantum dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.5.2 Triple quantum dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.6 Molecules and molecular complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 3.6.1 Fullerene molecules as quantum dots . . . . . . . . . . . . . . . . . . . . 93 3.6.2 Nanotubes as quantum dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.6.3 Single electron tunneling through metal organic complexes . 96 3.6.4 Vibrational degrees of freedom in single molecular tunneling 101

xi

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4

DYNAMICAL SYMMETRIES IN THE KONDO EFFECT . . . . . . . . 107 4.1 Kondo mapping and beyond (surplus symmetries) . . . . . . . . . . . . . . . 108 4.2 Kondo effect in quantum dots with even occupation . . . . . . . . . . . . . . 126 4.3 Kondo physics for short chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 4.3.1 Serial geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 4.3.2 Side geometry, Fano – Kondo effect . . . . . . . . . . . . . . . . . . . . . 147 4.3.3 Cross geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 4.3.4 Parallel geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 4.3.5 Multichannel Kondo tunneling . . . . . . . . . . . . . . . . . . . . . . . . . 162 4.4 Kondo physics for small rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 4.4.1 Kondo tunneling and Aharonov – Bohm interference . . . . . . 190

5

DYNAMICAL SYMMETRIES IN MOLECULAR ELECTRONICS . 197 5.1 Kondo effect in molecular environment . . . . . . . . . . . . . . . . . . . . . . . . 197 5.1.1 Chiral symmetry of orbitals and Kondo tunneling . . . . . . . . . 199 5.1.2 Kondo effect in the presence of Thomas-Rashba precession . 201 5.1.3 Scanning tunneling spectroscopy via Kondo impurities . . . . . 206 5.2 Kondo effect in molecular magnets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 5.3 Phonon assisted tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 5.3.1 Two-electron tunneling at strong electron-phonon coupling . 227

6

DYNAMICAL SYMMETRIES AND SPECTROSCOPY OF QUANTUM DOTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 6.1 Kondo effect in the presence of electromagnetic ﬁeld . . . . . . . . . . . . . 234 6.2 Excitonic spectroscopy of quantum dots . . . . . . . . . . . . . . . . . . . . . . . . 240

7

DYNAMICAL SYMMETRIES AND NON-EQUILIBRIUM ELECTRON TRANSPORT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 7.1 Dynamically induced ﬁnite bias anomalies in tunneling spectra . . . . 248 7.2 Dephasing and decoherence in quantum tunneling . . . . . . . . . . . . . . . 258 7.2.1 Vector Keldysh model in the time domain . . . . . . . . . . . . . . . . 276

8

TUNNELING THROUGH MOVING NANOOBJECTS . . . . . . . . . . . . 283 8.1 Conversion of coherent charge input into the Kondo response . . . . . . 286 8.1.1 Single-electron shuttling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 8.2 Time-dependent Landau-Zener effect . . . . . . . . . . . . . . . . . . . . . . . . . . 292

9

MATHEMATICAL INSTRUMENTATION . . . . . . . . . . . . . . . . . . . . . . . 309 9.1 SU(2) group for arbitrary spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 9.2 Kinematical constraints for systems with SO(n) and SU (n) symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 9.2.1 SO(4) group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 9.2.2 Noncompact groups SO(p, n − p) . . . . . . . . . . . . . . . . . . . . . . . 313 9.2.3 Groups of conformal transformations . . . . . . . . . . . . . . . . . . . 314 9.2.4 From SU (2) to SU(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 9.3 Bosonization and fermionization for arbitrary spins . . . . . . . . . . . . . . 320

Contents

xiii

9.3.1 9.3.2 9.3.3 9.3.4 9.3.5 9.3.6 9.3.7 9.3.8

Schwinger boson representation for the SU(2) group . . . . . . 321 Holstein – Primakoff boson representation for the SU(2) group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 Dyson – Maleev representation for the SU(2) group . . . . . . . 323 Pomeranchuk – Abrikosov spin fermion representation for the SU(2) group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 Spin-fermion representations for the SO(n) groups . . . . . . . . 325 Popov – Fedotov semi-fermion representation . . . . . . . . . . . . 326 Majorana fermionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 Mixed fermion-boson representations . . . . . . . . . . . . . . . . . . . 327

10 CONCLUSIONS AND PROSPECTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341

Chapter 1

INTRODUCTION

In modern theoretical physics the word-combination ”dynamical symmetry” is frequently used in the context of various mechanisms of dynamical symmetry breaking of vacuum expectation value (Higgs – Anderson mechanism in particle physics, Anderson – Nambu mechanism in superconductivity, etc, see [101] for basic references). This book is devoted to analysis of dynamical symmetries that arise when a group theoretical approach is used in a description of a contact between a few electron nanosystem S with deﬁnite symmetry GS and a macroscopic system B (”bath” or ”reservoir”). Due to this contact the symmetries of the system S and the corresponding conservation laws are violated. If the contact between the two systems is weak enough, the dynamics of interaction may be described in terms of transitions between the eigenstates of a system S belonging to different irreducible representations of the group GS generated by the operators which obey the algebra gS . If the operators describing transitions between these eigenstates together with generators of the group GS form an enveloping algebra dS for the algebra gS , one

may say that the system S possesses dynamical symmetry characterized by some group DS . Dynamical symmetry group offers mathematical tool for a uniﬁed ap-

proach to quantum objects, which allows one to consider not only the spectrum of a system S , but also its response to external perturbation violating the symmetry GS and various complex many-body effects characterizing interaction between the system S and its environment B. An initial impact to the study of dynamical symmetries of the above kind was given in a context of classiﬁcation of elementary particle multiplets. The ﬁrst representative example of dynamical symmetry was an attempt to construct hadron multiplets and transitions between states within this miltiplet by means of generators of the group SU(3) [126, 127, 296, 447]. This group of unitary matrices of K. Kikoin et al., Dynamical Symmetries for Nanostructures: Implicit Symmetries in Single-Electron Transport Through Real and Artificial Molecules, DOI 10.1007/978-3-211-99724-6_1, © 2012 Springer-Verlag/Wien

1

2

1 INTRODUCTION

the 3-rd order describes the states of 3-level system and all transition between these states. Three states were identiﬁed with three quarks labeled by u, d, s ”colors”. The concept of dynamical symmetry (the “eightfold way”), as an approximate symmetry generated by some operator algebra which describes transitions between the states belonging to various irreducible representations of the group GS as a result of dynamics was formulated quite distinctly in Ref. [81] (see also [39]). Following this paradigm, the dynamical symmetry group DS of a quantum mechanical system may be deﬁned as a ﬁnite-dimensional Lie group whose irreducible representations act in a Hilbert space of all states of a subsystem S in a given energy interval E which characterize the scale of interaction of this subsystem with its environment B. A short time later the energy spectrum of an integrable quantum mechanical system , namely the rigid rotator, also was described in terms of dynamical symmetry given by SO(4) group of 4-dimensional orthogonal matrices [46, 289]. The conventional symmetry group of rigid rotator is the usual SO(3) group of 3D rotations, and additional dimension arises when the ”supermultiplets” with different orbital moments l and ”selection rules” Δ l = 0, ±1 are included in the dynamical group. In parallel, it was recognized that the well-known fourth dimension hidden in the Schr¨odinger equation for a hydrogen atom and the Runge – Lenz vector related to this hidden symmetry can also be described in terms of dynamical symmetry: it was shown that all the discrete levels of an electron in a Coulomb potential form a multiplet of a conformal group SO(4, 2) [274, 294]. When treating the components of the Runge – Lenz vector as three more group generators together with the usual operators of angular moment, one sees that the enveloping o(4) algebra generates the SO(4) group of 4D rotations [387], which is the real symmetry of the Schr¨odinger equation for an electron in a Coulomb ﬁeld in accordance with the early quantum-mechanical solution of this problem [36, 111]. In this case, additional group operators do not describe transitions within the energy multiplet, and one may speak about the hidden symmetry of Schr¨odinger equation with a Coulomb potential ∼ 1/r. The ideas of dynamical symmetry have been applied also to other integrable systems, in particular to n-dimensional quantum oscillator [40, 145, 173], where the generators of SO(n, 1) group unite all levels of harmonic oscillator into a single irreducible representation, to non-relativistic electron in quantizing magnetic ﬁeld, and to some other problems. Further generalization of the ideas of dynamical symmetry includes also the non-stationary states of quantum systems not necessarily characterized by deﬁnite energy.

1 INTRODUCTION

3

The main achievements of the dynamical symmetry approach during the ”Sturm und Drang” period of its development are summarized in the monograph [275] [published in Russian]. Various facets of the Coulomb problem for the hydrogen atom treated in terms of hidden and dynamical symmetry approach are discussed in two more books [95, 201]. In the latter book the supersymmetry of hydrogen atom which is closely related to existence of the Runge – Lenz vector is also discussed. During the last decade of past century novel approximately symmetric few-body quantum objects became available for theoretical analysis due to rapid progress of nanotechnology and nanophysics. These nano-objects are quantum dots with countable number of electrons and controllable spin states incorporated in electric circuits, where metallic electrodes play part of a reservoir B for a quantum dot S [238, 359]. Another class of quantum objects with similar properties are molecular complexes which form bridges between metallic electrodes or between the metallic substrate and the tip of tunnel microscope [70, 295]. It was recognized [203] that the concept of dynamical symmetry is highly useful for the study of many-body effects which accompany tunneling through quantum dots and molecular bridges. In case of strong Coulomb blockade which suppresses charge ﬂuctuations in a quantum dot, the spin state of a dot with given number N of electrons is usually well deﬁned. Then electron tunneling through the dot which may be detected as a single electron tunnel current between the source and drain electrodes, breaks the spin symmetry of this dot. This symmetry violation as well as the many-body effects which accompany electron tunneling through quantum dots may be quite elegantly described within a framework of dynamical symmetry approach. Unlike the integrable systems with dynamical symmetries described in the monographs [95, 201, 275], the problems of complex quantum dots and molecules in contact with boson or fermion bath as a rule cannot be solved exactly. Moreover, the type of dynamical symmetry strongly depends on the characteristic energy scale E of the coupling between the nanoobject S and the bath B. Besides, this symmetry may be changed with decreasing temperature and varying control parameters, thus resulting in quantum criticality phenomena, which may be easily detected as variations of current-voltage characteristics in single-electron tunneling experiments. The dynamical symmetries are usually described by the Lie groups SO(n) with n 4 or SU(n) with n 3. Like in integrable systems mentioned above, these symmetries become a source of speciﬁc response of nanoobject S to external ﬁelds.

4

1 INTRODUCTION

Dynamical symmetries may be also discerned in time-dependent, non-equilibrium and stochastic effects. In this book all facets of dynamical symmetries of nanosystems are discussed both in terms of strict mathematical deﬁnitions and in a context of practical physical applications in nano- and molecular electronics. Some aspects of dynamical symmetries in the physics of complex quantum dots were brieﬂy considered in our reviews [32, 204, 206]. We start with an exposition of dynamical symmetries in exactly solvable models both mentioned above and newly found (Chapter 2), then give a short description of nanostructures which were practically realized during the last two decades (Chapter 3). The central part of the book is devoted to studies of dynamical symmetries in complex quantum dots and molecular complexes (Chapters 4 – 6) with a special accent on the Kondo-resonance tunneling regime. The latter regime is a salient example of many-body phenomenon, where the dynamical symmetry plays a decisive part. Non-equilibrium tunneling through nanoobjects is a special and vast enough branch of contemporary nanophysics which deserves a special monograph. In this book we concentrate only on those non-equilibrium effects which are directly related to dynamical symmetries of quantum dots and molecular complexes (Chapter 7). Special type of temporal phenomena in nanoobjects are adiabatic and nearly adiabatic effects induced either by classical motion (“shuttling”) of nanoobject or cyclic variation of the device parameters which result in periodic time-dependent level crossing (time-dependent Landau – Zener effect). Symmetry related aspect of these phenomena are discussed in Chapter 8. It is presumed that the readers of this book possess a basic knowledge of the main principles of the Group theory and its applications in Quantum mechanics within a framework of standard textbooks like [92, 130, 132, 151, 327, 428]. We also use where necessary the method of many-body Green functions. One may address to the monograph [106] as an introductory course to this ﬁeld. However we considered expedient to collect in the Mathematical Annex (Chapter 9) all relevant information about the characteristic properties of those Lie groups which are responsible for dynamical symmetries in nanosystems and to present other useful mathematical information related to the physical problems discussed in this book. In Chapter 10, which terminates the book, the implementation of dynamical symmetry ideas in nanophysics is summarized and possible future development of this approach is discussed.

Chapter 2

HIDDEN AND DYNAMICAL SYMMETRIES OF ATOMS AND MOLECULES

We concentrate in this book on the symmetry properties of nanoobjects (quantum dots, rings and short chains of quantum dots, molecular complexes) in a weak tunneling and/or capacitive contact with reservoirs (metallic electrodes attached to quantum dots, metallic substrates or edges of nanowires for molecular complexes deposited on these surfaces and points, etc). Before turning to these artiﬁcially engineered devices, we will review in brief the origin of dynamical symmetry in ”natural” quantum objects, i.e. in some integrable quantum systems with well deﬁned energy spectrum and quantum numbers. Conventionally the symmetry of such systems is considered in terms of the symmetry group GS of Schr¨odinger equation. This description is based on the fundamental Wigner theorem [428] which states that the eigenfunctions which belong to a given energy level E are transformed along the same irreducible representation of the group GS . Sometimes two or more energy levels coincide not because of symmetry demands but due to accidental degeneracy. Such a degeneracy will play important part in the following chapters of this book. Here we concentrate on two other aspects of the symmetry of quantum systems, namely on the dynamical and hidden symmetries inherent in some integrable quantum objects. Following the deﬁnition used in Ref. [274], we deﬁne the dynamical symmetry group DS as a Lie group characterized by the irreducible representations which act in the whole Hilbert space of eigenstates |l λ of a Schr¨odinger equation ˆ λ = El |l λ H|l

(2.1)

describing quantum system S . Here l is the index of irreducible representation and

λ enumerates the lines of this representation. Projection operators for an irreducible K. Kikoin et al., Dynamical Symmetries for Nanostructures: Implicit Symmetries in Single-Electron Transport Through Real and Artificial Molecules, DOI 10.1007/978-3-211-99724-6_2, © 2012 Springer-Verlag/Wien

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2 HIDDEN AND DYNAMICAL SYMMETRIES OF ATOMS AND MOLECULES

representation l λμ

X(l) = |l λ l μ |

(2.2)

play central part in the procedure of construction of irreducible representations of a group of Schr¨odinger equation GS . The basic property of these operators is given by the equation λμ (2.3) X(l) |l ν = δll δμν |l λ . These operators are useful for construction of basis functions for irreducible representations of GS . Group generators obeying algebra gS may be represented via operators (2.2) (see Chapter 9). To construct an algebra which generates a dynamical group, one should add to the set (2.2) the operators λμ

X(ll ) = |l λ l μ |

(2.4)

which project the states belonging to different irreducible representations (l = l ) of the group GS one onto another. Unifying the notations |l λ = |Λ , one may write the commutation relation ˆ = (EΛ − EΛ )Hˆ [X ΛΛ , H]

(2.5)

The right hand side of this relation turns into zero provided the states Λ and Λ belong to the same irreducible representation of the group GS . If one succeeds in constructing a closed algebra dS from the set of operators (2.2),(2.4) then it is possible to say that the system described by the Hamiltonian (2.1) possesses the dynamical symmetry DS . This algebra is conditioned by the norm

∑Xλλ = 1

(2.6)

λ

and the commutation relations for the operators X κλ . In general case these relations may be presented in the following form [170] [X κλ , X μν ]∓ = X κν δλ μ ∓ X μλ δκν

(2.7)

“General case” means that the Fock space includes states which may belong to different charge sectors, where changing the state λ for the state κ implies changing the number of fermions Nλ → Nκ in a many-particle system. If both Nλ − Nκ and Nν − Nμ are odd numbers(Fermi-type operators), the plus sign should be chosen

2 HIDDEN AND DYNAMICAL SYMMETRIES OF ATOMS AND MOLECULES

7

in Eq. (2.7). If at least one of these difference is zero or even number (Bose-type operators), one should take the minus sign. The operators X κλ were exploited by J. Hubbard as a convenient tool for description of elementary excitations in strongly correlated electron systems (SCES). His seminal model of interacting electron motion in a narrow band, known now as a Hubbard model [169, 170, 171] was the ﬁrst microscopic model of SCES for which the conventional perturbative approach based on Landau Fermi liquid hypothesis turned out to fail (see detailed discussion in Ref. [157]. Now the realm of SCES is really vast, and the most of artiﬁcial nanostructures belong to this realm. In particular, complex quantum dots under strong Coulomb blockade are typical examples of short Hubbard chains or rings (see Chapter 4). The Hubbard operators (2.4) obeying the commutation relation (2.5) is a convenient tool for construction of the algebras generating the dynamical symmetry groups of the resolvent operator Rˆ = (Hˆ − E)−1 or Schr¨odinger operator Rˆ −1 . We will use these operators in a systematic way to construct the irreducible tensor operators O (r) (scalars, r = 0, vectors, r = 1, and tensors r = 2) which transform along the representation of the dynamical group which characterizes the symmetry properties of the supermultiplet of the eigenstates of the Schr¨odinger equation: (r)

Oρ =

(r)

∑ Λ |Oρ

|Λ X ΛΛ .

(2.8)

ΛΛ

Here the index ρ stands for components of irreducible tensor operator of the rank r. On the one hand, it is clear that the operators X ΛΛ are able to generate all the eigenstates of the Hamiltonian Hˆ from any given initial state Λ . On the other hand, the components of the operators O (r) form a closed algebra, which characterizes the dynamical symmetry group provided the Hamiltonian Hˆ possesses such symmetry. Having in mind future applications to geometrically conﬁned nanoobjects, we restrict ourself mainly by discrete eigenstates. In the two following sections we discuss the symmetry properties of two integrable quantum mechanical systems (rigid rotator and hydrogen atom) and show how the dynamical symmetries DS emerge from the apparent symmetry SO(3) of the Schr¨odinger equation.

8

2 HIDDEN AND DYNAMICAL SYMMETRIES OF ATOMS AND MOLECULES

2.1 Rigid Rotator A simpliﬁed quantum-mechanical description of molecular motion in a framework of rigid rotator model implies quenching of vibrational excitations, whereas the rotational degrees of freedom are described as rotation of a ”solid body” around some axis n = nξ , nη , nζ which in turn precesses around a ﬁxed z axis in a 3D space. The Hamiltonian of symmetric rotator is h¯ 2 h¯ 2 2 h¯ 2 2 h¯ 2 Lξ + L2η + Hˆ = L2ζ = L + 2I⊥ 2I

2I⊥ 2

1 1 − I I⊥

L2ζ

(2.9)

Here the coordinates (ξ , η , ζ ) are bound to the rotation axis, (I⊥ , I ) are two component of the moment of inertia. Both types of rotations are quantized but the energy levels depend only on the quantum number l and the eigenvalues κ of the operator Lζ which change in the interval κ = −l, . . . + l E jκ

h¯ 2 h¯ 2 = l(l + 1) + 2I⊥ 2

1 1 − I I⊥

κ 2.

(2.10)

In case of fully symmetric rotator with I⊥ = I the levels lose dependence on κ and acquire 2l + 1-fold degeneracy. Additional degeneracy in projection of the angular momentum on the z-axis of ﬁxed reference frame results in total (2l + 1)2 -fold degeneracy of the level E j of spherically symmetric rotator. This additional symmetry is inessential for the level classiﬁcation, but it is meaningful from the point of view of the dynamical symmetry of rigid rotator [95, 275, 289]. Indeed, any rotation

z ζ

α

y Fig. 2.1 Rigid rotator precessing around the axis z.

x

of the coordinates is characterized by three Euler angles in the precessing system (ξ , η , ζ ) and one more angle α between the axes ζ and z (Fig. (2.1). Corresponding spherical coordinates are (r, ϕ , ϑ , α ). The basis functions for description of such a motion are the hyperspherical harmonics Ynlm (α , ϑ , ϕ ). On the other hand, these

2.1 Rigid Rotator

9

harmonics form the basis for the irreducible representations of the group SO(4) of rotations on a 4D sphere (see Section 9.2.1). The six generators L, K of this group obey o(4) algebra deﬁned in Eq. (9.14). One may then turn to linear combinations J(1,2) = (L ± K)/2 (9.16). These two vectors describe the two types of rotations mentioned above. Due to the kinematic constraint (9.19) these operators have the same eigenvalues j1 ( j1 + 1) = j2 ( j2 + 1) = l(l + 1) and their projections J1ζ and J2z have 2l + 1 values. Thus, the total degeneracy of an eigenstate with given l is (2l + 1)2. The operators L± , Lζ performing rotations around the axes nξ , nη , nζ generate the o(3) algebra for the subgroup SO(3) (invariance group), whereas the operators K ± , Kζ , which depend on all rotation angles (ϕ , ϑ , α ) (9.21) generate the dynamical algebra o(4), and thus deﬁne the dynamical symmetry group SO(4) of a rigid rotator which unites all the energy levels of rigid rotator in an inﬁnite “supermiltiplet” [289]. To show this, let us consider the vector K as an irreducible tensor of the 1-st rank and express its components Kτ1 via projection operators (2.4) following the pattern (2.8). Here τ = 0, ±1 stands for Kζ , K± , respectively. In this case, one should use the (ll )

operators Xmm describing transitions between states with given momentum l and its projection m and the states with other values l m of these quantum numbers, (1)

Kτ =

(1)

∑ lm|Kτ

ll |l m Xmm

(2.11)

lm,l m

Then, using the Wigner-Eckart theorem, we represent the coefﬁcients in this expansion as l 1 l (1) l−m (2.12) l K (1) l lm|Kτ |l m = (−1) −m τ m (cf. Eq. 2.8) The factors in the r.h.s. of this equation are the Wigner 3 j-symbol and the reduced matrix element of the vector K. Explicit form of these matrix elements may be found in [275, 289, 291]. It follows from the triangle rule for 3 j-coefﬁcients in Eq. (2.12) that the transversal components K ± = Kx ± iKy of the operator K, work as ladder operators which connect the states with Δ l = ±1 and thus unite all the energy levels of rigid rotator into an inﬁnite multiplet of the semisimple group SO(4) with generators L, K and two Casimir operators (9.18) or (9.19). One may perceive from the above procedure that the choice of dynamical symmetry is not a unique procedure. For example, one may change the signature in the metrics from {+, +, +, +} to {+, +, +, −} and introduce generators K¯ j (9.27) in-

10

2 HIDDEN AND DYNAMICAL SYMMETRIES OF ATOMS AND MOLECULES

stead of K j . These operators represent dynamical symmetry SO(3, 1). Usually the choice of enveloping group is determined by physical reasons (e.g., by the type of perturbation which actuates the dynamical symmetry). In particular, in case when this perturbation implies the selection rules Δ l = 0, ±1, ±2 for transitions between (2) different states, then one has to use the irreducible tensor Kτ of the 2nd rank in the expansion (2.8). Five components of this tensor together with the operators L j of the invariance group SO(3) form the set of generators of the dynamical group SU(3) (see Sections 2.3 and 9.2.4).

2.2 Hydrogen atom and Runge-Lenz vector The quantum mechanical problem of electronic spectrum of hydrogen atom inherited its peculiar properties from its classical analog, i.e. from the mechanical problem of rotation of one celestial body in the gravitational ﬁeld ∼ 1/r of another body (Kepler problem). It was realized three centuries ago that the orbital rotation of celestial body in such a ﬁeld is characterized by a speciﬁc constant of motion which is known now as a Laplace-Runge-Lenz vector (although two latter physicists only used this vector in their own tutorial and scientiﬁc texts). This vector arose anew in the analysis of the Schr¨odinger equation for an electron wavefunction ψ (r) in the potential ﬁeld created by a proton. This equation in atomic units (e = 1, h¯ = 1, m = 1) has the form: Δ 1 − − ψ (r) = E ψ (r) . (2.13) 2 r Here Δ is the 3D Laplacian. This equation obviously has the spherical symmetry SO(3) generated by operators of inﬁnitesimal 3D rotations, but the eigenlevels corresponding to discrete states with E < 0 depend only on the principal quantum number, En = −

1 2n2

(2.14)

and not on the orbital momentum l = n − 1, n − 2, . . . 1, 0, thus possessing the n2 fold degeneracy. All peculiarities of the behavior of an electron in a potential ∼ 1/r stem from the fact that the rotation group SO(3) is only a subgroup of the true symmetry group of Eq. (2.14) . To reveal this symmetry let us follow the approach used in Refs. [36, 111] and turn to the momentum representation of the Schr¨odinger equation (2.13)

2.2 Hydrogen atom and Runge-Lenz vector

1 p2 ψ (p) + 2 2π where

ψ (p) =

1 (2π )3/2

11

p20 ψ (q)dq = − ψ (p) (p − q)2 2

ψ (r)e−i(p·r) dr, −

(2.15)

p20 =E 2

(2.16)

Then we make a conformal mapping of each point (p, p0 ) onto a point on the surface of the 4D sphere of unit radius with the coordinates (ξ1 , ξ2 , ξ3 , ξ4 )

ξi =

p20 − p2 2p0 p (i = 1, 2, 3); ξ = ; i 4 p20 + p2 p20 + p2

∑

ξi2 = 1

(2.17)

i=1−4

Then by means of substitution

π Ψ (ξ1 , ξ2 , ξ3 , ξ4 ) = √ (p0 )5/2 (p20 + p2 )2 ψ (p) ≡ Φ (p) 8 Eq. (2.15) is transformed into

Ψ (ξ1 , ξ2 , ξ3 , ξ4 ) = 1 2π 2 p0

R4

(2.18)

Ψ (ξ1 , ξ2 , ξ3 , ξ4 )d 4 ξ 2 |ξ1 − ξ1 | + |ξ2 − ξ2 |2 + |ξ3 − ξ3 |2 + |ξ4 − ξ4 |2

The latter equation is invariant relative to rotations in a 4D space. Thus, we see that the real symmetry of an electron in a Coulomb ﬁeld is SO(4). One may construct the inﬁnitesimal rotation operators in the space {ξ1 , ξ2 , ξ3 , ξ4 }. There are six such operators describing rotations in six 2D planes (ξi ξ j ). Details of this construction may be found in the book [327]. Returning back from ξ -space to original variables ˆ where Hˆ is the Hamiltonian oper{p, p0 } and changing p0 for the operator −H, ator in Eq. (2.13), one eventually ﬁnds equations for these generators: r×p−p×r 2

1 p×L−L×p r − F= 2 r −2Hˆ

L=

(2.19)

The vector of orbital momentum L contains three generators of the group SO(3), which in this case is only a subgroup of the true symmetry group SO(4) [111, 324]. Three more generators of the latter group are given by the components of the vector F. The Runge – Lenz vector mentioned above is in fact ˆ (2.20) A = F −2H.

12

2 HIDDEN AND DYNAMICAL SYMMETRIES OF ATOMS AND MOLECULES

ˆ After replacing the operator Hˆ by its eigenIt commutes with the Hamiltonian H. value E in Eq. (2.19) for F, the commutation relations for the generators L, F acquire the form [Li , L j ] = iεi jk Lk , [Fi , Fj ] = iεi jk Lk ,

(2.21)

[Fi , L j ] = iεi jk Fk . which is exactly the algebra o(4) of the SO(4) group generators (see Section 9.2.1). Besides, these operators are subject to kinematic restrictions C1 = L2 + F2 = −1 −

1 , C2 = L · F = 0, −2Hˆ

(2.22)

which are in fact two Casimir operators for the group SO(4) [cf. Eq. (9.18)]. Using (2.22), one may express the Coulomb Hamiltonian via Casimir invariants: 1 1 1 1 =− Hˆ = − 2 C1 + C2 + 1 2 (L + F)2 + 1

(2.23)

Next, we make the rotation 1 1 J = (L + F), K = (L − F), 2 2 and transform the ﬁrst Casimir operator into 1 1 J2 = K2 = − 1+ 4 2Hˆ

(2.24)

(2.25)

The commutation relations for the last set of generator are [Ji , J j ] = iεi jk Jk , [Ki , K j ] = iεi jk Kk ,

(2.26)

[Ki , J j ] = 0. [cf. Eq. (9.19)]. As is discussed in Section 9.2.1, these relations points at the local isomorphism SO(4) = SO(3) × SO(3). Inserting the eigenvalues j( j + 1) = k(k + 1) of the operators J 2 = K 2 and the eigenvalues E of the Hamiltonian Hˆ into Eq. (2.25) one obtains E = −1/2(2 j + 1)2. Since j is integer, one may take 2 j + 1 = n = 1, 2 . . . and just come to the well known result (2.14). Thus, we see that the additional degeneracy of discrete spectrum of hydrogen atom is a direct consequence of an

2.2 Hydrogen atom and Runge-Lenz vector

13

additional ”hidden” dimension which results in appearance of additional Casimir operator. To reveal the dynamical symmetry of an electron in the Coulomb ﬁeld, one should notice that the solutions of Eq. (2.18) may be represented by means of harmonic polynomials ϕ ({ξ j }) of the vector ξ on the 4D sphere, which satisfy the equation Δ4 ϕ = 0 , (2.27) where

Δ4 =

4

∂2

∑ ∂ξ2

j=1

j

is the 4D Laplacian. One may construct operators of inﬁnitesimal transformations, which commute with Δ 4 [274]. These are ∂ , L j = i ∑ ε jkl ξk ∂ ξl k,l ∂ 2 ∂ R j = ∑ ξk − 2ξ j ξk − 2ξ j ∂ξj ∂ ξk k Pj = −i

∂ ∂ξj

I = 1 + ∑ ξk k

∂ ∂ ξk

(2.28)

or, in alternative notations Mkl : L j ≡ ε jkl Mkl

(2.29)

[see Eq. (9.12) in the Mathematical Annex (Chapter 9)]. These operators may be combined in antisymmetric linear combinations Lμν (μ , ν = 0, 1, 2 . . . 5), Lμν = Mμν (μ , ν = 1 . . . 4), Lμ 5 = (Rμ + iPμ )/2 (μ = 1 . . . 4), Lμ 0 = (Rμ − iPμ )/2 (μ = 1 . . . 4), L50 = = −I.

(2.30)

which form the algebra of the conformal group SO(4, 2) (see Section 9.2.2) and obey the commutation relations (9.32).

14

2 HIDDEN AND DYNAMICAL SYMMETRIES OF ATOMS AND MOLECULES

The operators Lμν in the ﬁrst line of Eq. (2.30) form the algebra o(4) for the subgroup SO(4) of dynamical group SO(4, 2). The representations of this subgroup are characterized by the principle quantum number n of discrete hydrogen levels En (2.14), whereas the radial quantum numbers, namely j, k and their projections related to the hidden symmetry determine the degeneracy of discrete electron states in a Coulomb potential. Like in the case of rigid rotator, the operators Lμ 5 and Lμ 0 involving two additional dimensions in the Fock space work as ladder operators in the basis | j, jz ; k, kz just connecting the states with different quantum numbers n and thereby realizing the dynamical symmetry SO(4, 2) [41, 274]. This dynamical symmetry is realized in electric dipole transition between neighboring energy states (n → n ± 1, j → j ± 1) [41]. It is worth noting that the dipole transitions realize the dilatation operation (9.30) of this conformal group. Although the above considerations are conﬁned to three-dimensional rigid rotator and Coulomb atom, the results can be generalized to any dimension n. In the general case the invariance group SO(n) generates the enveloping dynamical group SO(n + 1) or SO(n, 1) as a compact or non-compact dynamical group for n-dimensional rigid rotator [289]. Similar studies of the Coulomb problem [11, 212, 274, 289, 387] show that the invariance group of n-dimensional hydrogen atom is SO(n + 1) due to existence of additional invariant, i.e. Runge – Lenz vector. Its form is a natural extension of Eq. (2.20). Using the notation M jk for the deﬁnition of the inﬁnitesimal rotation in the plane (x j xk ) dissecting the n-dimensional sphere [see Eq. (9.12), the component A j of this vector reads A j = M jk

xj ∂ ∂ + M jk − . ∂ xk ∂ xk r

(2.31)

With the help of this equation one constructs the component j of the n-dimensional vector F Fj =

1 −2Hˆ

Aj

(2.32)

and repeat the above analysis in order to reveal the dynamical symmetry SO(n, 2) [274]. Another important aspect of hidden symmetry of the hydrogen atom is its supersymmetry intimately related to the projection of the Runge-Lenz vector on the electron spin direction [394]. We postpone discussion of this problem till Section 2.6.1, where this property will be considered together with other examples of supersymmetries.

2.3 Dynamical symmetries for spin systems

15

2.3 Dynamical symmetries for spin systems In Sections 2.1 and 2.2 we dealt with integrable systems where the dynamical symmetry emerges due to rotations Rn in real space and the invariance group is a group SO(n) of rotations on an n-dimensional sphere. Now we turn to the problems where the spinor structure of wave function predetermines both the symmetry group of Schr¨odinger equation and the dynamical symmetry group of supermultiplet. The source of spinor structure may be the spin variable alone, or some other discrete index (color), i.e. number of wells in a trap with several minima, or combination of both mechanisms. In any case the invariance group is the group SU(2) of unimodular matrices of 2nd rank (see Section 9.1 for mathematical deﬁnitions). To introduce the SU(2) symmetry group and its generalization SU(n), we ﬁrst consider a handbook problem of one or several particles in shallow enough quantum trap with two minima and assume that each of two wells contains a single discrete s-level with zero orbital momentum. This elementary quantum object known under the name of two-level system (TLS) is used as a constituent of various physical applications some of which will be discussed in subsequent chapters of this book. Since the TLS model may be solved exactly for any occupation of double quantum well, we use it as a toy model with an energy spectrum consisting of few levels which allows one to demonstrate how dynamical symmetry of the spectrum emerges from the symmetry of the Hamiltonian. We will consider here the cases of two-well trap with occupation N = 1, 2. The extension of this approach for multiwell traps and larger occupation numbers N is straightforward. Let us start with a Hamiltonian for a single particle in a double-well potential VTLS (r), (2.33) Hˆ = H1 + H2 + Ht . Two wells are labeled with indices 1 and 2. A particle in an individual well j is described by the Schr¨odinger equation Δr − + V j (r) ψ j (r) = ε j ψ j (r). 2

(2.34)

Then the last term in Eq. (2.33) which describes the tunneling barrier between two wells is deﬁned as Vt (r) = VTLS (r) − V1(r) − V2(r)

(2.35)

16

2 HIDDEN AND DYNAMICAL SYMMETRIES OF ATOMS AND MOLECULES

For the sake of simplicity it is assumed that the direct overlap between the wavefunctions of the particles in two wells is negligibly small and the width of the tunneling barrier is parametrized as W = ψ1 |Vt |ψ2 . Like in previous cases [see Eqs. (2.9) and (2.23)], our aim is to write the Hamiltonian of TLS in terms of invariant operators. For this case we reformulate the Schr¨odinger equation via spinor wave function ψˆ = {ψ1 χ , ψ2 χ }, where χ is a twocomponent spinor 1 0 , χ2 = (2.36) χ1 = 0 1 ˆ ψˆ acquires the matrix In this representation the effective Hamiltonian Hˆ eff = ψˆ |H| form Hˆ eff = ε0 τˆ0 + δ ε τˆz + W τˆx (2.37) where

ε1 + ε2 ε1 − ε2 , δε = 2 2 and the Pauli matrices τˆi are deﬁned in Eq. (9.3). Diagonalization of this Hamiltonian is an easy task. It gives the two-level spectrum (2.38) E(b,a) = ε0 ∓ δ ε 2 + W 2 ε0 =

The subindices b, a correspond to the bonding and antibonding combination of eigenfunctions ψ(b,a) .

ψb ψa

=

cos θ sin θ − sin θ cos θ

ψ1 ψ2

.

(2.39)

The mixing angle is deﬁned as tan θ =

2W . |ε1 − ε2 |

(2.40)

One should note that the above results are valid both for Bose particles and spinless Fermi particles. When one deals with electron or other fermion with spin 1/2 and the wave function ψ(b,a),σ , the energy levels are degenerate in spin index σ unless the tunneling is spin-sensitive. Then the energy spectrum of TLS consists of two spin doublets. Transitions within the spin doublet are determined by usual spin 1/2 matrices σ = τ /2, The set of spin 1/2 operators may be constructed by means of Hubbard operators X σ σ

2.3 Dynamical symmetries for spin systems

17

1 σz = (X ↑↑ − X ↓↓), σ + = X ↑↓ , σ − = X ↓↑ . 2

(2.41)

In order to construct the algebra of operators describing transitions between the levels which belong to the supermultiplet (2.38) consisting of two spin doublets, one may use the same Pauli matrices as in the Hamiltonian (2.37), whereas transitions between the bonding and antibonding states are given by the Hubbard operators X ab = |ab| and the like. One may construct the Pauli matrices acting in subspace {ψb , ψa } in the following way:

τ0 = (X bb + X aa ), τ3 = (X bb − X aa), τ + = X ba , τ − = X ab .

(2.42)

It is clear that all possible transitions between the energy levels (2.38) are described

by composite Hubbard operators of the type X σ σ X ab which can be represented as components of a direct product σ ⊗ τ of spin and pseudospin operators. These components provide 15 generators of su(4) algebra [99]: (σ0 , σ + , σ − , σ3 , ) ⊗ (τ0 , τ + , τ − , τ3 ) − σ0 ⊗ τ0 .

(2.43)

Thus, the dynamical symmetry of full TLS supermultiplet is SU(4). If one is interested only in spin conserving transitions between bonding and antibonding states, then two subspaces remain orthogonal and the matrix (2.43) reduces to

σ0 τ 0 0 σ τ0

(2.44)

where the ﬁrst block corresponds to tunneling transitions without spin ﬂips, and the second block is created by the spin-ﬂip processes within each spin doublet E. Then the dynamical symmetry reduces from SU(4) to SU(2) × SU(2). Next we turn to a double quantum well occupied by two electrons, N = 2. In the general case two more parameters enter the game, namely, the Coulomb and exchange energy of a two-electron system. In our case both electrons are in orbital s-states, and the Coulomb repulsion parameter U between the two electrons within the same well is enough to describe both Coulomb and exchange components of the interaction. The latter may appear in the problem, e.g. as an indirect exchange J ∼ W 2 /U induced by virtual interdot tunneling (see below). From the point of view of quantum-mechanical description, a doubly occupied two-well trap is a caricature of a hydrogen molecule, where all vibrational and rotational degrees of freedom are frozen and the Coulomb attraction of two protons is modeled by a trap with

18

2 HIDDEN AND DYNAMICAL SYMMETRIES OF ATOMS AND MOLECULES

two potential minima. However, unlike the “natural” hydrogen molecule, which is strictly symmetric relative to permutation of two hydrogen atoms and should be classiﬁed in terms of even and odd combinations of wavefunctions ψ j , the double quantum well is asymmetric in the general case. The measure of this asymmetry is the parameter δ ε . It controls the energy spectrum of doubly occupied trap together with parameters U and W . It is known that one may use two types of basis functions in a description of twoelectron states depending on the ratio between the tunneling amplitude W and the Coulomb repulsion U. If the tunneling is dominant, W U, the appropriate basis is a set ψ± of bonding/antibonding states (the Hund-Mulliken method). In the opposite limit of strong interaction, U W the natural basis is the Slater determinant of antisymmetrized products ψi (r)ψ j (r ) with i, j = 1, 2 (the Heitler-London method). In the latter case the weight of “polar” states with two electrons in the same well is suppressed by strong repulsive interaction. Of course, the spectrum contains the same number of energy levels (namely, six) in both limits. In any case the ground state is singlet, the ﬁrst excited state is triplet, and the two other states are charge transfer excitons [206]. In the forthcoming chapters various examples of this six-level spectrum will be considered in more detail. Here we conﬁne ourselves with general discussion of dynamical symmetry for a supermultiplet consisting of one spin triplet and three spin singlets. Let us present the Hamiltonian of a two-well trap in the diagonal form Hˆ = ∑ EΛ |Λ Λ | ≡ ∑ EΛ X ΛΛ Λ

(2.45)

Λ

[cf. Eq. (2.4)]. Here Λ = S, T μ , E2 , E3 stands for the ground state singlet, triplet with spin projection μ = 1, 0, 1¯ and two excitons, respectively. All possible transitions between these levels are presented in Fig. 2.2(a). It is immediately seen from this level scheme, that one may organize all interlevel transitions in the supermultiplet by means of three irreducible scalars Aα and four irreducible vectors, S, Rα (α = 1, 2, 3), namely A1 = i(X E2 E3 − X E3E2 ), A2 = i(X E3 S − X SE3 ), A3 = i(X SE2 − X E2S ), √ √ ¯ ¯¯ ¯ Sz = (X 11 − X 11 ), S+ = 2(X 10 + X 01 ), S− = 2(X 01 + X 10 ), √ √ ¯ 1S 1S S1¯ − S1 R1z = −(X 0S + X S0 ) R+ (2.46) 1 = 2(X − X ) R1 = 2(X − X ) √ √ ¯ 2 1E 0E2 E2 0 + 1E2 E21¯ − E2 1 R2z = −(X + X ) R2 = 2(X − X ) R2 = 2(X − X ) √ √ ¯ ¯ 1E3 E3 1 − X E 3 1 ) R− − X 1E3 ) R3z = −(X 0E3 + X E30 ) R+ 3 = 2(X 3 = 2(X

2.3 Dynamical symmetries for spin systems

19

Here S is the usual spin one operator, and the appearance of three more vectors R j reﬂects the extension of effective dimension of spin multiplet from 3D Fock space with the rotation group SO(3) to 6D space with the symmetry group SO(6). Each set of singlet/triplet transitions adds one more dimension to the effective spin space and the appearance of three scalars reﬂects permutation symmetry of the supermultiplet shown in Fig. 2.2(a). The operator algebra o(6) is given by the commutation relations (in Cartesian coordinates) [S j , Sk ] = iε jkl Sl [Rα j , Rα k ] = iε jkl Sl , [Rα j , Sk ] = iε jkl R1α l . [Rα j , Rβ k ] = iεαβ γ ε jkl Al (α = β ), [Rα j , Aβ ] = iεαβ γ Rγ j , [S j , Aα ] = 0, [Aα , Aβ ] = iεαβ γ Aγ .

(2.47)

Casimir operators which impose kinematic constraint on the excitations are C0 = S 2 + ∑ Rα2 + ∑ A2α = 5, Cα = S · Rα = 0 α

(2.48)

α

i.e. all vectors Rα are orthogonal to the spin vector. Besides, one may construct higher order invariants like in the case of the n-dimensional hydrogen atom [274].

E3 E2

T

E2

T S

SO(6)

S

SO(5) (a)

T S

SO(4) (b)

(c)

Fig. 2.2 (a) Scheme of the energy levels for SO(6) dynamical symmetry group (triplet T and three singlets S, E2 , E3 ); (b) the same for SO(5) group (triplet T and two singlets S, E2 ); (c) the same for SO(4) group (triplet T and singlet S). Alternative notation S1 , S2 , S3 for spin singlet states is used in subsequent chapters. Solid arrow denote vector generators S (transitions within spin triplet) and Ri (transitions between triplet and singlets. Dashed lines denote scalar generators Ai describing transitions between singlet states.

Let us now consider the well where only one exciton survives (the second one falls into the continuum spectrum and decays). Then we remain with a multiplet consisting of two singlets, (ground state S and exciton E2 ) and the spin triplet T ,

20

2 HIDDEN AND DYNAMICAL SYMMETRIES OF ATOMS AND MOLECULES

Fig. 2.2(b). In this case, three irreducible vectors S, R1 , R2 and one scalar A3 is enough to generate the supermultiplet from the ground state level. These operators give ten generators of the dynamical group SO(5) with the algebra o(5) which may be obtained from (2.47) by means of obvious reduction εαβ γ → εαβ and cancelling the last commutation relation. The Casimir constraint in this case is C0 = S 2 + R12 + R22 + A23 = 4.

(2.49)

If the double-well trap is too shallow to retain excitons, we are left with the supermultiplet consisting of the ground state singlet G and the spin triplet, T μ , Fig. 2.2(c). The dynamical symmetry of singlet/triplet supermultiplet is given by two vectors S and R1 , and we return back to the group SO(4), which describes its dynamical symmetry with the Casimir constraint C0 = S 2 + R12 = 3.

(2.50)

Here the vector S contains three generators of the symmetry group of the Schr¨odinger equation similarly to the vector L in cases of rigid rotator and hydrogen atom, whereas the vector R1 containing three operators of singlet/triplet transitions reveals the dynamical symmetry of spin supermultiplet and plays the same part as the vector K (2.11) in the problem of quantum rotator or the Runge – Lenz vector F (2.19) in the problem of Coulomb atom. Next, the Hamiltonian (2.45) may be rewritten in terms of invariant operators with the help of expansions (2.46) and Casimir constraints. In particular, in case of singlet/triplet system obeying SO(4) dynamical symmetry, the equality R12 = 1 + 2X SS follows from the normalization condition (2.6) and the Casimir constraint (2.50). Then the Hamiltonian may be represented in the form 1 Hˆ = ET S 2 + ES R12 + const 2

(2.51)

which reﬂects its SO(4) dynamical symmetry. In analogy with Eq. (9.16) one may introduce the operators S1 =

S+R S−R , S2 = 2 2

(2.52)

with Casimir constraints S12 + S22 = 3/2, S12 − S22 = 0.

(2.53)

2.3 Dynamical symmetries for spin systems

21

The simple quadratic form (2.51) for Hˆ exists only for n = 4. More universal representation uses the set of equalities ¯¯

S 2 = X 11 + X 00 + X 11 , Rα2 =

S2 + 3X Eα Eα 2

(2.54)

valid for any SO(n) with n = 4, 5, 6. These equalities are derived from Eqs. (2.46). Then the Hamiltonian (2.45), which includes the system of triplet and three singlets possessing SO(6) dynamical symmetry acquires the form

1 1 1 ET − ES + EE2 + EE3 S 2 + ES R12 + EE2 R22 + EE3 R32 . (2.55) Hˆ = 2 3 3 In case of SO(5) symmetry the terms ∼ E3 should be omitted, in case of SO(4), only the terms ∼ E1 should be retained. The scalars Aα do not enter explicitly in the Hamiltonian, but they play an essential part in the response of the system to perturbations which violate the dynamical symmetry of the Hamiltonian (see Chapters 7 and 8).

T2 S

T2 S2

S1

T1

SO(7)

(a)

SO(8)

T1 (b)

Fig. 2.3 (a) Scheme of the energy levels for SO(7) dynamical symmetry group (two triplet T1 , T2 and singlet S); (b) the same for SO(8) group (two triplets T1 , T2 and two singlets S1 , S2 ); The meaning of the arrows is the same as in Fig. 2.2.

If the supermultiplet loses its invariance relative to rotations in spin space, its dynamical symmetry changes accordingly. Let us consider, for example, the set of triplet and two singlets [Fig. 2.2(b)] which has been shown above to obey the SO(5) dynamical symmetry. If the nanoobject possesses an axial anisotropy, an additional term DSz2 added to the spin Hamiltonian results in splitting of spin triplet ¯

¯

ET,±1 − ET,0 = D. Due to this anisotropy, two more operators X 11 and X 11 arise in the set of Hubbard operators. One may organize 16 operators of this extended

22

2 HIDDEN AND DYNAMICAL SYMMETRIES OF ATOMS AND MOLECULES

set into 15 generators of the SU(4) group in accordance with the deﬁnition of Gell-Mann matrices λi for 4-dimensional spin space [397] [See Section 9.2.4, Eq. (9.41)]: ¯

¯

¯

¯¯

¯

X1 = X 11 + X 11, X2 = i(X 11 − X 11), X3 = X 11 − X 11 , ¯

¯

X4 = X 10 + X 01, X5 = i(X 10 − X 01), X6 = X 10 + X 01, 1 ¯ ¯¯ ¯ X7 = i(X 01 − X 10 ), X8 = √ (X 11 + X 11 − 2X 00), 3 ¯

(2.56)

¯

X9 = X S1 + X 1S, X10 = i(X S1 − X 1S ), X11 = X S1 + X 1S , ¯

¯

X12 = i(X S1 − X 1S ), X13 = X S0 + X 0S , X14 = i(X S0 − X 0S ), 1 ¯¯ X15 = √ (X 11 + X 11 + X 00 − 3X SS ). 6 In the absence of spin singlet the problem is reduced to the well known model of single-ion anisotropy for spin 1, which possesses the SU(3) symmetry [312, 313]. The algebra u(3) is determined by the ﬁrst eight operators X1 − X8 from the set (2.56), which are formed by means of the Gell-Mann matrices of the 3rd rank (9.35). (r) These operators may be organized in the irreducible tensors Oρ (2.8) in two different ways. In case of anisotropic spin 1 problem the physically reasonable way is to group these 8 operators in one vector and one tensor of 2nd rank [313] in accordance with recipe (9.40). The vector operator O (1) is nothing but the spin 1 operator S deﬁned in Eqs. (2.46), and the tensor operator O (2) is the operator of quadrupole momentum Qˆ (2) with components ¯ ¯ (2) (2) Qˆ +2 = X 11 ∼ (S+ )2 , Qˆ −2 = X 11 ∼ (S− )2 , ¯¯ (2) Qˆ 0 = (X 11 + X 11 ) − 2/3 ∼ Sz2 − 2/3,

¯ (2) Qˆ +1 = X 10 − X 01 ∼ (Sz S+ + S+Sz ), ¯ (2) Qˆ −1 = X 10 − X 01 ∼ (Sz S− + S−Sz ).

(2.57)

[see Eq. (9.40)]. Another mathematical possibility is to group them into two vectors and two scalars (see Section 9.2.4). Physical realization of such possibility was demonstrated in Ref. [245] for a three-level potential well occupied by 4 electrons. In that case the spin rotation invariance was broken by external magnetic ﬁeld (see Section 4.3 for further details). More realizations used in quantum electronics (interaction of 3-level atomic systems with light) may be found in Ref. [160]

2.4 Hubbard atom and Fulde molecule

23

To summarize the survey of possible SO(n) symmetries in spin systems described in the following chapters we present a table of representations of semisimple groups SO(n) with n from 4 to 8 via scalar and vector irreducible operators (2.8): n rank 4 6 5 10 6 15 7 21 8 28

V 2 3 4 6 8

A 0 1 3 3 4

S,T 2S,T 3S,T S,2T 2S,2T

(2.58)

In the third and fourth columns the number of vector (V) and scalar (A) operators is shown, the last column explains the structure of energy spectrum, namely the number of spin singlets (S) and spin triplets (T) entering the corresponding supermultiplet. The kinematic schemes of interlevel transitions corresponding to these groups are shown in Figs. 2.2 and 2.3. One may construct similar hierarchy of dynamical symmetries starting from the problem of an electron in a trap with three minima (three-level system), then adding spin variable, increasing the number N etc. In this hierarchy the invariance group of the Schr¨odinger operator is SU(3). Physical models possessing higher symmetries SU(n) groups are also described in current literature [161, 310, 397, 398].

2.4 Hubbard atom and Fulde molecule The method of projection operators for dynamical symmetry groups based on the expansion (2.8) allows one to construct operator algebras for more complicated cases where the states in the supermultiplet belong to adjacent charge sectors with electron numbers, N = N0 , N0 ± 1. Let us demonstrate its abilities for a simplest case of N0 = 1, using as an example the elementary cell of the Hubbard Hamiltonian [169]. Hˆ Hub = ∑ Hˆ i + ∑ Hˆ i j i

(2.59)

ij

where Hˆ i = εd

∑

σ =↑,↓

di†σ diσ + Unid↑nid↓

(2.60)

is the single-site Hamiltonian describing the states with variable occupation number N (“Hubbard atom”). In this “non-degenerate” version of the Hubbard model it is

24

2 HIDDEN AND DYNAMICAL SYMMETRIES OF ATOMS AND MOLECULES

supposed that each potential well contains only one level εd < 0. The second term is a sum of tunneling operators Hˆ i j = t ∑ di†σ d jσ

(2.61)

σ

It describes the electron motion in a narrow band which emerges from atomic levels εd due to intersite tunneling with the amplitude t in presence of strong shortrange repulsion U, which is a prototype of Coulomb blockade in tunneling through nanoobjects. Here we are interested in the dynamical symmetry of the Hamiltonian Hˆ i (2.60) diagonalized in accordance with (2.45). Since the tunneling motion changes the occupation numbers at any site i, this dynamical symmetry should include all states with N = 0, 1, 2. The energy spectrum consists of four states |Λ with Λ = 0, ↑, ↓, 2 corresponding to an empty site, N = 0, singly occupied site, N = 1 with electron spin projections ↑, ↓, and doubly occupied site, N = 2, respectively. The energy levels EΛ are E0 = 0, E1 ≡ E↑ = E↓ = εd , E2 = 2εd + U.

(2.62)

One can treat the parameter εd (electron binding energy in the Hubbard atom) as a control parameter regulating the occupation N . If the energy differences E01 = E0 − E1 = −εd , E21 = E2 − E1 = εd + U

(2.63)

are positive, then the Hubbard atom is singly occupied in the ground state. Since the Hubbard atom in SCES is embedded into a macroscopic electron ensemble, the chemical potential μ may be introduced in the Hamiltonian. It is convenient to use

μ as a reference level for addition/removal energies (2.63). In this case, changing occupation means change of the model parameters relative to the chemical potential. Practical realizations of this pattern in nanosystems are described in Chapter 3. In accordance with our general approach to SCES, the Hamiltonian of the Hubbard atom may be represented in the diagonalized form (2.45) (Hubbard representation by means of X -operators. All interlevel transitions described by the Hubbard operators are shown in Fig. 2.4(a). The Hubbard operators in the space (0, ↑, ↓, 2) arise as a result of expansion of electron creation annihilation operators dσ† = X σ 0 + σ X 2σ¯ , dσ = X 0σ + σ X σ¯ 2 . This transformation is non-linear. Its inverse reads

(2.64)

2.4 Hubbard atom and Fulde molecule

25

X σ 0 = dσ† (1 − nd σ¯ ), X 2σ = σ dσ†¯ nd σ , X σ σ = nd σ (1 − nd σ¯ ), X 00 = (1 − nd σ )(1 − nd σ¯ ), X 22 = nd σ nd σ¯ , X σ σ¯ = dσ† dσ¯ , X 20 = σ dσ† dσ†¯ .

(2.65)

Unlike the models discussed above, the X-operators for the Hubbard atom connect the states belonging to different charge and spin sectors. Hence the operators

X ΛΛ obey the superalgebra (2.7) which includes both commutation and anticom mutation relations, because the full set {X ΛΛ } contains operators which connect the states with different N , so that the difference NΛ − NΛ acquires the values 0, ±1, ±2. Besides, the spin variable of these excitations acquires both integer values 0,1 for Bose-like transitions and half-integer values ±1/2 for Fermi-like transitions. However, these operators may be grouped in combinations, which obey the u(4) algebra of the Gell-Mann matrices (9.41) similarly to the supermultiplet (S, 0, ±1) discussed above. To ﬁnd these combinations, one should change the in¯ 0, S, ) → (↑, ↓, 0, 2) in the system (2.56), that is dices (1, 1, X1 = X ↑↓ + X ↓↑, X2 = i(X ↓↑ − X ↑↓), X3 = X ↑↑ − X ↓↓, X4 = X ↑0 + X 0↑, X5 = i(X ↑0 − X 0↑), X6 = X ↓0 + X 0↓, 1 X7 = i(X 0↓ − X ↓0), X8 = √ (X ↑↑ + X ↓↓ − 2X 00), 3 2↑ ↑2 2↑ X9 = X + X , X10 = i(X − X ↑2 ), X11 = X 2↓ + X ↓2,

(2.66)

X12 = i(X 2↓ − X ↓2), X13 = X 20 + X 02, X14 = i(X 20 − X 02), 1 X15 = √ (X ↑↑ + X ↓↓ + X 00 − 3X 22). 6 In the limit of U → ∞ the “polar” state Λ = 2 is projected out from the effective Fock space and the dynamical symmetry SU(4) reduces to SU(3) in accordance with this isomorphism. Eight generators of this group are formed by means of GellMann matrices X1 - X8 in the subspace (↑, ↓, 0). Thus, the model of two-electron double quantum well with uniaxial spin anisotropy is isomorphous to the Hubbard atom model from the point of view of dynamical symmetry of their energy levels, although another type of expansion over the irreducible tensors should be used in this model (see below). Although part of the operators forming SU(4) and SU(3) groups are “Bose-like” (X1 − X3 , X8 , X13 − X15 ), and the rest are “Fermi-like (X4 − X7 , X9 − X12 ) in the sense of commutation relations (2.7), they commute in accordance with the Gell-Mann

26

2 HIDDEN AND DYNAMICAL SYMMETRIES OF ATOMS AND MOLECULES

algebra (9.37). To understand this mapping in more details it is convenient to express the Hubbard operators and the Hubbard Hamiltonian Hˆ i (2.60) via the irreducible operators T, U, V, W, Y, Z given by Eqs. (9.38) and (9.42): T+ = X ↑↓ , T− = X ↓↑ , Tz = X ↑↑ − X ↓↓ V+ = X ↑0 , V− = X 0↑ , Vz = X ↑↑ − X 00 U+ = X ↓0 , U− = X 0↓ , Uz = X ↓↓ − X 00 W+ = X ↑2 , W− = X 2↑ , Wz = X ↑↑ − X 22 Y+ = X ↓2 , Y− = X 2↓ , Yz = X ↓↓ − X 22 Z+ = X 02 , Z− = X 20 , Zz = X 00 − X 22

(2.67)

Equations (2.67) realize the general expansion scheme (2.8) for the irreducible vector operators in the group SU(4). The triad T is nothing but the set of spin 1/2 operators (S+ , S− , 2Sz , ) acting in the charge sector N = 1. The triad Z describes two-particle excitations (N = 0 ↔ N = 2). The rest four triads describe transitions between different charge sectors (N = 1 ↔ N = 0, 2). The dual nature of Hubbard operators manifested in the commutation relations (2.7) allows one to use them for construction of su(3) algebra formed by spin and pseudospin operators with commutation relations (9.43), (9.44). These commutation relations ensure complex dynamical properties of Hubbard-like SCES.

charge 0,2

0

0 spin

charge 0 or 2

1

0

1

spin (a)

(b)

Fig. 2.4 (a): Scheme of energy levels for a Hubbard atom with SU(4) dynamical symmetry describing transitions between the states with occupation N = 0, 1, 2. (b) The same for a reduced spectrum with SU(3) dynamical symmetry describing transitions between the states with N = 0 or 2 and N = 1. Bose-like transitions with even δ N = 0, ±2 are shown by dashed lines, Fermi-like transitions with odd δ N = ±1 are shown by solid lines.

It is expedient to rewrite the original Hamiltonian (2.60) in terms of generators of the group SU(4) in the case where all four eigenstates (2.62) shown in Fig. 2.4(a) are taken into account, and in terms of SU(3) generators in the case when the polar

2.4 Hubbard atom and Fulde molecule

27

states with N = 2 are frozen out [Fig. 2.4(b)]. Let us denote the Fock space in which the Gell-Mann matrices are deﬁned as Φ¯ 4 = ↑ ↓ 0 2 , Φ¯ 3 = ↑ ↓ 0 or ↑ ↓ 2

(2.68)

for 4-level and 3-level cases, respectively. In the general case of SU(4) symmetry the Hamiltonian (2.60) rewritten in terms of Gell-Mann generators (2.66) acting in the space Φ¯ 4 [see also (9.49)] reads √ E2 4 E0 2 1 − √ X8 + √ X15 + 1 − 6X15 Hˆ iSU(4) = 4 4 3 6 2 2 E1 1 + √ X8 + √ X15 + 2 3 6

(2.69)

Similarly √ E1 √ Ep 1 − 3X8 + 2 + 3X8 , Hˆ iSU(3) = 3 3

(2.70)

where p = 0, 2 for two cases of SU(3) symmetry in the subspace Φ¯ 3 where the states E2 or E0 , respectively, are frozen [see Eq. (9.51)]. In terms of the irreducible operators (2.67) the Hamiltonian Hˆ i acquires more transparent form. In a chosen representation the z-components of the irreducible operators may be diagonalized, so that the equations Hˆ i Oz = εz Oz are valid. In case of SU(4) space Φ¯ 4 there are three such operators that can be diagonalized simultaneously. One may choose these operators as Tz , Qz = Uz + Vz and Pz = Wz + Zz . Then adding the Zeeman term with external magnetic ﬁeld h = g μB B, we represent Hˆ i as E10 E12 Hˆ iSU(4) = · Qz + · Pz 8 8 E20 h · (Qz − Pz ) + · Tz + C4 · 1. + 4 2

(2.71)

Here E pp = E p − E p , the constant C4 = (E0 + E2 + 2E1 )/4 and 1 is the unit matrix of 4-th rank. Thus, the eigen operators for a general SU(4) Hubbard atom are Tz Qz Pz Qz − Pz , , , , 2 8 8 4

(2.72)

±h, ± E10, ± E12, ± E20 ,

(2.73)

with eigenvalues εz equal to

28

2 HIDDEN AND DYNAMICAL SYMMETRIES OF ATOMS AND MOLECULES

These eigenvalues correspond to spin-ﬂip excitations in the singly-occupied state N = 1, hole addition/extraction, electron addition/extraction and two hole excitations, respectively [see Fig. 2.4(a)]. In the degenerate SU(4) model with E0 = E2 only the two ﬁrst operators from (2.72) are involved. In the reduced subspace Φ¯ 3 only two operators Tz /2, Qz /3 enter the Hamiltonian of the Hubbard atom E1p h Hˆ iSU(3) = · Qz + · Tz + C3 · 1 3 2

(2.74)

where p = 0 or 2, C3 = (E0 + 2E1 )/3 and 1 is the unit matrix of 3-rd rank. Here the eigenvalues of charge excitations are ±E01 or ±E02 [Fig. 2.4(b)].

Another prototypical model of elementary cell that demonstrates dynamical symmetry from which various many-particle effects emerge, is the so called ”Fulde molecule“ originally introduced to illustrate the origin of Kondo paradigm (complete screening of localized spin immersed in the sea of free metallic electrons due to multiple creation of electron-hole pairs in this Fermi bath [79, 119]. This artiﬁcial molecule consists of two orbitals centered around two adjacent sites. One of these orbitals is localized in a deep energy level εd within a narrow potential well, another one occupies a shallow level εs in a wide potential well. The tunneling V between these levels is allowed. The Hamiltonian of the system is Hˆ FM = εd ∑ dσ† dσ + εs ∑ c†σ cσ + V ∑(dσ† cσ + c†σ dσ ) + Und↑ nd↓ . σ

σ

(2.75)

σ

The last term in the Hamiltonian (2.75) is the repulsion interaction introduced in the Hubbard model. It is assumed that this repulsion is strong for d-electrons and negligible for s-electrons. In fact this model is an extremely asymmetric version of the TLS with N = 2 considered in Section 2.3 [see Eq. (2.45) and discussion below]. In this limit the strong Hubbard repulsion suppresses the ”polar” states with two electrons in the d-well, In case of weak tunneling, V (εs − εd ), the twoelectron spectrum consists of ﬁve levels ES = εs + εd −

2V 2 2V 2 , ET = εs + εd , EE = 2εl + . εs − εd εs − εd

(2.76)

The second-order correction to the positions of singlet levels ES and EE is the indirect exchange of Anderson type, which favors antiparallel orientation of spins in the ground state of the Fulde molecule. Thus, we come to the familiar case of the supermultiplet with SO(5) dynamical symmetry which contains the subgroup

2.4 Hubbard atom and Fulde molecule

29

SO(4) of low-lying spin singlet – spin triplet set. The latter symmetry survives in the excitation spectrum of Anderson-Kondo model, where the localized d-electron is hybridized with non-interacting Fermi continuum [119].

2.4.1 Three-fold way for Hubbard atom As was noticed in the sixties [127], various families of hadrons are classiﬁed in accordance with the irreducible representations of the SU(3) group (see also [92]). In particular, 18 baryons form two multiplets corresponding to representations D(11) (octet of baryons with spin 1/2) and D(30) (decuplet of baryons with spin 3/2); octet of spinless mesons also transforms along the representation D(11) . The higher representations of SU(3) are realized in the physics of strong interaction because these ”elementary” particles possess not only spin and charge but also isospin and hypercharge quantum numbers. Really the elementary particles representing SU(3)color symmetries are quarks with their fractional charge and other quantum numbers. The SU(3) symmetry in the hadron multiplets under strong interaction is satisﬁed only approximately due other types of interaction, so that this symmetry may be treated as a dynamical symmetry in the original sense of this notion. The Hubbard atom with frozen doubly occupied states possesses only two quantum numbers, namely spin and charge. Therefore, the multiplet of Hubbard states is described by the lowest irreducible representation D(10) of SU(3) group. To show this one should recollect that only two of the eight Gell-Mann matrices can be diagonalized simultaneously. Following Eq. (2.74) (see also [92]), we choose the representation with diagonal matrices Tz and Qz . Then the set of allowed states is deﬁned by two integer numbers λ , μ so that the eigenstates are determined as 1 MT = λ + μ , MQ = (λ − μ ). 3

(2.77)

The whole set of eigenstates form a two-dimensional triangular lattice on the plane ¯ (MT , MQ ). Each irreducible representation Dλ μ¯ is marked by the indices λ¯ , μ¯ corresponding to the state with maximum eigenvalue M¯ Q and the maximum value of M¯ T possible at this M¯ Q . Then the other states forming this irreducible representation are constructed by means of the ladder operators T± , U± , V± acting on the state |M¯ Q , M¯ T .

This procedure results in construction of the stars of basis vectors Dλ μ and poly-

gons connecting the points generated by the ladder operators subsequently acting

30

2 HIDDEN AND DYNAMICAL SYMMETRIES OF ATOMS AND MOLECULES

MQ 1

T − (1,1/3)

(−1,1/3) −1

U+

0

1

V−

MT

(0,−2/3)

Fig. 2.5 Irreducible representation D(10) of SU(3) group for the Hubbard atom with inﬁnite U (see text for detailed explanation).

−1

on the point (M¯ Q , M¯ T ). In case of baryon family the corresponding multiplets are hexagon with doubly degenerate central point for representation D(11) and triangle with ten point in its vertices and on its sides for representation D(30) . In Hubbard atom the multiplet is represented by a triangle (Fig. 2.5) labeled in accordance with the state with highest quantum numbers λ = 1, μ = 0, which correspond to the state |N , σ = |1, ↑ of the Hubbard atom. Two other components of the multiplet Φ¯ 3 may be generated from the state |1, ↑ by means of the ladder operators T− = X ↓↑ and V− = X 0↑ . The ﬁrst of this operators corresponds to a ”Bose-like” excitation with spin 1, and the second one is the ”Fermi-like” excitation with spin 1/2. The triangle D(10) is closed by means of the operator U+ = X ↓0 . Interrelations between the values of the parameters λ , μ , the eigenvalues of the operators Tz and Q and the eigenvalues |Λ of the Hubbard Hamiltonian are presented in table (2.78):

λ μ MT 1 0 1 0 -1 -1 -1 1 0

MQ Λ 1/3 u 1/3 d -2/3 h

(2.78)

Here the notations u, d, h are used for the spin up, spin down and hole states, respec¯ ¯ tively. The basis vectors in the plane (MT , MQ ) are D10 , D01 , D11 . In spite of formal analogy with the SU(3) description of elementary particles, in case of the Hubbard atom where spin and charge are the only quantum numbers this symmetry is realized only as a dynamical symmetry of interlevel transitions induced due to interaction with the bath. This means that the diagonal components of the group generators Vz and Uz describe interlevel transitions (excitations) rather than the eigenstates of the Hamiltonian. The ladder operators V± and U± , describe

2.5 Fock – Darwin atom

31

excitations corresponding to transitions between adjacent charge sectors. These operators enter the interaction Hamiltonian which describes coupling between the nanoobject and the bath. Various forms of this interaction will be discussed in Chapter 4. In analogy with the term offered by Gell-Mann and Ne’eman for the multiplet of light hadrons, one may use the term ”three-fold” way for the SCES with approximate SU(3) symmetry. This formal analogy does not imply physical analogy. In case of of hadron octet the ladder operators describe transition from one set of quarks to another. Since SU(3) symmetry is only approximate the energy levels in the octet characterized by spin, charge, isospin and hypercharge are split (masses of “elementary” particles are different), and the lowest state is occupied by proton. Interlevel transitions (conversion of a hadron into another one or its decay) are induced by electroweak interactions. The basic symmetry is the SU(3)color symmetry of u, d, s quarks, In the Hubbard atom one deals only with spin and charge, and the ladder operators mean transitions between the charge sectors (N = 0) ↔ (N = 1) and between spin states within the charge sector N = 1, so its dynamical symmetry is described by the lowest irreducible representation D(10) . The kinematical scheme shown in Fig. 2.5 corresponds to the scheme of interlevel transitions for the SU(3) group [Fig. 2.4(b)]. Similar scheme for the group SU(4) [Fig. 2.4(a)] have a form of pyramid in the corresponding 3D space (see below).

2.5 Fock – Darwin atom The problem of discrete states of an electron in a parabolic (harmonic) potential was solved at the early stage of quantum mechanical studies by V.A. Fock [110] and C.G. Darwin [72]. 70 years later the Fock – Darwin atom became the physical reality because it turned out that the conﬁning electrostatic potential in so called vertical quantum dots [238] has nearly parabolic shape (see Section 3.3). Having in mind this physical realization, we consider here the two-dimensional Fock – Darwin model for an electron conﬁned in parabolic well in a perpendicular magnetic ﬁeld B with vector potential gauged as A = B2 (−y, x). The 2D Fock – Darwin Hamiltonian then has the following form 1 2 e 2 m 2 2 p − A + ω0 r − μB σ · B, Hˆ FD = 2m c 2

(2.79)

32

2 HIDDEN AND DYNAMICAL SYMMETRIES OF ATOMS AND MOLECULES

where ω0 is an eigenenergy of parabolic potential. Let us ﬁrst ignore the last term which describes the Zeeman splitting of electron levels in magnetic ﬁeld and rewrite the Hamiltonian as ωc2 mr 2 ωc p2 2 ˆ + ω0 + + lz . HFD = (2.80) 2m 4 2 2 where ωc = eB/mc is the cyclotron frequency. The ﬁrst two terms in (2.80) is the Hamiltonian of an electron in harmonic potential with the conﬁnement frequency Ω = ω02 + ωc2 /4 and the third one is the energy of angular motion in magnetic ﬁeld. Then it is clear that the dynamical symmetry of the Fock – Darwin atom is closely related with the dynamical symmetry of quantum oscillator. The latter was revealed in Refs. [40, 145, 173]. One may construct the algebra o(2, 1) of the non-compact Lorenz Group SO(2, 1) in 2+1 dimensions for the operators generating transitions between the eigenstates of harmonic 1D oscillator. This group is locally isomorphic to the unitary group SU(1, 1), i.e., SO(2, 1) = SU(1, 1)/Z2 , where Z2 = {1, −1}. The generators of the Lorenz group in D=2+1 act in the Hilbert space where the basis functions |Λ = |n (a† )n |n = √ |0 n!

(2.81)

are built from the ground state |0 = π −1/4 exp(−x2 /2) by means of creation oper√ ators a† = (x − ∂ /∂ x)/ 2. In accordance with the general deﬁnitions given in Section 9.2.2, the Lie algebra o(2, 1) is given by three generators L1 , K1 , K2 with commutation relations [K1 , L1 ] = −iK2 , [K2 , L1 ] = −iK1 , [K1 , K2 ] = iL1

(2.82)

and the Casimir operator C = L21 − K12 − K22

(2.83)

so that L1 is generator of geometric rotations, and K1 , K2 perform Lorenz transfor√ mations. The ladder operators are introduced as K ± = (iK1 ± K2 )/ 2. Then the o(2, 1) algebra may be realized on the basis of Bose operators a, a† in the following way [39]: a† † a √ K+ = − √ a a + 1, K − = √ a† a, 2 2 L1 = a† a + 1/2.

(2.84)

2.5 Fock – Darwin atom

33

The matrix elements of interlevel transitions described by the operators K ± , L1 (2.84) are n+1 n m|K + |n = − √ δm,n+1 , m|K − |n = √ δm,n−1 , 2 2 m|L1 |n = (n + 1/2)δm,n .

(2.85)

Another realization [145] of the same algebra is obviously given by K+ =

√

√ a† a a† , K − = a a† a, L1 = a† a + 1/2.

(2.86)

Thus, the non-compact group SO(2, 1) with the Casimir operator C = −1/4 is indeed the dynamical group, which realizes the single irreducible representation for all levels of one-dimensional harmonic oscillator. Discussion of dynamical symmetries which are realized in D-dimensional oscillators may be found in Refs. [40, 173]. Now we return to the Fock – Darwin atom with the Hamiltonian (2.80). This is the two-dimensional system, and the basis functions realizing the dynamical algebra should consist of two sets of oscillator states [152, 188, 274, 423]. For this sake, we turn from the Cartesian coordinates x, y to the complex coordinate ξ , ξ ∗ :

ξ=

x + iy √ x − iy √ mΩ , ξ ∗ = mΩ . 2 2

(2.87)

Then two pairs of boson operators are introduced as √ √ a = −(i/ 2)(ξ + ∂ /∂ ξ ∗ ), a† = (i/ 2)(ξ ∗ −∂ /∂ ξ ) √ √ b = (1/ 2)(ξ ∗ + ∂ /∂ ξ ), b† = (1/ 2)(ξ − ∂ /∂ ξ ∗ ).

(2.88)

As a result of this transformation the Fock – Darwin Hamiltonian is reduced to the Hamiltonian of two-dimensional anisotropic oscillator. The basis functions (a† )n (b† )m √ |0, 0 (2.89) n!m! are built from the ground state function |0, 0 = mΩ /2π exp(−|ξ |2 ). The Fock – Darwin Hamiltonian is readily rewritten in terms of boson operators 1 1 + Ω− nˆ b + (2.90) Hˆ FD = Ω + nˆ a + 2 2 |Λ = |n, m =

where Ω± = Ω ± ωc /2, nˆ a = a† a, nˆ b = b† b .

34

2 HIDDEN AND DYNAMICAL SYMMETRIES OF ATOMS AND MOLECULES

The energy spectrum of a conﬁned electron in terms of boson occupation numbers n, m is given by En,m = Ω (n + m + 1) +

ωc (n − m) 2

(2.91)

Thus, we come to a conclusion that the dynamical symmetry of the Fock – Darwin atom is the direct product SO(2, 1) × SO(2, 1) in accordance with general prescription of the theory of d-dimensional quantum oscillator [40]. The energy spectrum (2.91) interpolates smoothly between the two limiting cases. In zero ﬁeld, ωc = 0 it gives the eigenvalues of the 2D harmonic oscillator, and in the extremely strong magnetic ﬁeld ωc ω0 it describes the Landau levels of a free electron in magnetic ﬁeld, because in this limit the radius of magnetic conﬁnement is much smaller than that of the potential conﬁnement. But the dynamical symmetry of quantum oscillator is realized in all cases. Another way of treating this problem is to express the Fock – Darwin states in terms of effective momentum operators. Now we restore the last term in the Hamiltonian (2.79), so that the state vectors are determined by three quantum numbers, |Λ = |nmσ . Then the total angular momentum is a result of addition of the orbital momentum and the electron spin. The former is given by the operator lˆ = nˆa − nˆ b [cf. Eqs. (2.80) and (2.90)], the latter is the usual spin 1/2 operator. To build the total moment, one has to turn to the spinor representation for boson operators, a † † χ¯ = (a b ) , χ = . b Then two spinor operators may be constructed ¯ 0 χ , J = χσ ¯ χ, jˆ = χσ and the Fock – Darwin Hamiltonian acquires the form 1 Hˆ FD = Ω jˆ + + ωc Jz , 2

(2.92)

(2.93)

so that the spectrum of the Fock Darwin atom in magnetic ﬁeld is characterized by additional dynamical symmetry SU(2) given by the ladder operators J ± . Other possibilities of revealing dynamical symmetries of electron states in oscillator-like problems are discussed in the books [80, 274].

2.6 Dynamical symmetry and supersymmetry

35

2.6 Dynamical symmetry and supersymmetry The supersymmetry group is a group which unites in a special way the continuous Lie transformations with discrete reﬂection-like even-odd transformations. Speciﬁcally this is a group generated by an algebra, which covers the algebra of anticommuting fermionic operators { fi , f j† } = δi j

(2.94)

[bi , b†j ] = δi j

(2.95)

and commuting bosonic operators

with [bi , f j ] = 0. The supersymmetry algebra and the corresponding supergroup have been constructed in Refs. [141, 412, 424]. Within several years the supersymmetrical formulation of quantum mechanics was proposed [430]. Supersymmetric operations describe transitions between the states with different numbers of bosons and fermions and in this sense the supersymmetry is closed to dynamical symmetries discussed above. In this section we will present the basic ideas of supersymmetry following the review [128] and then apply these ideas to some of integrable models introduced in the preceding sections. The supersymmetric generators are deﬁned in a Fock space deﬁned by the basis vectors |nB , nF , nB = 0, 1, 2 . . . ∞, nF = 0, 1.

(2.96)

Transformation of a boson into fermion and v.v. is deﬁned by the supercharge operators Q+ = qb f † , Q− = qb† f

(2.97)

where q is c-number. In the space (2.96) these operators act as Q+ |nB , nF = q|nB − 1, nF + 1, Q− |nB , nF = q|nB + 1, nF − 1.

(2.98)

It is convenient to introduce two Hermitian combinations of the operators (2.97): Q1 = Q+ + Q− , Q2 = −i(Q+ − Q− ). These operators anticommute,

(2.99)

36

2 HIDDEN AND DYNAMICAL SYMMETRIES OF ATOMS AND MOLECULES

{Q1 , Q2 } = 0.

(2.100)

because the operators Q± are nilpotent, Q2+ = Q2− = 0.

(2.101)

Q21 = Q22 = {Q+ , Q− }.

(2.102)

Besides,

In order to construct closed supersymmetric algebras one should deﬁne the supersymmetric Hamiltonian written in terms of supersymmetric operators. This fundamental property of supersymmetric systems is formulated as ˆ Q] = 0, [H,

(2.103)

where Q is any of the operators from the set (2.97) or (2.99). The simplest toy Hamiltonian characterized by the supersymmetry group S(2) is Hˆ = Q21 = Q22 = {Q+ , Q− }.

(2.104)

In this way we come to the simplest Lie superalgebra s(2) including commutation and anticommutation relations: {Qi , Q j } = 2Hˆ δi j , ˆ = 0, i, j = 1, 2. [Qi , H]

(2.105)

Supersymmetric Lie algebras are Z2 graded, i.e. the constituent generators are classiﬁed in accordance with some parity as even operators Oe and odd operators Oo . The structure of the Lie superalgebra is as follows: [Oe , Oe ] → Oe , [Oe , Oo ] → Oo , {Oo , Oo } → Oe .

(2.106)

In the above example (2.105) the only even operator is the Hamiltonian Hˆ and two odd operators are Qi . To reveal the sense of “parity” ascribed to the supersymmetric operators, one has to deﬁne the variation of some operator O under transformation generated by another operator P. In case of conventional Lie algebras this variation is

2.6 Dynamical symmetry and supersymmetry

δ O ∼ [ε P, O],

37

(2.107)

where ε is the transformation parameter. In case of superalgebras ε is not a cnumber. This parameter should commute with Bose operators and anticommute with Fermi operators. Namely, choosing P = Q, one demands

δ b ∼ [ε Q, b] = ε [Q, b] ∼ ε f , δ f ∼ [ε Q, f ] = ε {Q, f } ∼ ε b.

(2.108)

This deﬁnition means that the parameters ε also possess deﬁnite parity: even parameters εe are c-numbers commuting with all operators and parameters, while odd parameters εo commute with even quantities and anticommute with odd ones. The above deﬁnition of supersymmetric operators and parameters may be uniﬁed. Let us change the normalization of the Q-operators: Qi qi = √ , E

(2.109)

ˆ Then Hˆ transforms into the unit where E is an eigenstate of the Hamiltonian H. ˆ operator 1 in the superspace (2.96), and the algebra deﬁned by Eqs. (2.105) reads {qi , qk } = 2δik

(2.110)

These relations deﬁne the Clifford algebra with generators qi . On the other hand, odd parameters ε anticommute with each other and each of them anticommutes with itself, which means ε 2 = 0. These relations may be summarized as {εi , ε j } = 0.

(2.111)

Thus, the odd parameters form a Grassmann algebra, and each Grassmann algebra with N constituents is associated with Clifford algebra for 2N constituents. Now let us ﬁnd out what kind of physical object is described by the toy Hamiltonian (2.104). Using the deﬁnition (2.97) , one may rewrite (2.104) as 1 1 2 † † 2 2 ˆ + q nF − . H = {Q+ , Q− } = q (b b + f f ) = q nB + 2 2

(2.112)

This is a sum of non-interacting systems of bosonic and fermionic oscillators with the same frequency Ω = q2 : Hˆ = Hˆ B + Hˆ F ,

(2.113)

38

2 HIDDEN AND DYNAMICAL SYMMETRIES OF ATOMS AND MOLECULES

1 1 2 † ˆ HB = q b b + , EB = Ω nB + , nB = 0, 1, . . . ∞, 2 2 1 1 Hˆ F = q2 f † f − , EF = Ω nF − , nF = 0, 1. 2 2 Thus, the energy spectrum of the supersymmetric system is non-negative and doubly degenerate, E{nB ,nF } = Ω (nB + nF ).

(2.114)

This spectrum is shown in Fig. 2.6. The operators Q± generate “rotations” boson ↔ fermion in a supersymmetric Fock space without changing the energy of the system, E{nB ,0} = E{nB −1,1} .

(2.115)

The ground state energy of the supersymmetric system is zero, because the zero vibration energies of bosonic and fermionic excitations compensate each other. One may say that the zero vacuum energy of the supersymmetric state is split into negative and positive inﬁnite vacuum energies of Fermi- and Bose-sectors of the system. Supersymmetric operators may be represented in a spinor form due to the fact that the fermionic wave function is a spinor of second rank in a subspace nF = 1, 0: ψ1 ψ= , (2.116) ψ0 where ψ1,0 corresponds to nF = 1, 0. Respectively, the fermionic creation and annihilation operators are realized as Pauli matrices (9.3) in this representation, f + = τ + , f = τ − . Then the supercharge operators acquire the form v Q1 = q(b† τ − + bτ + ) = B1 τ1 + B2τ2 , Q2 = −i(b† τ − − bτ + ) = −B2 τ1 + B1 τ2

(2.117)

where B1 = q(b + b†), B2 = iq(b − b†) In this representation the supersymmetric Hamiltonian (2.104) is 1 Hˆ = {B− , B+ } + [B−, B+ ]τ0 2 with B± = B1 ± iB2 .

(2.118)

2.6 Dynamical symmetry and supersymmetry

39

2.6.1 Manifestations of supersymmetry in atomic models Some of the model Hamiltonians with deﬁnite dynamical symmetries considered in the preceding sections also may be treated in terms of the group S(2) of supersymmetric transformations. In particular, the minimal model (2.104) with the spectrum (2.114) is related to the limiting case of the Fock – Darvin model (Section 2.5) with

ωc ω0 . As was mentioned above, in strong magnetic ﬁeld spatial conﬁnement of electrons in the Fock – Darvin atom is determined by the magnetic length, and then the electrons at Landau levels acquire the supersymmety [128].

Ω (n B+ n F)

n

n

B

4 4 3 2 1

Q+ Q−

B

5 3 2 1 0

0 n F= 0

n F= 1

Fig. 2.6 Supersymmetric spectrum (2.119) of Fock – Darwin atom at ω0 /ωc → 0. Transitions with changing occupation numbers nB and nF are marked by arrows, Ω = ωc .

Indeed, in the limit ω0 /ωc → 0, where Ω+ → ωc , Ω− → 0, instead of two-boson representation (2.90) one may turn to a mixed fermion-boson representation, where the spectrum of Landau states for spinful electrons may be represented as

1 1 ωc 1 E = ωc n + ± = ωc nB + + nF − . 2 2 2 2

(2.119)

In this representation bosonic degrees of freedom describe quantized orbital motion of Landau electrons and fermionic degrees of freedom correspond to electron spin states split due to the Zeeman effect. Looking at the spectrum of Landau levels rearranged in accordance with Fig. 2.6, we see that the supersymmetric transformation corresponds to a spin ﬂip accompanied by simultaneous transition from a given Landau level to adjacent one. Choosing one of two cartesian coordinates forming the basis {ξ , ξ ∗ } (2.87) as a continuous variable and spin projection as a discrete variable, we conclude that the supersymmetry of Landau electrons is a result of

40

2 HIDDEN AND DYNAMICAL SYMMETRIES OF ATOMS AND MOLECULES

interplay between continuous orbital motion and discrete spin “reﬂection” in accordance with general properties of supersymmetric systems. Intimate relation between dynamical symmetry and supersymmetry is clearly seen in this system: the supersymmetry is realized as interstate transitions generated by the supercharge operators Q± (2.97) involving both bosonic and fermionic sectors of Fock superspace. At the same time the dynamical SO(2, 1) symmetry is realized as interstate transitions within the bosonic sector generated by the operators K ± , L1 (2.84). At ﬁnite ratio ω0 /ωc the Fermi-Bose supersymmetry of the electron spectrum breaks because the conﬁnement potential of Fock – Darwin atom ∼ ω02 violates the equivalence of boson and fermion frequencies in the spectrum (2.119), but the dynamical SO(2, 1) symmetry still exists.

Supersymmetry is also a generic feature of an electron in the Coulomb potential (Section 2.2). To reveal this supersymmetry [236, 394], one has to add a discrete spin variable to the set of continuous operators L and F (2.19) responsible for the SO(4) symmetry of the spinless problem determined by Eqs. (2.21). The projection of the Runge – Lenz vector F onto the electron spin direction plays the part of the supercharges Q1,2 in this model. To ﬁnd these supercharges , we construct the invariants for the group SO(4) × SU(2) which arise as a result of adding the orbital and spin moments of an electron in the total moment J = L + S. The set of observˆ L2 , J2 , Jz with eigenvalues E, l(l + 1), j( j + 1), m, ables is given by the operators H, respectively. It is convenient to introduce the scalar operator K with eigenvalues ±κ in such a way that the states with ﬁxed j and l = j ± 12 correspond to the same |κ | but to opposite “projections” ±κ . This aim is achieved by introducing J2 − L2 = −(1 + K ) so that 1 L2 = K (K + 1) ⇒ l(κ ) = |κ | + (sgnκ − 1), 2 (2.120) 1 J = K − 1 ⇒ j(κ ) = |κ | − , 2 2

2

It is clear that the index κ labels transitions in graded superspace. Indeed, the scalar operator K plays the key part in the construction of supercharges. The parity operator with eigenvalues ±sgnκ is K Pκ = . (2.121) |κ | This operator controls the Z2 grading of superalgebra.

2.6 Dynamical symmetry and supersymmetry

41

The scalar (S · A) is an odd operator in the supersymmetric sense, and its square is an even operator connected with the even operator Hˆ entering the superalgebra by the following relation 1 ˆ 2 1 HK + . (2.122) (S · A)2 = 2 2 [see Eqs. (2.22)-(2.25)]. Restricting ourselves to the subspace of ﬁxed |κ | ≡ k, we rewrite (2.122) as 1 (S · A)2 = Hˆ + 2 = Hˆ . (2.123) 2 k 2k Then we introduce two supercharge operators 1 Q± = √ (S · A)(1 ∓ Pκ ), 2k

(2.124)

(see Eq. 2.20) and their Hermitian combinations are Q1 =

(S · A) i , Q2 = iQ1 Pκ = 2 (S · A) k k

(2.125)

These two operators together with “shifted” operator Hˆ (2.123) generate the s(2) algebra (2.100) - (2.105). Substituting A for F in Eqs. (2.123) - (2.125), we come to the Clifford algebra (2.110). The energy levels of a nonrelativistic particle with spin 1/2 in a Coulomb ﬁeld involved in supersymmetric degeneracy are those with ﬁxed quantum numbers j, m and l = j ± 1/2 (see Fig. 2.7). Like in the above example, the S(2) supersymmetry complements other symmetries characterizing the supermultiplet of the energy levels: the hidden symmetry of the atom is characterized by continuous SO(4) or SO(4, 2) groups describing transitions between the states with given parity κ , while transitions with changing parity initiated by the operator (S · A)/k are involved in the supersymmetry group S(2). More detailed analysis of this problem including description of irreducible representations of the S(2) group may be found in Ref. [128]. The supersymmetric quantum mechanical description of a spinful particle in a 1/r potential in arbitrary dimension d is given in Ref. [212].

To close the discussion of supersymmetric properties of real and artiﬁcial atoms, we consider in brief the prospects of supersymmetric description of SCES models with a Hubbard atom or a grid of Hubbard atoms as zero-order approximation.

42

2 HIDDEN AND DYNAMICAL SYMMETRIES OF ATOMS AND MOLECULES

Fig. 2.7 Supersymmetric spectrum of hydrogen atom. All levels of equal j and m are connected by an S(2) supersymmetry (after [394])

We have noted above that the dynamical symmetry SU(3) unites ”Bose-like” and ”Fermi-like” excitations into the a supermultiplet shown in Fig. 2.4(b). It is a tempting prospect to look for relation between this symmetry and supersymmetry S(2). The theoretical grounds of such prospect can be looked for in various fermioniza tion and bosonizations procedures available for Hubbard operators X ΛΛ . In particular, the operators from the triads U, V, W, Y (2.67) may be expressed via products of Fermi- and Bose operators in slave-boson [37, 60, 237, 350] and slave-fermion [30, 122, 433] representations. Spin operators from the triad T may be represented as products of spin-fermions [1, 335], Popov-Fedotov semi-fermions [214, 337] or Majorana fermions [271, 429]. Representations of these operators via products of Bose operators are also available [278]. Existing fermionization/bosonization procedures for Hubbard operators are described in Section 9.3. Here we are interested in mixed fermion-boson representations which formally remind those used in the deﬁnition of supercharge operators Qi (2.97), (2.99). In order to reveal the hidden extension of the Fock space of Hubbard atom (2.60) with the energy spectrum (2.62), we work in adjacent charge sectors N = 1, 2, Fig. 2.4(b). Following the general scheme of the s(2) algebra we represent the offdiagonal operators from the triads W, Y (2.67) as supercharge operators using slaveboson representation [cf. Eqs. (9.83)]

2.6 Dynamical symmetry and supersymmetry

43

2σ σ2 Q+ = b † f σ , Q− = b fσ† . σ =X σ =X

(2.126)

Respectively, − 2σ Q 1σ = Q + + Xσ2 σ + Qσ = X

(2.127)

− 2σ − X σ 2) Q2σ = −i(Q+ σ − Qσ ) = −i(X

Then − σσ + X 22 = nF σ + nB. Q21σ = Q22σ = {Q+ σ , Qσ } = X

(2.128)

Here Bose operators b, b† stand for “doublon” states in doubly occupied Hubbard atom [237]. The supercharge operators transforms the system from the charge sector N = 2 with zero spin into the charge sector N = 1 with spin 1/2, and thus connect fermionic and bosonic sectors of the Fock space Φ¯ 3 (↑, ↓, 2). The Hamiltonian Hˆ iSU(3) rewritten in terms of the supercharge operators (2.126), acquires the form Hi = ε1 ∑ X σ σ + (2ε1 + U)X 22 = ε1 ∑ Q2σ + UX 22 σ

(2.129)

σ

with Q2σ = Q21σ = Q22σ . Renormalizing the operator Hˆ i → ε1 ∑ Hˆ σ , σ

one arrives at two s(2) superalgebras (2.105) for the operators Hˆ σ , Q1σ , Q2σ in the limit r = U/ε1 → ∞, where two spin sectors are disentangled. At ﬁnite r any interaction with the bath B initiates spin-ﬂip transitions, which merge two spin subsectors via the anticommutation relation {Q1σ , Q2σ¯ } = −i( fσ† fσ¯ − fσ†¯ fσ ).

(2.130)

One should emphasize that the existence of s(2) superalgebra does not imply supersymmetry of the Hubbard atom in the classical meaning of this term, because the original Hamiltonian Hˆ iSU(3) describes fully discrete system with the constraint (2.6), which in the fermion-boson representation reads as

∑ nF σ + nB = 1. σ

so that the physical sector of phase space consists of three states

(2.131)

44

2 HIDDEN AND DYNAMICAL SYMMETRIES OF ATOMS AND MOLECULES

|nB , nF↑ , nF↓ = |1, 0, 0, |0, 1, 0, |0, 0, 1

(2.132)

instead of (2.96). Thus, the boson-fermion transformation Q+ ↑ |1, 0, 0 = |0, 1, 0 reveals only internal symmetry of the SU(3) Hubbard atom. To make the Bose states continuous or quasicontinuous, this toy model should be generalized. In particular, one may introduce “N-colored” model with N orbitals [5] and ascribe additional index m = 1, 2 . . . N to slave bosons, i.e. introduce Bose operators as X σ m,2 = fσ† bm etc. In case of N → ∞ this supersymmetric model with slave-boson continuum is exactly solvable, and one may hope that the supersymmetric approach will give reasonable description of excitations in Hubbard atom at ﬁnite N [61, 167]. This mathematical trick does not look too promising for the description of realistic physical models. More attractive is the use of slave-fermion approach, where the Hubbard operators are expressed via spinless charged fermion operators g and Schwinger boson operators a, b (see Section 9.3). For example, in the subspace Φ¯ 3 = 0, ↑, ↓ one introduces the supercharge operators Q†↑ = X ↑0 = a† g, Q†↓ = X ↓0 = b† g,

(2.133)

Then the Schwinger bosons describe local spin ﬂuctuations by means of bosonized operators from the triad T [see Eqs. (2.67),(9.56)]. Introducing N-component Schwinger representation, one transforms the Bose branch into continuous twodimensional harmonic oscillator at large N. Such continuous mode arises naturally, if the bath B is chosen in a form of local spin environment like in molecular magnets with large total spin (see Section 5.2). In any case, supersymmetric aspects of Hubbard-like and Anderson-like models need further investigation (see [98] for further discussion of supersymmetric description of periodic Hubbard model). The theory of mesoscopic objects also has beneﬁted from the supersymmetry ideas [88]. Supersymmetric approach reveals hidden symmetries in disordered and chaotic systems, including the fundamental problem of electron localization. This fructiferous offshoot of supersymmetry is, however, beyond the scope of our book.

2.7 Quasienergy spectrum for periodical time-dependent problems

45

2.7 Quasienergy spectrum for periodical time-dependent problems In some practical applications devices consisting of nanoobjects, electrodes and contacts are subject to periodic perturbation ˆ H(x,t) = Hˆ 0 (x) + Hˆ (x,t),

(2.134)

where x stand for spatial coordinates, t is real time, Hˆ (x,t + T ) = Hˆ (x,t) and T = 2π /Ω is a period of perturbing potential. There is no stationary solutions

Ψ (x,t) = ∑ cκ ψκ (x) exp(−iεα t) α

of the Schr¨odinger equation i

d Ψ (x,t) = Hˆ Ψ (x,t) dt

(2.135)

characterized by the energy states εα . However, in case of strictly periodic in time perturbation one may introduce a set set of functions ψα (x,t) and corresponding quasienergy states f α which inherit the quantum numbers α from the eigenstates of the time-independent Hamiltonian Hˆ 0 ,

Ψ (x,t) = ∑ cα ψα (x,t) exp(−i fα t)

(2.136)

α

Moreover, the spectrum of quasienergies possesses the dynamical symmetry in the same mathematical sense as the spectrum of eigenenergies εα for Hˆ 0 . Theoretical description of quasienergy states is based on appropriate modiﬁcation of the basis wave functions, which explicitly takes into account the periodicity in time domain [352, 378, 439, 440]. Indeed, the periodicity in time implies

ψα (x,t + T ) = e−i fα T ψα (x,t)

(2.137)

This means that the wave function may be represented in the form

ψα (x,t) = e−i fα t uα (x,t)

(2.138)

in accordance with the Floquet – Bloch theorem. The Floquet amplitude is periodic in time, uα (t + T ) = uα (t). The quantity fα plays the same part as the quasimomentum kα in the Bloch wave function periodic in space. Basing on this analogy,

46

2 HIDDEN AND DYNAMICAL SYMMETRIES OF ATOMS AND MOLECULES

Nikishov and Ritus [298] introduced the concept of four-dimensional quasimomentum, where the factor fα in the exponent in Eqs. (2.137), (2.138) plays part of quasienergy. It is natural to choose for the basis functions φα (x,t) for the quasienergy states those, which evolve from the eigenfunctions of the Hamiltonian Hˆ 0 . In principle, each eigenstate of the Hamiltonian Hˆ 0 may be put in correspondence to some quasienergy state. To verify this statement, one may introduce the perturbation λ (t)Hˆ p and trace the variation of the solutions when λ varies slowly from 0 to 1 during a time much longer than the period T . Due to periodicity in time, the variable t may be eliminated from the calculation of quasienergy spectrum by means of the expansion of the wave functions ψα (x,t) in a Fourier series,

ψα (x,t) =

+∞

∑

umα (x)e−i( fα +mΩ )t .

(2.139)

m=−∞

Then the Fourier transformation of the Schr¨odinger equation [id/dt − H(t)]ψ (t) results in an inﬁnite system of equations for the Fourier components umα : mn ( fα + mΩ − εα )umα = ∑ Hαβ u nβ , mn = Hαβ

1 T

+T /2 −T /2

(2.140)

nβ

ˆ dtα n|H(t)| β mi(m−n)Ω t

(2.141)

We learn from this system that the quasienergy is not conserved in a periodic ﬁeld: if the state fα is replaced by fα + m Ω , the secular equation mn |=0 |( fα − εα + mΩ )δαβ δmn − Hαβ

(2.142)

which determines the solutions of Eq. (2.140) remains unchanged. The satellites at mΩ arise in the quasienergy spectrum in the same sense as the Umklapp vectors mG accompany the quasimomenta in Bloch states, i.e. the quasienergy fα in accordance with the Fourier expansion (2.139) is deﬁned up to the additive constant, f α n − fα m = (n − m)Ω . ˆ The composite Hilbert space S for the Hamiltonian H(x,t) (2.134) is made up of the subspace R of square integrable functions in conﬁguration space and subspace

T of functions periodic in time, S = R T. The orthogonal basis functions for the temporal subspace T are the exponents exp(imΩ t), and the functions

Φα m (x,t) = uα (x,t) exp(imΩ t)

(2.143)

2.7 Quasienergy spectrum for periodical time-dependent problems

47

taken at a given moment t are orthogonal, Φα m |Φβ n =

1 T

T ∞ 0

−∞

Φα∗ m (x,t)Φβ n (x,t) = δαβ δmn .

(2.144)

They form a complete set of basis functions for a time-dependent Schr¨odinger equation with the Hamiltonian (2.134). One may treat conversion of the energy spectrum εα into the quasi-energy spectrum fα m as a manifestation of the dynamical symmetry SO(2) of Fourier transformation using the basis e−imΩ t in the Fock subspace T. indexdynamical symmetry groups! – SO(2) group In this special case we deal with real dynamics of quantum mechanical states in time. Due to Floquet theorem, this dynamics results in (a) transformation of stationary states with deﬁnite energies εα into quasienergy states fα m and (b) initiation of transitions between these states given by the non-diagonal matrix elements (2.141). The key features of quasienergy spectrum may be demonstrated on the elementary example of two-level systems discussed above in Section II.3. Let us assume that the tunneling term in the Hamiltonian (2.37) is periodic in time in accordance with a simple law W (t) = V cos Ω t. In this case, the spectrum of stationary states consists of two levels ε1,2 , perturbation Hˆ p (t) contains a single harmonics ±Ω , and the secular equation (2.142) has the following structure .. . 0 0 0 0 0 0 F2 − 2Ω V V F − Ω 0 0 V 0 0 0 1 0 0 F − Ω V 0 0 0 0 2 0 0 V F 0 0 V 0 1 = 0 (2.145) 0 V 0 0 F V 0 0 2 0 0 0 0 V F + Ω 0 0 1 + Ω V 0 0 0 V 0 0 F 2 0 0 0 0 0 0 V F + 2 Ω 1 . . . (Fi = fi − εi ). This equation illustrates transformation of the stationary spectrum Fi = 0 into a set of two interconnected ladders of quasienergy states. The temporal dynamics of a particle in such a double well is determined by the evolution operator U(t,t ) = Ψˆ (t)Ψˆ −1 (t ) This dynamics is given by transitions between two quasienergy ladders.

(2.146)

Chapter 3

NANOSTRUCTURES AS ARTIFICIAL ATOMS AND MOLECULES

3.1 Introductory remarks In the 60-es and the 70-es, when the basic concepts of dynamical symmetries have been formulated and elaborated, only a few physical realizations of these symmetries could be found in the realm of existing quantum mechanical objects. One may mention the hydrogen atom (Section 2.2) and the harmonic oscillator in various spatial dimensions (Section 2.5) as the systems for which the study of their dynamical symmetries revealed additional facets of excitation spectra and response to external ﬁelds. Rapid progress of nanotechnology and nanophysics in the recent two decades signiﬁcantly extended the ﬁeld of applicability of these concepts and enriched the theory with some new ideas. Contemporary nanophysics deals with artiﬁcial structures which consist of ﬁnite number of electrons conﬁned within a tiny region of space. If the electron de Broglie wavelength λ = h/p ∼ 13 nm (p is characteristic electron momentum) exceeds the conﬁnement radius, then the electron energy spectrum is discrete. As a result, such object can be treated as a ”zero-dimensional” artiﬁcial atom or molecule with spatially quantized discrete states, well deﬁned symmetry and controllable electron occupation. An artiﬁcial atomic cluster may be studied experimentally and even modiﬁed in a controllable way. In many real physical situations few-electron nanoobjects possess well deﬁned angular symmetry, and the discrete spectrum of conﬁned electrons is characterized by various dynamical symmetries. In order to investigate the excitation spectrum of conﬁned electrons, the nanoobjects are included in electrical circuits. Metallic wires or puddles forming ”leads” in these circuits become parts of quantum mechanical nanosystems. A distinguished feature of these systems is that the dynamical symK. Kikoin et al., Dynamical Symmetries for Nanostructures: Implicit Symmetries in Single-Electron Transport Through Real and Artificial Molecules, DOI 10.1007/978-3-211-99724-6_3, © 2012 Springer-Verlag/Wien

49

50

3 NANOSTRUCTURES AS ARTIFICIAL ATOMS AND MOLECULES

metries of the conﬁned electrons are manifested in the interaction with macroscopic environment: metallic leads play part of a ”Fermi bath” B of conduction electrons, and the coupling between conﬁned and free electrons activates excitation spectrum of nanoobject. This means that one cannot study the dynamical symmetry GS of nanoobject per se. Instead, one should write out the Hamiltonian of the whole system Hˆ = Hˆ d + Hˆ b + Hˆ db

(3.1)

where three terms describe nanoobject, Fermi-bath and their coupling, respectively. The coupling term Hˆ db in the general case includes direct coupling (quantum tunneling of electrons between two subsystems), direct interaction of Coulomb and exchange nature and indirect (kinematic) interaction induced by tunneling. If the symmetry of a nanoobject is well deﬁned, the Hamiltonian Hˆ d may be diagonalized by means of projection operators (2.2), and the generators of the dynamical symmetry group (2.8) arise in the interaction term Hˆ db in combination with operators describing excitation in the Fermi bath. In this respect nanostructures should be qualiﬁed as a special class of quantum systems possessing dynamical symmetries which cannot be treated in the same way as in the integrable systems described in Chapter 2, because the interaction not only activates this symmetry but also involves charge and spin degrees of freedom of the bath. This principal difference was pointed out in Refs. [203], where the quantum tunneling through artiﬁcial molecule with even electron occupation N = 2 in the presence of many-particle interaction of the Kondo type was described by means of the generators of SO(4) symmetry introduced in Sections 2.2 and 2.4. Before turning to description of dynamical symmetries in many-body interacting systems, it is worth to describe experimental realizations of nanoobjects which in fact initiated the avalanche of theoretical studies of many-body effects in artiﬁcial nanostructures. These are artiﬁcially fabricated structures known as quantum dots. Quantum dot is an object where certain number of electrons is conﬁned in a potential trap of characteristic size comparable with the de Broglie wavelength. Such dots may be fabricated from various materials by diverse methods including colloidal synthesis, nanolithography, growth on strained surfaces, etc. In our brief survey we limit ourselves only with nanocrystals possessing well deﬁned symmetry, controllable electron occupation and spin state for which the dynamical symmetry is the inherent characteristics.

3.2 Planar quantum dots

51

3.2 Planar quantum dots Planar (lateral) quantum dots are fabricated in semiconductor heterostructures, where the electrons conﬁned in a two-dimensional layer between two semiconductors (usually GaAs/GaAlAs) are locked in a nano-size puddle by electrostatic potential created by electrodes superimposed on the heterostructure (see [196, 351, 359] for a description of early stage and further development of the physics of planar quantum dots). Experimental setup realizing such electron conﬁnement is shown in Fig. 3.1(a). Electrostatic potential created by electrodes transforms a twodimensional electron layer into a tunnel structure [Fig. 3.1(b)]. A potential well formed by the electrodes constitutes an artiﬁcial atom (quantum dot) with electrons captured in this well ﬁlling discrete energy levels. Conduction electrons outside this trap form two-dimensional Fermi reservoirs. These electrons may tunnel through the barriers which form a quantum dot. An electric circuit arises when the bias voltage vb is created between two Fermi reservoirs [Fig.3.1(c)], which play parts of source (s) and drain (d) electrodes in this circuit. The character of electron tunneling is predetermined by parameters characterizing a quantum dot. These parameters are the radius of the dot and the size of the tunnel barriers. The ﬁrst of these parameters deﬁnes the average interlevel spacing

δ ε in a discrete spectrum ε j of a quantum dot and the capacitance C of this dot. The second parameter gives the tunneling amplitudes Ws and Wd between the electrodes and the dot. One may specify the generic Hamiltonian (3.1) for the tunneling problem in the following way: vgCg 2 † ˆ ˆ Hd = ∑ ε j d jσ d jσ + Hint + Q ndot − e jσ Hˆ b = ∑

∑

kσ l=s,d

Hˆ db ≡ Hˆ tun =

εkl c†lkσ clkσ

(3.2)

∑ ∑ Wl j (d †jσ clkσ + h.c.) jkσ l=s,d

Here we suppose that the quantum dot has arbitrary shape without any angular symmetry, so the index j simply enumerates the levels bottom-up. Hˆ int is the electronelectron interaction in the quantum dot. Usually the self-consistent Hartree term is included in the deﬁnition of the discrete levels ε j , and the relevant contribution to Hˆ int is the exchange between electrons occupying different levels of a neutral quan(ext)

tum dot. Q = e2 /2C is the capacitive energy of the dot, ndot = ∑ jσ d †jσ d jσ is the

52

3 NANOSTRUCTURES AS ARTIFICIAL ATOMS AND MOLECULES

(a)

(b) 1 4 2 3

(c)

Fig. 3.1 (a) Planar quantum dot. Two-dimensional electron gas formed on the interface GaAs/Alx Ga1−x As. Electrodes imposed on the surface of the heterostructure conﬁne the 2D electrons in the QD; (b) Planar QD, top view. Gate voltage vg is created between the electrodes 1 and 2. The small area conﬁned by the electrodes 1 – 4 indicates region where metallic 2D electrons playing part of a quantum dot are located. (c) Equivalent electric circuit of single electron transistor including quantum dot. Cg ,CL ,CR are capacitances of various contacts, gate voltage Vg applied to the QD regulates its occupation.

number of extra electrons which are injected in the dot due to tunneling described by the Hamiltonian Hˆ tun . Corrections to capacitive energy take into account the capacitance of the gate Cg and the gate voltage vg on electrode 2. The capacitive energy Q may be large enough for small enough quantum dots. If the hierarchy of the energy scales Q > (δ ε , J) Wl j

(3.3)

takes place (J is the exchange coupling constant), then one may assert that charge transfer through a quantum dot occurs in a regime of single electron tunneling (SET). To show this and to reveal the possible dynamical symmetries of planar quantum dots let us consider the excitation spectrum of the quantum dot in more details. First, we note that the electrostatic energy of and isolated quantum dot described by the Hamiltonian Hˆ d is a parabolic function of the occupation number,

3.2 Planar quantum dots

53

Eel (N ) = Q(N − ν )2 ,

(3.4)

where ν = vgCg /e and N = N − N0 is the deviation of occupation number from its value for a neutral state of the quantum dot with occupation N = N0 . At zero gate voltage N = 0, and the energy ”cost” of electron injection or ejection is δ εm + Q, where δ εm = |εm − εtop | is the difference between the energy of the state occupied by the extra electron or hole and the energy of the highest occupied discrete state in the dot. One says that the tunneling between the electrodes and the quantum dot is suppressed due to the Coulomb blockade effect. The theory of the Coulomb blockade was pioneered in the work [137] based on the previous relevant studies [243, 262].

0

εF

1

11 00 00 11 00 11 00 11 00 11 00 11 00 11 00 11

s

2

11 00 00 11 00 11 00 11 00 11 00 11 00 11 00 11

d

0

11 00 00 11 00 11 00 11 00 11 00 11 00 11 00 11

1

2

11 00 00 11 00 11 00 11 00 11 00 11 00 11 00 11

0

00 11 11 00 00 11 00 11 00 11 00 11 00 11 00 11

1

2

00 11 11 00 00 11 00 11 00 11 00 11 00 11 00 11

0

1

11 00 00 11 00 11 00 11 00 11 00 11 00 11 00 11

2

11 00 00 11 00 11 00 11 00 11 00 11 00 11 00 11

3

1

2

00 11 11 00 00 11 00 11 00 11 00 11 00 11 00 11

3

000 111 111 000 000 111 000 111 000 111 000 111 000 111 000 111

Fig. 3.2 Upper panel: Variation of the energy of quantum dot Eel (N ) as a function of gate voltage. Lower panel: corresponding variation of addition energies for electron and hole excitations relative to the Fermi level in the leads. See text for further discussion.

Varying the gate voltage vg , one may change the occupation number minimizing Eel from N = 0 to N = ±1, ±2 . . .. Fig. 3.2 illustrates this change as well as the reconstruction of the discrete spectrum of the conﬁned electrons. Five subsequent parabolas in the upper part of this ﬁgure correspond to ﬁve stages of dot recharging from N = 1 to N = 2. The chemical potential of the system is pinned to the Fermi level εF of the leads is in the mid of the Coulomb blockade gap. In the ﬁrst three diagrams the dot is singly occupied, With increasing vg the states with N = 0 and N = 1 approach to the degeneracy point (third diagram), in the point of Coulomb resonance Eel (N = 2) = Eel (N = 1) (fourth diagram) the gap vanishes and resonance source-deain tunneling is possible as is shown in lower part of the ﬁgure. In the rightmost diagram the dot is occupied by N + 1 electrons and small Coulomb gap exists for the hole excitation. It is worth noting that the curves depicting the

54

3 NANOSTRUCTURES AS ARTIFICIAL ATOMS AND MOLECULES

energy (3.4) deserve the name ”Hubbard parabolas” since J. Hubbard was the ﬁrst to introduce them for explanation of the origin of subbands for fermionic excitation in various charge sectors of his model of strongly correlated non-Fermi liquid [169, 170, 171] (see also [157]).

v

b

0

odd

even

odd

even

vg

vg

G

v

g

Fig. 3.3 Upper panel: Tunnel conductance G (lines) as a function of bias vb and gate voltage vg at T → 0. Rhombic Coulomb windows correspond to suppression of tunnel current due to the Coulomb blockade. Each Coulomb window sets an integer N . Middle panel: “Coulomb staircase” - stepwise change of occupation number N with increasing gate voltage vg at vb → 0 and T → 0. At ﬁnite T the steps are smoothed. Lower panel: The tunneling conductance G as a function of vg at vb → 0. Peaks in G correspond to transitions from a Coulomb window with occupation N to the next one with occupation N + 1.

The Coulomb blockade in the SET regime is usually represented by means of the so called ”Coulomb diamond diagrams” which show tunnel conductance as a function of gate voltage vg and source-drain bias vb (see Fig. 3.3). Dark lines in the upper panel of this ﬁgure correspond to nonzero conductance G(vg , vb ) due to inelastic cotunneling. Empty ”Coulomb diamonds” bound the regions where the

3.2 Planar quantum dots

55

tunneling is suppressed by the Coulomb blockade. At zero bias vb = 0 the occupation number N (vg ) is a stepwise function (mid panel): each step corresponds to the crossover from a given charge sector N to the adjacent sector N ± 1. Each step in the occupation corresponds to a certain Coulomb resonance peak in the differential conductance (lower panel). Having in mind this general picture, we now turn to our main goal, namely the description of dynamical symmetries of the excitation spectrum of the generic Hamiltonian (3.1). It is evident that the last term Hˆ db in this Hamiltonian is responsible for transitions between different energy states of electrons in the dot. Since the electron interaction in the dot is dominant in case of strong Coulomb blockade (3.3), it is worth to write the Hamiltonian Hˆ d in the form (2.45) where the interaction is built in the eigenstates |Λ :

Hˆ d = ∑ EΛ X ΛΛ

(3.5)

Λ

As before, the operators O forming the algebras for the dynamical symmetry groups should be constructed by means of the Hubbard operators X ΛΛ [see Eq. (2.8)], and tunneling processes may intermix the states |Λ , |Λ belonging both to the same and to different charge sectors of the Coulomb diamond diagram (Fig. 3.3). The tunneling Hamiltonian Hˆ db may be transformed by means of the rotation operation [135]

cekσ cokσ

1 = Wj

Ws j Wd j −Wd j Ws j

cskσ cdkσ

,

(3.6)

Ws2j + Wd2j . Due to the axial symmetry of the device, only the even linear combination of band orbitals cekσ enters the tunneling Hamiltonian Hˆ db (3.2), where W j =

so that the two other terms in the Hamiltonian (3.2) may be rewritten as Hˆ b =

∑ ∑ εk c†lkσ clkσ

l=e,o kσ

Hˆ db =

∑ W j (d †jσ cekσ + h.c.)

(3.7)

k jσ

provided εks = εkd = εk . The Hamiltonian Hˆ db may be rewritten in terms of ”Fermilike” Hubbard operators X Λ λ , where the states |Λ and λ | belong to adjacent charge sectors: Hˆ db = ∑ W jΛ λ (X Λj λ cekσ + h.c.) (3.8) k jσ

As was emphasized in Chapter 2, dynamical symmetry in the general case is not a universal characteristics of the system: it depends on the type of perturbation

56

3 NANOSTRUCTURES AS ARTIFICIAL ATOMS AND MOLECULES

which induces the interlevel transitions and on the energy scale of this perturbation. In case of quantum dot under strong Coulomb blockade the perturbation is the tunnel coupling with the Fermi bath, and the energy scale is assigned by the inequalities (3.3). Inasmuch as the lead-dot tunneling changes the electron occupation, N → N ± 1, the excitation spectrum involved in the supermultiplet includes these three adjacent charge sectors. Electronic levels in a quantum dot are ﬁlled up one by one. Dot recharging is controlled by varying gate voltage, so that the electron occupation changes from odd to even and then again to odd in the adjacent Coulomb windows. Since the ground states and excitation spectra are different for even and odd occupation, the type of dynamical symmetries also varies from one Coulomb window to another. When the dot occupation N is odd, the ground state is spin degenerate: an electron occupying the highest discrete level may be in two spin states σ =↑, ↓. Only an electron occupying the highest discrete level is involved in interaction with the Fermi bath B, so one may exclude the 2n ”passive” states from electron counting, and ascribe the number N = 1 to the state in the minimum of the Hubbard parabola for odd occupation. In a symmetric conﬁguration (second Hubbard parabola in Fig. 3.2) the lowest excited states with N ± 1 = 0, 2 are also degenerate. If we are interested only in excitations within the energy range 0 ε Q, the Hamiltonian Hˆ d (3.2) is reduced to the Hamiltonian of the Hubbard atom (2.60) where εd is the highest occupied level in the dot and the parameter U is substituted for Q. In accordance with the symmetry analysis carried out in Section II.42.4, the basic dynamical symmetry within the above energy range is SU(4). The structure of the multiplet and interlevel transitions for odd occupation N = 2n + 1 is represented in Fig. 2.4(a). The operator algebra is determined by the system of Gell-Mann operators (2.66) acting in the space Φ¯ 4 (2.68). As was noticed in Section 2.4, this set includes both Bose-like operators X1 − X3 , X8 , X13 − X16 and Fermi-like operators X4 − X7 , X9 − X12 . The corresponding transitions are marked by horizontal and slanted arrows in Fig. 2.4(a), respectively. Following the symmetry analysis of the Hubbard atom Hˆ d we turn to the representation of SU(4) group which diagonalizes this Hamiltonian. In this representation the Hubbard operators are grouped in six triads (2.67). The symmetric conﬁguration shown in the second diagram of Fig. 3.2 corresponds to degenerate SU(4) symmetry with the Hamiltonian

Ep Hˆ d = (Qz + Pz ) + C4 · 1 8

(3.9)

3.2 Planar quantum dots

57

[see Eqs. (2.71) and discussion below]. Here E p is the addition energy E p = E(N = 0) − E(N = 1) = E(N = 2) − E(N = 1).

σ

σ’

σ σ’

(a)

σ

(b)

σ’

(c)

σ’ σ

(d)

Fig. 3.4 Electron cotunneling through quantum dot under strong Coulomb blockade. (a) electron σ leaves the singly occupied QD for one lead and electron σ from another lead replaces it in the dot; (b) similar process involving electrons from the same lead; (c) electron σ from one lead tunnels into the singly occupied QD and electron σ leaves QD for another lead; (d) similar process involving electrons from the same lead.

If the energy scale ε¯ established by interaction with the bath B is reduced to the interval ε¯ Q, a quantum dot with odd occupation loses its dynamical symmetry at ε < ε¯ . Transitions between adjacent charge sectors (N = 1 → N = 0, 2) are quenched, and only Bose-like spin-ﬂip processes described by the operators X σ σ

are possible. This means that only three linearly independent operators, namely X1 , X2 , X3 , or Tz , T± from the whole set (2.67) remain in the set of group generators, and the original SU(4) symmetry is reduced to SU(2) symmetry of spin 1/2. In terms of physical processes such reduction means that within this energy range only elastic cotunneling with and/or without spin ﬂip is possible. Various electron cotunneling processes are illustrated in Fig 3.4. Due to strong Coulomb blockade electron with spin σ may tunnel from any lead to the dot only when another elec-

58

3 NANOSTRUCTURES AS ARTIFICIAL ATOMS AND MOLECULES

tron with spin σ leaves the dot for another lead (a,c) or for the same lead (b,d). There exist two channels of cotunneling: the intermediate virtual states with N = 0 or N = 2 arise on equal footing in a process of electron injection from the source and departure to the lead. Besides, cotunneling ends with creation of an electronhole pair in the bath B. In case (a,c) creation of such pair means charge transfer from the source to the drain electrode. In any case cotunneling processes may be described in a framework of reduced effective Hamiltonian Hˆ eff valid within the energy interval ε¯ (Q, εd − εF ). This Hamiltonian known as the Schrieffer – Wolff (SW) Hamiltonian may be obtained by means of integrating out the high-energy states in second-order perturbation theory taking the ratio W /Q as a small parameter [158, 372], or, equivalently, by means of the canonical transformation ˆ −U , Hˆ eff = eU He

W W U =∑ X σ 0 c kσ + σ X 2σ¯ ckσ − H.c. εk − εd − U kσ εk − εd

(3.10)

where only the terms up to the second-order in W are retained. More detailed discussion of various cases of electron cotunneling will be presented in Chapter 4. Here we only stress that the X-operators entering the tunneling Hamiltonian (3.8) and the matrix U (3.10) are in fact the ladder operators from the set of generators of the Lie group SU(4): X σ 0 = (V+ , U+ ), X 2σ¯ = (Y− , W− ) [see Eqs.(2.67)]. In accordance with the multiplication/commutation rules (9.44) their products generate spin-operators S = T/2 in the second-order expansion of Hˆ eff (3.10) in the charge sector N = 1 of the space Φ¯ 4 (2.68). As a result the SW Hamiltonian has the form Hˆ eff = Hˆ d + Hˆ b + Hˆ ex , J Hˆ ex = S · s, 2

where J ∼ W2

1 1 + εd + Q − εF εF − εd

(3.11) (3.12)

is indirect antiferromagnetic exchange coupling parameter and s = ∑ ∑ sek,ek ≡ c†σ τ σ σ cσ kk σ σ

(3.13)

3.2 Planar quantum dots

59

Due to the positive sign of the effective exchange coupling, the cotunneling mechanism favors antiparallel orientation of the localized spin S of the dot and the itinerant spin s of the bath. Since both of them are spins one half, this means that the ground state of the whole system is obliged to be a spin singlet. Increasing the gate voltage, we turn to strongly asymmetric conﬁguration shown in the third diagram of Fig. 3.2, where E(N − 1) − E(N ) E(N + 1) − E(N ). In our convention this means that the conﬁguration N = 0 is quenched at low energies and only the transitions from the ground state with N = 1 to the excited doubly occupied states N = 2 are involved in the dynamical symmetry activated by tunneling processes. Similar situation arises when vg is decreased and the conﬁguration N = 2 is quenched (leftmost diagram in Fig. 3.2. In both these cases the dynamical symmetry is reduced from SU(4) to SU(3) with algebra of group generators given by the ﬁrst eight operators from the set (2.67). To describe the cotunneling via virtual doubly occupied states, one has to change in these equations the index ’0’ (zero occupation) to ’2’ (double occupation), in accordance with the transition shown by the arrows in Fig. 2.4(b). As before, the horizontal arrows in this ﬁgure are related to the Bose-like operators X1 − X3 , X8 and the slanted arrows designate the Fermi-like operators X4 − X7 . According to (2.74), the Hamiltonian Hˆ d contains the operators Uz and Vz , whereas the tunneling term Hˆ db contains the ladder operators U± and V± . In the low-energy regime ε < ε¯ these terms generate operators from the triad T, and electron cotunneling is equivalent to indirect SW exchange with J ∼ W 2 /(εd + Q − εF ). In a symmetric conﬁguration E(N + 1) = E(N ) (fourth Hubbard diagram of Fig. 3.2) the gap between the states in adjacent charge sectors of Fig. 2.4(b) vanishes, so that the SU(3) group characterizes the dynamical symmetry of quasielastic cotunneling at ε < ε¯ Q. The values of the gate voltage vg corresponding to this resonance cotunneling regime determine the boundaries of the corresponding Coulomb diamond in the upper panel of Fig. 3.3. At zero bias further increase of vg means transition to the next Coulomb window with even number of electrons in the ground state of the quantum dot. As a rule, supermultiplets of low energy states of quantum dots with even occupation N = 2 obey SO(n) symmetries. The origin of these symmetries is illustrated in Fig. 3.5. In the singlet ground state ES the highest occupied discrete level is ﬁlled with two electrons with antiparallel spins. One of these electrons may be moved to the next discrete level, and the energy cost of this excitation is the level spacing δ ε . The energy of this singlet excitation is EE2 = ES + δ ε . Due to negative exchange

60

3 NANOSTRUCTURES AS ARTIFICIAL ATOMS AND MOLECULES

ε

δε

S

T

E2

F

E3

Fig. 3.5 Ground state and ﬁrst excitations in planar QD with even occupation. In the ground state the system is in a spin singlet state. When one or two electrons are moved to the ﬁrst excited level above εF both singlet and triplet excitons may be formed.

interaction Hˆ int the triplet excited state with the energy ET = EE2 − I arises (I is the exchange coupling constant for two electrons occupying neighboring discrete levels in the dot). One more excited state with energy EE3 = ES + 2δ ε corresponds to double occupation of the next discrete level. The supermultiplet containing all these states is characterized by the SO(6) dynamical symmetry group formed by 15 operators X ΛΛ describing transitions between all levels within this sextet [see Fig. 2.2(a)]. In all practically interesting situations the characteristic energy range ε¯ δ ε includes only the states ES , EE2 and ET [Fig. 2.2(b)]. The symmetry SO(5) of this supermultiplet was analyzed in Section 2.4 in connection with the spectrum of Fulde molecule [see Eq. (2.76)]. Ten operators packed into three vectors and one scalar form the generating algebra o(5). If the inequality

δε −I δε

(3.14)

is valid, then further decrease of the characteristic energy interval down to ε¯ δ ε −I results in quenching the ”orbital” degrees of freedom and leaves us with the SO(4) singlet/triplet spin supermultiplet [Fig. 2.2(c)] discussed in Sections 2.2 and 2.4 and described in detail in the Mathematical Appendix (Section 9.2.1). This is the minimal dynamical symmetry group for a quantum dot with even occupation in case when the Coulomb blockade suppresses the charged states with odd occupation. Like in the odd-occupation charge sectors, the Hamiltonian Hˆ d + Hˆ db may be rewritten in terms of the SO(n) group generators. These expressions may be found in Section 4.2. A way to realization of dynamical symmetries in measurable physical effects is paved by the form of the effective cotunneling Hamiltonian. If the lead electrons are

3.3 Vertical quantum dots

61

presented in Hˆ ex only by their spin degrees of freedom, the cotunneling processes involving dynamical degrees of freedom are described by the Hamiltonian containing not only spin invariant like in Eq. (3.11) but also by the invariants including other generators of the dynamical symmetry group [245], J Jα Hˆ ex = S · s + ∑ Rα · s 2 α 2

(3.15)

Scalar operators from the corresponding algebras do not enter the effective Hamiltonian directly, but marginally inﬂuence the dynamical properties of the tunneling structure via kinematic constraints imposed by Casimir operators. In some conﬁgurations of tunneling devices the orbital degrees of freedom of band electrons are also affected by tunneling processes. Then the effective Hamiltonian acquires more complicated form. Various types of cotunneling Hamiltonian will be derived and used in subsequent chapters of this book.

3.3 Vertical quantum dots From planar quantum dots possessing dynamical symmetries predetermined primarily by Coulomb blockade, we turn to nanoobjects of deﬁnite geometrical shape, where the angular symmetry of electron wave functions results in the formation of orbital shells. These are vertical quantum dots, which may be treated as true artiﬁcial atoms which differ from genuine elements of the Mendeleev Periodic table by dimensionality [238]: they have cylindrical symmetry instead of spherical one, so that the effective spatial dimensionality of these object is reduced from 3 to 2. The conﬁguration of a trap for conﬁned electrons in vertical dots is predetermined by their cylindrical form: these nanoobjects may be fabricated from an AlGaAs/InGaAs heterostructures: circular InGaAs nanosize pillar conﬁnes few electrons in a nearly parabolic potential. A disc-like dot is sandwiched between source and drain reservoirs made of n-GaAs, gate in a form of a rim envelops the dot and the whole structure form a cylinder (Fig. 3.6). Electrons in parabolic potential well are described by the Fock-Darwin model exposed in Section 2.5. Their energy spectrum is given by Eq. (2.91). It is convenient to redeﬁne the quantum numbers in the eigenfunctions (2.89) as m = n + |l| and to rewrite the eigenenergy as En,l = Ω (2n + |l| + 1) −

ωc |l| 2

(3.16)

62

3 NANOSTRUCTURES AS ARTIFICIAL ATOMS AND MOLECULES

Fig. 3.6 (a) Vertical quantum dot fabricated from AlGaAs/InGaAs heterostructure, schematic setup. Electrons are conﬁned in Inx Ga1−x As quantum well; (b) Electron micrograph of a device [117]; (c) Occupation of 1s and 2p levels in the ground and excited states of Fock-Darwin atom with N = 1; (d) Occupation of 1s and 2p levels in the ground (singlet) and excited (triplet) states of Fock-Darwin atom with N = 2.

so that n = 0, 1, 2, . . . and l = 0, ±1, ±2, . . . are the radial and azimuthal quantum numbers respectively (see also [273, 413]). This equation points out the order in which the shells are occupied by electrons in this artiﬁcial atom. In zero magnetic ﬁeld electrons subsequently occupy s, p, d . . . shells with n = 0, 1, 2 . . ., and occupation of their orbitally degenerate states is determined by the Pauli principle (Fig. 3.7). The shell structure is illustrated in Fig. 3.8(b). This structure is observable in current – gate voltage characteristics, where the one by one injection of electrons from the Fermi reservoir is seen as a series of discrete peaks in the tunneling current [Fig. 3.8(a), upper panel]. Conﬁgurations with complete shells are characterized by ”magic” occupation numbers N¯ = 2, 6, 12, . . . Experimentally measured addition energy E(N ) − E(N − 1) for the next in turn electron injected into vertical dot shown in the inset demonstrates a relative stability of the ﬁlled shells similar to that in ”natural” atoms.

3.3 Vertical quantum dots

l

−2

63

−1

0

+1

+2

n 2 1 0

Fig. 3.7 Scheme of consecutive occupation of energy shells in vertical QD (no interaction, zero magnetic ﬁeld).

In ﬁnite magnetic ﬁeld discrete orbital levels undergo diamagnetic (Larmor) shift characterized by the cyclotron frequency ωc . Usually the Zeeman splitting is negligibly weak in comparison with diamagnetic shift due to small value of g factor in 2D electron gas in GaAs, and we neglect this splitting here and below in this section. At certain values of the magnetic ﬁeld the levels belonging to different values of n (and thereby to different occupation numbers) may cross [Fig. 3.8(c)]. Due to this crossing the order of occupation of the shells at given magnetic ﬁeld may change with varying gate voltage. The same mechanism is responsible for injection of extra electrons in the dot when the state with higher n becomes energetically favorable with growing magnetic ﬁeld. This is how the dynamical symmetry SO(2, 1) mentioned in Section 2.5 is realized in vertical quantum dots. Up to this point we neglected the interaction between the electrons in the quantum dots with cylindrical symmetry. Meanwhile, this interaction affects the level structure and the order of level occupation in the same way as in the Periodic table of natural chemical elements. As a result, dynamical symmetries of vertical dots change appropriately. The simplest object where the Coulomb and exchange electron-electron interaction is relevant is a vertical dot with ﬁlled s-shell (N = 2). Like natural para- and ortho-helium, this artiﬁcial atom can be in singlet and triplet spin state [117] (see Fig. 3.6, right panel). In the ground state two electrons with opposite spins occupy the 1s level. In the triplet state one of these electrons is excited into the 2p state, and exchange interactions favors parallel spin conﬁguration. These states are denoted as 2 S0 and 2 T M , where the left and right superscripts correspond to the electron number N and orbital number M, respectively. In ﬁnite magnetic ﬁeld the 2p level shifts downwards and the singlet/triplet transition takes

64

3 NANOSTRUCTURES AS ARTIFICIAL ATOMS AND MOLECULES

(c)

Fig. 3.8 (a) Current peaks corresponding to subsequent occupation of energy levels in vertical QD. Inset: addition energies corresponding to electron injection (see the text); (b) Shell structure of Fock-Darwin atom and consecutive occupation of these shells with corresponding addition energies consisting of capacitive energy e2 /C and the level spacing Δ E (cf. Fig.3.7); (c) Energy level crossing in Fock-Darwin atom due to the Larmor shift in external magnetic ﬁeld [see Eq. (3.16)] (after [238]).

place at some critical value of cyclotron frequency ωc = ω¯ c [413, 189]. This level crossing observed in [117] is illustrated in Fig. 3.9 (panels a,c). In accordance with classiﬁcations discussed above, this crossover is characterized by SO(4) dynamical symmetry. Direct singlet-triplet transitions are forbidden, but the spin-dependent interaction with the leads lifts this symmetry suppression (see Section 4.2). Similar ﬁeld induced level crossing is possible in the charge sector N = 3 [Fig. 3.9 (panels b,d)]. In accordance with numerical calculations [189], in this case a triple level crossing is possible, where two spin doublet states 3 D1 , 3 D2 and one spin quartet state 3 Q3 compete. One may speak in this case about reducible dynamical group SU(2) × SU(2) × SU(2) describing transitions within this supermultiplet. Of course, these dynamical symmetries characterize transitions only within the scale comparable with the interlevel distance near the level crossing. Extension of this scale involves higher levels from the same charge sector (dashed lines in Fig. 3.9), and the order of the dynamical symmetry groups increases accordingly. However, as will be shown in the following chapters, the scale of tunnel coupling and concomitant many-body effects is small enough, so that as a rule one may take into account only the ground state and one or two lowest excited levels of the vertical quantum dot.

3.3 Vertical quantum dots

65

Fig. 3.9 Energy level crossing due to the Larmor shift in the external magnetic ﬁeld in FockDarwin atom with N = 2 [panels (a) and (c)] and N = 3 [panels (b) and (d)] (after [189]).

Like in the case of planar quantum dots, dynamical symmetries of vertical quantum dots may involve states from adjacent charge sectors. One such supermultiplet arising due to diamagnetic shift in the magnetic ﬁeld [134] is shown in Fig. 3.10. This ﬁgure presents the low-lying part of the energy spectrum of vertical quantum dot with N = 5. Two crossing levels in this charge sector form spin and orbital doublet 5 D4 ,5 D6 . Tunneling between the dot and the leads involves the spin singlet states from adjacent charge sectors with N = 4, 6, namely the states 4 S4 and 6 S6 . The scheme of interlevel transitions within this supermultiplet corresponds to the SU(6) group (provided the transitions with Δ N = 2 are excluded). If the charged states are quenched, we still remain with SU(4) symmetry of spin plus orbital doublet. It will be shown in Section 4.3, how this dynamical symmetry inﬂuences the Kondo effect in quantum dots.

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3 NANOSTRUCTURES AS ARTIFICIAL ATOMS AND MOLECULES

Fig. 3.10 Energy level crossing due to the Larmor shift in the magnetic ﬁeld in the Fock-Darwin atom with N = 5 and shell occupation schemes. Virtual states in adjacent charge sectors N = 4, 6 which contribute in the SW exchange are also shown (after [134]).

3.4 Self-assembled quantum dots Characteristic energy scales in artiﬁcial quantum dots fabricated in epitaxial heterostructures described above are several orders smaller than those in real atoms. Typical level spacing and Coulomb blockade energies in planar dots are ∼ 0.1 meV and ∼ 0.5 meV, respectively in comparison with the corresponding values of ∼ 1 eV and ∼ 5 eV in ”natural” atoms. However, there exists a special class of heterostructures in which the atomic energy scale is partially conserved. These are self-assembled quantum dots, i.e. nanoislands formed on the interface of two semiconductor layers in the process of growth. An InAs layer grown by molecular beam epitaxy on a GaAs substrate tends to form islands of regular (pyramidal) shape in order to minimize the energy of strains arising due to the mismatch of lattice parameters of two zinc blende semiconductors (Fig. 3.11). The dimensions of such a pyramid are 15-25 nm across and several nm height (see [43] for a review). The energy spectrum of electrons conﬁned in the

3.4 Self-assembled quantum dots

67

InAs

GaAs Fig. 3.11 InAs self-assembled quantum dot in a strained heterostructure GaAs/InAs.

pyramidal dot inherits the structure of bulk semiconductor: it consists of spatially quantized valence and conduction states with level spacing ∼ 0.01 eV. These two quasi continua are separated by the energy gap exceeding 1 eV [42]. The surface electrons form a 2D wetting layer which plays part of metallic bath for these quantum dots. d p s Δ s Fig. 3.12 Energy spectrum of InAs self-assembled quantum dot in a strained heterostructure GaAs/InAs.

p d

Both theoretical calculations and experimental studies have shown that the conﬁning potential for electrons and holes in self-assembled quantum dots is nearly parabolic, and the low-energy spectrum may be approximated with two-band Fock – Darwin model [42, 153, 349]. Although in bulk InAs the electron states near the top of the valence band and the bottom of the conduction band have different angular symmetry (p- and s-type, respectively), this asymmetry is practically negligible

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3 NANOSTRUCTURES AS ARTIFICIAL ATOMS AND MOLECULES

in conﬁned structures. Both conduction and valence states retain the conventional shell structure of the Fock – Darwin model where the low-energy shells are classiﬁed in accordance with the ﬁlling scheme of Fig. 3.7. Thus, the energy spectrum of self-assembled quantum dot possesses additional particle-whole symmetry with good experimental accuracy (see Fig. 3.12). In the ground state all valence levels are occupied by electrons in accordance with the Pauli principle. The Hamiltonian of self-assembled dot in external magnetic ﬁeld is Hˆ d = ∑ Ωvp a†vpσ avpσ + Ωcp a†cpσ acpσ + Hˆ int

(3.17)

pσ

Here the particle representation is used for the conduction electrons acpσ and the hole representation is introduced for the valence states avpσ , so the two excitation branches are positive and identical,

Ω vp = Ωcp = Δ /2 + E p

(3.18)

where p = n, l are radial and azimuthal quantum numbers, E p is deﬁned in Eq. (3.16), Δ is the offset energy comprising the semiconductor energy gap and vertical conﬁning energy. Interaction terms take into account Coulomb repulsion between the particles belonging to the same branch and Coulomb attraction between electrons and holes. We discuss here only the symmetry properties of optically active subspace with zero values of orbital momentum L = 0 and spin momentum S = 0 (singlet excitons) and neglect Zeeman splitting in comparison with the diamagnetic shift. Then the spin index may be ignored and the interaction term is reduced to Hˆ int = Hˆ cv + Hˆ cc + Hˆ vv = −

∑ i j|ucv |mka†ci a†v j avk acl

i jkm

1 1 i j|ucc |mka†ci a†c j ack acm + ∑ i j|uvv |lka†vi a†v j avk avm 2 i∑ 2 jkl i jkm

(3.19)

Interactions between electrons and holes are also symmetric im|ucv | jk = i j|ucc |mk = i j|uvv |mk.

(3.20)

It was noticed [431] that the reduced Hamiltonian (3.17), (3.19) with the interaction (3.20) belongs to a class of exactly solvable models with hidden symmetry studied in Ref. [86]. Indeed, there exists a closed vector algebra for the operators P = {P+ , P− , Pz }, which are creation and annihilation singlet exciton operators and population inversion operator, respectively:

3.4 Self-assembled quantum dots

69

1 P+ = ∑ a†cp a†vp , P− = ∑ avp acp , Pz = (Nc + Nh − N0 ) 2 p p

(3.21)

Nc = ∑ a†cp acp , Nv = ∑ a†vp avp ,

(3.22)

p

p

(N0 is the total number of orbitals). These operators obey the SU(2) algebra [P+ , P− ] = 2Pz , [Pz , P± ] = ±P± .

(3.23)

Commutation of the operator P+ with the Hamiltonian (3.17), (3.19) under the condition (3.20) results in [Hˆ d , P+ ] = ∑(Δ + 2Ei )a†ci a†vi − ∑i j|vcv |kka†ci a†v j i

(3.24)

i jk

One may introduce the operator Pn for a given degenerate shell p = n, l with −n ≤ l ≤ n and N0 = 2n − 1. These operators still obey the algebra (3.23). Then within a (n) subspace of given shell Hˆ , the commutator d

(n)

(n)

[Hˆ d , Pn+ ] = EX Pn+

(3.25)

determines an equation of motion for the exciton creation operators with given radial quantum number n and binding energy (n)

EX = Δ + 2En −

1 (n) ll|vcv |l l . 2n − 1 ∑ ll

(3.26)

Thus, we see that the low-energy spectrum of self-assembled quantum dots possesses surplus SU(2) symmetry in addition to other dynamical symmetries of the Fock-Darwin atom discussed in sections 2.5 and 3.3 due to the electron-hole symmetry of its states. Using the bosonic operators (2.88) one may construct coherent many-exciton states (2.89). In the external magnetic ﬁeld some of the many-exciton levels belonging to the shells n and n + 1 may intersect due to the Larmor shift similarly to electronic levels (Fig. 3.9). The last term in Eq. (3.24) is responsible for hybridization of such many-exciton states. Electrons from the wetting layer may be injected in a self-assembled quantum dot, and charged excitonic states with two electrons and one hole (trions) form in the dot as a result of electron tunneling [417]. Two-dimensional electrons from the wetting layer play part of a reservoir B for the self-assembled quantum dot, and cotunneling processes between the wetting layer and the dot pave the way to many-

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3 NANOSTRUCTURES AS ARTIFICIAL ATOMS AND MOLECULES

electron phenomena of the Kondo type [146]. All these phenomena will be discussed in detail in Chapter 6.

3.5 Complex quantum dots In the three previous sections we analyzed dynamical symmetries of individual quantum dots with discrete electronic spectra, which liken them to few-electron atoms. The next logical step is the extension of this approach to molecule-like structures. For example, a pair of coupled quantum dots usually called double dot (DQD) remind a molecular dimer, triple quantum dot (TQD) may be considered as a molecular trimer, etc. These complex quantum dots (CQD) also may be incorporated into electric circuit, and the number of electrons in CQD as a whole and in each of its constituents may be varied by means of the gate electrodes applied to these nanoobjects. Besides, the traps may have different radii, and therefore different values of Coulomb blockade parameters; the electron afﬁnity may be regulated by means of gate voltage changing the relative depth of the discrete levels in adjacent dots; the heights of the inter-dot tunnel barrier which determines the covalency is also variable. Playing with all these parameters, the whole zoo of artiﬁcial molecules and molecular ions consisting of identical ”atoms” (e.g. H2 -like or He2 -like dimers) or complexes containing several components imitating, e.g., CH dimers of CH2 trimers, etc may be fabricated and studied.

s W

V l

r

W d (a)

(b)

(c)

(d)

Fig. 3.13 Serial (a) and parallel conﬁgurations of double quantum dots.

We are interested pre-eminently in the symmetry properties of quantum dots. From this point of view the novel features introduced in the theory by extra complexities stem from spatial molecular symmetry, e.g. axial symmetry of dimer, tri-

3.5 Complex quantum dots

71

angular symmetries of trimers and so forth. Arrangement of CQD in a circuit is also meaningful. Basic conﬁgurations of double and triple quantum dots in tunnel contact with two or three electrodes are shown in Figs 3.13 - 3.15. Discussion of various experimental and theoretical approaches to these nanoobject may be found, e.g., in Refs. [12, 13, 23, 48, 57, 75, 99, 163, 165, 176, 203, 245, 246, 247, 264, 265, 284, 287, 308, 320, 325, 332, 363, 385, 393, 409, 436].

(a)

(e)

(b)

(f)

(c)

(g)

(d)

(h)

(i)

Fig. 3.14 Chain (a-d, i) and closed loop (e-h) conﬁgurations of triple quantum dots.

Four basic conﬁgurations of DQD between two metallic leads shown in Fig. 3.13 may be classiﬁed as serial (a), parallel (c,d) and T-shaped (b ). In case of parallel geometry two-channel (d) and ”which pass” (c) geometry arrangements are possible. In the latter case an electron injected from the source to the dot has an ”option” to tunnel either through the left or through the right dot. More possibilities exist for TQD set-ups, Fig. 3.14. In addition to three above geometries (a,b,d) linear TQD may be arranged in cross geometry in the two-terminal circuit (c) and in V-shaped geometry in three-terminal circuit (i). Besides, TQD may be fabricated in a form of triangle which in particular may be isosceles or equilateral. In two-terminal case triangular TQD may be coupled via two channels with source and via one channel with drain or v.v. (e). Two more geometries are Δ -like and ∇-like geometries (f,g). Three-terminal conﬁguration as a whole may preserve the symmetry of isosceles and equilateral triangle (h). Vertical dots also may be

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3 NANOSTRUCTURES AS ARTIFICIAL ATOMS AND MOLECULES

(a)

(b)

(c)

(d)

Fig. 3.15 Serial (a), parallel (b,d) and closed loop (c) conﬁgurations of double and triple vertical quantum dots.

arranged in serial and parallel geometries [Fig. 3.15(a,b)]. In case of triple vertical quantum dot one may fabricate three-channel device [Fig. 3.15(c)]. If one of the three channels is blocked than the object behaves as a two-channel Δ -like structure, Fig. 3.14(f). If two channels are blocked, then the vertical TQD behaves similarly to planar ∇-like TQD, Fig. 3.14(g). The two-channel trimer geometry shown in Fig. 3.15(d) is isomorphous to the planar conﬁguration of Fig. 3.14(d). Even more possibilities open for quadruple quantum dots, but we conﬁne our study only with DQD and TQD because the basic manifestations of dynamical symmetries may be traced by the example of these conﬁgurations. One should note that complex quantum dots may be fabricated both in planar and in vertical design.

3.5.1 Double quantum dots Introductory notes towards the theory of DQD may be found in Section 2.4. So called Fulde molecule described there is a particular case of asymmetric DQD. We begin study of dynamical symmetries in these dimers with planar DQD. The most general form of the Hamiltonian of “Hubbard dimer” taking into account both interdot tunneling and Coulomb blockade effects is † Hˆ DQD = Hˆ d1 + Hˆ d2 + Q12 δ n1 δ n2 + ∑ ∑(Vi j d1i σ d2 j σ + H.c.). ij σ

(3.27)

3.5 Complex quantum dots

73

Here the dots are marked by the index p = 1, 2. The ﬁrst two terms in this Hamiltonian are the Hamiltonians of the individual dots deﬁned in the ﬁrst line of Eq. (3.2), Q12 is the energy of the capacitive interaction between two dots, δ n p stands for the excess electrons or holes in dot p. The last term in the Hamiltonian (3.27) describes the interdot tunneling. If the level hierarchy of type (3.3), namely Q (δ ε ,V ) W

(3.28)

takes place, If only the highest occupied levels in both dots are taken into like in the case of single quantum dots (Section 3.2), then the Hamiltonian may be easily diagonalized and rewritten via Hubbard projection operators, Hˆ DQD = ∑ EΛ X ΛΛ

(3.29)

Λ

Since in all experimental realizations of DQD only the states belonging to few adjacent charge sectors are involved, the relevant symmetries may be described for few representative occupation states with N = 1, 2, 3 and the pair of discrete states

ε pi in each dot. Then the symmetry of DQD is isomorphous to that of potential trap with two minima described in Section 2.3. DQD in the charge sector N = 1 where one electron is shared between two wells may be treated as a conventional two-level system with two orbital and two spin degrees of freedom. Its energy states are given by Eq. (2.38), and the generators of its dynamical symmetry group SU(4) may be presented as a direct product of Pauli matrices σ and τ for spin and orbital pseudospin, respectively [see Eq. (2.43) and Section 9.2.4]. This dynamical symmetry may be realizes in cotunneling processes when an electron with the given spin projection ejected from one dot to one of the two leads is replaced by another electron with the same or opposite spin projection injected to another dot from the same or different lead. The general form of the cotunneling Hamiltonian is [436] ΛΛ ΛΛ † cρ cρ Hˆ ex = ∑ Jρρ X

(3.30)

where the operators X ΛΛ describe transitions between the electron states in DQD, and the index ρ stands for combinations of lead, orbital and spin indices of band electron and hole states involved in cotunneling processes.

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3 NANOSTRUCTURES AS ARTIFICIAL ATOMS AND MOLECULES

Operators Bρρ = c†ρ cρ also describe transitions between the states within a supermultiplet of electron-hole pair states involving two spin projections and several orbital indices. Therefore they obey some SU(2n) symmetry depending on the number of orbitals n. The latter quantity is predetermined by the geometry of a device and the energy scale of the lead-dot interaction. Leaving discussion of interaction scale for subsequent chapters, we now discuss in general terms the dependence of the group invariants entering the effective Hamiltonian (3.30) on the index n. In case when the DQD possesses SU(4) dynamical symmetry, one should discriminate between three different situations: 2n < 4, 2n = 4 and 2n > 4. The value of the index n is predetermined by the lead-dot conﬁguration. In the serial, T-shape and ”which pass” conﬁgurations shown in Fig. 3.13(a-c) DQD is coupled with one orbital from each lead, whereas in the parallel geometry [Fig. 3.13(d)] two orbitals from each lead are involved in tunneling. To derive the effective cotunneling Hamiltonian let us ﬁrst rewrite the tunneling Hamiltonian Hˆ tun in variables incorporating explicitly the symmetry of a device. We begin with highly symmetric devices possessing both the bottom-top and the left-right symmetry. This condition implies that two dots are identical and all tunneling channels are also identical, so that the single parameter W describes all tunneling processes. Then two √ √ † † combinations of lead and dot operators, (c†akσ ± c†bkσ )/ 2 and (d1k σ ± d2kσ )/ 2 appear in the tunneling Hamiltonian. Here the label a, b stand for the source/drain electrodes and the label 1, 2 enumerates ﬁrst/second tunneling channel. Let us denote the even and odd states relative to the bottom-top reﬂection by the indices e, o, and mark the even and odd states relative to the left-right reﬂection by the indices g, u. In these variables the tunneling Hamiltonian is written as (a) Hˆ tun = W ∑[c†ekσ deσ + c†okσ doσ + H.c.] kσ

(b) Hˆ tun = ∑[c†ekσ (Wg dgσ + Wu duσ ) + H.c.] kσ

(c) Hˆ tun

= 2W ∑[c†ekσ dgσ + H.c.] kσ

(d) Hˆ tun = W ∑[c†gkσ dgσ + c†ukσ duσ + H.c.]

(3.31)

kσ

It follows from (3.31) that the lead-dot tunneling in conﬁgurations (a) and (d) involves the full discrete (”orbital”) symmetry, in conﬁguration (b) the dot parity is not conserved and in conﬁguration (c) the odd states are completely discarded. In

3.5 Complex quantum dots

75

case (b) the tunneling parameters are different for even and odd dot states because only dot 1 contributes the matrix elements. One may say that in the sector N = 1 conﬁgurations (a) and (d) the effective number of dot and lead orbitals is the same: nd = nl = 2, in conﬁguration (b) nd = 2, nl = 1, and in conﬁguration (c) nd = nl = 1, so that in this case the SU(4) symmetry is reduced to SU(2). In no case the regime with nl > nd can be realized in this charge sector. Then using the Hubbard representation [169] d †pσ = X σ p,0 + σ X 2,σ p

(3.32)

for dot states with parity p, one may derive the cotunneling (indirect exchange) Hamiltonian by means of the Schrieffer-Wolff transformation for all four tunnel Hamiltonians (3.31): (a,d) σσ † Hˆ ex = J ∑ ∑ ∑ X pp c p k σ c pkσ pp kk σ σ

(b) Hˆ ex =

σσ † ck σ ckσ ∑ ∑ ∑ Jpp Xpp pp kk σ σ

(c) Hˆ ex (a,d)

The Hamiltonian Hˆ ex

σσ † = ∑ ∑ Jg Xgg cgk σ cgkσ

(3.33)

kk σ σ

may be rewritten in terms of Pauli matrices σ and τ of

SU(4) group [see Section 9.2.4, Eq. (9.45)]. In these conﬁgurations the pseudospin matrix describes transitions between even and odd channels. Pseudospin operators may be incorporated in the Hamiltonian in several ways. In particular, the interplay between spin and pseudospin variables may be described by means of the Kugel – Khomskii exchange originally introduced in a context of spin and orbital ordering in ternary oxides of transition metals [242]. Neglecting the difference between the components of the matrix J of exchange parameters, J pp = J, it is convenient to write it in the fermionic representation [99] (see Section 9.3): (a,d) Hˆ ex = 4J(F · f)

(3.34)

where the 4-vectors are deﬁned via the generators λˆ (9.46) of the SU(4) group: F=

1 1 fl† λˆ lm fm , f = ∑ ∑ c†kl λˆ lm ckm 2∑ 2 k lm lm

In the general case this Hamiltonian may be presented via spin operators

(3.35)

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3 NANOSTRUCTURES AS ARTIFICIAL ATOMS AND MOLECULES

S pp = |pσ σ p σ |, s pp = ∑ |kpσ σ k p σ |

(3.36)

(a,d) Hˆ ex = J ∑ Sˆ pp · sˆp p .

(3.37)

kk

pp

The same representation for the case (b) reads (b) Hˆ ex =

∑ Jpp Sˆpp · s.ˆ

(3.38)

pp ,

After transforming the quadratic form containing the matrix elements of exchange Hamiltonian and the localized spin in the r.h.s. of Eq. (3.38) to the principal axes, this Hamiltonian turns into (b) Hˆ ex = Jα Sα · s + Jβ Sβ · s.

(3.39)

so that the effective symmetry of the DQD in this case is SU(2) ⊗ SU(2). Finally the cotunneling in conﬁguration (c) is described by the conventional spin Hamiltonian (c) (3.40) Hˆ ex = Jg Sg · s [cf. Eq. (3.11)]. In this case, the lead electrons with effective SU(2) symmetry impose this symmetry onto two-orbital DQD, so that the only manifestation of orbital degrees of freedom in cotunneling processes is a numerical factor 4 in the effective exchange constant reﬂecting the orbital degeneracy of the tunneling Hamiltonian (c) Htun . The above derivation procedure may be generalized for the case of asymmetric DQD. In that case the bonding and antibonding TLS orbitals form the basis of dot electron states instead of even and odd electron states, and similar conbination of source and drain electrons enter the tunnel Hamiltonian after appropriate rotation in the Fock space [99, 180, 345]. Next we turn to the case of double occupation of a DQD, N = 2. The dynamical symmetry of a doubly occupied DQD may be mapped on the corresponding symmetry of a TLS investigated in Section 2.3. This symmetry is predetermined by the structure of the multiplet EΛ in the Hamiltonian Hˆ DQD (3.29), which in the general case consists of three spin singlets and spin triplet. Thus, the basic dynamical symmetry group for doubly occupied DQD is SO(6) [see Eq. (2.55)]. The relative position of these levels may be varied by changing the parameters of the DQD. Various modiﬁcations of the one- and two-electron energy levels are

3.5 Complex quantum dots

77

ε+ Q

ε 2+ Q

ε1

εF

δ12

ε

ε2

Ee

E1

Eo

Q −δ 12

Q

2ε (a) symmetric

T S

ε + ε2

ε 1 + Q1

ε 1 + Q1

ε1+ Q

δ 12

1

E2

ε2 + Q2 δ 21

ε1 Q − Q −δ 1

2

(b) biased

ε2 + Q2 ε2

E1

21

E2 δ +Q 21

T S

ε1

ε2

2

ε + ε2 1

(c) asymmetric

S δ −Q

T S

21

2ε2+ Q2

2

T S (d) charge polarized

Fig. 3.16 Upper panels: Single electron levels for symmetric (a), biased (b), asymmetric (c) and charge polarized (d) DQD under strong Coulomb blockade Q j . Low-energy inter-well tunneling processes are shown by dashed arrows. Lower panels: Splitting of two-electron levels due to interwell tunneling in all four cases.

presented in Fig. 3.16. The most symmetric conﬁguration shown in panel (a) is realized for a fully symmetric DQD where the left and right wells are identical. The parity of the electronic states is conserved in this conﬁguration and the SO(6) multiplet is split into two groups of levels separated by the charge transfer gap ∼ Q. In the lower part of this multiplet one ﬁnds the familiar spin singlet (even state of two electrons located in the left and right wells) and the spin triplet (odd state of the same electrons). Two high-energy states are the even and odd charge transfer excitons, i.e. the states with two electron either in even or in odd combination of polar two-electron states where both electrons are located in the same quantum well. It should be noted that two even states form bonding/antibonding pair, whereas two odd states remain non-bonding. In lowest order in W , the energy levels are given by the following equations: ES = 2ε − Δ ES ET = 2 ε EEo = 2ε + Q EEe = 2ε + Q + ΔES.

(3.41)

Here ΔES = 2W 2 /Q. Within the energy scale E Q the full dynamical symmetry is reduced from SO(6) to SO(4), so that Hˆ DQD is given by Eq. (2.51).

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3 NANOSTRUCTURES AS ARTIFICIAL ATOMS AND MOLECULES

Due to the non-trivial dynamical symmetry activated by dot-lead cotunneling, the effective exchange Hamiltonian contains two local vectors [203]. One of them is the usual vector S for spin 1, and another is the vector R1 containing Hubbard operators responsible for singlet-triplet transitions (see Eq. (2.46) for its deﬁnition, where the index G should be substituted for S. Hˆ ex = J0 S · s + J1R1 · s

(3.42)

If the gate voltage vg is applied to one of the two dots (say, to the right one labeled by the index p = 2) the even/odd symmetry is broken because the electronic levels in the right dot are shifted downwards [see Fig. 3.16(b)]. Then the energy difference

δ12 = ε1 − ε2 is the measure of asymmetry. In such a conﬁguration the labels 1, 2 should be used for classiﬁcation of electronic states instead of e, o and two charge transfer singlet excitons may be classiﬁed as E1 and E2 , respectively. One of these excitons, namely E2 , is relatively soft: at large enough δ12 but still in perturbative interdot tunneling regime its energy EE2 becomes comparable with ET . The second exciton is quenched, so that the multiplet (2)

ES = ε1 + ε2 − ΔES ET = ε1 + ε2

(2)

EE2 = 2ε2 + Q + ΔES

(3.43) (2)

determines the SO(5) dynamical symmetry of the DQD. Here ΔES = 2W 2 /(Q − (2)

δ12 ). The energy gap between two excited states is ΔET = Q + ΔES − δ12 . Another reason for reﬂection symmetry violation is the difference in the radii r p of two dots. Due to such difference the Coulomb blockade parameters are different as well. In particular, if r2 r1 then Q2 Q1 . The energy level ordering in such DQD is shown in Fig. 3.16(c). The energy of the soft charge transfer exciton is (2)

EE2 = 2ε2 + Q2 + ΔES .

(3.44)

(2)

The energy shift ΔES = 2W 2 /(δ12 + Q2 ). From the point of view of dynamical symmetry this regime is similar to that of Fig. 3.16(b). The form of the effective cotunneling Hamiltonian in these two cases is dictated by the generating algebra o(5) discussed in Section 2.3. This algebra contains three vectors S, R1 , R2 and one scalar A3 deﬁned in Eq. (2.46) (with corresponding change of indices):

3.5 Complex quantum dots

79

Hˆ ex = J0 S · s + J1 R1 · s + J2 R2 · s.

(3.45)

The Hamiltonian (3.45) was derived in Ref. [245]. Although the scalar A3 does not enter the Hamiltonian, it inﬂuences the dynamical properties of the DQD implicitly due to kinematic constraint imposed by the Casimir operators (2.48). We will discuss mechanisms of such indirect inﬂuence in Chapter 7. Further increase of the gate voltage applied to the right dot leads eventually to charge polarization of DQD where both electrons reside in this dot and only virtual tunneling transitions to the left dot should be taken into account [see Fig. 3.16(d)]. In this extreme case the order of singlet states is reversed, so that the former charge transfer exciton becomes the ground state level of DQD. (2)

EE2 = 2ε2 + Q2 + ΔES ET = ε1 + ε2 (2)

ES = ε1 + ε2 − ΔES

(3.46)

(2)

In this case, the energy shift is ΔES = 2W 2 /(|δ12 | − Q2). Reduction of the scale E of the dot-lead interaction results in the corresponding reduction of dynamical symmetry from SO(5) to SO(4) in cases (b,c) and in its complete quenching in the extreme case (d).

E E2

E

E

S

ET ES

EE2 0

− δ 12

Fig. 3.17 Evolution of the levels ES , ET and EE2 in DQD as a function of the single energy level difference δ12 (see text for the detailed explanation).

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3 NANOSTRUCTURES AS ARTIFICIAL ATOMS AND MOLECULES

Figure 3.17 illustrates the evolution of effective dynamical symmetry of supermultiplet of doubly occupied states in asymmetric DQD as a function of a control parameter, namely the gate voltage converted in the energy difference δ12 . The curves are drown in a reference frame ε1 + ε2 = const. Evolution of δ12 in this frame is shown by dashed line. In the limits of strongly negative or strongly positive δ12 the perturbative equations (3.44),(3.46) are valid. In the degeneracy point δ12 = Q2 the energy splitting for two spin singlet states is 4W . From this avoided level crossing picture we conclude that within the energy scale E W the evolution of dynamical symmetry as a function of diminishing δ12 is SO(4) → SO(5) → U(1), whereas in a smaller scale E ∼ Δ T S the evolution is simply quenching of spin degrees of freedom SO(4) → U(1). Mechanisms of S/T crossover within the SO(4) multiplet are also available. We will discuss them in the next chapter.

3.5.2 Triple quantum dots Among many conﬁgurations of TQD with different occupation numbers we choose several combinations which promise interesting effects in tunneling spectra for detailed consideration. We begin with linear trimers which possess the reﬂection symmetry s − c − s. The central dot (c) may differ from the side dots (s) by its radius so that the Coulomb blockade Qc = Qs . In conﬁgurations presented in Fig. 3.14(a,c,d) this axial symmetry is supported by the geometry of dot-lead links: in serial (a) and parallel (d) arrangements the central dot is disconnected from the source and drain, whereas in cross geometry (c) the side dots are disconnected from the leads. Like in DQD the relative position of the single electron levels εc and εs may be controlled by the gate voltage applied, say, to the dot c. All possible variants of few-electron mutiplets in TQD may be obtained by adding the discrete levels belonging to a third valley to the two-well conﬁgurations presented in Fig. 3.16. The energy spectra and the corresponding dynamical symmetries for centrosymmetic three-well conﬁgurations may be obtained from those shown in Fig. 3.16(b,c,d). These level schemes are presented in Fig. 3.18. Positive values of the level asymmetry parameter δsc = εs − εc > 0 correspond to panels (a), (c), (e) and negative value δsc < 0 gives the level orders shown in panels (b) and (d). In the panels (c,d,e) the condition Qc = Qs is realized. It is seen from these schemes that the orbital degeneracy is an integral feature of the electronic spectra of TQD: there are two side states which form even and odd

3.5 Complex quantum dots s

c

81

s

ε s+ Q

ε c+ Q

ε c+ Q εs

ε s+ Q εc

εc

εs

ε s + Qs εs ε c + Qc εc (b)

(a)

(c)

ε c+ Qc εs + Qs εs

εs + Q ε c+ Q εc

εs

εc (d)

(e)

Fig. 3.18 Energy states for TQD . Charge transfer transitions due to inter-well tunneling for occupation N = 2 are shown by dashed arrows (see the text for further discussion).

√ orbitals |sb,a = (|s1 ± |s2)/ 2. In the charge sector N = 1 this degeneracy is removed due to interdot tunneling, because only the bonding orbital |sb hybridizes with the central state |c. The antibonding (odd) combination remains intact. Thus, the basis states for centrosymmetric trimers are [247] ⎛ ⎞ ⎛ ⎞⎛ ⎞ d+ c cos θ sin θ 0 ⎝ d− ⎠ = ⎝ − sin θ cos θ 0 ⎠ ⎝ sb ⎠ 0 0 1 da sa

(3.47)

[θ is the mixing angle for TLS deﬁned in Eq.(2.40)]. The corresponding three-level system is deﬁned by the equations

1 2 2 ε± = (εc + εs ) ∓ δcs + 4W 2 εa = εs . (3.48) Evolution of this level system as a function of δsc at ﬁxed position of εs is shown in Fig. 3.19. The basis (3.47) is valid also for isosceles triangular quantum dots (see below). Only the modes |d± contribute to the lead-dot tunneling Hamiltonian in centrosymmetric geometries Fig. 3.14(a,c,d)]. Like in the case of DQD we denote them as (de , do ) in conﬁguration (a) and as (dg , du ) in conﬁgurations (c),(d). Then the tunneling Hamiltonian acquires the same form as those for the cases (a),(c),(d) in

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3 NANOSTRUCTURES AS ARTIFICIAL ATOMS AND MOLECULES

Eq. (3.31) and the Schrieffer-Wolff transformation in charge sector N = 1 generates the same cotunneling Hamiltonians (3.34),(3.39), (3.40) as in DQD. The non-centrosymmetric conﬁguration of Fig. 3.14(b) should be considered separately. In this case, two dots are attached from one side to the left dot which is coupled with the source and drain leads. Let these side dots be identical and the inequality W /|εl − εs | 1 be valid. Due to the absence of mirror symmetry all three states are involved in tunneling due to admixture of side states to the left one. Three levels form a singlet (shifted level εl ) and quasi doublet (bonding and antibonding combination of two side levels): εl ≈ εl +

W2 εls

W2 εls W2 εu ≈ εs + W − εls

εg ≈ εs − W −

(3.49)

(εls = εl − εs ). Depending on the sign of εls , either a singlet (εls < 0) or a quasi doublet (εls > 0) form a ground state of this TQD. In this conﬁguration all three states of an electron located in TQD may be involved in the lead-dot tunneling processes. Varying the parameter εls from negative to positive values one my control the crossover from purely spin cotunneling (3.40) to spin tunneling enhanced by the orbital quasidegeneracy (3.39). A TQD in a form of triangle may have additional discrete symmetry provided this triangle is equilateral. Equilateral means that all three dots are entirely identical and all three tunneling channels are identical as well, W jl = W. A perfect triangular TQD is in fact a potential well with three equivalent minima and equal barriers between the wells. The electron states form a basis for two irreducible representations Γ = A, E of the point group C3v or the permutation group P3 which is isomorphous to C3v . Let us enumerate the wells as j = 1, 2, 3 in a clockwise direction and introduce creation operators for an electron in equilateral TQD as the corresponding linear combinations of the dot operators di† : √ † † † † dA, σ = (d1σ + d2σ + d3σ )/ 3 ,

√ dE† ± ,σ = (d1†σ + e±2iϕ d2†σ + e±iϕ d3†σ )/ 3 Here the phase ϕ = 2π /3. The eigenlevels corresponding to this basis are

(3.50)

3.5 Complex quantum dots

83

EA = ε + 2W,

EE = ε − W.

(3.51)

Usually the tunneling parameter W < 0, so that the ground state of equilateral TQD with N = 1 is an orbital singlet/spin doublet |A, σ and the orbital/spin doublet |E± , σ is an excited state. Looking at the TQD setups which contain a triangle as an element of their geometry [Fig. 3.14(e-h)], we see that the equilateral triangular TQD retains the C3v symmetry only in the three terminal geometry (h). If one of the inter-dot tunneling channels is broken like in the V-shape three-terminal geometry (i), the symmetry of the device is that of isosceles triangle. In all other cases (e,f,g) the point symmetry of the triangular TQD reduces from C3v to C2d (or from P3 to P2 ) even if the dot remains equilateral. The two terminal device (e) possesses only the left-right reﬂection symmetry and conﬁgurations (f) and (g) with one and two side dots retain only the bottom-top reﬂection symmetry. One should analyze the form of the effective cotunneling Hamiltonians for all these cases in the same way as it was done above for centrosymmetric linear trimers using the basis (3.47) for representation of electronic states in triangular TQD. Starting with the most symmetric conﬁguration (h), we note that the electronic states in the three terminals should be classiﬁed in the same way as the electronic states in TQD (3.50): √ c†A,kσ = c†1kσ + c†2kσ + c†3kσ / 3 , √ c†E± ,kσ = c†1kσ + e±2iϕ c†2kσ + e±iϕ c†3kσ / 3 .

(3.52)

The point symmetry of the equilateral triangle is conserved also in the tunneling Hamiltonian: (h) (3.53) Hˆ tun = W ∑ ∑(cΓ† kσ dΓ σ + H.c.). kσ Γ

In actual physical situations [75, 246] either orbital singlet state EA or orbital doublet EE is involved in cotunneling through equilateral TQD. In the former case the general cotunneling Hamiltonian (3.30) may be reduced to Hˆ ex = JE S · sE+ E+ + S · sE− E− + JA S · sAA

(3.54)

where S ≡ SA = |A, σ σ A, σ |. The spin operator for lead electrons is diagonal in orbital indices because the irreducible basis (3.52) is used for this subsystem. The Hamiltonian (3.54) belongs to a new class of multichannel cotunneling Hamiltoni-

84

3 NANOSTRUCTURES AS ARTIFICIAL ATOMS AND MOLECULES

ans where the order of symmetry group for subsystem B exceeds that of subsystem S . In this speciﬁc case the number of channels is 3. If the order of the levels in the orbital triplet (3.51) is inverted, then the ground state of the TQD is the orbital doublet |Γ = |E± . Such an inversion may be realized by means of external magnetic ﬁeld applied normally to the plane of the triangle [246]. We come in that way to the case where the relevant degrees of freedom may be presented as spin 1/2 plus pseudospin 1/2 and the actual dynamical symmetry is SU(4). As was mentioned above, the cotunneling Hamiltonian in this case acquires the Kugel-Khomskii form (3.34). One may rewrite this Hamiltonian in another way by constructing the 15 generators of the SU(4) group as four spin vectors and one pseudospin vector [246]. In accordance with Eqs. (3.36), the spin vectors are SEα Eβ with α , β = ± and the components of the pseudospin vector T are T+= Tz =

∑ |E+ , σ E− , σ |, σ

T − = ∑ |E− , σ E+ , σ |,

(3.55)

σ

1 (|E+ , σ E+ , σ | − |E−, σ E− , σ |) . 2∑ σ

Their counterparts for lead electrons are

τ + = ∑ c†E+ ,kσ cE− ,kσ , kσ

τ − = ∑ c†E− ,kσ cE+ ,kσ , kσ

1 τz = ∑(c†E+ ,kσ cE+ ,kσ − c†E−,kσ cE− k,σ ). 2 kσ

(3.56)

Then the cotunneling Hamiltonian acquires the form Hˆ ex =

∑

Γ Γ =E±

JΓ Γ SΓ Γ · sΓ Γ + Jτ T · τ

(3.57)

This form of SU(4) Hamiltonian may be particularly useful for description of processes, where the pseudospin cotunneling without spin ﬂips is dominant. Continuing study of the charge sector N = 1, we turn to the V-shape geometry Fig. 3.14(i) where the three leads are equivalent, but the TQD possesses only the left-right reﬂection symmetry due to suppression of tunneling between the 1st and 2nd dots. Referring to the case of linear trimer considered above, we denote the dots 1,2 as side dots. Then the single electron levels in this trimer are given by Eq. (3.48) and the orbital degeneracy is removed even if εc = εs [the mixing angle θ = π /4 in Eq. (3.47)]. The three-channel tunneling Hamiltonian has the form

3.5 Complex quantum dots

85

(i) Hˆ tun = V ∑ c†bkσ dsbσ + c†akσ dsaσ + c†ckσ dcσ

(3.58)

kσ

√ Here the states |cb,a = (|c1 ± |c2)/ 2 are deﬁned in analogy with the states |sb,a in Eqs. (3.47), (3.48) and the label 3 changed for c. Unlike the case of two-terminal trimer Fig. 3.14(d) all 3 dot states are involved in cotunneling in three-terminal Vshape geometry and the effective exchange Hamiltonian has the form (3.37) with p, p = a, b, c when the energy scale E 4W . In this scale cotunneling charge transfer between all three leads is allowed.

ε−

E

εc εa 4W

ε+

Fig. 3.19 Upper panel: energy levels ε± , εa,c (3.48) (bold and dashed lines, respectively) as a function of level asymmetry parameter δsc ; lower panel: mixing angle θ as a function of δsc .

− δ sc cos 2 θ 1

0

1/2 0

If the orbital degeneracy is removed due to gate voltage applied to the central dot 3, the evolution of the three-level system follows Eq. (3.48) and Fig. 3.19. Within the same energy scale E 4W at negative δsc and |δsc | W the orbital doublet (EΓ =

εa , ε+ ) is effective in the exchange Hamiltonian, and we return to the case of SU(4) symmetry (3.57). This Hamiltonian describes two equivalent cotunneling channels (1-3) and (2-3). In the opposite limit of positive δsc and |δsc | W the orbital degrees of freedom are frozen (only even combination of the two channels is effective), and the electron cotunneling is described by the usual SU(2) Hamiltonian (3.11). Thus, the evolution of the energy spectrum controlled by the parameter δsc (Fig. 3.19) results in the crossover of dynamical symmetries SU(4) → SU(6) → SU(2). At

86

3 NANOSTRUCTURES AS ARTIFICIAL ATOMS AND MOLECULES

E 4W 2 /|δsc | dynamical symmetries are frozen out, and electron cotunneling is determined by the spin symmetry SU(2). The discrete symmetry C2d of the conﬁgurations (f) and (g) in Fig. 3.14 is characterized by a single reﬂection axis perpendicular to the tunneling direction. The symmetry axis is common for dot and lead electrons. These two conﬁgurations may be treated as generalization of conﬁgurations Fig. 3.13(b) and Fig. 3.14(a), respectively, for the case of triangular quantum dots. The source and drain states are clas√ siﬁed as |kσ b,a = (|skσ ± |dkσ )/ 2. Following Eq. (3.31) and using the basis (3.47), we conclude that (f) Htun = W ∑ sin θ c†bkσ d+σ + cos θ c†bkσ d−σ + c†akσ daσ + H.c. kσ

(g) Htun

= W ∑ cos θ c†bkσ d+σ − sin θ c†bkσ d−σ + H.c.

(3.59)

kσ

Like in the previous case, the symmetry SU(6) of isosceles triangular quantum dot may be reduced to SU(4) or SU(2) depending on the value of the control parameter δsc in accordance with (Fig. 3.19). It may be concluded from Eqs. (3.59), that within the energy scale E 4W in case (f) one deals with the two-channel Hamiltonian with nl = 2, and nd = 2, 3, 1 when varying δsc . In case (g) nl = 1, the state εa is eliminated from cotunneling processes, so that nd = 1, 2, 1 within the same energy scale. All these regimes may be described by means of the effective Hamiltonians derived above. Next we turn to description of effective Hamiltonians which can be realized for TQD in the charge sector N = 2. Doubly occupied trimers are characterized by the same orbital symmetry as singly occupied ones, but this symmetry is combined with the spin symmetry SO(n) characteristic for dots with even occupation. In the centrosymmetric geometries of Fig. 3.14(a,c,d) we ﬁnd the familiar mechanism of double orbital degeneracy of side states removed due to tunnel contact with the central dot. Now two of the three orbital states |s, σ b , |s, σ a , |c, σ , may be occupied. Interdot tunneling in the presence of Coulomb blockade generates indirect exchange which results in spin singlet/spin triplet splitting. The corresponding virtual transitions responsible for indirect exchange are shown by the dashed arrows in Fig. 3.18. The even side state forms two singlets with the central state, whereas the two-electron triplet state is formed as a product of |s, σ a and |c, σ . Thus, we arrive at the same structure of supermultiplet as in DQD [see Eqs. (3.43), (3.44), (3.46) where the levels ε1 and ε2 should be changed for εs and εc in accordance with

3.5 Complex quantum dots

87

the level scheme of Fig. 3.18]. Relative position of the levels as a function of control parameters also reproduces that for DQD (see Fig. 3.17). Like in the single electron charge sector, the Hamiltonian Hˆ ex acquires new features when the channel index (e, o) or (g, u) cannot be eliminated by canonical transformation of lead/dot coordinates. This is, for example, the case of parallel geometry Fig. 3.14(d), where either two side dots or the central dot is subject to strong Coulomb blockade (level schemes Fig. 3.18(d) and Fig. 3.18(e), respectively). Similarly to DQD with N = 2, the SO(5) supermultiplet is characterized by the vectors S, R1 , R2 , but some of these vectors may acquire an orbital index. In case of Fig. 3.18(e) the triplet and singlet state is formed by two electrons in accordance with the conventional spin addition scheme using Clebsch – Gordan Sμ coefﬁcients Cσ σ : |T μ =

1μ

∑ Cσ σ |c, σ ; sa , σ ,

σσ

|S = ∑ Cσ00σ¯ |c, σ ; sb , σ¯

(3.60)

σ

The even charge-transfer exciton Eg is separated by the energy gap from the lowenergy quartet like in Fig. 3.16(c), so we suppose that this degree of freedom is quenched. The lead-dot electron tunneling is carried out via the even and odd side electron state similarly to the N = 1 case (3.31). Both parity and spin are conserved in these processes: (d) Hˆ tun = W ∑[c†gkσ dbσ + c†ukσ daσ + H.c.]

(3.61)

kσ

As a result of the Schrieffer-Wolff transformation, which intermixes states with different parities we obtain the effective Hamiltonian Hˆ ex = Ju Su su + Jug

∑

[ fu†σ fgσ c†k gσ ckuσ + H.c.].

(3.62)

kσ ,k σ ,

In this speciﬁc case spin 1 operator Su preserves the odd parity, whether the tripletsinglet transitions intermix odd and even states. As a result, it is impossible to introduce the operator R1 from the set (2.46) and construct the scalar products like R1 · sug . Instead one has to resort to fermionization procedure for spin 1/2 electrons in the orbital states |su, σ , |sg, σ , namely to introduce operator Sug = fu†σ σ fgσ only for an electron in the side position (see Section 9.3 for mathematical details). Similar approach is apt to the case of strong blockade illustrated by Fig. 3.18(a). Next we turn to the conﬁguration shown in Fig. 3.18(d) with strong Coulomb blockade in the side dots and weak blockade in the central dot, Qs Qc and δsc < 0.

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3 NANOSTRUCTURES AS ARTIFICIAL ATOMS AND MOLECULES

In this case, both electrons are located in side dots and the indirect exchange between them is possible due to virtual tunneling in the central dot in the same way as in Zener-type double exchange [442] between localized spins via intermediate states in the dot c. This exchange favors antiparallel orientation of the side spins, so the even singlet is the ground state of the isolated TQD and the odd triplet is the low-energy excitation. This means that the lead-dot tunneling Hamiltonian has the form (3.61), cotunneling processes intermix even singlet and odd triplet states, thus activating the SO(4) symmetry of spin multiplet and Eq. (3.62) is valid for the cotunneling exchange Hamiltonian. The level conﬁguration shown in Fig. 3.18(d) may evolve into that of Fig. 3.18(c) under the change of the gate voltage applied to the central dot. In the latter conﬁguration the ground state of TQD is a spin singlet with two electrons in the central dot |c ↑, c ↓ with small admixture of conﬁguration |S from Eq. (3.60). The energy of this state is 2W 2 . (3.63) ESc = 2εc + Qc − δsc The charge transfer excitons form the singlet/triplet pair separated by the gap δsc . This is the case of U(1) symmetry corresponding to the extreme right limit of the crossover diagram shown in Fig. 3.17. Double electron occupation of equilateral triangular TQD should be described in terms of the full permutation group P3 characterized by three representations A1 , A2 , E or three Young tableaux [3], [13 ], [21]. A1,2 are one-dimensional representations with symmetric and antisymmetric basis functions, respectively, and E is the two-dimensional representation with basis functions, which are symmetric or antisymmetric relative to mirror reﬂections. In the singly occupied equilateral TQD considered above the antisymmetric state A2 cannot be constructed due to the well known frustration property of triangular cells, but in the sector N = 2 this state enters the spin multiplet which now consists of twelve states |Γ = |SA1 , |SE, |TA2 , T E.

(3.64)

In the limiting case |W | Q the set of eigenvalues for the eigenstates (3.64) is 8W 2 , ET E = ε2 + W, Q 2W 2 , ETA2 = ε2 − 2W. = ε2 − W − Q

ESA1 = ε2 + 2W − ESE

(3.65)

3.5 Complex quantum dots

89

Like in the singly occupied triangular TQD, the level ordering is sensitive to the sign of the tunnel integral W . It is seen from (3.65) that in case of W < 0, the singlet ESA1 is the ground state of triangular TQD, and the lowest excitation is a triplet ET E . If the tunnel integral is positive, W > 0 the level ordering is reverse. In that case the ground state is a triplet ETA2 and the lowest excitation is a doubly degenerate singlet ESE . This reversibility provides us additional possibilities of inducing S/T crossover in quantum tunneling (see below). But in all practical cases the maximum dynamical symmetry SO(12) reduces to the familiar SO(4) symmetry of singlet-triplet pair when the energy scale is E W (see, e.g., [75]). Analysis of cotunneling through doubly occupied isosceles triangular TQD [Fig. 3.14(e,f,g,i)] follows the lines of above study of centrosymmetric TQD, and the relevant dynamical symmetries arise as a result of the interplay between even spin singlet and odd spin triplet states (3.60) and the evolution of this spectrum shown in Fig. 3.17 resulting in crossover SO(4) → U(1) due to admixture of charge transfer singlet exciton. We leave the discussion of this regime till Chapter 4, where the speciﬁc physical manifestations of such cotunneling will be discussed. Here we focus on an unusual cotunneling regime which involves only spin singlet states and orbital doublet states. This regime may be realized in a two-channel trimer geometries shown in Figs. 3.14(d) and 3.15(d). Let us consider this geometry in the situation illustrated by Fig. 3.18(e). Occupation N = 2 means that one of the two electrons always resides in the central dot whereas the second one is poised between the two side dots. Virtual tunneling of an electron from the central dot to the side dots [dashed lines in Fig. 3.18(e)] favors antiparallel spin orientation in accordance with the level crossing scheme of Fig. 3.17 (see its left half), so that in case when the double occupation of the central dot is completely blocked (Qc → ∞), the spin multiplet consists of doubly degenerate spin singlet and doubly degenerate spin triplet. At ﬁnite but large Qc the orbital degeneracy is removed and one comes to the SO(8) multiplet formed by odd and even singlets and triplets Su,g and Tu,g . Due to the contact with the leads all these levels are renormalized and at certain conditions the pair of singlet levels falls well below the pair of triplet levels. The two-channel tunneling Hamiltonian Hˆ tun = W ∑(c†le,kσ dlsσ + c†re,kσ drsσ + H.c.)

(3.66)

kσ

generates the effective indirect exchange Hamiltonian which intermixes all channels and all states Su,g , Tu,g , but within the extremely low energy scale E < ΔT S only the

90

3 NANOSTRUCTURES AS ARTIFICIAL ATOMS AND MOLECULES

singlet states of the TQD contribute to the cotunneling processes: Hˆ ex = J ∑(X Sl,Sl c†le,kσ cle,k σ + X Sr,Sr c†re,kσ cre,k σ σ Sl,Sr † X cre,kσ cle,k σ

+

+ X Sr,Sl c†le,kσ cre,k σ )

(3.67)

were J ∼ W 2 /|εF − (εs + Qs )|. Here we deal with the unique situation, when the electron in the central dot with spin σ¯ does not participate in cotunneling, but controls the spin projection σ of an electron in a side dot, so that the cotunneling processes are not accompanied by spin ﬂips. However processes are possible, where one electron with spin σ leaves the left dot for the left lead and another electron with the same spin σ arrives from the right lead in the right dot or v.v. Then the Hamiltonian (3.67) may be reformulated in terms of pseudospin operators: Hˆ ex = J ∑ T · τ σ .

(3.68)

σ

where 1 = X Sr,Sl , Tz = (X Sl,Sl − X Sr,Sr ), 2 τσ+ = ∑ c†lg,kσ crg,kσ , τσ− = ∑ c†rg,kσ clg,kσ , T + = X Sl,Sr , T k

τz,σ

1 = 2

−

k

∑ k

c†le,kσ cle,kσ

−∑ k

c†re,kσ cre,kσ

.

(3.69)

In the exchange interaction (3.68) the dynamical variable is the orbital index l, r, whereas the spin index enumerates ”channels”. If the splitting ESu − ESg is taken into account, one should redeﬁne the pseudospin operators (3.69) in term of the parity indices g, u instead of l, r, and the two-channel pseudospin cotunneling Hamiltonian is written as Hˆ =

∑ ∑ εk c†pkσ c pkσ + ΔhTz + J ∑ T · τ σ

p=g,u kσ

(3.70)

σ

√ where c(g,u)kσ = (clekσ ± crekσ )/ 2. The second term in this Hamiltonian plays part of the Zeeman splitting of the dot states in the effective magnetic ﬁeld Δ h . The paradigm of inverted roles of spin and channel indices in tunneling problems was originally formulated in the theory of interaction between two-level impurities and metallic electrons [234, 410]. Later on various modiﬁcations of this idea were proposed in the theory of tunneling through quantum dots [244, 253, 279, 314]. The Hamiltonian (3.70) is, apparently, one of the simplest realizations of this idea in

3.6 Molecules and molecular complexes

91

complex quantum dots. We will continue the discussion of two-channel tunneling through CQD in Section 4.3.5 The brief review of dynamical symmetries of complex quantum dots presented in this section demonstrates that manifestation of these symmetries in tunnel junctions depends on the spatial symmetry of both quantum dots and tunnel contacts. Effective Hamiltonians which describe the low-energy cotunneling processes generalize the Schrieffer-Wolff Hamiltonian (3.11) in various ways depending on the energy scale and spin statistics (integer or half-integer spins for even or odd occupation of CQD). The basic dynamical symmetries of quantum dimers and trimers are described by the SU(6) group for N = 1 (spin doublet + 3 orbital ”colors”) and the SO(12) group for N = 2 (spin singlet/triplet pair + 2 orbital colors). Reduction of the relevant energy scale E of the interaction between the CQD and the bath results in lowering of the supermultipet symmetries. The minimal dynamical symmetries realized at low energies are SU(4) for N = 1 and SO(4) for N = 2. The general cotunneling Hamiltonian (3.30) in many cases may be expressed in terms of operators generating the relevant dynamical groups [see, e.g. Eqs. (3.34), (3.39), (3.42), (3.45), (3.54), (3.68). In the following chapters we will consider several interesting physical situations involving higher occupations N = 3, 4, where the second order cotunneling is accompanied by many-body effects which deeply modify the basic features of quantum tunneling spectra of CQD. Higher symmetries SO(n) with n = 5, 6, 7 are involved in these effects.

3.6 Molecules and molecular complexes From fully artiﬁcial structures where both complex quantum dot and bath are fabricated artiﬁcially we now turn to composite devices where ”natural” molecules and molecular complexes are integrated in the electrical circuits as nanoobjects and various few electron tunneling regimes involving molecular states may be realized. Among various possibilities of creating molecular bridges connecting two metallic bathes (see Refs. [76, 121, 154, 328] for review) the most commonly used setups are a nanotip of scanning tunneling microscope in a tunnel contact with a molecule adsorbed on a metallic layer (STM geometry) and a molecule deposited in a narrow gap between two edges of etched and broken metallic wire (break-junction geome-

92

3 NANOSTRUCTURES AS ARTIFICIAL ATOMS AND MOLECULES

try). In principle, STM tip may be used in the latter geometry as a third terminal or as a gate. Although it is difﬁcult to pick out contribution of a single molecule in a real tunneling experiments, we consider here the ideal bridge ”source-molecule-drain” and concentrate on similarities and differences between artiﬁcial and real molecules from the point of view of their dynamical symmetries manifested in single electron tunneling and two-electron cotunneling. Other effects speciﬁc for molecular tunneling electronics will be discussed in Chapter 5. The simplest molecule which can be attached to metallic electrodes is a hydrogen molecule H2 . Such tunneling structure was realized in break-junction geometry [78] where the tunneling current between two platinum electrodes via a bridge created by a single H2 molecule was measured. From the point of view of single electron tunneling the hydrogen molecule is a prototype of symmetric DQD occupied by two electrons, and the possible dynamical symmetry effects may be connected with the interplay between para- and ortho-hydrogen which forms the spin singlet/spin triplet pair. However the exchange gap in H2 is large in comparison with the tunneling energy, so that the most important dynamical effect in this case is the phonon assisted tunneling. Injection of extra electron into this small molecule induces excitation of vibration modes which are seen in tunneling conductance. The necessity of taking into account molecular vibrations in the theoretical consideration of molecular single-electron transistors adds additional dimensions to dynamical symmetries involved in electron tunneling through these natural nanoobjects (see Section 5.3 for further discussion). Recent development of molecular electronics have shown that the most promising candidates for composite nanodevices are macromolecules from the carbon family, namely fullerenes and carbon nanotubes. In comparison with artiﬁcial CQD, where the high spatial symmetry of multivalley structure may be achieved only within some experimental accuracy, one may relay upon perfect point symmetry of a molecular device and then play with the difference of tunneling transparencies of the barriers between the molecule and the two electrodes, e.g., use this difference as a control parameter changing the symmetry of the device as a whole. Probably, because of the perfectness of these nanoobjects the best results on such ﬁne effects as Kondo-type resonance tunneling, have been obtained in measurements using carbon macromolecules as quantum dots.

3.6 Molecules and molecular complexes

93

3.6.1 Fullerene molecules as quantum dots Icosahedral fullerene molecule C60 (”buckyball”) is a natural candidate to molecular quantum dot. Single electron molecular transistors using C60 were realized in multiple experiments (see, e.g., [93, 260, 295, 322, 323, 353, 354, 435]). The electron spectrum of C60 is a discrete set of occupied and unoccupied levels belonging to various molecular orbitals which obey the point symmetry group Yh [168]. There is an energy gap ∼ 0.5 eV between the highest occupied molecular orbital (HOMO)

εh and the lowest unoccupied molecular orbital (LUMO) εe . As a rule, HOMO and LUMO orbitals are orbitally degenerate. If a neutral molecule C060 is deposited between two electrodes (Fig. 3.20), it may be recharged by means of gate voltage in the same way as an artiﬁcial quantum dot (Fig. 3.2). The addition energy δ ε + Q 2− for transition C− 60 → C60 is estimated as ≈ 90 meV [93], and the SW cotunneling regime W Q discussed above may be realized in these adjacent charge sectors.

Fig. 3.20 Fullerene (upper panel) and endofullerene with magnetic ion inside (lower panel) as a quantum dot in a single-electron molecular transistor.

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3 NANOSTRUCTURES AS ARTIFICIAL ATOMS AND MOLECULES

The buckyball molecular quantum dot is characterized by spin 1/2 in the charge state C− 60 , so that the cotunneling Hamiltonian has the form (3.11) and there is no room for dynamical symmetry effects. In the doubly charged state in the presence of d-fold orbital degeneracy one deals with the supermultiplet consisting of four spin states (singlet+ triplet) and d orbital states. If d = 2, the tunneling problem is isomorphous to that of two-channel DQD in parallel geometry [Fig. 3.13(d)] with (d) the Hamiltonian Hˆ tun from Eq. (3.31). However, the Hund’s rule in the presence of orbital degeneracy favors the parallel spin orientation, so that the triplet should be the ground state and singlet should play part of a soft excitation. If the orbital degeneracy is somehow removed, the triplet-singlet transition may occur as a function of a control parameter, e.g., the gate voltage. This crossover was observed experimentally [353, 354]. Further details will be discussed in a context of Kondo effect in Chapter 4. In any case, following considerations presented in Section 3.5.a1, we conclude that at low-energy the resulting dynamical symmetry of this nanoobject is SO(4). Buckyball molecule is a representative of the huge fullerene family. Other, less symmetric molecules like C80 , C82 , C84 may be doped by transition or rare-earth metals which carry localized spin (see [223] for a review). From the viewpoint of tunneling conﬁguration such endofullerenes may be treated as analogs of artiﬁcial double quantum dots in a side geometry [Fig. 3.13(b)]. Being placed between two electrodes, an endofullerene molecule form a tunneling structure where only the carbon ”cage” is in contact with the electronic bath, whereas an encapsulated 4f - or 3d-ion is spatially separated from the leads (Fig. 3.20). Strongly correlated electrons in partially ﬁlled shells of encapsulated ion are coupled with molecular orbitals in a cage by exchange interaction (in case of rare earth ions) or by hybridization (in case of transition metal ions). If only spin degrees are taken into account, such structure may be considered as an analog of the Fulde molecule described in Section 2.4 with a spectrum shown in Fig. 3.16(c), where the carbon cage plays part of weakly correlated dot l and the metal ion is an analog of a side dot with Qr Ql . This analogy was ﬁrst pointed out in Ref. [203], where the family of lanthanocene molecules were referred as an example of caged strongly localized electrons. The actual dynamical symmetry in this case is the SO(4) symmetry of singlet-triplet pair provided the effective spin of the inner ion is 1/2 and the LOMO of carbon cage is singly occupied. The dimetallofullerenes M2 @C80 and M2 @C84 are also discussed in the current literature [223]. In these complex molecules two lanthanide ions are encapsulated in

3.6 Molecules and molecular complexes

95

a fullerene cage. Such molecules can be considered as analogs of triple double dot in side geometry [Fig. 3.14(g)]. We will continue the discussion of complex molecules with cage structure in Section 3.6.3.

3.6.2 Nanotubes as quantum dots Carbon nanotubes played an exclusive role in the studies of single electron tunneling through spatially quantized nanostructures. These allotropes of carbon with a cylindrical nanostructure can be synthesized in metallic and semiconductor modiﬁcations and the length of cylinders can reach millimeter or even centimeter scale [83, 309]. Single-wall carbon nanotubes (SWCN) have been shown to behave as model quantum wires where the electron transport is one-dimensional because of spatial quantization of the transversal motion [44]. Due to large linear size of SWCN it is relatively easy to connect it to metallic leads, i.e. to deposit it on a pair of gold wires. These wires form tunneling contacts with the SWCN and thereby on the one hand conﬁne electron motion in the inter-wire space and on the other hand play part of source and drain electrodes [306, 307, 395]. As a result electron states become fully conﬁned in this space and a segment of SWCN between two electrodes behaves as a quantum dot. Single electron tunneling through SWCN quantum dots is characterized by very distinct steps at the tunneling current – voltage characteristics, and the high quality of nanotubes allows observation of multiple Coulomb diamond windows with alternating even-odd occupation [33, 183, 272]. The most convincing evidences in favor of attainability of Kondo regime in single electron tunneling through quantum dots alse were found in these nano-devices [307]. The speciﬁc feature of nanotube quantum dots is their symmetry protected double orbital degeneracy. It is known that the spectrum of SWCN inherits the generic properties of 2D graphene layers where carbon atoms are packed in a two-sublattice hexagonal crystalline structure [83]. Two equivalent high symmetry points K1 and K2 on the boundary of the Brillouin zone of graphene generate two types of closed transversal orbitals both for valence and conduction electrons in SWCN. These orbitals are associated with clockwise and anticlockwise motion of electrons and holes [283] (see Fig. 3.21). This double degeneracy was detected in tunneling experiments [183, 272]. As usual, interplay between spin and orbital degrees of freedom results in SU(4) dynamical symmetry which is clearly seen in many-body effects charac-

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3 NANOSTRUCTURES AS ARTIFICIAL ATOMS AND MOLECULES

Fig. 3.21 Origin of spirality in nanotube quantum dots (see text for explanation)

teristic for this tunneling in charge sectors N = 1, 2, 3 [15]. These effects will be discussed in more details in Section 5.1.1. One more nanoscale carbonic system which may be interesting from the point of view of its similarity with complex quantum dots is carbon nano-peapod C60 @SWNT [240, 309]. Carbon peapod is a SWNT ﬁlled with several buckyball molecules. Using the above mentioned isomorphism between real and artiﬁcial molecules built in a tunneling device, one may associate C60 @SWNT with CQD in the T-shaped geometry generalizing that shown in Figs. 3.13(b) and 3.14(b). In this case, fullerene molecules inside the tube may be treated as a chain of side dots attached to the main dot formed by SWCN. These ”defects” should affect the single electron tunneling through the main tube [91]. However, the large number of internal degrees of freedom, including mechanical ones, make nano-peapods essentially more complex nanoobject than double or triple quantum dots.

3.6.3 Single electron tunneling through metal organic complexes One may say that endofullerenes considered in the end of the previous section belong to the class of metal doped organic complexes. These complexes contain one or

3.6 Molecules and molecular complexes

97

several metallic ions coupled chemically with some organic ligands. Having in mind an analogy with quantum dots under strong Coulomb blockade we discuss in this section tunneling properties of various transition or rare-earth metal doped organic complexes (TMOC or REMOC).

Fig. 3.22 Molecules from the metallocene family: ferrocene (upper panel), cerocene and ytterbocene (lower panel)

Studies of charge and energy transport through molecular bridges is a well developed branch of contemporary nanoelectronics [70, 295, 299]. These studies have taken new dimension when many-particle effects due to strong Coulomb and exchange interaction have been detected in tunneling transport in a breakjunction geometry with TMOC bridges connecting two edges of broken metallic wire [260, 323]. Several typical examples of TMOC and REMOC are shown in Figs. 3.22, 3.23. The ﬁrst three molecules belong to metallocene family. Ferrocene based molecule, namely sandwich-like ferrocene Fe(C5 H5 )2 in an organomolecular phenylethynyl framework, is shown in Fig. 3.22 (upper panel). In this TMOC molecular orbitals form channels for single electron tunnel transport between source and drain leads, whereas Fe ion sandwiched between two cyclopentatetraene rings could play part of strongly correlated subsystem. Measurement of single-electron tunneling through this molecular complex [131] demonstrates only Lorentzian-like

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3 NANOSTRUCTURES AS ARTIFICIAL ATOMS AND MOLECULES

resonances. This observation is the evidence of strong hybridization between molecular orbitals of ferrocene rather then presence of localized Fe-related spin. However such localized spins have been detected in other TMOC containing Co and V ions in organomolecular frameworks [260, 323].

Fig. 3.23 2 x 2 and 3 x 3 molecular grids secluding magnetic ions from iron group (after [415, 416]).

The next example is a cerocene molecule Ce(C8 H8 )2 where the rare-earth Ce ion is sandwiched between two cyclooctatetraene (COT) rings (Fig. 3.22, lower panel, left). In this molecule the degree of hybridization between the strongly correlated 4 f -state of the ion Ce+3 (f1 ) and molecular orbital (COT)2 is relatively weak, so that the actual valence of the cerium ion is ∼ 3.2 [47, 79]. The ground state of this molecule is a spin singlet combination 1 A1g ( f π 3 ) of an f electron and molecular π orbital, and the ﬁrst excited state is a triplet spin state 3 E2g . Another member of this family, ytterbocene (hole counterpart of cerocene) has more complicated molecular structure with two pentagonal and two hexagonal CH rings attached to Yb3+ ( f 13 ) (Fig. 3.22 lower panel, right). The ground state of this molecule is spin triplet, and the gap for a singlet excitation is tiny, ∼ 0.1 eV. Such structure of spin multiplet

3.6 Molecules and molecular complexes

99

makes cerocene and ytterbocene good candidates for manifestation of SO(4) dynamical symmetry, provided one ever succeeds to fabricate a molecular transition with these REMOCs as tunneling elements [203]. The molecular structure of metal organic complexes discussed hitherto may be represented as a ”cradle” upon which a ion with unﬁlled 3d- or 4 f - shell reposes. In other family of TMOC molecular cage plays part of scaffold or grid which sustains several molecular ions and simultaneously forms bridges attaching these ions to metallic electrodes. Two examples of 2 × 2 and 3 × 3 grids containing four or nine transition metal ions, respectively, [415, 416] are presented in Fig. 3.23. In this case, a nanoobject S is a system of localized spins correlated due to Heisenberglike exchange interaction I jl in the presence of single-ion magnetic anisotropy D j . The effective spin Hamiltonian for n × n grid is n×n = ∑ I jl S j · Sl + ∑ D j S2jz + gμ B · S. Hex jl

(3.71)

j

S

M

2 1 0 −1 −2

0

h Fig. 3.24 Energy spectrum of 2 x 2 molecular grids of spin 1/2 centers with antiferromagnetic Heisenberg interaction as a function of the magnetic ﬁeld h = gμB B. M is total spin projection.

The last term in the Hamiltonian (3.71) describes Zeeman splitting of spin states EΛ in the external magnetic ﬁeld B. These states form a supermultiplet including different values of the total spin S which arise as a result of vector adding of individual spins of transition metal ions. Due to Zeeman splitting, these levels possess rich dynamics induced by multiple level crossing [414] (see Fig. 3.24). Dynamics of spin reversal in time dependent magnetic ﬁeld is usually described in terms of quantum spin tunneling of Landau – Zener type [35, 58, 340] (see Section 8.2). If such molecular grid is attached to metallic leads, tunneling between the localized d-

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3 NANOSTRUCTURES AS ARTIFICIAL ATOMS AND MOLECULES

states and lead electrons should result in effective exchange of SW type ∼ Jex S j · sl in accordance with Eq. (3.11) and in additional indirect exchange of RudermanKittel-Kasuya-Yoshida (RKKY) type between localized spins.

Fig. 3.25 Molecular magnets Fe8 (a) and CsFe8 (b).

In fact molecular grids may be considered as representatives of a more wide class of TMOC, namely single-molecule magnets [125]. These are molecular complexes of various shapes and sizes consisting of an inner magnetic core formed by few transition metal ions and surrounding shell of organic ligands. Two examples of molecular magnets are shown in Fig. 3.25. Molecular magnets may be trapped in a nanogap between two electrodes [155] and investigation of rich spin dynamics together with Kondo tunneling which has speciﬁc features for this type of spin supermultiplet [355] is a rapidly developing branch in molecular spintronics using single-molecule magnets [45] . Dynamical symmetry aspects of this phenomenon will be discussed in Section 5.2.

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101

3.6.4 Vibrational degrees of freedom in single molecular tunneling Electron tunneling through molecular complexes is accompanied by excitation of vibrational degrees of molecular motion. The mechanisms of molecular vibration involvement in single melecule/single electron tunneling crucially depend both on the geometry of devices and the symmetry of molecules. In the process of single electron tunneling an extra electron from the source enters the molecular cage and then leaves it for the drain (Fig. 3.20). Recharging of a molecule may be assisted by the change of the shape of a molecule and its position. Both these processes are described in terms of excitation of molecular vibration modes. Distortion of the organic ligand framework may in turn affect the relevant parameters of the generic Hamiltonian (3.1), (3.2), namely change the electron level positions EΛ in Hˆ d and modify the tunneling integrals W in Hˆ db . In fullerene and nanotube based single molecular transistors only ligand electron levels are involved in cotunneling processes, so the basic Hamiltonian describing interaction with molecular vibration modes has the conventional form, but the input parameters of the Anderson Hamiltonian depend on conﬁguration coordinates uα corresponding to the normal distortion modes α . In the most general case the contribution of molecular distortions in the Hamiltonian has the form Hˆ d = ∑ ∑(EΛ + DΛα uα )|Λ Λ | +

α

Λ

= ∑∑∑ Hˆ db

∑

j kσ α l=s,d

[Wl j (uα )d †jσ clkσ

∑ ∑ BΛΛ α uα |Λ Λ |

ΛΛ α

+ h.c.]

(3.72)

Vibronic correction in the ﬁrst term of Hˆ d describes renormalization of molecular levels due to ligand framework distortions. The eigenvectors |Λ include HOMO/LUMO orbital indices and all interaction corrections including Coulomb blockade term. The prime in the corresponding sum means that the ground state energy of molecular complex is taken as a reference point, so that the sum over Λ includes only states with extra electrons or extra holes in the molecular ”cage”. The second correction in Hˆ takes into account ”molecular excitons” which can arise tod

gether with the distortions of molecular framework, e.g., in the case when one extra electron occupies some molecular orbital on the one end of the molecular structure close to the source lead, and another electron leaves the molecular orbital on the other end of the structure close to the drain lead. Vibronic contribution in Hˆ db takes

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3 NANOSTRUCTURES AS ARTIFICIAL ATOMS AND MOLECULES

into account change in the tunneling integrals due to shifts of the geometrical center of the molecule and distortions of its shape. Electron levels of transition or rare-earth ions in TMOC and REMOC are also sensitive to the distortion of molecular cages via crystal ﬁeld and ligand ﬁeld effects. Such distortions lower the symmetry of the molecular environment and induce level shifts and level splitting in the unﬁlled 3d or 4 f shells, which are also incorporated in the ﬁrst term of the Hamiltonian Hˆ d (3.72). Thus, one may use the technique of vibronic states which was developed within a framework of the Anderson impurity model in a context of chemisorption theory of mixed valence states [159] or in description of phonon broadening of optical spectra [384]. This program was realized in many papers on phonon assisted tunneling through molecular complexes (see e.g., [9, 34, 49, 136, 208, 225, 226, 286, 419]). In accordance with the conventional Born – Oppenheimer scheme, the wavefunction of a molecule may be represented as

Ψ {r, R} = ∑ ψΛ {r, R}χΛ α {R}.

(3.73)

Λα

where the electronic component ψΛ depends parametrically on the set of nuclear

coordinates {R} which should be expanded over normal modes uα = Rα − R0 corresponding to small deviations of nuclei from equilibrium positions {R0 }. Vibra-

tional wavefunctions χΛ α in turn depend on the state Λ of the electronic subsustem. Born-Oppenheimer approximation is based on the fact that the faster electronic subsystem readjusts itself adiabatically to the slow motion of the nuclei in the molecular complex. This readjustment results in renormalization of the levels EΛ (uα ) due to reconstruction of potential induced by the nuclear motion corresponding to the mode α . Besides, non-adiabatic processes of phonon emission and absorption accompany recharging of the molecule in the course of electron cotunneling. Both adiabatic and non-adiabatic components of vibration distortion essentially modify the tunneling spectra and add new dimensions to dynamical symmetry effects. These processes may be presented by means of conﬁguration diagrams. For example, when an electron leaves the molecule for the metallic bath, the charge state of the molecule changes from N to N − 1. If this process occurs adiabatically, the molecule remains in the ground vibronic state. In the non-adiabatic regime an electron escape is accompanied by phonon emission [Fig. 3.26(a)]. As a result on top of the step on the Coulomb staircase characteristics Fig. 3.3 a series of smaller steps appears in accor-

3.6 Molecules and molecular complexes

103

dance with multiphonon replicas around the given Coulomb resonance N /(N − 1) [Fig. 3.26(b)].

Fig. 3.26 (a) Non-adiabatic tunneling of electron from molecular complex to the metallic bath accompanied by phonon emission; (b) Modiﬁcation of a Coulomb step in the tunneling currentvoltage characteristics due to multiphonon processes.

If a single vibration mode u characterized by the frequency Ω0 is active in electron-phonon interaction, one may retain the electron-phonon term only in Hˆ . d

Introducing phonon operators b0 , b†0 for the displacement ﬁeld, we rewrite this term as

1 Hˆ d = Hˆ d + λ Ω0 ∑ X ΛΛ (b† + b) + Ω0 b† b + 2 Λ

(3.74)

Here λ is the dimensionless constant of electron-vibron coupling. This coupling may be transferred from Hˆ into the tunneling term by means of the Lang d

– Firsov transformation [250] using the transformation matrix exp S with S = (λ /Ω0 ) ∑Λ X ΛΛ (b† − b). Lang-Firsov canonical transformation results in polaron shift of the dot levels EΛ (N ) → EΛ − N λ 2 Ω 0 and in appropriate modiﬁcation of Hˆ db :

104

3 NANOSTRUCTURES AS ARTIFICIAL ATOMS AND MOLECULES † = ∑ ∑ ∑[Wl j e−λ (b −b) d †jσ clkσ + h.c.] Hˆ db

(3.75)

j kσ l

Now the displacement operator is incorporated in the tunneling matrix elements and the multiphonon broadening of the tunneling rate may be presented as a convolution of electron tunneling rate and phonon line-shape function [159]. In a single mode regime the resonance tunneling rate for transitions shown in Fig. 3.25 is

Γ (ω ) = ∑ Gnm

Γ (ω )

[ω − εd − (n − m)Ω0]2 + Γ 2 n−m S m! n−m {Lm (S)}2 Gnm = e−S n!

,

nm

(3.76)

where Γ (ω ) = π |W |2 ∑k δ (ω − εk ), εd is the position of the molecular level involved in tunneling transition, n, m are the numbers of phonons absorbed and emitted in the are the associated Laguerre polynomials, S is the so called tunneling process, Ln−m m Huang – Rhys factor, which is deﬁned via the overlap of displaced phonon wave functions, χ (0)|χ (u) = exp(−S/2). Far from direct resonance (deep in the Coulomb window) only the cotunneling mechanism is effective. In this case, the cotunneling rate J(ω ) should be convoluted with the phonon line shape function Gnm in the same way as it has been done with the Breit-Wigner linewidh Gamma in Eq. (3.76) (see, e.g., [207]). The SchriefferWolff indirect exchange integral J(ω ) in this case should be calculated at ﬁnite frequencies.

Fig. 3.27 Singlet-triplet excitation in a molecular complex with N = 2 accompanied by phonon emission.

We see that the phonon subsystem introduces its own dynamical symmetry SO(2, 1) in the tunneling problem (see Section 2.5). If the molecular spectrum also

3.6 Molecules and molecular complexes

105

has non-trivial dynamical symmetry, the full dynamical group is the direct product of two dynamical symmetry groups in the Born – Oppenheimer paradigm. The simplest example of such interplay is the case when the molecule with even electron occupation is characterized by the singlet/triplet spin multiplet, Λ , Λ = S, T in the Hamiltonian Hˆ d (3.72), Fig. 3.27. Then the adiabatic component of the distortion potential results in periodic modulation of the spin gap Δ T S (u), whereas the phonon emission/absorption processes contribute in indirect exchange interaction. Then the Hamiltonian (3.42) describing exchange between the electron bath and singlet-triplet spin multiplet would be modiﬁed appropriately [207, 208]. The dynamical symmetry of the system in this case is SO(4) × SO(2, 1). Various manifestations of this dynamical symmetry including many-body effects in cotunneling will be studied in detail in Section 5.3. In any case vibration assisted single molecule tunneling is essentially dynamical problem, and the most interesting effects are essentially non-equilibrium phenomena. Slow vibration motion of a molecule anchored at the metallic electrodes may be utilized also for realization of the ”shuttling” regime where molecular vibrations may be treated as purely classical mechanical motion of nanoobject between two metallic banks. Electrons captured by this shuttle from one bank are transported to another bank. Mechanical motion may be induced by external electromotive or magnetomotive forces. Such molecular device is one of the realizatons of nanoelectromechanical single-electron tunneling transistor (NEM-SET) [143, 376]. If a shuttling molecule possesses its own dynamical symmetry (e.g. due to existence of low-energy singlet-triplet pair in the spin spectrum) shuttling dynamics is enriched by involvement of this symmetry [220]. Manifestations of dynamical symmetry in shuttling electron transport will be discussed in Section 8.1.1.

Chapter 4

DYNAMICAL SYMMETRIES IN THE KONDO EFFECT

We have shown in the preceding chapter how dynamical symmetries of quantum dots and molecular complexes are uncovered in cotunneling processes which induce transitions between states belonging to different irreducible representations of the low-energy part of the spatially quantized energy spectrum of these nanoobjects. Cotunneling is a second order perturbation in the lead-dot tunneling coupling W which intermixes low-energy states of a bath B and a nano-object S . As a rule, the second-order SW-like description of interaction between two subsystems is insufﬁcient because few-electron nanoobjects coupled with Fermi (or Bose) sea belong to the class of strongly correlated electron systems (SCES) in which the energy scale of the interaction essentially exceed the characteristic kinetic energy. To be more exact, these tunneling structures belong to a speciﬁc subclass of SCES where a few-level subsystem S with strong correlation and internal degrees of freedom is coupled with a weakly interacting subsystem B with continuous spectrum and macroscopic number of states N (see [118] for the general discussion). It is known that in the weak coupling limit W Q electron tunneling induces infrared orthogonality catastrophe in SCES, where excited states are orthogonal to the ground state in the asymptotic limit N → ∞ [18]. This peculiar property makes perturbation theory inefﬁcient at low temperature and energy because of infrared singularities in the perturbation series. Even at high energy not only the basic cotunneling processes shown in Fig. 3.4 but also the relevant higher-order corrections should be taken into account. In our case these corrections are related to multiple creation of low-energy electron-hole pairs in the bath B in a process of dot-lead cotunneling. Since in each cotunneling act an excited state of the bath nearly orthogonal to all preceding states is activated, one may expect that the true ground state of the system differs radically from that of the uncoupled dot-bath system. K. Kikoin et al., Dynamical Symmetries for Nanostructures: Implicit Symmetries in Single-Electron Transport Through Real and Artificial Molecules, DOI 10.1007/978-3-211-99724-6_4, © 2012 Springer-Verlag/Wien

107

108

4 DYNAMICAL SYMMETRIES IN THE KONDO EFFECT

Among numerous manifestations of the universal Anderson’s orthogonality catastrophe theorem, the Kondo effect is one of the most striking phenomena. The necessary precondition of Kondo singularities is the minimal SU(2) symmetry, which describes the internal degrees of freedom in a subsystem S . Two spin projections or two orbital states in the nanoobject provide us with this symmetry. We have seen, however, that complex quantum dots and molecular complexes are, as a rule, characterized by additional degrees of freedom which may be described in terms of dynamical symmetries SU(n) with n > 2 and/or SO(n) with n > 3. In this chapter we focus our efforts on the search for manifestation of these symmetries in the Kondo-effect which is inherent in tunnel spectroscopy of artiﬁcial and natural atoms and molecules.

4.1 Kondo mapping and beyond (surplus symmetries) The profound analogy between the Kondo scattering of electrons in simple metal doped by magnetic impurities and the Kondo tunneling of electrons between metallic leads through a quantum barrier or quantum well conﬁning localized spin was ﬁrst revealed theoretically [17, 27, 135, 297]. The substantiation of this analogy is quite simple: both systems are described by the Anderson impurity model with the Hamiltonian (3.1) and in case of strong on-site Hubbard repulsion U the SchriefferWolff transformation projects it on the effective exchange Hamiltonian (3.11) which is a generic Hamiltonian for describing the Kondo effect [233] Small quantum dots with odd number of electrons under constraints of strong Coulomb blockade Q were recognized as natural objects for experimental conﬁrmation of this analogy. The capacitive energy Q of a quantum dot plays the same part as the Coulomb repulsion energy U of localized impurity electrons. First unambiguous experimental evidences of Kondo tunneling through planar quantum dots were obtained only ten years later [69, 139, 371]. The most convincing manifestations of resonance Kondo tunneling have been observed in metallic single-wall carbon nanotubes [307]. The experimental observations of these Kondo features are exempliﬁed in Fig. 4.1. In the left panel (a) one sees two neighboring Coulomb windows on the diagram G(eV, vg ) (tunnel conductance as a function of bias and gate voltage). These windows correspond to two subsequent occupations of the dot (even, E) and odd (O). In the odd window the zero bias anomaly (ZBA) in the conductance is observed as a bridge in the middle of the Coulomb window (see also the inset

4.1 Kondo mapping and beyond (surplus symmetries)

109

Fig. 4.1 Nanotube as a quantum dot: tunneling spectrum. (a) Tunneling conductance as a function of bias eV and gate voltage vg in two adjacent Coulomb windows with even and odd electron occupation. Inset: Kondo-type zero bias anomaly at given vg ; (b) Scheme of inelastic cotunneling process involving excited discrete state in the quantum dot; (c) Universal behavior of G(T /TK ) for two different Coulomb windows with odd occupation (after [307]).

where the peak in the conductance is marked). Nothing of this sort is seen in the even occupation window. Only the ﬁnite bias anomaly (FBA) corresponding to inelastic cotunneling process shown in the middle panel (b) crosses this window. The temperature dependence of a ZBA peak in tunnel conductance is quite universal. This dependence is scaled by the Kondo temperature [right panel (c)]: the curves G(T )/G0 extracted from different odd windows are scaled by some temperature TK ∼ 1 K. This universal feature correlates with predictions of the theory of Kondo effect (see below). The Anderson – Kondo problem is one of few exactly solvable models in the theory of SCES [21, 402]. Like in other problems with infrared singularities, one may speak about weak coupling limit at high energy/temperature and strong coupling limit at (E, T ) → 0. The Kondo temperature TK establishes the energy scale at which the crossover from weak coupling to strong coupling regime takes place. In the weak coupling regime (E, T ) TK the perturbation approach using the cou-

110

4 DYNAMICAL SYMMETRIES IN THE KONDO EFFECT

pling W /E as a small parameter is legal. Since the renormalization corrections to the model parameters are logarithmic functions of (E, T ), one may use the renormalization group (RG) approach which is essentially equivalent to the weak-coupling perturbation theory. The RG method allows one to ﬁnd the ﬁxed points of a strong coupling regime starting from the weak-coupling limit. If one uses the Anderson Hamiltonian (3.1) as a generic model of tunneling device, the two-stage scaling procedure should be used [148, 158, 185]. This two stage scaling is intimately connected with the dynamical symmetry of our nanoobject. Let us describe the RG procedure using the simplest version of the Anderson model [16] with the Hamiltonian Hˆ And = Hˆ d + Hˆ b + Hˆ db =

εd ∑ σ

dσ† dσ

(4.1)

+ Und↑nd↓ + ∑ kσ

εk c†kσ ckσ

+∑ kσ

(W dσ† ckσ

+ W ∗ c†kσ dσ ),

where magnetic impurity or isolated quantum dot described by the ﬁrst two terms possesses the SU(4) dynamical symmetry of ”Hubbard atom” (see Section II.4 2.4). The energy spectrum EΛ (Λ = 0, σ , 2) of the Hubbard atom is given by Eqs. (2.62), and the single electron excitations correspond to transitions between states belonging to adjacent charge sectors N ↔ N − 1:

εd = Eσ − E0 , εd + U = E2 − Eσ .

(4.2)

The single-electron states of the Hamiltonian (4.1) are shown in Fig. 4.2(a) . The above mentioned SU(4) dynamical symmetry may be explicitly revealed in the form of the Anderson Hamiltonian after expansion of the electron creation and annihilation operators in Hˆ d via Hubbard operators: dσ = X 0σ + σ X σ¯ 2 , nd σ = X σ σ + X 22

(4.3)

These Hubbard operators may be rewritten via Gell-Mann matrix representation X1 − X15 generating the su(4) algebra in the Fock space {↑, ↓, 0, 2} [see Eqs. (2.67), (9.39), (9.49),(9.42)]: d↑ = V− + Y+ , d↓ = U− − W+ ,

(4.4)

nd↑ = V+ V− + Y− Y+ , nd↓ = U+ U− + W− W+

(4.5)

If the second Hubbard level εd + U is above the upper edge of the conduction band, the dynamical symmetry is reduced to SU(3) in accordance with Fig. 2.4.

4.1 Kondo mapping and beyond (surplus symmetries)

111

εd +U

0

0

D0 ε’d

ε’d

εd

εd

D

D0 (a)

D

0

(b)

Fig. 4.2 Jefferson-Haldane renormalization of the energy levels of quantum dot in the Kondo regime. (a) Initial energy level positions for N = 1; (b) Scaling of εd due to reduction of the energy ¯ Renormalization of the levels εd and εd corresponds to localized moment and scale from D0 to D. mixed valence regimes, respectively.

Then instead of (4.3) - (4.5) we have d↑ = X 0↑ = V− , d↓ = X 0↓ = U− , nd↑ = V+ V− , nd↓ = U+ U−

(4.6)

The Kondo effect is inherent in the asymmetric Anderson model where the level εd falls deep in the lower (occupied) part of the half-ﬁlled conduction band with the bandwidth 2D. In this case the spin excitations in the sector N = 1 are described by the operators T = {X1 , X2 , X3 } belonging to the ﬁrst triad of Gell-Mann matrices (2.67), and the charge excitations with unrenormalized energies (4.2) are described by the triads U and V [see also Eqs. (9.39)]. Only charge excitations enter the Anderson Hamiltonian explicitly, while the spin operators T appear in the effective SW exchange Hamiltonian with projected out charge excitations. Explicit form of the Hamiltonian (4.1) in terms of Gell-Mann matrices of the SU(3) group is

εd Hˆ SU(3) = Hˆ b + (Vz + Uz ) + C3 · 1 3 +W ∑ (V+ ck↑ + U+ ck↓ ) + H.c. k

(4.7)

112

4 DYNAMICAL SYMMETRIES IN THE KONDO EFFECT

Here the operators from the triads V and U are responsible for transitions (N = 1) ↔ (N = 0).

The RG method is based on the idea of renormalization of the model parameters which are relevant at low energy as a result of the change of the scale of the high energy excitations. If the model is renormalizable, any such parameter P(ε ) may be represented as P(ε ) = P[(1 + λ )ε ] − λ ε P(ε )

(4.8)

where λ is positive inﬁnitesimal and P (ε ) is the derivative with respect to frequency. The quantity −λ ε P (ε ) is the contribution to P(ε ) from high-energy states which are to be integrated out, preserving the form of P(ε ) but changing its scale. Adopting this paradigm, we immediately notice the inevitability of the two stage RG procedure as a direct consequence of the two energy scales inherent in the Anderson model and the dynamical SU(3) symmetry of its excitation spectrum. Indeed, the characteristic scale of charge excitations which arises as a result of rescaling is |εd − εF |, and the Gell-Mann matrices U and V contain the Hubbard operators responsible for these excitations. On the other hand, the energy scale of spin excitations described by the matrices from the triad T is zero in the absence of tunneling and ∼ TK |εd − εF | in the presence of tunneling at T → 0. This means that at the ﬁrst stage of the RG procedure the charge excitations should be integrated out. Then at the second stage the renormalization of the states in the spin sector with SU(2) symmetry will lead us to the ﬁxed point of the Kondo effect. To realize this program [148, 158] let us calculate the renormalization of the bare states EΛ due to reduction of the energy scale of electron and hole excitations in the bath from D to D − δ D. If the particle and hole states in the cutoff region are labeled k+ and k− , respectively, then the renormalized energies EΛ are E0 = E0 −

∑−

k=k ,σ

|W |2 |W |2 , E1 = E1 − ∑ |εk | + εd |ε | − εd k=k+ k

(4.9)

The spin summation is absent in the second equation due to the fact that each state |σ is renormalized separately. From here one estimates 1 δ εd Γ 2 − (4.10) ≈ |δ D| π D − εd D + εd where

4.1 Kondo mapping and beyond (surplus symmetries)

Γ (ε ) = π W 2 ∑ δ (ε − εk ) = π W 2 ρ( ε ).

113

(4.11)

k

ρ (ε ) is the density of states in the band continuum. Within the actual energy scale one may take it exactly at the Fermi level, ρ (ε ) ≈ ρ0 . Iterating this cutoff process one obtains the scaling equation Γ d εd =− . dη π

(4.12)

where η = ln D is a scaling variable. The second scaling parameter Γ (ε ) may be kept constant because the derivative dΓ /d η = O(Γ /D) is negligibly small in comparison with (4.10). Integrating the differential equation (4.10) under the condition of constant Γ , we ﬁnd the scaling invariant Γ πD = εd∗ . εd + ln (4.13) π 2Γ With the two scaling invariants Γ and εd∗ at hand one may analyze the scaling trajectories. It turns out [148] that for εd∗ −Γ scaling stops at some D¯ Γ α D¯ D¯ = εd∗ − ln (4.14) π Γ where the d-level falls beyond the current energy scale D¯ [Fig. 4.2(b)]. Below this scale at D < D¯ charge degrees of freedom generated by the vectors U and V from the set (9.39) are frozen, and then the states E0 and E1 as well as their difference εd may be integrated out of the low-energy part of the excitation spectrum. The remaining states | ↑, | ↓ belong to the charge sector N = 1, and the interlevel transitions are generated by the components of the triad T. Thus we are left with the conventional SU(2) symmetry of spin 1/2. Then the SW transformation leads to the Hamiltonian Hˆ ex from (3.11) where the enhanced coupling constant J¯ is |W |2 J¯ = ¯ D

(4.15)

√ instead of (3.12). The high-energy cut-off parameter D¯ may be estimated as ∼ DΓ . This family of scaling trajectories is parted by a separatrix from another family with scaling invariant |εd∗ | Γ . In the latter domain the charge degrees of freedom are not quenched till the energy D ∼ Γ where this scaling procedure terminates. In this regime of ”intermediate valence” localized impurity moment is not formed and the Kondo effect is absent [see upper ﬂow trajectory in Fig. 4.2(b)].

114

4 DYNAMICAL SYMMETRIES IN THE KONDO EFFECT

Returning back to the ﬁrst regime, we continue the RG procedure in accordance with scenario elaborated for the Kondo problem [19]. At this stage the scaling parameter is the SW exchange coupling J generated at the ﬁrst stage. The scale contraction D → D − δ D calculated in the same way as in Eq. (4.10) gives the second order corrections to this coupling constant, J → J + δ J

δ J = −2ρ0 J 2

|δ D| E −D

(4.16)

From here one derives the ﬂow equation for J d(ρ0 J) = −2(ρ0 J)2 dη

(4.17)

¯ = J.¯ (The regular perturbative approach to derivation of with initial condition J(D) the scaling equations will be presented in Section IV.3.5). Solution of Eq. (4.17) is a scale dependent effective exchange parameter J(E, T ) =

J¯ . ¯ 1 − 2ρ0J ln(D/max{E, T })

(4.18)

Equation (4.18) deﬁnes the stable ﬁxed point J → ∞ of scaling theory and the characteristic energy/temperature TK =

√ Γ De−π |εd |/2Γ

(4.19)

of a crossover from the weak coupling regime, where the above approach is valid to the strong coupling regime. In the low-energy domain one should turn either to the numerical RG procedure [239] or to the exact Bethe Ansatz [21, 402], which provides us with information about the ground state and low-energy excitations at T TK as well as with accurate description of transient and high-energy regimes. Although the excitation spectrum found by means of exact methods is not too helpful in calculation of the tunneling spectra of quantum dots, one may rely on these results in order to predict the behavior of Kondo-related anomaly in currentvoltage characteristics of tunnel structures. Indeed, it is known [149, 251, 300] that the famous Friedel rule for the scattering phase shift δ (ε ) is satisﬁed for the manybody ground state of magnetic impurity screened by the excitations in the Fermi sea. The inﬁnite ﬁxed point of the scaling theory signiﬁes that the localized spin S = 1/2 is fully ”screened” in the ground state, so that the phase shifts δσ (ε = 0) =

π /2 for both spin projections, whereas in a weak coupling regime at high energy δ↑ (0) = π (occupied level E1↑ ) and δ↓ (0) = 0 (empty level E1↓ ) in inﬁnitesimally

4.1 Kondo mapping and beyond (surplus symmetries)

115

small magnetic ﬁeld. Thus the net contribution of Kondo scattering to the total phase shift given by both charge and spin scattering at the Fermi level is

π π δs↑ = − , δs↓ = + . 2 2

(4.20)

This unitarity limit for elastic Kondo scattering transforms into unitarity limit for Kondo transmission when the problem of tunneling through the quantum dot with odd occupation is mapped on the Kondo Hamiltonian (3.11) with spin of delocalized electrons given by Eq. (3.13). Like in the Kondo impurity problem, the diagonal σ have the form elements of the transition matrix (T -matrix) tek,ek σ tke,ke (εk ) ∼ π −1 e±iδσ (εk ) sin δσ (εk ),

(4.21)

and the transmission probability of elastic electron tunneling between source and drain at T → 0 is

W (0) = lim ∑ sin2 δσ (ε ) = 1 ε →0 σ

(4.22)

This limit corresponds to the formation of Abrikosov – Suhl resonance on the Fermi level of the electron gas in the leads which is responsible for perfect transparency of a quantum dot at zero T. The tunneling conductance is deﬁned as the derivative of the tunnel current I over source-drain bias eV . Based on the above exact result (4.22), one may apply the Breit-Wigner formula for resonance transmission coefﬁcient in the t-matrix and tunneling conductance G0 = dI/d(eV )V →0 . In the general case where Ws = Wd , 2e2 4|Ws |2 |Wd |2 . (4.23) G0 = h (|Ws |2 + |Wd |2 )2 In a symmetric case G0 is equal to the conductance quantum 2e2 /h. At ﬁnite but small T TK impurity-related triplet excitations from the ground state of the Kondo singlet are responsible for decrease of the conductance. These Fermi liquid like excitations give corrections to the ZBA [300, 344]: πT 2 G = G0 1 − . TK

(4.24)

In the weak coupling limit T TK the tunneling conductance may be coupled by means of the Golden rule with the renormalized exchange coupling J from Eq. (4.18) used in the tunneling amplitude. Then G(T ) = G0

3π 2 . 16 ln2 (T /TK )

(4.25)

116

4 DYNAMICAL SYMMETRIES IN THE KONDO EFFECT

In the intermediate region one may refer to the results of numerical RG calculation of the t-matrix [66] which can be ﬁtted by an empirical form s Tκ2 (4.26) G(T ) = G0 T 2 + Tκ2 with Tκ = TK / 21/s − 1 and s = 0.22 ± 0.01 for spin S = 1/2. Experimental data presented in the panel (c) of Fig. 4.1 ﬁt these theoretical curves quite well.

In a two-stage Jefferson-Haldane/Anderson RG scenario it is tacitly implied that the charge degrees of freedom are completely quenched at the second stage of scaling procedure. This conjecture in fact oversimpliﬁes the real situation. In accordance with the exact solution of the Anderson problem [402] charge and spin excitations form two branches of the excitation spectrum above the singlet ground state of the Kondo impurity (holon and spinon branches, respectively) at (E, T ) TK . It follows from exact solution of the Anderson model [197] that in the limit U Δ the density of local spinons may be represented as a narrow peak centered at the Fermi level εF = 0 1 TK . (4.27) Ds (ε ) = π ε 2 + TK2 In the same limit the density of holon states is given by a Lorentzian centered around the level εd and smeared by the hyperbolic secant function which has a tail at the Fermi level: Dh (ε ) =

1 2πΓ

∞ −∞

dx

Γ 1 cosh(π x/2Γ ) (ε − εd − x)2 + Γ 2

(4.28)

(see Fig. 4.3, left panel). Meanwhile in practical calculations of the tunneling spectra of CQD at ﬁnite temperatures and energies the commonly used tool is a Green function method. Calculation of Green functions for strongly correlated electron systems is a mathematical challenge because conventional tools like Feynman – Dyson diagrammatic technique cannot be used without radical modiﬁcation. Even in the reduced version of asymmetric Anderson model given by the Hamiltonian (4.1), (4.7) the operators dσ and nd σ are represented via generators of the SU(3) group which form the complicated non-abelian algebra which includes both Fermi-like and Bose-like commutation relations and intermixes charge and spin degrees of freedom (see Sections 2.4,

4.1 Kondo mapping and beyond (surplus symmetries)

117

D s (ε)

D (ε)

Dh (ε) εd

ε

εF

εd

(a)

εF (b)

Fig. 4.3 (a) Density of holon and spinon states, Dh (ε ) and Ds (ε ), respectively, in accordance with exact solution of the Kondo problem. (b) Density of states in the mean-ﬁeld slave boson approximation, which merges the two branches of excitations.

2.6.1) and the Hubbard operators in expansions (4.3) are expressed via Gell-Mann matrices forming SU(3) group (4.7). Perturbation theory and diagram techniques for operators with complex su(n) and so(n) algebras are available (see, e.g., [158, 181]). However, these techniques are quite cumbersome, and summation of perturbation series is possible in rare cases not necessarily related to the demands of real physical situations. In view of these predicaments it is highly desirable to modify the Hamiltonians including Hubbard operators in such a way that the advantages of Feynman rules standardizing calculation of diagrams as well as Dyson ”skeleton dressing” standardizing series summation could be used, but the implicit dynamical symmetries could be preserved. In principle these goals may be achieved by means of bosonization and fermionization methods which map the original Fock space with generic SU(n) and/or SO(n) symmetries to auxiliary spaces containing only Bose and Fermi operators. The price of this standardization is the complicated structure of vertices in transformed Hamiltonians and additional constraints imposed in order to exclude unphysical states from the partition function. One of these methods summarized in Section 9.3 is the slave-boson transformation [37, 60, 237]. If the Hubbard operators in Eqs. (4.3) are substituted for combinations of spinon and holon operators (9.84), the tunneling term in the Hamiltonian (4.7) acquires the form Hˆ db = ∑(V fσ† hckσ + H.c.). kσ

(4.29)

ε

118

4 DYNAMICAL SYMMETRIES IN THE KONDO EFFECT

The kinematic constraint imposed on Hˆ SU(3) in this representation has the form L = h† h + ∑ fσ† fσ = 1.

(4.30)

σ

It is convenient to include this constraint in the Hamiltonian by means of Lagrange multiplier λ , thus turning to the grand canonical ensemble, Hˆ → Hˆ − λ (L − 1). We see that in the slave-boson representation the tunneling Hamiltonian is represented by a three-tail vertex. In addition, one should care about the dynamics of auxiliary charged bosons represented by the ﬁeld h. Holons also mediate indirect exchange between localized spins and conduction electrons in the SW effective Hamiltonian. It is tempting to start with the mean-ﬁeld approximation introducing ”anomalous average” h¯ = b. In this case the Hamiltonian is reduced to a quadratic form with renormalized matrix element V = bV . The Green function of the system is deﬁned via the resolvent operator ˆ −1 G(ω ) = (ω Iˆ − H) In the mean ﬁeld approximation this Green function reads Gcc Gcd G= Gdc Gdd

(4.31)

(4.32)

Here the indices c, d denote chain and side dot states, respectively. Then the Green function may be calculated analytically with the parameter b deﬁned by means of minimization of the free energy. One gets for the component Gdd (ω ) at small ω TK : Gdd (ω ) =

b2 . ω − εd − ∑k |V |2 /(ω − εk )

(4.33)

εd is a ”replica” of the deep level εd in the exponentially close vicinity ∼ TK Here to the Fermi level of the leads. Appearance of this replica is the artefact of the mean ﬁeld slave-boson approximation. Indeed, the tunneling Hamiltonian taken in the form ∑k (V fσ† ckσ + H.c.) violates the U(1) gauge invariance of the prime Hamiltonian Hˆ db (4.29) because the operator ckσ annihilates charged quasiparticle, while the operator fσ† creates neutral one [190]. To restore the electromagnetic gauge invariance one has to ascribe the corresponding gauge phase φ to spinon operator, fσ → fσ exp(−iφ ) and thereby to ascribe charge to this variable. As a result some part of spin spectral density is transferred to the charged sector. Thus one may conclude that the generic SU(4) symmetry of quantum dot is violated as well. This part of charge spectral density is also exponentially small because the corresponding

4.1 Kondo mapping and beyond (surplus symmetries)

119

residue of the Green function is deﬁned by the parameter b2 ∼ ρ0 TK [cf. Eq. (4.27)]. With all these reservations one may state that the mean ﬁeld description reproduces some features of the Abrikosov – Suhl resonance, although this resonance is artiﬁcially transferred from the spin sector to the charge sector and moved downwards from the Fermi level (see Fig. 4.3, right panel). Neglecting this small shift, one may rewrite Gdd (4.33) in the resonance form Gdd (ω → 0) ≈

b2 ω − iΓK

(4.34)

where ΓK ∼ ρ0 b2 is the imaginary part of the self energy. This Green function should be placed in the exact equation for the t-matrix which has the following structure Gcc (ω ) = gcc [1 + T (ω )gcc ] = gcc (1 + gcc|V 2 |Gdd )

(4.35)

Here gcc = (ω − Hˆ b )−1 is the bare Green function of the conduction electrons in the leads. Since the second term of Eq. (4.35) determines the tunneling transparency of the quantum dot, one may say that in spite of all the shortcomings and inconsistencies, the slave boson model gives at least semi-phenomenological description of the Kondo-resonance tunneling regime. The form of the t-matrix T (ω ) ∼

|V 2 | ω − Σ (ω )

(4.36)

with Kondo singularity near the Fermi level is used in many analytical theories of Kondo tunneling (see Ref. [281] for general discussion). Numerical RG calculation of the excitation density of states also correlates with the qualitative picture shown in Fig. 4.3. One should note that the mean-ﬁeld slave boson approximation nevertheless becomes asymptotically correct in a model where the double valued spin subindex σ in the operators c and d is changed for a ”color” m with inﬁnite number of ”tints”, so that the symmetry group of the model is SU(N) with N → ∞. The reason for appearance of SU(N) groups (with ﬁnite N) in the theory of Kondo tunneling will be elucidated below. Diagrammatic technique for the slave-boson representation allows one to calculate the Green function (4.35) without appealing to the mean-ﬁeld truncation in the so called non-crossing approximation [311]. However, strict calculation of ﬂuctuation corrections ∼ 1/N to this asymptotic mean ﬁeld theory remains an ill-deﬁned mathematical and physical problem.

120

4 DYNAMICAL SYMMETRIES IN THE KONDO EFFECT

We have seen that from the point of view of dynamical symmetries the procedure of reduction of the generic Anderson model to an effective spin Hamiltonian is in fact the reduction of the original SU(4) symmetry of Hˆ d in the Anderson Hamiltonian (4.1) to purely spin symmetry SU(2) by means of quenching the charge degrees of freedom in the process of contraction of the relevant energy scale. Meanwhile in the conventional theory of the Kondo effect some ”surplus” dynamical symmetries survive in the low-energy part of the excitation spectra in various physical systems. The realm of ”exotic” Kondo effects is a signiﬁcant part of modern Kondo physics [67]. Among natural generalizations of the standard SU(2) model (3.11) the extension of the Kondo problem to multichannel systems with several orbital states of conduction electrons and multisite systems with several localized spins should be mentioned ﬁrst of all. We will discuss here in brief dynamical symmetries related to these two models and refer to the review [67] for description of more exotic conﬁgurations related to speciﬁc properties of magnetic impurities (rare-earth or actinide group metal ions) with strong spin-orbit interaction. In these ions the form of the supermultiplet is predetermined by the interplay between the relativistic ﬁne structure and the crystal and ligand ﬁeld effects. In the original Kondo model the orbital degrees of freedom of the conduction electrons are ignored under the assumption that the metallic Fermi surface is formed by s-electrons. Then the crystal may be approximated by a sphere with impurity ion in its center. The angular variables are irrelevant and we are left with a onedimensional single channel integrable model. In the general case an impurity may preserve orbital degeneracy and higher spherical harmonics of the conduction electrons could be taken into account. If some partial wave l with harmonics |lm is chosen, where the orbital projection m acquires 2l + 1 values −l ≤ m ≤ l , the degenerate Anderson Hamiltonian for these harmonics is written as (l) Hˆ And = E0 X 00 + E1 ∑ X mσ ,mσ +

+

∑

mkσ

mσ mσ ,0 (V X cmkσ

+ H.c.)

∑ εk c†mkσ cmkσ

mkσ

(4.37)

This Hamiltonian is derived for a subspace N = (0, 1) under the assumption U > D like in the previous case. The SW transformation reduces this Hamiltonian to the form Hˆ CS = Hˆ d + Hˆ b + ∑ X mσ ,m σ c†mσ cm σ (4.38) mσ ,m σ

4.1 Kondo mapping and beyond (surplus symmetries)

121

known as a Coqblin – Schrieffer Hamiltonian [62]. There is still a selection rule Δ σ = 0, ±1 for spin indices, but all possible changes of orbital projections are allowed in the effective exchange term of the Hamiltonian (4.38). This means that one may ascribe 2l + 1 ”colors” to each projection of spin operators s (l) and S(l) with matrix elements smm , Smm . The symmetry group corresponding to the Lie algebra of these operators is SU(N) = SU(4l + 2). The net result of orbital degeneracy is the appearance of the factor 2l + 1 in the right hand side of the scaling equation (4.17) and appropriate enhancement of TK [301], TK → gTK , where the enhancement factor is

2l . g = exp (2l + 1)J If the spin-orbit interaction is taken into account, then the impurity and the conduction band states are characterized by the full moment and its projection | jm j , so that the indices mσ should be substituted for the quantum number m j in the Hamiltonian (4.38). In this situation we return back to the basic SU(2) symmetry but instead of spin operators S and s the full moment operators J and j enter the effective exchange Hamiltonian. The next step in generalization of the exchange Hamiltonian is explicit introduction of crystal ﬁeld effects in the problem of Kondo screening. In a crystal ﬁeld the continuous spherical symmetry of homogeneous space gives way to a discrete symmetry of the point crystalline group. Now the eigenstates of the Hamiltonian Hˆ d in the degenerate Anderson model should be expanded in cubic Harmonics |γ μ , where μ stands for the lines of irreducible representation γ of the point group of discrete rotations in an elementary cell containing impurity ion. In this representation Hˆ And acquires the form [65] (l) Hˆ And = E0 X 00 + ∑ ∑ Eγ X γ μσ ,γ μσ + ∑ ∑ εk c†γ μ kσ cγ μ kσ

γμ σ

+

∑ (V X

γ μ kσ

γ μσ ,0

γ μ kσ

cγ μ kσ + H.c.)

(4.39)

From the point of view of dynamics of Kondo screening it is crucially important that the crystal ﬁeld splitting of single electron level E1 → Eγ inserts another energy scale in the spectrum of charge excitations. If, e.g., one deals with 3d states of the iron group impurities in a cubic crystalline environment, then the cubic harmonics are γ = t2g , eg , and this new scale is Δ c f = Eeg − Et2g . On the other hand, the conduction electron spectrum is practically not affected by the crystal ﬁeld splitting. One simply has to re-expand the plane waves in cubic harmonics instead of the spherical

122

4 DYNAMICAL SYMMETRIES IN THE KONDO EFFECT

ones. The main effect of these additional channels of charge excitations is that the Kondo effect loses its universality in case when the magnitudes of Δc f and TK are comparable [302].

E

D0

E

εd +U

Δ cf

D

0 εd +Δ cf εd

(a)

(b)

Fig. 4.4 Scaling procedure in the presence of crystal ﬁeld splitting Δcf .

The loss of universality may be understood in terms of the relevant dynamical symmetries as a three-stage renormalization. Indeed, starting from the same point as in Jefferson – Haldane scaling procedure described above, we reach the limit where the charge excitations described by operators X σ 0 , X 0,σ are quenched in accordance with Eq. (4.14), but the crystal ﬁeld excitations with the energy Δ c f are still alive ¯ one may neglect this energy in the Anderson’s ”poor man’s Fig. 4.4. While Δc f D, scaling” ([19]) and continue the scaling procedure for the Hamiltonian (4.37) with the full SU(2[l]) symmetry, where ([l] = 2l + 1) is the full orbital degeneracy. This procedure ends with the Kondo temperature [l] − 1 TK = TK0 exp [l]

(4.40)

i.e. with the Kondo temperature enhanced with the factor g introduced above in comparison with its value TKO for spin 1/2 Kondo effect. If the inequality TK Δc f is valid, this is the end of the story, and the crystal ﬁeld splitting is irrelevant for the Kondo effect. In the opposite limit Δc f TK the scaling procedure should be continued, and eventually one comes from the Coqblin – Schrieffer Hamiltonian (4.37) to the Cornut – Coqblin Hamiltonian (4.39), where the crystal ﬁeld excitations are

4.1 Kondo mapping and beyond (surplus symmetries)

123

quenched and the effective orbital degeneracy is given by the rank [γ ] of representation γ of cubic point group. This means that the effective symmetry is reduced to SU(2[γ ]) and the true low-energy Kondo temperature is [γ ] − 1 . (4.41) TK = TK0 exp [γ ] In case of l = 2 and cubic symmetry of the environment [γ ] = 3 so that the enhancement factor diminishes from g = exp(4/5J) to g = exp(2/3J) as a result of reduction of the dynamical symmetry SU(10) → SU(6) due to quenching of some interlevel transitions within a crystal ﬁeld multiplet. The possibility of deviation from monotonous temperature behavior of the resistivity due to two Kondo scales was noticed for the ﬁrst time in Ref. [276]. If TK ≈ Δ c f , the problem loses the logarithmic universality and the Kondo temperature becomes an explicit function of

Δc f . Up to now we have found two mechanisms of involvement of dynamical symmetries in the multistage character of formation of the Kondo singlet. In the conventional Anderson – Kondo scenario where the energy scales of charge and spin degrees of freedom are essentially different, the role of the former excitations is in contraction of the energy interval for the collection of logarithmic contribution in ¯ This contraction reduces the pre-exponential the Kondo screening from D0 to D. factor in the Kondo temperature and thus the Kondo temperature itself: if the charge degrees of freedom are quenched from the very beginning (both levels εd and εd +U ¯ All these proin Fig. 4.2(a) are outside the band continuum), this factor is D0 > D. cesses develop within the basic SU(4) multiplet. In the orbitally degenerate model additional charge excitations with small energy scale arise due to the crystal ﬁeld splitting, and the Kondo stage of scaling evolution splits into two stages itself. Reduction of symmetry in this case also takes place in a framework of SU(2N) group scenario. The net effect of orbital degeneracy is enhancement of TK (4.40),(4.41). It worth mentioning that in the presence of strong spin-orbit interaction the role of crystal ﬁeld splitting in the Kondo scattering signiﬁcantly changes. As was mentioned above, in this case the generic Kondo Hamiltonian has the form KJi ji and the crystal ﬁeld splits the atomic levels EJ where J is the full moment. Representative example is the ion Ce3+ (4 f 1 ) with J = 5/2 in the ground state conﬁguration. Crystal ﬁeld splits the sextet E5/2 into the doublet EΓ7 ground state and the quartet of EΓ8 excitations. Since in the (J j)-coupling scheme the spin and orbital degrees of freedom are merged into a common manifold, crystal ﬁeld excitations should be treated

124

4 DYNAMICAL SYMMETRIES IN THE KONDO EFFECT

by means of moment operator algebra with the corresponding selection rules, so that quenching of a crystal ﬁeld exciton in a process of RG contraction means effective reduction of a moment from J = 5/2 to J = 1/2 within representations of a symmetry group SU(2). This scenario has no direct analogs in quantum dots, so we refer to the Refs [67, 292] for further discussion. Hitherto we discussed only the localized moments of singly occupied magnetic ions. In conﬁgurations with more than one atom one has to take into account the Hund’s rules for spin and orbital moments, so the spin S varies from 1/2 to 5/2 and back to 1/2 with increasing occupation of the 3d shell of magnetic ion from 3d 1 to 3d 9 . Effective orbital moment in the crystal ﬁeld varies less regularly and may be completely quenched in some conﬁgurations. As a result the number of orbital degrees of freedom n in the general case does not correlate with the spin S. One should discriminate between the cases of fully screened local spin n = 2S, underscreened spin n < 2S and overscreened spin n > 2S [302]. The latter case is especially interesting because the behavior of the system does not follow the Fermi liquid scenario of the conventional Kondo effect developed in Ref. [300]. In this case the inﬁnite ﬁxed point is unstable and an extra ﬁnite ﬁxed point arises on the scaling trajectory [302]. This point in turn should be classiﬁed as a saddle point, because it is unstable against orbital anisotropy which is practically unavoidable in real physical systems. The search of this elusive ﬁxed point is a great challenge both in natural systems with Kondo scattering and in artiﬁcial systems with Kondo tunneling. We will pay special attention to this problem in Section 4.3.5, where the prospects for observation of multichannel Kondo effect in CQD will be discussed and the corresponding RG procedure for ovescreening regime will be described. Now we turn to a brief survey of dynamical symmetries in the Kondo physics for multisite impurity conﬁgurations, which are obviously related to those of complex quantum dots (Section 3.5). Two-impurity Kondo model was a subject both of scaling RG analysis [184] and numerical RG computations [186, 187]. The Hamiltonian of this model with antiferromagnetic exchange coupling constant is Hˆ = Hˆ b + Hˆ ex = ∑ εk c†kσ ckσ − J k,σ

∑

Si si ,

(4.42)

i=1,2

τˆ si = ∑ c†kσ ck σ ei(k−k )·ri . 2 kk The model is characterized by the 1 ↔ 2 mirror symmetry, so it is convenient to turn to even-odd variables like in the case of DQD considered above. Then

4.1 Kondo mapping and beyond (surplus symmetries)

Hˆ ex = (S1 + S2 ) · (Je se + Jo so ) + Jm (S1 − S2 )seo

125

(4.43)

where the combinations of conduction electron operators c†ekσ = c†kσ cos k · r, c†okσ = c†kσ i sin k · r

(4.44)

form the basis for the spin operators se , so and seo . The corresponding exchange coupling constants are √ J sin kF r Je,o = (4.45) 1± , Jm = Je Jo . 2 kF r These constants are taken at k = kF . The zero coordinate is chosen at half a distance 2r = |r1 − r2 | between impurities. The indirect exchange interaction of RudemanKittel-Kasuya-Yosida (RKKY) type via conduction band states is generated to second order in J: Hˆ RKKY = −I S1 · S2 , I = 2 ln 2(Je − Jo )2 = 2 ln 2(Je2 + Jo2 − 2Jm2 ).

(4.46)

This interaction is always ferromagnetic, so it favors parallel orientation of two localized spins. There is a formal analogy between the Hamiltonian Hˆ ex + Hˆ RKKY (4.43), (4.46) and the Hamiltonian Hˆ DQD (3.29). Being diagonalized and expressed in Hubbard operators, the two-site exchange Hamiltonian describes the same spin triplet-spin singlet supermultiplet as Hˆ DQD . In this model the ground state of the localized spin subsystem is triplet. If one resorts to the representations of the SO(4) group, the vectors S1 and S2 transforms to six generators of the SO(4) group (2.52) with kinematic constraints (2.53). In terms of these generators, the ﬁrst parity conserving term in the Hamiltonian (4.43) is responsible for transitions within spin triplet described by the generator S = S1 + S2 , the second term which mixes even and odd terms contains the generator R = S1 − S2 and thus describes the singlet-triplet transitions. One should, however, emphasize the signiﬁcant differences between the two models. First, unlike the model of DQD adopted in Section 3.5.1, the intersite coupling I is essentially less than the site-band coupling parameters Ji , so one should be cautious with the use of the representation {Λ } = {S, T } especially because the coupling constants Ji are enhanced in the leading logarithmic approximation, whereas the constant I acquires logarithmic divergences only in higher-order perturbation theory [186]. Besides, the induced RKKY interaction is ferromagnetic, whereas the indirect interdot exchange in DQD is antiferromagnetic.

126

4 DYNAMICAL SYMMETRIES IN THE KONDO EFFECT

Due to anisotropy Je > Jo the Kondo screening in the two-site model is a twostage effect with two Kondo temperatures TK (Je ) > TK (Jo ). At the ﬁrst stage half of the initial spin 1 is screened, and one comes to an intermediate unstable spin 1/2 ﬁxed point characterized by a Kondo temperature TK (Je ). If both these temperatures exceed ΔST = I then this two stage regime results in the inﬁnite strong coupling ﬁxed point with both Je and Jo → ∞ at ε → 0. This means that the phase shift of the conduction electrons at the Fermi level is asymptotically π /2 in both even and odd parity channels. Although the RKKY interaction derived from the original Hamiltonian (4.43) is ferromagnetic, one may add to the Hamiltonian additional antiferromagnetic coupling I to complete the formal study of the ﬂow diagram (J, I ). It turns out [187] that at the given TK ∼ exp(−1/|J|) there is a critical point Ic = −2.2TK where Kondo effect disappears and the phase shifts are zero in both channels. The ground state at I < −2.2TK is the state of locked impurity singlet. One may expect that the basic features of the two-site Kondo effect will be reproduced in Kondo tunneling through CQD with low energy spectrum consisting of spin singlet and triplet. We will describe this regime in terms of the generators of the dynamical SO(4) group in the following sections.

4.2 Kondo effect in quantum dots with even occupation Our next task is to ﬁnd out novel facets of Kondo physics which may be uncovered in studies of the Kondo regime in conductance of quantum dots. Isomorphism between the Kondo scattering in magnetically doped metals and the Kondo tunneling through Coulomb blockaded quantum dots noticed in the pioneering works [135, 297] exists for quantum dots with odd occupation in a standard geometry shown in Fig. 3.1(c). The ﬁrst experimental studies of tunneling conductance conﬁrmed this scenario (see Fig. 4.1). The immediate question which arises is whether it is possible to ﬁnd conditions where the Kondo tunneling regime is realized for even occupation. In a search of an answer to this question the idea of dynamical symmetries arises in a natural way. The simplest path toward the Kondo effect in a quantum dot with even occupation with a singlet ground state is to apply an external magnetic ﬁeld inducing the Zeeman splitting in the excited spin triplet state of doubly occupied dot [341]. It is known that the Zeeman splitting is detrimental for the conventional Kondo effect be-

4.2 Kondo effect in quantum dots with even occupation

127

cause it suppresses inelastic spin-ﬂip processes which are the main ingredient of the Kondo screening [21, 402]. Paradoxically, the same Zeeman splitting switches on Kondo tunneling in a doubly occupied dot possessing basic SO(6) dynamical symmetry in a two-level approximation taking into account the highest occupied ground state level and the lowest excited levels ε1,2 left from the whole discrete level of quantum dot. These levels are divided by the level spacing δ ε . The dot spectrum consists of doubly occupied levels ε1 and ε2 and two electron states with one electron at each level. This system of three singlet and one triplet states is described in Section 2.3, and the effective exchange Hamiltonian has the form (3.15). The lowenergy part of this spectrum consists of the ground state singlet and the triplet spin exciton ES = 2ε1 , ET = ε1 + ε2 − δex

(4.47)

where δex is the energy shift due to direct ferromagnetic exchange between electrons in different orbitals. Thus in the absence of magnetic ﬁeld the symmetry of a dot is reduced to SO(4) for the energy scale E Δ T S = δ ε − δex . Further decrease of E results in freezing of triplet excitations, so that the Kondo screening is suppressed in the singlet ground state.

−1

ε2

δε

ε1 S (a)

ET

0

ES

1

Bc

T

B

(b)

Fig. 4.5 (a) Quantum dot with double occupation N = 2 in singlet and triplet state; (b) S/T multiplet in magnetic ﬁeld.

The mechanism of magnetic ﬁeld induced Kondo effect is elucidated in Fig. 4.5. In the external ﬁeld B the triplet level ET is split in accordance with the Zeeman effect, ΔZ (B) = g μB B: ET → E μ , so that E±1 = E0 ± Δ Z. When the magnitude of the ﬁeld B approaches the critical value Bc corresponding to full compensation of the singlet/triplet gap, Δ Z (Bc ) = ΔT S , new energy scale Em ∼ ET 1 − ES arises. Within

128

4 DYNAMICAL SYMMETRIES IN THE KONDO EFFECT

this scale the symmetry of the semisimple group SO(4) = SU(2) × SU(2) is reduced to an SU(2) symmetry of the quasidegenerate {S, T 1} doublet. This reduction can be described in terms of generators P1+ =

√ √ 2|T 1S|, P1− = 2|ST 1|, P1z = |T 1T1 | − |SS|,

(4.48)

which generate the SU(2) subgroup describing the doublet ET 1 , ES . The complementary vector P2 deﬁned as P2+ =

√ √ ¯ P− = 2|T 1T ¯ ¯ 1|, ¯ 2|T 0T 1|, 0|, P2z = |T 0T 0| − |1 2

(4.49)

generates a second subgroup SU(2) of the dynamical SO(4). Transitions described by the generators P2 are quenched at Em Δ T S (B = 0). Within the subspace {S, T 1} the effective spin Hamiltonian may be written Hˆ m = Em P1z + JP1 s.

(4.50)

Thus the SW projection resulted in the mapping of original SO(4) problem to the effective Hamiltonian for the conventional Kondo effect in external ﬁeld Em . If the Kondo temperature TK |Em |, then the Kondo tunneling is effective and ZBA in tunnel conductance may be observed. This means that tuning magnetic ﬁeld B in the vicinity of the critical value Bc , one should see very sharp peak in tunneling conductance in the middle of a Coulomb diamond with even occupation. Precisely this effect was observed in measurement of tunneling conduction through a SWNT quantum dot [307]. The Kondo-type ZBA in the even window was absent at zero B, then suddenly appeared at B = Bc = 1.38 T and disappeared again at B > Bc . In the adjacent odd window the tunnel conductance behaved in full agreement with prescriptions of conventional Kondo model. ZBA was observed in zero magnetic ﬁeld, then it was split in two ﬁnite bias peaks in ﬁnite ﬁeld and practically dissolved in inelastic continuum at B ∼ Bc (see Fig. 4.6). This magnetic ﬁeld induced Kondo effect was the ﬁrst experimental manifestation of dynamical symmetries in tunnel conductance of quantum dots with even occupation (see also [346]. The possibility of tuning the parameters of the Anderson Hamiltonian describing the quantum coupled to the metallic leads by means of external ﬁeld may be exploited in other ways. In particular, one may vary the exchange gap ΔT S in a vertical quantum dot by changing the Larmor shifts of the doubly occupied singlet and triplet (see Fig. 3.9). Again, at some critical ﬁeld Bc these levels cross and another type of magnetic ﬁeld induced Kondo effect should be observed

4.2 Kondo effect in quantum dots with even occupation

129

Fig. 4.6 Magnetic ﬁeld induced Kondo effect in quantum dot with even occupation. (a) Tunnel conductance phase diagram for two adjacent Coulomb windows with even (E) and odd (O) occupation at a series of magnetic ﬁelds; ((b) Evolution of current-voltage characteristics as a function of bias eV at ﬁxed gate voltage vg ; (c) Temperature dependence of Kondo-type ZBA at the critical ﬁeld Bc where the Zeeman splitting compensates the S/T exchange gap for the state |1, 1; (d) ZBA peak as a function of log T (after [306]).

[133]. In the vicinity of the critical value of the control parameter B ∼ Bc , the full SO(4) symmetry of the doubly occupied quantum dot is involved in the formation of the Kondo peak. The appropriate RG theory has been formulated in Refs. [100, 133, 203] (see [364] for the ﬁrst experimental observation of singlet/triplet crossover). As was shown in Section 3.5.1, the effective spin Hamiltonian for this problem may be formulated in terms of generators of the SO(4) group: J1 J2 1 ET S2 + ES R21 + S · s + R1 · s. Hˆ = Hˆ d + Hˆ ex = 2 2 2

(4.51)

130

4 DYNAMICAL SYMMETRIES IN THE KONDO EFFECT

Generalization of the ”Poor man’s scaling” ﬂow equation (4.17) for the model with two vectors gives the following system of equations: d j1 /d η = − j12 + j22 , d j2 /d η = −2 j1 j2

(4.52)

Here ji = ρ0 Ji . The solution of this system is sensitive to the magnitude of the external ﬁeld ﬁeld B. In a triplet sector at B Bc and |Δ T S | TK1 = D¯ exp(−1/ j1 ) the system is in the asymptotic limit of the conventional underscreened Kondo effect for S = 1. When B → Bc + o, the spectrum becomes quasidegenerate and the system (4.51) is reduced to a single equation for the effective integral j+ = j1 + j2 ; d j+ /d η = −( j+ )2 .

(4.53)

In this regime the SO(4) symmetry survives down to the lowest energies and the Kondo temperature is maximum: TKm = D¯ exp(−1/ j+ ).

(4.54)

TK(Δ) / TK (0) 1

Fig. 4.7 Kondo temperature for doubly occupied QD as a function of triplet/singlet gap ΔT S .

0

Δ TS

When the exchange gap Δ T S changes sign at B < Bc the ground state of the dot is singlet and TK rapidly tends to zero. The full curve TK (B) obtained as a result of numerical solution of the system (4.52) is shown in Fig. 4.7. In the intermediate asymptotic regime for negative Δ T S and TKm ≤ |ΔT S |, the power law

TK (B) TKm

ρ (4.55)

4.2 Kondo effect in quantum dots with even occupation

131

is valid, where ρ < 1 is a universal constant. The physical explanation of this behavior follows the general paradigm of multistage quenching of dynamical symmetry: at high enough energy E ΔT S the full SO(4) symmetry is involved in the Kondo effect. Reduction of the scale E to the values E < Δ T S results in quenching the triplet/singlet transitions and only the SU(2) symmetry of spin 1 survives. However the Kondo scenario loses its universality at Δ T S ∼ TK because the magnitude of TK in the inﬁnite ﬁxed point depends on the scale Δ T S , where half of the spin degrees of freedom are frozen out:

ΔT S ∼ TK (ΔT S ).

(4.56)

Non-universal behavior of tunneling conductance G(T ) in the vicinity of the S/T crossover have the same reasons as the energy scale dependence of TK . The character of deviations from the universal scenario described by Eqs. (4.24)–(4.26) depends on the position of the system relative to the critical point of singlet-triplet degeneracy.

On the triplet side of the crossover the whole S/T multiplet is involved in

Kondo cotunneling at T |Δ T S | and the curve G(T ) follows the Kondo scenario with TK given by Eq. (4.54). In the opposite limit T |ΔT S | the singlet-triplet transitions are quenched, and G(T → 0) approaches the unitarity limit for undersreened Kondo effect with standard Kondo temperature TK < TKm where the constant j1 enters the exponent instead of j+ . In this regime a non-monotonous temperature dependence of G(T ) with a maximum in the low-temperature part of the curve is expected. On the singlet side the above mentioned two-stage screening process results, ﬁrst, in the increase of G with reducing T at T ΔT S where the spin degrees of freedom are still active. This behavior is scaled by the larger Kondo temperature TKm . At T ΔT S both vectors S and R are quenched, and the Kondo scattering vanishes with T approaching zero. The slope G(T → 0) should be the steeper, the closer is the system to the critical point. In the critical region B ≈ Bc where |ΔT S | TK , all spin degrees of freedom constituting the SO(4) dynamical symmetry group survive till the lowest attainable temperature, and the ZBA in the tunneling conductance should survive as well. Experimental ﬁndings on magnetic ﬁeld driven two-stage Kondo effect in AlGaAs/GaAs quantum dos with even occupation [426] correlate well with this picture (see Fig. 4.8). Similar results are obtained for a single fullerene molecule quantum dot [353]. However, in this type of quantum dots the singlet/triplet crossover scenario is enriched by possible contribution of orbital degrees of freedom because the singlet and triplet states are related to different orbital states of the dot [Fig. 4.5(a)],

132

4 DYNAMICAL SYMMETRIES IN THE KONDO EFFECT

Fig. 4.8 Amplitude of ZBA peak in the conductance of lateral DQD as a function of temperature at various magnetic ﬁelds. Upper curve (circles) corresponds to B ≈ Bc , next curve (squares) is measured at B < Bc , the rest three curves are obtained for B > Bc (after [426]).

so one should pay special attention to transition from generic two-orbital Anderson model to effective exchange model [99, 100, 165, 166, 180, 338, 343, 344, 345]. We already approached the two-orbital model at double occupation in the discussion of SO(n) symmetries in DQD (Section 3.5.1). Unlike the symmetric DQD, the electron states in planar quantum dots are not characterized by any parity, so the conﬁguration under consideration is close to that in Fig. 3.16(d). There is a possibility to manipulate the S/T spectrum (4.47) of a planar dot, e.g., by tuning the parameters ε1,2 [224]. Making the excitation energy ε21 = ε2 − ε1 small enough by means of appropriate variation of the conﬁnement potential, one may ﬁnd the regime where the exchange energy gain in the triplet state overcomes the energy loss ε21 . The basis for the generators of the SO(4) group is constructed in the following way:

† † |S = d1↑ d1↓ |0 ;

(4.57) 1 † † † † † † † † |T 1 = d1↑ d2↑ |0, |T 0 = √ d1↑ d2↓ + d1↓ d2↑ |0, T 1¯ = d1↓ d2↓ |0 . 2 Unlike the generic model [135], there is no universal framework in which the odd combination of lead states is eliminated, so that one has to deal with the matrix of tunnel elements and the problem of Kondo tunneling is implicitly the two-channel

one W=

Ws1 Wd1 Ws2 Wd2

,

(4.58)

and the appropriate dynamical symmetry of the Kondo problem is SO(6). However, one of the eigenmodes of this matrix is zero, provided Ws1 W = s2 = α ≡ eiϕ tan θ . Wd1 Wd2

(4.59)

4.2 Kondo effect in quantum dots with even occupation

and the transformation cos ϑ sin ϑ cdkσ c+kσ iϕ =e c−kσ − sin ϑ cos ϑ cskσ

133

(4.60)

eliminates the channel ”-” from the tunnel Hamiltonian. Then the Kondo tunneling problem is reduced to the single channel SO(4) case discussed above. Another way to reduce the problem to the SO(4) case is to keep the mirror reﬂection symmetry of the lead states. This symmetry is restored for a special set of tunneling parameters [180] W1 W1 . (4.61) W= W2 −W2 Then the tunneling Hamiltonian acquires the form Hˆ tun = ∑[W1 c†ekσ d1σ + W2 c†okσ d2σ + H.c.]

(4.62)

kσ

[(cf. Eq. (3.31)], so that the even and odd components of the bath electron wavefunctions are coupled to the lower and upped dot levels respectively. Looking at the basis (4.57), we conclude that the cotunneling through the level ε1 couples the leads with the singlet state, cotunneling through the level ε2 couples the leads with the triplet state, and the cotunneling which changes occupation from |11 to |12 and back intermixes even and odd states in the leads and generates |S ↔ |T states in the dot. One may say that in this speciﬁc case an effective parity (u, g) may be ascribed to the dot states (1,2). Then the SW projection procedure transforms the (a) tunneling term into cotunneling Hamiltonian Hˆ cotun (3.33). Since the triplet and the singlet states of doubly occupied dot are also bound to even and odd two-electron states [Fig. 4.5(a)], we come to the situation described by the effective SW Hamiltonian (3.62). The second cotunneling term in Eq. (3.62) is responsible for singlet-triplet mixing. Although the dynamical symmetry of the problem is the same SO(4) symmetry of the S/T multiplet, one cannot introduce generators S and R and has to use spin-fermion representation instead. The weak coupling scaling theory reformulated in these operators, naturally, gives the same scaling equations (4.52) for the exchange parameters ju = ρ0 Ju and jug = ρ0 Jug in the Hamiltonian (3.62) and predict similar evolution of Kondo temperature as a function of the control parameter as is shown in Fig. 4.7. As was mentioned above, the control parameter is Δ T S itself. The two-stage Kondo screening procedure may be described also in terms of successive screening of T/S transitions

134

4 DYNAMICAL SYMMETRIES IN THE KONDO EFFECT

at E > |ΔT S | and then elimination of spin Su at low energies. It is clear, however, that these spin operators cannot be identiﬁed with spins of the two electrons in a quantum dot, and there is no isomorphism between the Kondo problem for a dot with double occupation and the two-site Kondo problem discussed in Section 4.1. One may derive the scaling equations in the weak coupling regime even without simplifying assumption (4.61). In this case [99] the evolution of parameters in the intermediate energy scale is slightly more complicated, but eventually the diagram Fig. 4.7 for the evolution of TK as a function of the control parameter remains the same. The temperature dependence of tunneling conductance G(T ) should follow the law (4.25) in the appropriate temperature interval T TK . In order to judge about tunneling related conductance in the strong coupling regime, we again resort to estimates of the scattering phase shift. In case of even-odd symmetry of the leads the tunneling conductance is given by the equation [129, 281] G=

e2 sin2 (δeσ − δoσ ) h ∑ σ

(4.63)

As usual one should appeal to the Friedel sum rule which tells that

∑ ∑ δnσ = 2π

(4.64)

n=1,2 σ

because the dot is doubly occupied. On a triplet side far from the critical value of

ΔT S = 0 both levels are occupied and screened due to the Kondo effect, so that δnσ = δn = π /2

(4.65)

in the unitarity limit. On the singlet side there is no Kondo screening and both electrons occupy the level 1. As a result

δ2σ = δ2 = 0, δ1σ = δ1 = π

(4.66)

In both limits the transition probability is zero in accordance with Eq. (4.63). Therefore, the Kondo effect induced tunneling transparency and hence the tunneling conductance at zero T should reach some maximum in the S/T crossover region. NRG calculations [166] verify this reasoning (see Fig. 4.9). It should be noted, however, that the deviation from the universal behavior (4.63)(4.66) is possible provided the difference in magnitudes of the two Kondo temperatures corresponding to two-stage Kondo screening is large enough [338]. If one

4.2 Kondo effect in quantum dots with even occupation

135

Fig. 4.9 Numerical RG calculation of the phase shifts δσ (upper panel) and ZBA peak in conductance (lower panel) in a two-orbital DQD with N = 2 as a function of interlevel distance ε12 (after [166]).

denotes the Kondo temperatures related to the high energy SO(4) regime (4.52) and the low-energy region of SU(2) regime as TK2 and TK1 , respectively, then under the condition T 1 1 1 (4.67) ln K2 = − TK1 j1 j+ there exists an exponentially broad energy/temperature range where the underlying physics is that of a one-channel spin 1 Kondo model with partially screened spin 1 (reduced to a spin 1/2). In this energy interval the residual spin is ferromagnetically coupled to the conduction sea and this coupling gives logarithmically singular corrections to the phase shift δ (ε ). Such behavior results in the corresponding deviations of the conductance G(B, T ) from the above scenario (see [338] for further discussion). Two-orbital conﬁguration may also be realized in vertical dots, where the Larmor shift induces crossing of two orbital levels with different main quantum numbers n of the Fock – Darwin spectrum together with S/T crossover of many-electron state

136

4 DYNAMICAL SYMMETRIES IN THE KONDO EFFECT

[365]. This crossing is also marked by signiﬁcant enhancement of Kondo related conductance. The mechanism of magnetic ﬁeld induced Kondo effect on the singlet side of the phase diagram due to Zeeman splitting works with some modiﬁcation also in the two orbital Anderson model (see [343] for details). We will return to this problem in Section 4.3.5.

4.3 Kondo physics for short chains In the early period of theoretical studies of tunneling through nanoobjects, the general model of a short chains of quantum dots under strong Coulomb blockade restrictions in a contact with leads was formulated in terms of the Mott – Hubbard picture [222, 382]. The theory was based on the idea that electrons injected from the source do not lose coherence when propagating through the sequence of quantum dots until they leave the chain for the drain electrode. This approach is valid only for short enough “Hubbard chains”, where the tunneling V between the adjacent valleys exceeds the tunneling W between the dot and the metallic leads. Using the generalized Landauer method, one describes the tunneling transparency in terms of the Green functions of a nanoobject in contact with the leads [281]. Such a procedure starts with diagonalization of the Hamiltonian of a nanoobject with subsequent calculation of renormalization of the spectrum of quantum dot due to tunneling contact with the leads. Early studies of this problem concentrated on the calculations of the Coulomb blockade peaks which arise with changing the occupation of the valleys (Coulomb staircases, Fig. 3.3). In terms of the Hubbard model, the Coulomb resonances are the Hubbard ”minibands” [382], which arise as a result of collective Coulomb blockade [138] (Hubbard repulsion). It is known, however, that the spectral function of the Hubbard model contains also the central peak of predominantly spin origin. The analog of this peak is responsible for zero-bias anomaly in the tunneling conductance, which is at the center of our attention in this chapter. As was pointed out in Section 3.5, nano-chains may be arranged in serial, parallel, T-shaped and cross geometries. Some examples of these geometries for symmetric dimer and trimer chains are shown in Figs. 3.13(a-d) and 3.14(a-d). In this section we will describe various manifestations of dynamical symmetries in Kondo physics of these symmetric conﬁgurations. In case of DQD these manifestations have much in common with those discussed in Section 4.2 in case of even occupation where

4.3 Kondo physics for short chains

137

the basic dynamical symmetry is SO(4). In case of odd occupation new features are related to SU(4) and SU(3) symmetries. More intriguing is the situation with linear trimers (TQD) where complex interplay of several SO(n) symmetries makes Kondo tunneling even more multifarious. Additional effects arise in T-shape geometry, where the side dots themselves may be arranged in various conﬁgurations.

4.3.1 Serial geometry A DQD in serial geometry [Fig. 3.13)a)] is the simplest experimental realization of complex quantum dot. Various manifestations of Kondo effect in this geometry have been investigated by means of all the available theoretical methods both analytically and numerically [25, 48, 99, 129, 178, 331, 391, 235]. Here we formulate the problem in terms of dynamical symmetries and concentrate on the symmetry related aspects of Kondo tunneling through serial DQD. In all cases we conﬁne ourselves with the limit V > W , where the manifestations of dynamical symmetries are especially distinct. As was shown in Section 3.5.1, the Hamiltonian Hˆ DQD (3.27) describes the two-level system with N electrons in the double quantum well. In the sector N = 1 two-level spectrum of the symmetric DQD is characterized by the energy gap ε12 = |W | [see Eq. (2.38)] between two levels and the effective cotunneling Hamiltonian for the serial DQD with single electron occupation is given by Eq. (3.34). Dynamical symmetry of spin and orbital excitations in this charge sector is given by the SU(4) group. Since the gap ε21 introduces additional energy scale in the weak coupling limit, the two-stage scenario of Kondo screening is realized [99]. At E ε12 all orbital degrees of freedom are involved in the Kondo effect together with spin degrees of freedom and the scaling equation for the exchange coupling is d j/d η = −4 j2 .

(4.68)

Solution of this equation gives the Kondo scale TK1 = D¯ exp(−1/4 j).

(4.69)

If the gap is wide enough, ε12 TK1 , then the scaling contraction can be continued till the energy interval E ε12 where the orbital excitations are frozen and the low-energy Kondo temperature is deﬁned only by the spin degrees of freedom:

138

4 DYNAMICAL SYMMETRIES IN THE KONDO EFFECT

TK2 = D¯ exp(−1/2 j),

(4.70)

where D¯ ∼ ε12 . Like in previous cases of the two-stage Kondo screening, the system eventually reaches the unitarity limit δ↑ = δ ↓= π /2, but the temperature dependence of the conductance is not universal, i.e. depends on the scale ε12 characterizing SU(4) → SU(2) crossover.

TK( ε12) / TK (0)

1

Fig. 4.10 Kondo temperature for doubly occupied QD in serial geometry as a function of the energy gap ε12 .

0

ε 12

If ε12 TK1 , then the second stage does not occur down to the lowest temperatures and in this respect the character of Kondo screening resembles that for a single-channel Kondo effect near the S/T crossover point. The case ε12 ∼ TK1 reminds the situation with non-universal Kondo temperature described by Eq. (4.56). The SU(4) symmetry of Kondo scattering survives in the strong coupling regime and the phase shift in the unitarity limit is π /4 per each channel in accordance with the Friedel sum rule for N = 1. Evolution of TK (ε12 ) is presented in Fig. 4.10. The Zeeman splitting in ﬁnite magnetic ﬁeld is detrimental for the spin channel but it does not affect the orbital pseudospin variables. Thus in the regime ε12 TK1 magnetic ﬁeld induces SU(4) → SU(2) crossover, where the spin degrees of freedom are suppressed at B TK2 while the pseudospin ﬂuctuations are responsible for the Kondo effect. The studies of Kondo tunneling in the charge sector N = 2 on the whole follows the scenario of the two-level quantum dot discussed above, and the analogy with the two-impurity Kondo model is frequently used in these discussions. We refer to the original papers cited above for details. Next we turn to manifestations of the Kondo effect in a serial TQD. This geometry provides more possibilities for interplay between orbital and spin degrees of freedom, primarily because the central dot is not coupled with the leads but rather

4.3 Kondo physics for short chains

139

Fig. 4.11 TQD with open central dot (after [68]).

plays part of mediator between the two side dots [Fig. 3.14(a)]. The mediation mechanism may be controlled, e.g. in the devices, where the central dot is large enough so that the Coulomb blockade in it does not suppress charge ﬂuctuations like in the experimental realization shown in Fig. 4.11. In this device [68] the central dot is practically open metallic box due to connection with the reservoir, so that it mediates the RKKY interaction I between the spins localized in the side dots (provided the occupation of these dots is odd). Each of these side spins is coupled with its own lead by means of antiferromagnetic effective exchange, so that the Kondo effect in the side dots competes with the trend to parallel orientation of their spins (I < 0) or net zero spin of the TQD (I > 0). In both cases the tunneling is tuned due to nonlocal spin control. Theoretical studies of this conﬁguration [380, 406, 411] conﬁrmed the resemblance of this type of TQD to the two-site Kondo model. The minimal fewelectron model which mimics this type of Kondo effect in TQD is a doubly occupied dot with spectrum shown in Fig. 3.18(d). In this model with N = 2 the effective interaction I Sl · Sr is provided by the double exchange due to virtual hopping of the side electrons to the central site shown by the dashed arrows. More interesting from the point of view of dynamical symmetries is the case of TQD with strong Coulomb blockade in the central dot with the spectrum shown in Fig. 3.18(e). We will repeatedly turn to this nanoobject in view of various Kondo regimes for various occupations, which may be realized in this conﬁguration. Here we start the study of serial TQD in the charge sector with even occupation N = 4 [245]. The four-electron multiplet is formed by two singlets, two triplets and charge transfer singlet exciton with doubly occupied side dots, so that the basic dynamical symmetry is SO(9). Even neglecting the high energy excitonic state, we are left with

140

4 DYNAMICAL SYMMETRIES IN THE KONDO EFFECT

SO(8) symmetry which promises rather rich phase diagram for the Kondo effect. The energy levels EΛ are given by the following equations ETs = εc + εs + 2εs¯ + Qs¯, ESs = εc + εs + 2εs¯ + Qs¯ − 2Ws βs

(4.71)

Here the indices s = l, r and s¯ = r, l stand for the side dots, Qs is a Coulomb blockade parameter in side dots, Qc → ∞ excludes double occupation of the central dot, the central level is deep so that βs = W /(εs − εc ) 1. In a symmetric conﬁguration with mirror symmetry axis εs = εs¯ so that ETs = εc + 3εs + Qs , s = l, r ES+ = εc + 3εs + Q − 4W βs , ES− = ETa = εc + 3εs + Qs ,

(4.72)

In this conﬁguration non-bonding triplet and bonding singlet levels are degenerate and form a septet. In all cases one electron occupies the central dot, the next two electrons occupy the two side dots, and the fourth electron is distributed between the side dots. The analysis of Kondo screening begins with Jefferson-Haldane procedure introduced in Section 4.1. In this case instead of single scaling invariant εd∗ (4.13), we have four invariants EΛ∗ = EΛ (D) − π −1ΓΛ ln(π D/ΓΛ ),

(4.73)

and four scaling equations indexrenormalization group (RG)! – scaling equations

π dEΛ /d lnD = ΓΛ .

(4.74)

where ΓΛ = πρ0 |WΛ |2 is the tunneling rate for the last electron in a 4-electron state |Λ . Here we deal with one more example of decisive inﬂuence of dynamical symmetries available at high energies on the low-energy properties of complex quantum dots. Looking at the spectra (4.71) and (4.72), we note that the singlet states ESs are renormalized due to hybridization with the corresponding charge transfer excitons (last terms in the corresponding equations). This hybridization means appropriate transfer of electron density and reduction of tunneling rates,

ΓSs = (1 − βs2)ΓTs ,

(4.75)

4.3 Kondo physics for short chains

141

so that ΓSs < ΓTs . As a result the non-bonding triplet levels (and antibonding singlet state ES− in the symmetric conﬁguration) ﬂow downwards faster than the hybridized singlet states. Thus the difference in tunneling rates may result in level crossings. The initial conditions for the ﬂow equations (4.74), i.e. the initial positions of the levels εs , εc may be tuned in such a way that this level crossing will take place near the termination point D = D¯ [see Eq. (4.14)]. The levels which are degenerate or quasi degenerate with the accuracy of ∼ TK at D ∼ D¯ determine the dynamical symmetry group for the Kondo effect in complex quantum dot described by the Anderson ”Poor man’s scaling” procedure. Flow trajectories for different dynamical symmetries achievable in serial TQD are shown in Fig. 4.12. In all cases the Jefferson-Haldane renormalization begins with the singlet ground state and terminates in a crossover regime. The most symmetric conﬁguration with mirror plane and energy spectrum described by Eq. (4.72) demonstrates the highest dynamical symmetry P × SO(4) × SO(4) where P stands for the l − r permutation. Here all scaling trajectories cross in the same point [Fig. 4.12(a)]. In the less symmetric conﬁgurations where the level degeneracy is absent in the initial conditions, one may tune the parameters in such a way that two triplets and one singlet or two singlets and one triplet intercross at D = D¯ and remain nearly degenerate at lower energies. These are the cases of SO(7) [Fig. 4.12(b)] and SO(5) [Fig. 4.12(c)] symmetries, respectively. The familiar cases of SO(4) symmetry (one singlet and one triplet) and SO(3) symmetry (one triplet, S=1) are also achievable. The full phase diagram of available dynamical symmetries as a function of two parameters x = Γl /Γr and y = (εl − εc )/(εr − εc ) is presented in Fig. 4.13. This diagram contains the whole zoo of SO(n) symmetries and even narrow sectors of survived singlet ground state (dashed areas). Crossovers between adjacent phases with appropriate change of TK may be tuned by means of the control parameters x, y. To ﬁnd TK for exotic SO(n) symmetries one has to write out effective SW Hamiltonians and then derive and solve the ﬂow equations for the coupling parameters (see [245] for detailed calculations). Since the Kondo effect for the SO(4) symmetry is already discussed in Section 4.2, we turn now to the SO(5) case, where the three vectors S, R1 , R2 (2.46) are responsible for Kondo tunneling and the ”passive” scalar A1 is involved in the Casimir operator (2.48). In accordance with the ﬂow diagram Fig. IV.11(c), the generators of the SO(5) group should be written in terms of the states |Sl , |Sr , |Tl ν and the triplet |Tr ν is detached. Then the spin vector S is related to the left dot, and the generators R1 and R2 are deﬁned in accordance with (2.46) by means of the fol-

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4 DYNAMICAL SYMMETRIES IN THE KONDO EFFECT

Fig. 4.12 Jefferson-Haldane ﬂow trajectories for energy levels in serial TQD with N = 4 corresponding to various effective dynamical symmetries of low-energy multiplets. Upper panel: P × SO(4) × SO(4); middle panel: SO(7); lower panel: SO(5) (after [245]).

4.3 Kondo physics for short chains

143

Fig. 4.13 Full phase diagram of dynamical symmetries involved in Kondo tunneling through serial TQD with N = 4 as a function of the parameters x = Γl /Γr and y = (εl − εc )(εr − εc ). Dashed sectors correspond to the singlet state with the symmetry U(1) (after [245]).

lowing substitution in Hubbard operators: S → Sl , E2 → Sr . Then the general SW Hamiltonian (3.15) in this speciﬁc case reads as ˜ 1 · srl + R ˜ 2 · slr ) + J4S · sr . Hex = J1 S · sl + J2 Rl · sl + J3(R

(4.76)

The last term in this Hamiltonian is generated on the second (Anderson’s) stage of the scaling procedure, so the scaling equations indexrenormalization group (RG)! – scaling equations d j1 /d η = − j12 + j22 + j32 /2, d j2 /d η = −2 j1 j2 , d j3 /d η = − j3 ( j1 + j4 ), d j4 /d η = − j42 + j32 /2

(4.77)

should be solved with the boundary conditions ¯ = ρ0 Ji (i = 1, 2, 3); j4 (D) ¯ = 0. ji (D) Solution of Eqs. (4.77) gives the Kondo temperature for the SO(5) multiplet:

(4.78)

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4 DYNAMICAL SYMMETRIES IN THE KONDO EFFECT

⎛ TK = D¯ exp ⎝−

⎞ 2 ⎠, j1 + j2 + ( j1 + j2 )2 + 2 j32

(4.79)

so that the last coupling parameter j4 remains irrelevant. Similar procedure for the SO(7) symmetry generates the effective Hamiltonian of the type (3.15), but the set of group generators consists of two spin vectors Sl and Sr and four vector Ri (i = 1 − 4) describing transitions between two triplet states and two singlet states. The cumbersome system of six scaling equation has been derived and solved in Ref. [245]. The resulting TK has the same form as Eq. (4.79) but with its own combination of exchange parameters in the exponent. In the symmetric case P × SO(4) × SO(4) even and odd combinations of the generators Ss and Rs arise in the effective Hamiltonian and the scaling equation, but the resulting Kondo temperature contains only even coupling constants j1g and j2g : 4 ¯ TK = D exp − √ . ( 3 + 1)(3 j1g + j2g )

(4.80)

The remaining states with SO(3) and P × SO(3) × SO(3) symmetries possess only trivial dynamical symmetries and need no special comments. To summarize, the investigation of this multistage and multiphase Kondo problem has uncovered a complicated phase diagram shown in Fig. 4.13. Each of the phases is characterized by its own TK , and in each ”critical” region the change of the symmetry is accompanied by nonuniversal behavior of TK of the same type as in the case of S/T crossover discussed in section IV.24.2. Several examples of sharp variation of TK as a function of the control parameter in the crossover regions are presented in Fig. 4.14. The corresponding routes are pointed out by dashed arrows in Fig. 4.13. As everywhere, the change of TK should be accompanied by the change of the tunneling conductance at ﬁnite temperature in accordance with the laws exposed in Section 4.1 [Eqs. (4.25),(4.26)]. One may expect peculiar modiﬁcations of the phase diagram in presence of the Zeeman splitting. If a TQD resides in a singlet domain (dashed sector which adjoins to the SO(5) line in Fig. 4.13), then the Zeeman splitting may compensate the energy gap ET 1l − ESr . In the SO(4) scenario for the S/T multiplet this compensation resulted in restoration of SU(2) symmetry in accordance with Eq. (4.26) and Fig. 4.5(b). In this case the actual subspace contains the states {T 1l , Sl , Sr }, i.e. it belongs to the class of three-level systems (see Fig. 4.15), so that the symmetry restored in magnetic ﬁeld is SU(3). All 15 generators of this group adjusted to the parameters

4.3 Kondo physics for short chains

145

TK êTK0 1 SO H7L SO H5L0.8

P×SO H4L×SO H4L

0.6 P×SO H3L×SO H3L

0.4 SO H4L

-4

-3

-2

0.2 -1

1

2

3

4

δrlêTK0

Fig. 4.14 Variation of TK in serial TQD with N = 4 as a function of the difference of the gate voltages δrl = vgr − vgr regulating the differences of the energy levels εl − εc and εr − εc . TK0 is the value of TK at the point of highest degeneracy P × SO(4) × SO(4) (after [245]).

of the model TQD Hamiltonian and expressed via Gell-Mann matrices (9.39) may be found in Ref. [245].

−1

Fig. 4.15 Magnetic ﬁeld induced SU(3) symmetry of Kondo tunneling in TQD with N = 4.

ET E Sl E Sr

0

B

1

Even more peculiar situation arises in the charge sector N = 3, whose low-energy spin multiplet contains two spin 1/2 doublets |B1,2 and a spin quartet |Q, 3 EB1 = εc + εl + εr − [Wl βl + Wr βr ] , 2 1 EB2 = εc + εl + εr − [Wl βl + Wr βr ] , 2 EQ = εc + εl + εr ,

(4.81)

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4 DYNAMICAL SYMMETRIES IN THE KONDO EFFECT

where βs = Ws /(εs − εc ) ≡ Ws /Esc . There are also four charge-transfer excitonic counterparts of the spin doublets separated by the charge transfer gaps ∼ εl − εc + Ql and εr − εc + Qr from the above states. We do not take these states into account. Like in the four-electron case, the scaling equations (4.74) may be derived with different tunneling rates for different spin states (ΓQ for the quartet and ΓBi (i = 1, 2) for the doublets). ΓQ = πρ0 Vl2 + Vr2 ,

ΓB1 = γ12 ΓQ , ΓB2 = γ22ΓQ ,

(4.82)

3 2 1 2 (4.83) γ1 = 1 − βl + βr , γ2 = 1 − βl2 + βr2 . 2 2 Since ΓQ > ΓB1 , ΓB2 , the scaling trajectories cross in a unique manner: This is the with

complete degenerate conﬁguration where all three phase trajectories EΛ intersect [EQ (D ) = EB1 (D ) = EB2 (D )] at the same point D . This happens at bandwidth D = D (Fig. 4.16). If this degenerate point occurs in the SW crossover region, ¯ the SW procedure involves all three spin states, and it results in the i.e., if D ≈ D, following cotunneling Hamiltonian Hex =

∑ (J1s S + J2sR) · ss ,

(4.84)

s=l,r

E 0.2

0.4

-3.8 -3.9 Fig. 4.16 Scaling trajectories resulting in the SO(4) × SU(2) symmetry of TQD with N = 3 (after [245]).

-D

0.6

0.8

D 1 Q B2

B1

-4 -4.1

D*

– ﬂow trajectories This is a somewhat unexpected situation where Kondo tunneling in a quantum dot with odd occupation demonstrates the dynamic symmetry features characteristic of dot with even occupation. The reason for this scenario is the speciﬁc structure of the wave function of TQD with N = 3. The corresponding wave functions |Λ are vector sums of states composed of a ”passive” electron

4.3 Kondo physics for short chains

147

sitting in the central dot and singlet/triplet (S/T) two-electron states in the l, r dots. Constructing the eigenstates |Λ , one concludes that the spin dynamics of such TQD is represented by the spin 1 operator S corresponding to the l − r triplet, the operator R realizing T/S transitions, and the spin 1/2 operator sc of a passive electron in the central well. The latter does not enter the effective Hamiltonian Hex (4.84) but inﬂuences the kinematic constraint via the Casimir operator C = S 2 + R 2 + s 2c = 15/4.

(4.85)

The dynamical symmetry is therefore SO(4) × SU(2), and only the SO(4) subgroup is involved in Kondo tunneling. The scaling equations have the form, 2 2 + j2s ], d j1s /d η = −[ j1s

d j2s /d η = −2 j1s j2s ,

(4.86)

From Eqs. (4.86) we obtain the Kondo temperature, TK = max{TKl , TKr },

(4.87)

with TKs = D¯ exp [−1/( j1s + j2s )]. This is the ﬁrst but not the last example of physical situation, where CQD with odd occupation mimics the properties of CQD with even occupation. We will meet below other cases of such behavior in other conﬁgurations of tunneling devices. Serial TQD with mirror symmetry and occupations N = 3 also demonstrate remarkable properties in tunneling Kondo regime, namely, the closeness to non-Fermi liquid two-channel Kondo effect discovered by Nozieres and Blandin [302]. We postpone discussion of this case till Section 4.3.5 where all possibilities of realizations of multichannel Kondo effect in CQD with dynamical symmetries will be considered.

4.3.2 Side geometry, Fano – Kondo effect Effective exchange Hamiltonians for the simplest case of DQD and TQD in sideconnected (T-shape) geometry [Figs. 3.13(b) and 3.14(b)] have been derived in Section 3.5.1 . Let us start our discussion of the Kondo effect for T-shape conﬁgura(b) tions with singly occupied dimer. The tunneling Hamiltonian Hˆ tun and the exchange

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4 DYNAMICAL SYMMETRIES IN THE KONDO EFFECT

(b) Hamiltonian Hˆ ex for this case are presented in Eqs. (3.31) and (3.33)], respectively.

Now we will discuss the problem in more details from the point of view of Kondo physics. As we already know, the symmetry of DQD with odd occupation is SU(4). Two ingredients of the operator algebra are electron spin 1/2 and pseudospin 1/2. The latter variable describes position of an electron in a double-well potential of twolevel systems [bonding and antibonding combinations (2.39)]. In the general case the tunneling Hamiltonian acquires the form (b) Hˆ tun = ∑[c†ekσ (Wb dbσ + Wa daσ ) + H.c.]

(4.88)

Wb = W cos θ , Wa = W sin θ .

(4.89)

kσ

with

Correspondingly, the exchange Hamiltonian expressed in terms of spin operators (3.33) is Hˆ ex = ∑ J pp (S pp, · s) (4.90) pp

Here p, p = a, b, index e is omitted because only the even combinations of the lead states are involved in cotunneling. Then the matrix JS may be diagonalized in the orbital space by means of standard frame rotation (2.40). The mixing angle is deﬁned as tan Θ = tan 2θ = 2Jab /(Jaa − Jbb).

(4.91)

This transformation excludes the antibonding states from the exchange Hamiltonian, and eventually we come to the standard form Hˆ ex = J(Sb · s)

(4.92)

with SU(2) symmetry. It should be noted that DQD with single electron occupation is a limiting case of the general model with one electron and two orbitals studied in Refs. [99, 343]. If two orbitals are independent, then the ratio (4.91) is not valid and both orbital modes survive in exchange Hamiltonian (3.39). In this general case two Kondo temperatures TK(α ,β ) corresponding to two coupling constants J(α ,β ) arise, and we return

to the two-stage SU(4) → SU(2) Kondo effect discussed above for other physical realizations of dynamical symmetries.

4.3 Kondo physics for short chains

149

Next we turn to the case of doubly occupied DQD in the side geometry [203]. This model was the ﬁrst physical system where the methods of dynamical symmetry groups have been used for discussion of the Kondo effect in quantum dots with even occupation. The energy levels of the DQD in the charge sector N = 2 are presented in Fig. 3.16. In all cases the ground state of the isolated DQD is spin singlet. However, in case of biased and asymmetric conﬁgurations S/T crossover may be induced by coupling with the lead states. The mechanism of this crossover was already discussed in Section 4.3.1 in connection with the Kondo effect in TQD with N = 4. In both cases shown in Fig. 3.16(b,c) the low-energy spectrum contains soft charge transfer exciton EEr hybridized with the ground state singlet ES , while the triplet state remains non-bonding [see Eqs. (3.43), (3.44)]. As a result the tunneling rates for triplet and singlet level are

ΓT = πρ0W 2 , ΓS = (1 − β 2 )ΓT

(4.93)

where β = 2W /(Q− δ12) and β = 2W /(δ12 + Q2 ) for cases (b) and (c), respectively. Then two Jefferson – Haldane scaling equations for ET and ES have the same form as (4.74) and the ﬂow trajectories intercross under conditions similar to those for TQD (Fig. 4.12). Thus we arrive at the single channel two-stage Kondo scenario for the SO(4) symmetry discussed in Section 4.2 [(see Eq. (4.51) and ff.].

Up to this moment we considered DQD in the side geometry under the conditions where the Coulomb blockade is strong in both constituent dots. New physical effects arise when one of these dots is large enough and the Coulomb blockade in this dot is weak. Two possible conﬁgurations of this type are presented in Fig. 4.17. In the ﬁrst of these conﬁgurations shown in panel (a) the source and drain leads are linked via nearly open metallic large dot whereas the side dot is supposed to be Coulomb blockaded. In the second case [panel (b)] Coulomb blockade is supposed to be weak in the side dot and the localized spin responsible for Kondo correlation sits in the dot coupled with the leads. Manifestations of the Kondo effect in these two geometries are strikingly different. In case (a) one may assume that electron transport between source and drain is not subject to Coulomb blockade and practically ballistic at zero coupling with a side dot. The a question arises what kind of Kondo correlations may develop in this regime. Qualitative answer may be given immediately if one recollects, that the Kondo effect results in appearance of the Abrikosov – Suhl resonance at the Fermi

150

4 DYNAMICAL SYMMETRIES IN THE KONDO EFFECT 1111111 0000000 0000000 1111111 0000000 1111111 0000000 1111111

Fig. 4.17 Two possible Tshape conﬁgurations of DQD consisting of large and small dots. (a) side dot with strong Coulomb blockade; (b) side dot with weak Coulomb blockade

1111 0000 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111

11111111 00000000 00000000 11111111 00000000 11111111 00000000 11111111 00000000000 11111111111 11111111111 00000000000 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000 11111111 11111111 00000000 00000000 11111111 00000000 11111111

1111111 0000000 0000000 1111111 0000000 1111111 0000000 1111111

(a)

(b)

level. Then the interference of this resonance onto the band states results in an antiresonance in this continuum which is known as a Fano effect originally discovered in inelastic scattering of electrons on He atoms [102]. Microscopic calculations for various models of T-shape DQD [96, 194, 211, 374, 390, 400, 401, 444] indeed show that the destructive interference of ballistic source-drain tunneling and Kondo resonance scattering results in the appearance of characteristic Fano-shape anomaly in the tunneling conductance.

Fig. 4.18 One-dimensional conducting chain connecting source and drain electrodes with side dot under strong Coulomb blockade

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s

111 000 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111

d

In numerical calculations of Fano – Kondo effect a simpliﬁed model is often used where the large dot is approximated by a periodic half-ﬁlled metallic 1D chain and the side dot is treated as an extra site attached to this chain and occupied by one electron in the presence of Coulomb blockade (on-site Hubbard interaction U preventing double occupation of this site). The wire is connected with source and drain electrodes (Fig. 4.18), and this contact transform the device into a nearly open nanoobject. The Hamiltonian of quasi 1D system with small and big dot built in a chain is Hˆ = Hˆ ch + Hdot + Hˆ coupl = −t ∑ a†n,σ an−1,σ + a†n,σ an+1,σ n,σ

4.3 Kondo physics for short chains

151

+ ∑ ε0 nd σ + Und↑nd↓ + ∑(V dσ† a0σ + H.c.) σ

(4.94)

σ

Here the chain site coupled with the side dot is labeled by the index n = 0. The energy levels of electrons localized at the chain sites coincide with the Fermi energy which is taken as the reference level εF = 0. To understand the origin of Fano effect in this conﬁguration let us assume that the dot spectrum contains a resonance level Er of any origin at zero energy corresponding to the Fermi level of the metallic subsystem. Let us formulate the problem in terms of Green functions (4.31) for chain and dot electrons. The side dot is coupled only with the site n = 0 in the chain. Therefore the component G0 of the chain Green function Gc located at the central site is connected with Gd (ω ) by the T-matrix equation (4.35), G0 (ω ) = g0 (ω ) [1 + T(ω )g0 (ω )] .

(4.95)

Here g0 is the local Green function at the central site of the chain without decoration, and the t-matrix is T (ω ) = V 2 G d ( ω )

(4.96)

The local Green function may be represented in the form g0 (ω ) = (ω + iΓ )−1 where Γ is the coupling strength of the central site with the other parts of the wire and the leads. The resonance Green function is assumed to have the Breit – Wigner form Gd (ω ) =

1 ω − Er + iΓ

(4.97)

where Γ is the resonance width of the quantum dot level [cf. Eq. (4.34)]. Then the Breit – Wigner-like expression for the phase shift related to the resonance level may be written [150]: Γ (4.98) tan δ (ω ) = − ω − Er As a result the defect Green function Gd (0) near the resonance acquires the form Gd (0) =

1 . −Γ cot δ + iΓ

(4.99)

Eventually using the ratio Γ = V 2 /Γ we put Eqs. (4.95)-(4.99) into the transmission probability W (0) = Γ 2 |G0 (0)|2 of an electron at the Fermi level propagating through the central site and calculate the tunneling conductance Gw of the wire with defect at zero temperature by means of Landauer-type formula

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4 DYNAMICAL SYMMETRIES IN THE KONDO EFFECT

Gw =

2e2 2e2 W (0) = cos2 δ . h h

(4.100)

Comparing this result [194] with the unitarity limit formula (4.22) for transmission probability of the standard Kondo resonance we see that instead of perfect transmission at the Fermi level we deal here with full suppression of transport. This dip in the tunneling conduction is a ﬁngerprint of Fano-type destructive interference between the resonance scattering and the coherent transport. Calculation of the Green function G(ω ) (4.32) may be performed in a framework of Anderson model with side connected impurity at arbitrary energy, and this calculation allows one to get an analytical expression for a Fano – Cooper factor in the spectral density of electrons in the wire with given spin projection which is determined by the imaginary part of the second term in Eq. (4.95):

δ ImG0σ (ω ) = Img0 (ω )

(q + εσ )2 1 + εσ2 (4.101)

ω − Er σ Reg0 (ω ) q(ω ) = , εσ (ω ) = Img0 (ω ) Γ The factor fσ (ω ) = (q + εσ )2 /(1 + εσ2 ) necessarily turns into zero at some ε , which in the general case does not coincide with the resonance position.

LDOS

Fig. 4.19 Left panel: Local density of states (LDOS) at the site n = 0 with attached dot (Fig. 4.18). Right panel: Zero bias dip in the tunneling conductance in presence of Fano – Kondo effect

G

0

ε

0

eV

Returning to the Kondo effect in the conﬁguration of Fig. 4.18, we note that the defect marked by the oval may be treated as a DQD with single electron. According to our previous ﬁndings, the basic SU(4) symmetry of this defect reduces to the conventional SU(2) spin symmetry with exchange Hamiltonian (4.92). If the level Er is identiﬁed with the Abrikosov – Suhl resonance at the Fermi, then the Fano suppression of transmission should occur at the Fermi level for both spin projections,

4.3 Kondo physics for short chains

153

so that the dip in the Kondo-type ZBA in the tunneling conductance should be observed instead of conventional peak (see Fig. 4.19 where the local density of states (LDOS) related to the site n = 0 and the corresponding dip in tunnel conductance are presented in the left and right panels, respectively). Numerical calculations for the ﬁnite chain with Coulomb blockaded side dots conﬁrm this picture [400].

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s

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d

0000 1111 1111 0000 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111

s

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d

Fig. 4.20 Double quantum dot attached to metallic chain in side geometry (left panel) and in cross geometry (right panel).

The temperature dependence of Fano antiresonance should follow the temperature dependence of the Kondo peak in the side dot, so that this anomaly weakens at T > TK and the dip in tunnel conductance becomes less pronounced and eventually disappears. The Fano - Kondo lineshape f (ω ) may be revealed in conductance at ﬁnite T by varying the gate voltage applied to the side dot. The natural generalization of this type of device is an attachment of double quantum dot to metallic chain [392] (see Fig. 4.20, left panel). This geometry is a speciﬁc case of side-connected TQD [Fig. 3.14(b)] with large left dot which provides metallic tunneling channel interfering with the Kondo tunneling channel related to the side DQD. In this case one may expect Fano effect induced by SU(4) or SO(4) Kondo effect in the side DQD with odd and even occupation, respectively. Following the general scheme described above, one may state that all variations of the Kondo resonances in the side dots should be reﬂected in the corresponding variations of the Fano antiresonances in the conductance of the metallic wire. Experimental manifestations of the Kondo – Fano effect in complex planar quantum dots are also reported (see, e.g., Refs. [144, 358, 366]), although this phenomenon is better seen in molecular nanoobjects. The Kondo effect and related phenomena in molecular complexes will be considered in Chapter 5. Now we turn to the second type of T-shape conﬁgurations where the weakly correlated dot is side-connected to the strongly correlated one [Fig. 4.17(b)]. If the

154

4 DYNAMICAL SYMMETRIES IN THE KONDO EFFECT

side dot is large enough so that the discreteness of its spectrum is negligible at ﬁnite temperatures, i.e. the average interlevel distance δ¯ ε T , then this dot will serve as K

one more source of Kondo correlations. This conﬁguration may be treated as a threeterminal device with one passive electrode. The corresponding effective exchange Hamiltonian for odd occupation of the small dot acquires the form Hˆ ex = ∑ J pp (s pp · S)

(4.102)

pp

where the index p enumerates lead channels [cf. Eq. (4.90)]. Then the matrix J s may be diagonalized in the same way as the matrix J S above, and the Hamiltonian acquires the form

∑ Jr (sr · S).

(4.103)

r

In the general case the ”color” index r equals 1,2,3, and we deal with three-channel exchange Hamiltonian. In the degenerate cases like (4.59) the number of channels is 2. If the occupation of the small quantum dot is even, then similar terms ∼ Jr (sr · R) should be added to the effective Hamiltonian. On the surface of it, the increase of the number of cotunneling channels does not change the universality class (inﬁnite ﬁxed point) of the Kondo effect, and its only result is the corresponding increase of the number of stages on the way to the low-energy SU(2) limit: in a weak coupling limit each color generates its own Kondo temperature TKr and establishes the hierarchy TKr1 > TKr2 > TKr3 . The channel r1 with largest Kondo temperature survives at low energy, and the crossover SU(2) × SU(2) × SU(2) → SU(2) takes place for odd occupation. In case of even occupation the picture is more complicated, because each TKr splits into two Kondo temperatures in accordance with the scenario discussed in Section 4.2, but the ﬁnal conclusion is the same: the channel with highest Kondo temperature suppresses all other channels. However, this simple scenario breaks if at least two of coupling constants Jr coincide [302]. In this case the universality class of Kondo effect is different. The ﬁxed point may become ﬁnite and the low energy thermodynamics of Kondo effect is characterized by non-Fermi liquid behavior. Accidental degeneracy of two or more coupling constants in general case is hardly probable, and this regime may be considered only provided there are some special reasons for channel isotropy. Such reasons were found for the T-shape geometry under consideration [314], but we postpone discussion of this peculiar Kondo effect till Section 4.3.5 where this

4.3 Kondo physics for short chains

155

regime will be considered together with other possible realizations of multichannel Kondo effect.

s

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d

Fig. 4.21 Three metallic chains coupled with quantum dot by means of tunnelling channels.

The T-shape geometry provides other combinations of many-body effects related to dynamical symmetries of complex quantum dots. In particular, a one-dimensional quantum wire may play part of side dot (Fig. 4.21). The Kondo effect for magnetic impurity attached to the edge of such wire has been considered by means of exact methods elaborated for the integrable 1D models [104, 114, 326]. These methods (Bethe Ansatz and Luttinger liquid bosonization) are beyond the scope of our book, so the reader is referred to the original papers for details. The T-shape geometry opens the way to formulation of the multichannel Kondo effect in Luttinger liquid in the case if the source and drain electrodes are also quantum wires (Fig. 4.21) or to superposition of Kondo effects for 1D and 2D reservoirs in case of planar quantum dots with attached metallic wire. These problems are still open.

Fig. 4.22 Side-connected TQD. Open large dot M between two small dots S1 , S2 under strong Coulomb blockade (after [406]).

A side-connected large dot may be used as a medium for exchange interaction between the chain and side spins localized in the small dots (Fig. 4.22). Such geometry

156

4 DYNAMICAL SYMMETRIES IN THE KONDO EFFECT

[406] is closely related to the Harvard experiment [68]. One may treat this conﬁguration as a two-site Kondo defect controlling tunneling transport between source and drain electrodes. Changing the sign of the RKKY interaction I between the localized spins S1 and S2 in the Hamiltonian (4.46), one may induce the crossover from underscreened Kondo effect for total spin S = 1 to the non-Kondo regime in case of singlet state of T-shaped TQD accompanied by appropriate changes in the tunnel conductance.

4.3.3 Cross geometry Kondo tunneling for a cross geometry [Fig. 3.14(c)] has much in common with Kondo tunneling for a side geometry, but it has its own speciﬁc features due to the mirror symmetry of a TQD [248]. Effective exchange Hamiltonians for TQD in the cross geometry in charge sectors N = 1, 2 were analyzed in Section 3.5. (c) The Hamiltonian Hˆ ex (3.40) for a singly occupied TQD promises nothing new in comparison with the conventional S=1/2 Kondo effect. In case of double occupation the system preserving both left-right and top-bottom parity is mapped on the Kondo problem of two-orbital/two-lead exchange Hamiltonian discussed in Sections 4.2 and 4.3.1, so here we will not linger round these cases. Some unexpected features of Kondo tunneling are seen in the charge sector N = 3 with weakly correlated central dot and strongly correlated side dots [see Fig. 4.5(d) for the structure of energy levels]. In a triply occupied TQD the lowest energy states are two spin doublets, (even and odd relative to the l ↔ r permutation), a spin quartet state, and a doubly degenerate charge transfer exciton (with two electrons in the central dot). The corresponding energies are EDu = 2εs + εc − 3W 2 /Δ , EDg = 2εs + εc − W 2 /Δ , EQ = 2εs + εc ,

(4.104)

EEx = εs + 2εc + Qc + 2W 2 /Δ . Here Δ = Δ + Qc and the inequality β = W /Δ 1 is assumed to be valid as usual. The ground state of the TQD, spin doublet EDu is odd relative to the left-right permutation of orbitals.

4.3 Kondo physics for short chains

157

After performing the SW-like canonical transformation, one gets the effective exchange Hamiltonian Hex = −Ju Su · s,

(4.105)

where Su is the spin 1/2 vector operator with components Su j = 12 |Duσ τ j Duσ |, and s is the spin operator of lead electrons deﬁned via even combinations of lead electrons, s = 12 ∑k c†ekσ τ cekσ . Remarkably, the exchange constant Ju has ferromagnetic sign! To understand this anomalous sign one has to look at the structure of the three electron wave function |Duσ :

(b†cr dl†σ − b†cl dr†σ ) † † † √ |Du σ = cos θu + sin θu bcc (dl σ − drσ ) |0. (4.106) 3 √ † † † † d j↓ − di↓ d j↑ (1 − δi j )]/ 2, and the rotation angle is given by θu = Here b†i j = [di↑ √ arcsin( 3W /Δ ). The exchange coupling constant equals 2 cos2 θuV 2 1 1 Ju = − (4.107) + 3 |εc | |εc + Qc | The reason for an unconventional sign of the exchange interaction is that only one of the three electrons in the TQD is involved in the exchange interaction with the leads, and the overlap of the two other electrons wave function entering the state |Du σ gives the factor −1. As is known, ferromagnetic exchange coupling is irrelevant for the Kondo effect because the scaling procedure (4.18) renormalizes it into zero at low energies. Thus, in this geometry we encounter a unique situation, where the Kondo screening is ineffective for a quantum dot with odd occupation. It is the second (but not the last) example of unusual behavior of TQD in the Kondo regime [cf. Hamiltonian (4.84) for triply occupied TQD is serial geometry]. One more peculiar feature of this conﬁguration is the possibility of level crossing due to different tunneling rates of the states in the octuplet (4.104). Indeed the tunneling rate for the nonbonding quartet |Q is given by the usual equation

ΓQ = πρ0W 2 . The rate for mixed states |Du and |Dg are ΓD(u,g) ≈ πρ0 cos2 θ(u,g)V 2 , where the angle θg = arcsin(W /Δ ) and the angle θu is deﬁned above. As a result the hierarchy ΓQ > ΓDg > ΓDu arises. Due to this hierarchy a level crossing is feasible in the Jefferson – Haldane ﬂow diagram determined by Eqs. (4.73), (4.74). It turns out that all three ﬂow trajectories cross in the same point (Fig. 4.23). Thus we arrive at a situation where the basic dynamical symmetry SU(2) × SU(2) × SU(2) varies depending on the initial conditions (positions of the levels ¯ of Jefferson – Haldane scal(4.104) which deﬁne the termination point D¯ ≈ EΛ (D)

158

4 DYNAMICAL SYMMETRIES IN THE KONDO EFFECT

Fig. 4.23 Jefferson-Haldane ﬂow trajectories for low-energy levels in TQD with N = 3 in a cross geometry with mirror symmetry.

ing). If this condition is fulﬁlled in the vicinity of the crossing point (D¯ = D¯ cr ), the octuplet as a whole contributes the exchange Hamiltonian, which acquires the form Hˆ ex = Ju Su · s + Jg Sg · s + JQSQ · s + JRR · s,

(4.108)

It is expressed in terms of operators for localized spin 1/2, Su,g , spin 3/2, SQ , and the vector R which induces transitions between the quartet |Q and the doublet |Du . There are no transitions between |Q and |Dg since these states have opposite l − r parity. Both to the right and to the left of this point some of the states in the spin multiplets are quenched at T → TK , and TK depends on the energy gaps Δ Qg = ¯ − EDg (D) ¯ and Δ gu = EDg (D) ¯ − EDu (D). ¯ To ﬁnd the function TK (ΔQg , Δ gu ), EQ (D) one should solve the scaling equations for the coupling constants in the Hamiltonian (4.108) d jg d ju = −[ ju2 + 2 jR2 ], = − jg2 , d lnd d ln d jR d jQ d jR 2 = −[ jQ = − (5 jQ − ju ), − jR2 ], d lnd d ln d 4

(4.109)

The procedure is self-consistent because TK itself predetermines the characteristic energy interval for states in the spin Hamiltonian involved in its formation. Varying D¯ in Fig. 4.23 from the ferromagnetic non-Kondo regime (D¯ ∼ D¯ u ) to the crossing point D¯ cr , one reaches the point TK > 0 which arises due to inﬂuence of the excited

4.3 Kondo physics for short chains

159

¯ and EDg (D). ¯ Just then, TK sharply increases, reaching its maximum states EQ (D) value in the point of maximum degeneracy D¯ cr . Moving further to the left, the level EDu freezes out. This means that the vector R in the Hamiltonian (4.108) does not contribute anymore to Kondo co-tunneling, and the Kondo effect is determined by the pair of states EQ and EDg , with the dynamical symmetry of the TQD being SU(2) × SU(2). Further decrease of D¯ eventually results in the quenching of EDg . The system then exhibits an under-screened Kondo regime of a localized spin 3/2 moment. Fig. 4.24 illustrates these crossover effects on TK . Tunneling conductance follows variations of dynamical symmetry in the usual way.

TK TKcr 1 0.8 0.6 0.4

Fig. 4.24 Non-universal behavior of TK in TQD with N = 3 in the cross geometry with mirror symmetry.

0.2

DQ

0 0

0.25

Dcr 0.5

0.75

Du 1

D

The novelty of the present scenario is that the type of non-universal behavior of TK known for dots with even occupation with S/T crossover [99, 166, 203, 343, 345] is manifested in a quantum dot with odd occupation where the absence of Kondo effect occurs due to ferromagnetic exchange coupling with localized spin doublet. This ferromagnetic exchange competes with two anti-ferromagnetic exchange interactions (with doublet and quartet localized moments), so that, in some sense, one deals here with a ”three-stage” Kondo effect. A cross geometry was used also in the description of Fano – Kondo interference [315] (see Fig. 4.20, right panel). If the energy levels and the tunneling integrals for two dots 1,2 are different, ε1 = ε2 , W1 = W2 , each of these dots generates its own Fano – Kondo resonance. Besides, the indirect exchange between spins localized in sites 1,2 via metallic quasicontinuum in the wire may order them in a singlet or triplet. If the RKKY coupling constant I has ferromagnetic sign, then the Kondo impurity is underscreened, and the Green function Gd (ω ) in Eq. (4.95) contains ad-

160

4 DYNAMICAL SYMMETRIES IN THE KONDO EFFECT

ditional non-resonant term induced by residual scattering involving underscreened spin degrees of freedom. This means that the the phase shift δ (0) = π and the Fano destructive interference not completely suppresses the tunneling transmission probability. In case of antiferromagnetic sign of I , the RKKY exchange competes with two Kondo resonances, and the existence and the shape of the Fano dip depends on the results of this competition.

4.3.4 Parallel geometry Various conﬁgurations of CQD in the parallel geometry are shown in Figs. 3.13(c,d) and 3.14(d). The effective exchange Hamiltonians for the charge sectors N = 1, 2 are derived in Section 3.5. The ”Leitmotif ” of Kondo physics in these conﬁgurations is a two-stage screening of localizes spins in a framework of SU(4) and SO(4) dynamics of spin Hamiltonians for odd and even occupation, respectively. Various theoretical approaches have been used for the description of Kondo tunneling in parallel DQD [165, 179, 195, 280, 332, 391, 437]. One should distinguish between the ”which pass” geometry of Fig. 3.13(c) and the two-channel geometry of Fig. 3.13(d). Physically, this difference means that if the coherence length for the conduction electrons in the leads exceeds the distance between the ”entrances” to the left and right lead-dot channels then the which pass regime for tunneling electrons is realized. If the leads are dirty enough so that the electron mean free path is less than the interchannel distance, then tunneling through the left and right dot are independent processes which interfere only due to the tunneling and/or capacitive contact between two dots. As is shown in Section 3.5.1, the exchange Hamiltonian for parallel DQD with odd occupation in the which pass regime has conventional SW form (3.40), so that this case needs no special discussion. One should note, however, that the behavior of this system changes radically in a perpendicular magnetic ﬁeld which breaks the chiral symmetry of left and right channel as a result of the Aharonov – Bohm effect [179, 195]. This effect will be discussed in Section 4.4.1 for various geometries where the Aharonov – Bohm loops may be realized. Two-channel DQD with odd occupation possesses SU(4) symmetry, the cotunneling is described by the Hamiltonian (3.35), the same as in case of serial geometry, and the Kondo problem is mapped on the two-orbital/two-channel model described in Section 4.3.1. In case of even occupation tunneling through parallel DQD demonstrates characteristic fea-

4.3 Kondo physics for short chains

161

ture of the two-stage Kondo effect and singlet/triplet criticality described in previous sections both in which pass [165, 391] and two channel [437] conﬁgurations. The easiest way to fabricate triple quantum dots in parallel geometry with two terminals is to exploit the vertical dot conﬁguration shown inFig. 3.15(d). The twochannel regime shown in Fig. 3.14(d) provides especially rich phase diagram. The number of possible models of Kondo tunneling for various combinations of the input parameters responsible for disposition of the electron levels (Fig. 3.18) and occupation numbers N = 2, 3, 4 . . . is enormous, so we content ourself with the model of strongly correlated central dot and weakly correlated side dots, Fig. 3.18(d) with few electrons. We have already seen in Section 4.3.1 that the Kondo tunneling in serial TQD in the charge sector N = 4 is characterized with the whole bunch of dynamical symmetries from SO(3) to SO(7). All these states arose as a result of solution of the scaling equations (4.74) and (4.77). The same picture with minor modiﬁcations emerges in parallel geometry. Modiﬁcation is necessary because the Jefferson – Haldane scaling procedure generates a new vertex. In addition to renormalization of the dot energies exempliﬁed in Eqs. (4.9) new vertex MlrΛΛ X ΛΛ arises provided the high energy states k+ are extended enough to envelop both the left and the right tunneling channel. Here

MlrΛΛ =

αlr |W 2 | ∑ 2 k=k+

1 1 + εk − EΛ + Eλ εk − EΛ + Eλ

,

(4.110)

αlr is measure of the overlap of the two channels via Bloch states in the leads, the states ΛΛ are either two triplets or two singlets from the set (4.72). The ﬂow trajectory of this vertex intermixing left and right tunneling channels is given by the equations of the type dMlr γ (4.111) =− dη D with a ﬂow rate γ = αρ0W 2 . The mixing vertices induce avoided crossing effect in ¯ < TK , this the ﬂow trajectories of Fig. 4.12 [see insets in panels (a,b,c)]. If Mlr (D) effect does not change the general phase diagram shown in Fig. 4.13. We have seen that a wide variety of phases with different dynamical symmetries TQD in the parallel conﬁguration of Fig. 3.14(d) arises due to the parity related left-right permutation. This parity ”doubles” the number of singlet and triplet states at even occupation. In the charge sector N = 2 within the energy scale E < 2εs the spectrum consists of two degenerate triplet states and two nearly degenerate

162

4 DYNAMICAL SYMMETRIES IN THE KONDO EFFECT

singlet states with basic dynamical symmetry SO(8) like in the sector N = 4 considered above. Different ﬂow rates for non-bonding triplets and bonding singlets are also expected, so the singlet-triplet level crossing is also possible in this case and the phase diagram should also be rich enough. As is already noticed in Section 3.5.2, remarkable effective Hamiltonian (3.70) arises at low energies in the singlet segment of the phase diagram which is usually of no interest for Kondo tunneling. However in doubly occupied TQD the multistage Kondo effect starts with maximum SO(4) × SO(4) symmetry of two quasidegenerate S/T multiplets. The triplet components of these multiplets become frozen at low energy, and the remaining degrees is SU(2) × SU(2) symmetry of orbital doublet with blocked spin ﬂip processes. Due to this blockade the Hamiltonian (3.70) looks very promising for the search of nonFermi liquid two-channel Kondo regime. In the next section we will consider this model together with other candidates for non-Fermi liquid singularities in Kondo tunneling.

4.3.5 Multichannel Kondo tunneling Nozieres and Blandin in their seminal paper [302] have noticed that in case when the number of scattering channels n (tunneling channels in our case) satisﬁes the inequality n > 2S, the low-energy properties of Kondo model change radically. In a model with n replicas of metallic electron band possessing SU(n) × SU(2) symmetry the RG ﬁxed point is ﬁnite and its position depends on the ratio n/2S. The physical reason of the radical change of scaling picture in comparison with the standard case n = 1 is that in this regime the local spin S = 1/2 is overscreened by n spins of neighboring conduction electrons, interaction of the resulting spin S with the next neighboring electrons is antiferromagnetic, so the overscreening multiple scattering repeats once more, and so forth. This means that the inﬁnite ﬁxed point is in fact unstable in the multichannel case. To treat this problem formally, one may use the weak coupling RG approach [19] and represent the results of the renormalization procedure by means of perturbation expansion of the S-matrix operator S (∞) ˆ S (∞) = T exp(−i

∞

−∞

Hˆ ex (t)dt

(4.112)

4.3 Kondo physics for short chains

163

where the operator Hˆ ex (t) describing the effective exchange between the dot and the band spin states should be written in terms of the generators of the appropriate symmetry group and taken in the interaction representation ˆ ˆ Hˆ ex (t) = eiH0 t Hˆ ex e−iH0 t

(4.113)

with Hˆ 0 = Hˆ b + Hˆ d . Using the pseudofermion representation for group generators (see Section 9.3) and taking into account the commutation relations for generators in time-ordered products, (see, e.g. [2, 67, 113] for technical details) one may derive the Gell-Mann – Low equations for the renormalized coupling constant and write out Feynman diagrams for the terms in their right-hand sides, i.e. ﬁnd the β – functions β ({ ji }) which predetermine the form of the ﬂow trajectories. Here { ji } is the set of dimensionless vertices in the exchange Hamiltonian.

j0

8

0

0

jc

8

j

j

Fig. 4.25 Upper panel: Bare vertex j0 and diagrams representing RG corrections to j0 for twochannel Kondo effect. Solid lines correspond to electron propagators, dashed lines stand for spinfermion propagators, which describe spin excitations in a QD. Lower panel: Flow trajectories for single channel and two channel Kondo effect .

All β – functions discussed up to now [see Eqs. (4.17), (4.52), (4.77), (4.86), (4.108)] correspond to a single loop approximation illustrated by the second diagram in the upper panel of Fig. 4.25. The solid lines in the diagrams shown in this ﬁgure stand for propagators for the band electrons (only the lines corresponding to states k+ are shown). The dashed lines denote pseudofermion propagators representing the generators Si and Ri of the corresponding SU(n) or SO(n) group. In the case under consideration only the spin operator S enters the bare and the renormalized vertices in Fig. 4.25. It was noticed in Ref. [302] that adding the 3-rd order diagram with closed electron-hole loop (right diagram in the upper panel of Fig. 4.25) radically changes

164

4 DYNAMICAL SYMMETRIES IN THE KONDO EFFECT

the properties of the β – function. The scaling equation now has the form dj n = − j 2 + j3 . dη 2

(4.114)

It is seen from Eq. (4.114) that unlike the single loop approximation (4.17), scaling stops not at inﬁnity but at ﬁnite scaling point jc (ηc ) = 2/n. The inﬁnite ﬁxed point is indeed unstable in accordance with the above qualitative arguments. The single channel and two channel ﬂow trajectories are compared in the lower panel of Fig. 4.25. The scattering phase δ (0) = π /4 in the unitarity limit. One may derive the effective Hamiltonian intended to describe elementary excitations and correlation functions in the nearest vicinity of the ﬁxed point by the methods available in the theory of integrable 1D systems [Bethe Ansatz, Tomonaga - Luttinger bosonisation and the current algebras of conformal ﬁeld theory (see [3] for a review and references). Now the spectrum may be decomposed into charge, spin and channel components. The main physical result of this approach is that the leading operators at the ﬁxed point have non-trivial scaling dimensions, so that ηc is a non-Fermi liquid ﬁxed point. As a result low temperature corrections to physical observables also have non-Fermi liquid scaling dimension ∼ (T /TK )α . In particular, corrections to the tunneling conductance should have the form

δ G(T )/G0 ∼ (T /TK )1/2 .

(4.115)

Fig. 4.26 RG ﬂow diagram for the two-channel Kondo effect with channel anisotropy.

The challenging goal of the experimental detection of the non-Fermi liquid regime in Kondo tunneling through quantum dots meets almost insurmountable stumbling block: the ﬁxed point jc is unstable against channel anisotropy [302].

4.3 Kondo physics for short chains

165

In fact it is a saddle point in the plane ( j1 , j2 ) in the two channel conﬁguration (see Fig. 4.26). The problem with derivation of the two channel Kondo Hamiltonian should start with a two channel Anderson Hamiltonian with tunneling term like Hˆ db = ∑ ∑ Wa dσ† craσ + H.c. ,

(4.116)

kσ a

where the index a = l, r enumerate channels. In the general case there is no mechanism of channel conservation in the process of cotunneling, and the effective SW Hamiltonian has the form Hˆ ex = ∑ Jab sba · S

(4.117)

ab

where Jab ∼ WaWb /Δ is the matrix of exchange parameters, and sab is the corresponding matrix of itinerant spin operators. The product of this matrices may be diagonalized in a usual way by means of transformation matrix (2.39). As a result one comes to anisotropic two-channel Hamiltonian Hˆ = ∑

∑

kσ α =1,2

where J(1,2) =

εkσ c†α kσ cα kσ + ∑ Jα sα · S,

(4.118)

α

1 (Jll + Jrr ) ∓ (Jll − Jrr )2 + 4Jlr2 . 2

(4.119)

Thus the channel anisotropy is a generic attribute of the two-channel cotunneling Hamiltonian. The general strategy in search of elusive non-Fermi liquid regime includes three directions: (i) to minimize the anisotropy and thus approach the saddle point so close that the critical non-Fermi liquid behavior will be experimentally achievable; (ii) to design a multichannel geometry in such a way that the interchannel exchange is suppressed at least for one pair of channels; (iii) to make channel conservation symmetry protected. Direction (i) is exempliﬁed by the study of a serial TQD in a conﬁguration shown in Figs. 3.14(a), 3.18(e) with triple occupation N = 3 [244]. In this geometry there are two tunneling channels through the side dots Hˆ db = ∑

∑

kσ s=l,r

Ws ds†σ csσ + H.c. .

(4.120)

The energy level mismatch δsc and the strong Coulomb blockade Qc in the central dot hamper indirect interchannel exchange. The energy spectrum of such TQD with

166

4 DYNAMICAL SYMMETRIES IN THE KONDO EFFECT

triple occupation consists of two spin doublets, spin quartet and two charge transfer doublet excitations where an electron from the central dot is moved to any of the side dots. The lowest of these states is the doublet with the energy ED1 = εc + 2εs −

3 |Wl2 | + |Wr2 | 2 δsc

(4.121)

and the wave function √ βs 6 (|exl σ − |exrσ ) |D1σ = γ |d1σ − 2

(4.122)

where the charge transfer excitons |exsσ are admixed to the doublet |d1σ : 1 † † † † † † † ds↓ − dc↓ ds↑ )ds†¯σ |0, |exsσ = ds↑ ds↓ ds¯σ |0. (4.123) |d1σ = √ ∑(dc↑ 6 s Here γ = 1 − 3βs2, other deﬁnitions are the same as in Eqs. (4.71), (4.72). We suppose that the left-right symmetry is preserved and retain the indices l, r in tunneling matrix elements Ws only for the sake of clarity. As was expected, the interchannel exchange in the SW Hamiltonian is possible only via the charge-transfer excitons admixed to the ground state doublet, so that the additional small parameter enter the coupling constant Jlr in Eq. (4.119): Jlr =

4VlVr βl βr . 3εF − εc

(4.124)

To reveal a two-channel Kondo physics encoded in the Hamiltonian (4.117) (a, b = s, s¯ = l, r we write out the systems of scaling equations for the β - function in accordance with diagrams shown in Fig. 4.25: 2 2 d js = − jss + js2s¯ + jss jss + js2¯s¯ + 2 jss¯ /2 dη 2 d jss¯ = − jss¯ ( js + js¯) + jss¯ jss + js2¯s¯ + 2 jss¯ /2 dη

(4.125)

These equation determine the ﬂow trajectories in a three-dimensional phase space { jl , jr , jlr } (Fig. 4.27). On the symmetry plane jl = jr = j Eqs. (4.125) reduce to a couple of RG equations for j1,2 = j ± jlr d ji /d η = − ji2 + ji ( j12 + j22 )/2 (i = 1, 2).

(4.126)

4.3 Kondo physics for short chains

167

Fig. 4.27 RG ﬂow diagram for the two-channel Kondo effect in serial TQD (after [244]).

These are the well-known scaling equations for the anisotropic two-channel Kondo model [302] with the ﬁxed point (1,1,0). On the plane jlr = 0 we have the scaling map of Fig. 4.26 with the same ﬁxed point. The two projections shown in Fig. 4.27 determine the three-dimensional ﬂow diagram. Although the non-Fermi liquid ﬁxed point remains inaccessible at any inﬁnitesimally small channel anisotropy, one may approach close enough to jc starting from the initial condition jlr0 jl0 , jr0 . Apparently, the critical non-Fermi liquid regime of the tunnel conductance cannot be achieved in this geometry. However one may scan experimentally the ﬂow diagram of Fig. 4.27 by measuring G(t) in a perturbative regime where the 3-rd order correction to the Born approximation G(3) (T ) = G0 | jlr |2 [ jl (T ) + jr (T )]

(4.127)

is valid at T > TK , where the Kondo temperature is max{TKl , TKr }, and TKs = D¯ exp(−1/Js) are the Kondo temperatures for the two independent channels. Varying T implies moving on a curve {Jl (T ), Jr (T ), Jlr (T )} in the three-dimensional parameter space. Besides, by varying the gate voltages vgs applied to the two side dots it is possible to tune the initial condition ( jl0 , jr0 ) from the highly asymmetric case jl0 jr0 to the fully symmetric case jl0 = jr0 . For a ﬁxed value of jl0 jr0 the conductance shoots up (logarithmically) at a certain

temperature T ∗ which decreases towards TK with | jl0 − jr0 | and jlr0 . The closer T ∗ is to TK , the closer is the behavior of the conductance to that expected in a generic two-channel situation. Thus, although the strong-coupling limit is unachievable, its precursor might show up in the intermediate coupling regime. G(T ) at various gate voltages and G(vg ) at various temperatures are presented in the two panels of Fig.

168

4 DYNAMICAL SYMMETRIES IN THE KONDO EFFECT

4.28. The dependencies are sharp enough and apparently detectable experimentally in perfectly symmetric TQD.

Fig. 4.28 Left panel: Conductance G in units of G0 as a function of temperature τ = T /TK at various gate voltages vgl and vgr applied to the left and right side dots of the TQD. The curves correspond to the symmetric case vgl = vgr (a) and the asymmetric cases with increasing difference vgl − vgr from (b) to (d). Right panel: Conductance G/G0 as a function of asymmetry of the gate voltages at different temperatures (after [244]).

The next nano-object where the two-channel effect following the direction (ii) was predicted [314] and then observed [339] is the T-shape double quantum dot depicted in Fig. 4.17(b). It was mentioned above that the large side dot (labeled below by the index ρ ) may serve as an additional channel of Kondo screening for a small dot (labeled by the index ς ) provided TK essentially exceeds the mean interlevel distance δ ε in a big side dot. Then the effective exchange Hamiltonian acquires the form (4.102) and the diagonalization procedure transforms it into orbitally anisotropic three-channel Hamiltonian (4.103). The parameters of the side dot ρ may be tuned in such a way that it is large enough to provide the inequality

δ ε TK but not too bulky, so that the Coulomb blockade Qρ in the side dot ﬁxes the number of electrons Nρ . This blockade suppresses the interchannel cotunneling ∼ Jl ρ and ∼ Jrρ in the Hamiltonian (4.102). Then the transformation used in Eq. (3.47) eliminates the odd channel from the matrix J s. As a result we remain with the two channel Hamiltonian (4.118) but without the additional condition (4.119), so that the coupling constants J1 and J2 are independent: J1 = Jρ stands for exchange between the spins in the small and big dots, J2 = Je couples spin in the small dot and that for the even combination of lead electrons. Due to this independence the constants J1,2 may be tuned by means of gate voltages applied to the two dots.

4.3 Kondo physics for short chains

169

Fine tuning of the experimental setup allows experimentalist to grope for the critical region where J1 = J2 [339]. It should be noted, however, that the infrared cutoff ε > δ ε does not allow one to reach the genuine unitarity limit, but this restriction is soft enough from the practical point of view.

Fig. 4.29 Experimental realization of planar DQD with large side dot, after [339].

In the experimental device shown in Fig. 4.29 the level spacing and the Coulomb blockade in the side dot are estimated as δ ε ρ ≈ 2μ eV and Qρ ≈ 100μ eV ≈ 1.2K. The Kondo temperature varies within the interval 200-300 mK (depending on the gate voltages), so the necessary preconditions for observation of the two-channel Kondo effect are satisﬁed. Transition from the single-channel to the two-channel tunneling regime may be detected by measuring the energy/temperature dependence of the tunneling conductance which is ∼ T 2 in the former case [see Eq. (4.24)] and ∼ T 1/2 in the latter case [see Eq. (4.115)]. Experimental data indeed demonstrate such a crossover (Fig. 4.30). The measurements were performed at ﬁnite source-drain bias and the data are scaled with the ratio eVds /T . In accordance with predictions of conformal ﬁeld theory [3] the square root behavior in this scale establishes only at eVds 3T . The experimental data approximately follow this trend. Although the measurements and tuning procedure are extremely reﬁned and more experimental data are desirable, one may say that the sought crossover is observable. The next question which naturally arises is whether the two-channel Kondo effect may be realized at even occupation. Seemingly, the answer is negative because to overscreen spin 1 one needs at least 3 Kondo channel and the theoretical and experimental difﬁculties which could arise in search of realistic condition of such a

170

4 DYNAMICAL SYMMETRIES IN THE KONDO EFFECT

Fig. 4.30 Fitting of experimental data in the setup shown in Fig. 4.29: the two-channel scaling (left panel) and the single-channel scaling (right panel), after [339].

regime are insurmountable. However one may resort to the same trick as in observation of Kondo effect in the SO(4) quantum dot with ground state singlet, namely to exploit the Zeeman splitting of the excited triplet state [209]. In this case we deal with a two-orbital small dot, whose properties were discussed in Section 4.2. It was noted in this discussion that the system with two reservoirs and single dot with two orbitals generically belongs to the class of two-channel Anderson models, however one of the two channels may be eliminated under the condition (4.59) imposed on the source-drain tunneling constants. We adopt this simpliﬁed model in anticipation that the second screening channel should be created due to coupling with the side dot. Then the two-channel Kondo regime for the small dot ς in the singlet ground state may be ignited in an external magnetic ﬁeld. To describe the SO(4) energy spectrum of the small dot one should use the basis (4.57). In external magnetic ﬁeld the splitting of the triplet results in the level crossing shown in Fig. 4.24(b) so that only the subgroup SU(2) of the dynamical group SO(4) is involved in Kondo tunneling. Then the basis {S, T1 } with generators (4.48) should be used for description of the ﬁeld induced Kondo effect with effective spin 1/2. The same mechanism works in the T-shaped conﬁguration shown in Fig. 4.31(a). The SW transformation applied to the Anderson Hamiltonian for the T-shaped model with dots ς , ρ gives the anisotropic exchange Hamiltonian in an effective magnetic ﬁeld:

J⊥r + − − + ˆ (P sr + P sr ) + Jz Pz srz H= ∑ 2 r=ρ ,e

4.3 Kondo physics for short chains

171

Fig. 4.31 (a) DQD consisting of a two-orbital small dot d with even number of electrons coupled to a large dot C and two leads L and R. (b) TQD in cross geometry with two small dots and one large dot (after [209].

+∑ r

J0r z P (nr − n¯ r ) + BZ Pz + Hˆ b . 2

(4.128)

Here n¯ r is the average electron density in channel r, the components of exchange interaction are deﬁned as ↑↓ ↑↑ ↓↓ ↑↑ ↓↓ , Jzr = Jrr + Jrr , J0r = Jrr − Jrr J⊥r = 2Jrr

(4.129)

The ”ﬁeld” BZ is small in comparison with the critical Zeeman ﬁeld Bc = Δ T S (0), BZ =

ET 1 (B) − ES J0r + ∑ n¯ r . 2 r 2

(4.130)

The exchange anisotropy may be eliminated from the Hamiltonian (4.128) if the experimental setup is modiﬁed in accordance with Fig. 4.24(b). In this modiﬁcation of cross geometry the small dot is split into two valleys coupled by the tunneling element Wd . Each of these valleys is assumed to be occupied by odd number of

172

4 DYNAMICAL SYMMETRIES IN THE KONDO EFFECT

electrons. If the highest unoccupied levels are equivalent, then the levels ε1,2 , which enter the Hamiltonian Hˆ d are ε1,2 = ε ∓Wd ; the singlet-triplet splitting in the SO(4) multiplet is given by the indirect exchange J = 2Wd2 /Qς instead of direct exchange J; and the SW procedure described above gives an effective Hamiltonian in which only one tunnel matrix element, Wρ , enters all parameters Jrσ σ (4.129). Besides the parameter Jor turns to zero. In contrast to the model of Fig. 4.24(a), in this cross geometry we obtain the standard two-channel Hamiltonian with isotropic exchange coupling in an effective magnetic ﬁeld (and no potential scattering), Heff = ∑ Jr P · sr + BZ Pz

(4.131)

r

where BZ is the effective Zeeman ﬁeld given by the ﬁrst term in Eq. (4.130). The ﬂow diagram for the spin-isotropic Hamiltonian (4.131) is determined by the usual two-channel scaling equations of the type (4.126). The system of scaling equations for the anisotropic Hamiltonian (4.128) with dimensionless coupling constants jrι is more complicated. In this case the third-order corrections in the r.h.s. of the scaling equations contains the sums jrι ∑r ∑κ =ι ( jrκ )2 , where r, r are the channel indices and the indices ι , κ mark the Cartesian components x, y, z in spin space. Besides, additional scaling equations should be written for the integrals j0r . Analysis of these equations shows [209] that the parameters j0r scale to zero. As to the anisotropic third order terms in the scaling equations for jri , it is known from predictions of conformal ﬁeld theory [4], that the exchange anisotropy is irrelevant for the two-channel regime. So we conclude that the ﬂow diagram for the two-channel Hamiltonian (4.128) with anisotropic exchange coupling retains the ﬁnite isotropic ﬁxed point and only the channel anisotropy is relevant. As to manifestation of this magnetic ﬁeld induced two channel Kondo effect, the following scaling behavior for the tunneling conductance should be observed under certain experimental conditions (see [209] for more details): |eV | g(0, T ) − g(V, T ) G0 Y = 2 πT π T /TK where Y (x) is the scaling function with the asymptotic behavior c x2 for x 1 Y (x) √3 √ x − 1 for x 1 π

(4.132)

(4.133)

with c = 0.748336. This function determines the transition between the small bias and the large bias regimes in the experimental curves shown in Fig. 4.30.

4.3 Kondo physics for short chains

173

Unlike the single channel case, the magnetic ﬁeld is a relevant parameter in the two-channel case. We anticipate that the effective magnetic ﬁeld in the even N two-channel Kondo model operates in a way similar to the action of the ordinary magnetic ﬁeld in the odd N two-channel Kondo model [342]. Notice though that the effective magnetic ﬁeld BZ [see Eq. (4.130)] is controlled by the real external magnetic ﬁeld, by the position of the two levels and by the coupling constants. The control parameter δ j = jρ − jς and the crossover temperature Tδ =

δj j¯2

2 TK

(4.134)

should be introduced to describe the scaling of the conductance near the critical point δ j = 0. Here j¯ = ( jρ + jς )/2 and TK is estimated as a crossover scale for the weak coupling regime on the ﬂow diagram of Fig. 4.26, ¯ −C arctan(π j TK ∼ De

∗ /4)

(4.135)

(see [67]). Here j ∗ stands for the corresponding value of current coordinate j(η ) in this ﬂow diagram, C < π /4 is a numerical constant. In these terms the tunneling √ conductance G(V = 0, T, B) for TK T BZ Bδ = δ j / jσ2 TK is determined by the relation g(0, T, 0) − g(0, T, BZ ) = −G0 sign (δ j ) . (4.136) (BZ /Bδ )2 For Bδ BZ TK the conductance is given by the Bethe-Ansatz solution [22] B BZ TK G(0, T, BZ ) − 1/2 = a sign (δ j ) δ − b log . G0 BZ TK B

(4.137)

with a and b of O(1). The three latter equations demand an additional comment. In the current studies of the singlet/triplet crossover in SO(4) CQD with even occupation, the singlettriplet energy gap Δ T S is frequently called a control parameter for quantum criticality responsible for transition from the Kondo-singlet state to a trivial singlet state of non-magnetic impurity. This wording is not too strict, because the genuine criticality implies some universality where the dynamic or the thermodynamic variables are measured with respect to a single energy scale. In this case the point ΔST = 0 is distinguished only by the maximal value of TK (ΔT S ) [see Fig. 4.7 and Eq. (4.55)]. There is no single energy scale TK for dynamic variables. Instead one has two interdependent scales TK and ΔST in accordance with Eq. (4.56)]. Besides, there is no change of dynamical symmetry in this point: within the scale ∼ TK it is the same

174

4 DYNAMICAL SYMMETRIES IN THE KONDO EFFECT

SO(4) symmetry both to the left and to the right of the ”critical” point Δ ST = 0. Real transition from the Kondo singlet to the nonmagnetic regime occurs at some negative ΔST ∼ TKm like in the two-site Kondo model [187]. Similar situations take place in the crossover SU(4) → SU(2) in the two-orbital single electron model (Fig. 4.10) and in all crossovers in the phase diagram of the TQD (Figs. 4.13 and 4.14). Unlike these crossovers, in case of transition from the non-Fermi liquid twochannel Kondo regime to the Fermi-liquid single channel Kondo regime we deal with genuine criticality. In all regimes the system ﬂows to the ﬁxed point independent of the length scale and the universal energy scale Tδ may be pointed out. One may draw a phase diagram in the plane {T, δ j } where the Fermi liquid and the non-Fermi liquid domains are separated by the boundaries T ∼ Tδ and the critical quantum point (QCP) exists at T = 0 [342, 434]. This diagram is sketched in Fig. 4.32.

T Fig. 4.32 Phase diagram describing transitions between single channel and two channel Kondo effect as a function of a control parameter δ . See text for further explanation.

T=Tδ NFL FL

FL 0

δ

In the ideal case (T = 0, δ ε = 0) the conductance should have a step-like form G(δ j ) = G0 θ (δ j ) which reﬂects the quantum phase transition from the Fermi liquid regime where the Kondo resonance arises due to the spin screening in the side dot ρ and does not result in resonance tunneling between the electrodes, to the Fermi liquid regime where the resonance in the source-drain channel provides perfect tunnel transparency of the DQD. This jump in the conductance would take place in the quantum critical point (QCP). In real situation at ﬁnite temperature the step is smeared, and its effective width is ∼ T /TK . In order to estimate the energy scale Tδ , one should derive and solve the RG equations for the relevant parameters [342], which are δ j and j¯. These equations read d j¯/d η = j¯2 , d δ j /d η = 2 j¯δ j . (4.138) Usual initial conditions

4.3 Kondo physics for short chains

175

¯ ≡ j¯0 , δ j(η = 0) = δ j (D) ¯ ≡ δ j0 j¯(η = 0) = j(D) are imposed on these equations. Looking at the ﬂow diagrams, one ﬁnds the region where the evolution towards the strong coupling single channel limit makes j(η ) ∼ 1 at D = TK . The corresponding value of δ j (η ) is

δ j∗ (TK ) = δ j0 / j¯02 .

(4.139)

If δ j∗ (TK ) 1, then one expects that the broad enough range of energies is still governed by the vicinity to the two-channel QCP point where the scaling dimension of the channel anisotropy parameter is 1/2 [3]. Hence the scaling behavior of the parameter δ j is given by the law

δ j∗ (D) ∼ δ j∗ (TK )

TK D

1/2 .

(4.140)

From Eqs. (4.139) and (4.140) we evaluate the characteristic temperature Tδ (4.134) which determines the boundaries of the critical region on the phase diagram of Fig. 4.32. Similar reasoning allows one to estimate the effective crossover magnetic ﬁeld and ﬁnd the crossover parameter Bδ for transition between the two-channel

B∗ (TK )

and the single channel Kondo tunneling regimes, Bδ ∼

Tδ TK ∼ (|δ j |/ j¯2 )TK .

(4.141)

On this route one comes to the laws (4.136) and (4.137). More about the phase diagram of the anisotropic multichannel Kondo model may be found in Ref. [369] Up to this point we discussed the approaches to the two-channel regime in a framework of the standard Kondo model, where the basic multichannel symmetry SU(n) × SU(2) is modiﬁed in such a way that the transversal pseudospin operators T ± responsible for the inter-orbital exchange and orbital anisotropy are somehow eliminated from cotunneling processes. It is really challenging to invert the problem and make the subgroup SU(n) responsible for the Kondo effect, while the spin SU(2) subgroup with suppressed operators S± labels tunneling channels. The idea of swapping the functions of charge and spin variables was formulated in connection with ”natural” TLS, namely heavy particles as substitutes for the Kondo scatterers in metallic Fermi liquid [67, 234, 410]. Since the scattering of electrons in TLS is purely potential, the spin ﬂip processes are strictly forbidden

176

4 DYNAMICAL SYMMETRIES IN THE KONDO EFFECT

in the effective ”exchange” , pseudospin is the only source of Kondo singularities, and the two projections of the electron spin may be treated as two channels for pseudospin Kondo scattering. However, the idea of accessibility of non-Fermi liquid two-channel Kondo regime in ”natural” TLS for heavy particles was subject to harsh criticism [7, 8, 191]. In those systems transition between two states (pseudospin ﬂip processes) is a real tunneling process with characteristic time ttun unlike the genuine spin ﬂip processes which are practically instantaneous. As a result the ultraviolet cutoff for the Kondo effect is the energy ∼ h¯ /ttun . This energy is of the order of the distance to the next excited level in a two-well potential. The latter interval is much less than the energy scale εF for the ”light” electrons. As a result the energy interval available for the formation of the logarithmic singularity is too narrow and the resulting Kondo temperature TK Δ . Thus the strong coupling regime in TLS remains in fact unattainable. Complex quantum dots provide their own mechanisms of two-channel Kondo tunneling assisted by pseudospin excitations. The ﬁrst of such mechanisms was proposed for a ”quantum box” connected to a lead by a single-mode point contact [253, 279]. This quantum box is a big quantum dot of the same type as the side dot ρ discussed above. Unlike the Kondo regime emerging deep in the Coulomb window, this mechanism is realized on the boundary of this window where the charge excitations N ↔ N + 1 are relevant dynamical variables. The key point of this model is the treatment of the tunneling Hamiltonian which is postulated in the form Hˆ db = t

∑

kκ ,σ

† dκσ ckσ + H.c.

(4.142)

† where the operator dκσ creates electron in a quasidiscrete state κσ in the quantum

box. Then the Hubbard-like operators for charge ﬂuctuations are introduced: Qˆ ± = ∑ |n ± 1n, Qˆ = −e ∑ n|nn|. n

(4.143)

n

and the model Hamiltonian is written as Qˆ 2 † Hˆ = Hˆ b + + VBQˆ + ∑ Qˆ + dκσ ckσ + H.c. . 2C0 k κ ,σ

(4.144)

Here C0 and VB are the capacitance of the quantum box and the external component of electric potential (gate voltage). These parameters control the excess electric charge inside the box ρ . The Hamiltonian (4.144) describes two ”conduction bands” coupled to an impurity with a macroscopic number of parabolically dispersed en-

4.3 Kondo physics for short chains

177

ergy levels. The mapping procedure is not too strict, but the resulting Hamiltonian has the form of effective two-channel exchange Hamiltonian without any kind of SW-like canonical transformation. As was mentioned above, the spin index plays part of a channel number. Then the NRG technique may be used for a numerical solution of the problem [254]. The relevant operator which logarithmically scales the crossover from the hight temperature to the low-temperature region is the capacitance C(T ) ∼ ln(T /TK ). Returning back to the genuine Kondo regime, we next mention the possibility of the two-channel Kondo effect in a vertical quantum dot advocated in Ref. [134]. As is shown in Fig. 3.10, in the charge sector N = 5 two orbital states with different orbital numbers l = 4, 6 cross at some critical value of the magnetic ﬁeld B∗ . Then transitions between the states |5, 4, Sz and |5, 6, Sz may result in inelastic SW cotunneling. Since the possible change of the orbital moment δ l = 0, 1, 2, the effective pseudospin of this cotunneling is 3/2. This value implies the underscreened Kondo effect. The hope is that the ﬁeld B∗ is large enough to provide not only the Larmor renormalization but also the Zeeman splitting which leaves only the states with δ l = 2 shown as the initial and the ﬁnal states of electron cotunneling processes in Fig. 3.10. In this case the effective spin is 1/2. Since the intermediate states in the adjacent charge sectors with N = 4, 6 available for SW exchange are spin singlets, the conservation of spin projection in this cotunneling is symmetry protected and the transition to the two-channel regime is possible in principle. The shortcomings of the model are related to the complexity of the many-electron spectra in the relevant charge sectors. First, the conservation of spin projection does not guarantee the equivalence of the two spin channels. The intermediate state with N = 6 is the state with half-ﬁlled shells of the Fock – Darwin atom whereas the singlet state with N = 4 contains a hole which may be ﬁlled by an electron with either up or down spin (see Fig.3.10). As a result the coupling constants for the states |5, l, ↑ and |5, l, ↓ are different, and ﬁne tuning of these constants by means of changing, say, the Larmor frequency Ω is necessary. Second, the low-lying triplet excitation induces spin-ﬂip component in the effective cotunneling, which results in the anisotropy of the spin channels and thus makes the window in the phase space for NFL Kondo regime even more narrow. The authors demonstrate the phase diagram where such window exists, but experimental search of this window is not an easy task. At least to the present day experimental support of this scenario is unavailable. Noticeably more realistic possibility of achieving the two-channel Kondo tunneling regime exists in the parallel TQD [Fig. 3.14(d)] with double occupation. This

178

4 DYNAMICAL SYMMETRIES IN THE KONDO EFFECT

possibility was mentioned already in Section 3.5.2. Here we will continue the discussion of Kondo cotunneling in a system described by the Hamiltonian (3.70). As was explained in Section 3.5.2, the low-energy part of the spectrum of doubly occupied TQD with single-electron levels shown in Fig. 3.18(e) consists of two spin doublets ESs , and two spin triplets ET s (s = l, r). In the limit Qc → ∞ these states are orbitally degenerate, but at ﬁnite but still large Qc δsc the orbital degeneracy is lifted so that the SO(8) supermultiplet is given by the equations ES(g,u) = εc + εs − Δ S(g,u), ET (g,u) = εc + εs ∓ −Δ T (g,u) ,

(4.145)

where ΔΛ (g,u) is the renormalization of the two-electron state due to the interdot tunneling W and the coupling with the bath. The rate of these scaling (JeffersonHaldane) renormalization is different for different states, as was discussed above for various SO(8) supermultiplets (see, e.g.,Fig. 4.13 and corresponding discussion). Now we are interested in the limiting case presented in Fig. 4.33, where the pair of spin singlet is separated from the pair of spin triplets by an exchange gap .

Tg

Fig. 4.33 Basic SO(8) set of energy levels of serial TQD with small central dot transformed into spin singlet/orbital doublet lowenergy multiplet after Jefferson - Haldane renormalization.

Tu Su Sg

Then the RG procedure starting at the energy ∼ D¯ Δ T S is controlled by the effective exchange Hamiltonian which obeys SO(8) dynamical symmetry including transitions between the components of the two triplets and the two singlets. These are 28 generators combined in 8 vector and 4 scalar operators [see Table (2.58)]. Four of these operators are spin operators Sl , Sr , Slr , Srl , for the two triples ET (l,r) , four operators R j describe transitions between the two triplets and two singlets,

4.4 Kondo physics for small rings

179

and four scalars Ai arise as a result of action of the left-right permutation operator Pˆlr on the basis (4.145) (see Fig. 2.3 where the general kinematic scheme of all these transitions is shown). At this high energy stage the Kondo screening occurs in accordance with Kondo scenario similar to that discussed above for SO(5) and SO(7) symmetries (Section 4.3.1). In the low-energy region, ε Δ T S , RG scenario changes drastically. Quenching of triplet states means also quenching of spin-ﬂip transitions. As a result the sole vector T deﬁned in Eq. (3.69) arises instead of the above eight vectors at the last stage of the scaling procedure, and the sourcedrain cotunneling at this stage is described by the effective Hamiltonian (3.70). It is important that the effective constant J in this Hamiltonian inherits the logarithmic enhancement accumulated by the high-energy ﬂow diagram, because the singletsinglet transitions which form the components of the vector T are generated by singlet-triplet transitions: e.g. X Sl,Sr = X Sl,Ti X Ti Sr , where |Ti is any triplet state generated by the operators Ri from Sl| in the high-energy region. In this way we arrive at the two-channel Hamiltonian with pseudospin operator as a source of Kondo screening and spin indices enumerating the screening channels. The channel isotropy is protected by the spin-rotational symmetry in the singlet state. At Qc → ∞ the non-Fermi liquid ﬁxed point Jc of the ﬂow diagram of Fig. 4.25 should be achieved. Large but ﬁnite Qc generate the term (ESu − ESg )Tz in the Hamiltonian, which is equivalent to the Zeeman ﬁeld in the convenient two-channel Kondo effect. The Zeeman operator is relevant for the two-channel Kondo effect, and its inﬂuence on the scaling behavior in the nearest vicinity of the QCP may be described by means of conformal ﬁeld machinery described above [see Eqs. (4.136), (4.137)]. This proposal still waits for its experimental check. Thus we see that the NFL Kondo effect in complex quantum dots is really an elusive phenomenon and more efforts are necessary to ﬁnd the path to the critical point on the isotropic bisector of the ﬂow diagram of Fig. 4.26.

4.4 Kondo physics for small rings New features of Kondo resonance regime in charge and spin transport through complex quantum dots arise if closed loops are present in the tunneling component of the electric circuit. Several highly symmetric conﬁgurations where the loop has a form of triangle are shown in Figs. 3.14(e-h) and 3.15(c). A triangle is the simplest loop-like tunnel structure. Well-shaped triangular triple quantum dots are fab-

180

4 DYNAMICAL SYMMETRIES IN THE KONDO EFFECT

ricated in vertical [12, 13] geometry. In principle such loops may be shaped in forms of quadrat, pentagon and so forth. The limiting case is a quantum ring where the nanoobject has a form of circular gutter. Such quantum rings were also realized experimentally [115] as well as more reﬁned objects where the quantum dot molecule is embedded in a quantum ring [176]. In all examples of linear conﬁgurations presented in Section 4.3 the only ”geometrical” symmetry involved in the Kondo tunneling was the mirror symmetry. In case of regular polygons involvement of discrete point symmetry groups adds new quantum numbers in the energy spectra of CQD. Violation of this symmetry by means of tuned gate voltages provides new methods of controlling the tunnel conductance. One more new element inherent in the loop geometry is a chiral symmetry: breaking of chiral symmetry makes clock-wise and counterclockwise paths in the loop nonequivalent. This nonequivalence results in peculiar interference phenomena. Let us start with the Kondo effect in perfectly symmetric CQD having the form of equilateral triangle built in a device which preserves this symmetry. In two-terminal arrangements this symmetry is preserved in the complex vertical dots shown in Fig. 3.15(c). In planar devices the equilateral symmetry is sustained only in the threeterminal conﬁguration of Fig. 3.14(h). In the studies of this conﬁguration the most basic features of tunneling through closed loops may be uncovered. The simplest conﬁguration for such Kondo tunneling is the equilateral triangular TQD (TTQD) in the charge sector N = 1 [246]. The effective exchange Hamiltonian for the latter case was derived in Section 3.5.2. The basis functions for dot and lead states transform along the irreducible representations Γ = A, E± of the point group C3v or the permutation group P3 [see Eqs. (3.50), (3.52)] and the energy multiplet is formed by the ground state EA and the doublet exciton EE (3.51). As was noted above, the ground state of the TQD is an orbital singlet due to the fact that the tunneling parameter (inter-dot tunneling integral) is usually negative, W < 0. To reveal the abilities of the dynamical symmetry mechanisms, we exploit the fact that the sign of this parameter changes in an external magnetic ﬁeld B directed normally to the plane of the triangle. The electromagnetic gauge phase should be added to the phase φ in Eqs. (3.50) so that the interdot tunneling matrix element acquires an additional phase W jl (B) = W jl e2π iφ jl . The cyclic boundary conditions are imposed on the phase in a ring,

∑ φ jl = Φ

jl

4.4 Kondo physics for small rings

181

where Φ is the number of magnetic ﬂux quanta threading the system and the nearest neighbors jl are enumerated in the clockwise direction along the ring. In case of equilateral triangle φ jl = Φ /3. This means that the energy spectrum of singly occupied TQD acquires the following form in magnetic ﬁeld: Φ . EDΓ (p) = ε − 2W cos p − 3

(4.146)

Here the index D stands for spin 1/2 doublet, and the values p = 0, 2π /3, 4π /3 correspond, respectively, to Γ = A, E± . Thus the orbital degeneracy is removed in the magnetic ﬁeld, but the energy levels are periodic functions of magnetic ﬂux. As a result the level crossing takes place at Φ = (2n + 1)π , (n = 0, ±1, ...) (see the upper panel of Fig. 4.34). The accidental orbital degeneracy of spin states induced by the magnetic phase Φ introduces new features in the Kondo effect. At B = 0 the basic dynamical symmetry SU(6) of spin doublet/orbital triplet spectrum of TTQD transforms into spin symmetry SU(2) at ε W , so that the orbital degrees of freedom of localized spin are quenched at low energies, while the orbital degeneracy of band states remains unaffected by magnetic ﬁeld. Thus the two stage RG evolution ends with the three-channel spin 1/2 Kondo Hamiltonian introduced in Section 3.5.2, Hex = JE S · sE+ E+ + S · sE− E− + JA S · sAA

(4.147)

The exchange vertices JΓ are −1 /3, JE = −2V 2 ΔQ−1 − ΔQ −1 2 −1 JA = 2V 3Δ1 + ΔQ + 2ΔQ−1 /3,

(4.148)

with Δ1 = εF − ε , Δ Q = ε +Q− εF , ΔQ = ε +Q − εF . Here Q and Q are the on-site and intersite Coulomb blockade parameters. Note that JA > 0 as in the conventional SW transformation of the Anderson Hamiltonian. On the other hand, the exchange constant for orbital doublet has ferromagnetic sign, JE < 0 due to the inequality Q Q . Thus, two out of the three available exchange channels in the Hamiltonian (4.147) are irrelevant and the multichannel Kondo effect eludes in this conﬁgurations like in many others. As a result, the conventional Kondo regime emerges with the doublet DA channel and the Kondo temperature, (A)

TK

= D exp {−1/ jA } ,

(4.149)

182

4 DYNAMICAL SYMMETRIES IN THE KONDO EFFECT

Fig. 4.34 Upper panel: Energy levels of equilateral triangular TQD with N = 1 as a function of the magnetic ﬂux Φ . Solid curve - level EA , dashed and dot-dashed curves - levels EE± , respectively. Lower panel: Corresponding oscillations of the Kondo ZBA in the tunneling conductance G/G0 (after [246]).

where jA = ρ0 JA . At the point of accidental degeneracy |Φ | = (2n + 1)π the ground state of the TTQD is spin and orbital doublet and the dynamical symmetry of the nanoobject is SU(4). The 15 generators of the SU(4) group include four spin vector operators SEa Eb with a, b = ± and one pseudospin vector T deﬁned in Eq. (3.55). Its counterpart for the lead electrons is the vector τ deﬁned in Eq. (3.56). The SW Hamiltonian is

4.4 Kondo physics for small rings

183

HSW = J1 (SE+ E+ · sE+ E+ + SE− E− · sE− E− ) +J2 (SE+ E+ · sE− E− + SE− E− · sE+ E+ ) +J3 (SE+ E+ + SE− E− ) · sAA +J4 (SE+ E− · sE− E+ + SE− E+ · sE+ E− )

(4.150)

+J5 (SE+ E− · (sAE− + sE+ A ) + H.c.) + J6T · τ , where the coupling constants are J1 = J4 = JA , J2 = J3 = J5 = JE deﬁned in (4.148) and J6 = V 2 (Δ1−1 + ΔQ−1 ). Thus, spin and orbital degrees of freedom of TTQD interlace in the exchange terms. The indirect exchange coupling constants include both diagonal ( j j) and nondiagonal ( jl) terms describing reﬂection and transmission co-tunneling amplitudes. The interplay between spin and pseudospin channels is described by the RG ﬂow equations d j1 /d η = −[ j12 + j42 /2 + j4 j6 + j52 /2], d j2 /d η = −[ j22 + j42 /2 − j4 j6 + j52 /2], d j3 /d η = −[ j32 + j52 ], d j6 /d η = − j62 , d j4 /d η = −[ j4 ( j1 + j2 + j6 ) + j6 ( j1 − j2 )], d j5 /d η = − j5 [ j1 + j2 + j3 − j6 ]/2.

(4.151)

Analysis of the solutions of Eqs. (4.151) shows that the symmetry-breaking vertices j3 and j5 are irrelevant, but the vertex j2 , whose initial value is negative evolves into the positive domain and eventually enters the Kondo temperature, (E)

TK

√ ! = D exp − 2 ( jA (1 + 2) + jE + 2 j6 ) .

(4.152)

We conclude from this equation that both spin and pseudospin exchange constants contribute to the Kondo effect on an equal footing. It follows from Eq. (4.146) and Fig. 4.34 that the crossover SU(2) → SU(4) → SU(2) occurs three times within the interval 0 Φ 3Φ0 and each level crossing results in transformation of TK from (4.149) to (4.152) and back. It should be noted that the real periodicity of the energy diagram in Φ is 2π : each crossing point may be considered as a doublet E± by the regauging phases ϕ in (3.50). Periodicity of crossovers induced by the magnetic ﬁeld distinguishes this type of SU(4) → SU(2) crossovers from those studied in Section 4.3.1, where the symmetry change was induced by varying gate voltages [see Eqs. (4.69), (4.70) and Fig. 4.10]. Conduc-

184

4 DYNAMICAL SYMMETRIES IN THE KONDO EFFECT

tance measured between any pair of electrodes follows the change of TK at high temperatures in accordance with Eq. (4.25) (see Fig. 4.34, lower panel).

Fig. 4.35 Energy levels of equilateral triangular TQD with N = 2 as a function of the magnetic ﬂux Φ . Solid curves: spin triplet levels ETA , ET E± , dashed curves: spin singlet levels ESA , ESE± , respectively.

In the charge sector N = 2 the energy spectrum of doubly occupied TTQD consists of twelve states (3.64). These states also vary periodically in the external magnetic ﬁeld [75] (see Fig. 4.35). Since the basic dynamical symmetry in this case is SO(12), the crossover diagram is more complicated than in the SU(4) representations of the singly occupied TTQD. Among the four intercrossing levels there are two spin triplets and two spin singlets. Looking at the grid shown in Fig. 4.35, we see that within the single period 0 Φ Φ0 the symmetry of the ground state changes as SO(3) → SO(4) → U(1) → SO(4) → SO(3). The Kondo temperature changes appropriately (see Fig. 4.36) as well as the tunneling conductance. The problem of Kondo effect in the charge sector N = 3 looks differently ﬁrst of all because the oscillations of the single electron levels nearly compensate each other in triply occupied states (see [75] for detailed explanation). The main effect of the magnetic ﬁeld is the Zeeman splitting. In zero ﬁeld the lower part of the spectrum is formed by eight levels: two orbital doublets with Sz = ±1/2 and orbital singlet with Sz = ±3/2, ±1/2. Only one level crossing in the ground state takes place in the interval 0 Φ Φ0 and the Kondo processes around the crossing point resemble those discussed in Section 4.2 in connection with the scenario of magnetic ﬁeld-induced Kondo effect. We will return to the discussion of Kondo tunneling in triangular three-spin conﬁguration in Chapter 5 devoted to the Kondo effect in molecules absorbed on metallic surface.

4.4 Kondo physics for small rings

185

TK

Φ/Φ 0 0

0.5 SO(4)

SO(3)

1 SO(4)

U(1)

SO(3)

Fig. 4.36 Oscillations of Kondo temperature in equilateral triangular TQD with N = 2 as a function of Φ /Φ0 .

In a two-terminal conﬁguration the third lead is qualiﬁed as ”passive”: it provides the reservoir for Kondo screening but seemingly does not participate in a charge transport. The latter statement is not quite accurate [407]. Due to the ”amplitude leakage” to the passive lead, the partial conductance of the left and right dots may differ noticeably, although the Kondo ZBA is sensitive mainly to the change of symmetry of the TTQD from equilateral to isosceles triangle.

E A ,E + E−

Fig. 4.37 Diamagnetic Larmor shift of energy levels in equilateral triangular TQD with N = 2 .

E+ , E− EA 0

1

Φ/Φ0

Having in mind the experimental realization of vertical TQD [12, 13], one may think about a single-channel regime of Kondo tunneling where all three tunneling paths start in the same point of the source electrode and end in the same point of the drain electrode. In this case the tunneling Hamiltonian simpliﬁes, Hˆ db = W

∑ (d †jσ cekσ + H.c.)

(4.153)

jkσ

so that only the fully symmetric partial component with Γ = A is involved it cotunneling processes, and we return to the conventional SU(2) exchange Hamiltonian

186

4 DYNAMICAL SYMMETRIES IN THE KONDO EFFECT

JSA · se . It worth mentioning that in vertical quantum dots with cylindrical symmetry the diamagnetic Larmor shift of the electron levels is not negligible. It results in general upward shift of energy levels and in general narrowing of the energy interval where level crossing takes place [75] (see Fig. 4.37). In the two-terminal conﬁguration [Fig. 3.14(e)] as well as in the Δ -type and ∇type conﬁgurations with vertical mirror reﬂection axis [Fig. 3.14(f,g)] the symmetry of the tunneling system is that of isosceles triangle (point symmetry group C2d ). If the violation of the ”ideal” C3v symmetry in the Δ -type geometry is weak, its inﬂuence on the evolution of the electronic levels EΓ (Φ ) in the magnetic ﬁeld results in an avoided crossing effect near the points Φ = nΦ0 /2 (Fig. 4.38). This means that the symmetry crossovers in the charge sectors N = 1, 2 discussed above may be observed only at energy/temperature exceeding the gap Δ ± = E+ |Hˆ d |E− . At low energies E Δ± the orbital degrees of freedom are quenched, and we return to the two-stage SU(4) → SU(2) scenario of Kondo screening discussed above.

Fig. 4.38 Avoided crossing effect in the diamagnetic Larmor shift of energy levels in triangular TQD with N = 2 in Δ -type or ∇-type geometries .

Quite interesting critical regimes arise in triangular geometries in the charge sector N = 3 at zero magnetic ﬁeld It is well known that magnetic frustrations are inherent to localized spins in triangular conﬁgurations. The mechanism of frustration is clearly seen if the effective Hamiltonian for the TTQD is rewritten in the Heisenberg-like form [284, 285]. Hˆ d = ∑ I jl S j · Sl + ∑ K jkl S j · (Sk × Sl ) jl

(4.154)

jkl

with antiferromagnetic coupling constants. The last term in the Hamiltonian (4.154) arises only in external magnetic ﬁeld: the three-site coupling constant K jkl ∼ V 3 /Q2 sin(2πΦ /Φ0 ).

4.4 Kondo physics for small rings

187

It is clear that the antiparallel orientation of spins in triangle is impossible for geometrical reasons, and the total energy is optimized by means of a compromise between parallel and antiparallel orientations of adjacent spins. The level structure with two nearly degenerate spin doublets and the high energy spin quartet mentioned above reﬂects this frustration motif which is especially distinct when all the coupling constants I jl are of the same order. The wavefunctions of two competing three-electron spin doublets have the following form: 1 |σ + = √ c†cσ c†l↑ c†r↓ − c†l↓ c†r↑ |0 2 # 1 " † † † |σ − = √ ccσ cl↑ cr↓ + c†l↓ c†r↑ − 2c†cσ¯ c†l σ c†rσ |0 6

c

s

000 111 111 000 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111

V

l

V’

r

l

V

r

000 111 111 000 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111

d

0000 1111 1111 0000 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111

(4.155)

c

000 111 111 000 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111

Fig. 4.39 Isosceles triangular TQD in Δ -type (left panel) and ∇-type (right panel geometries.

Here two equivalent vertices of isosceles triangle are marked as l, r (see Fig. 4.39, left panel). The side spins are locked in odd (triplet) conﬁguration in the |σ − doublet and even (singlet) conﬁguration in the |σ + doublet, so that the parity number ± may be ascribed to the spin 1/2 in the central dot c. The energy separation of these two doublets is

Δ ± = E+ − E − = I − I ≡

4V 2 4V 2 − . Q Q

(4.156)

The levels intersect in the fully symmetric case I = I (”complete frustration”). Manifestations of magnetic frustration are different for ∇-type and Δ -type geometries. In the former geometry far from the degeneracy point either even or odd doublet form the ground state of the isosceles TTQD depending on the sign of Δ± . In both cases the SW Hamiltonian has the standard form Hˆ ex = J p S · se

(4.157)

188

4 DYNAMICAL SYMMETRIES IN THE KONDO EFFECT

with effective constants J+ =

8W 2 8W 2 , J− = − . Q 3Q

(4.158)

This means that the Kondo screening is effective only for the even channel, whereas in the odd channel the ferromagnetic exchange is irrelevant for Kondo renormalization and the spin 1/2 of isosceles TTQD remains unscreened (”local moment” regime in terms of Ref. [239]). Tuning the parameter V one may induce crossover from the Kondo singlet regime to the local moment regime. In the vicinity of the crossover ”critical point” V = V where the states E+ and E− are nearly degenerate, the effective Kugel-Khomskii-like Hamiltonian may be derived by means of projection operators X ++ = ∑ |σ +σ + |, X −− = ∑ |σ −σ − |, σ

(4.159)

σ

namely 1 crit = J+ (1 + 4Tz ))S · se + Δ ±Tz (4.160) Hex 3 where Tz is the longitudinal component of the pseudospin vector T deﬁned as 1 Tz = (X ++ − X −− ), T + = X +− = ∑ |σ +σ − |, T − = X −+ . 2 σ

(4.161)

The results of NRG calculations for ∇-type TTQD reveal the change of the dynamical symmetry with decreasing energy/temperature scale [284]. Near the critical ﬁxed point with |Δ ± | < T < TK the |σ + doublet is screened due to the antiferromagnetic Kondo coupling even when it is not the ground state, while the ferromagnetically coupled |σ − remains a local moment. Thus the critical point comprises both a free moment S=1/2 and a Kondo singlet. This regime is a peculiar realization of SU(4) dynamical symmetry where the transversal components of the pseudospin T are strictly eliminated due to l − r parity conservation in the mirror symmetric TTQD setup. Such Z2 symmetry protected suppression of orbital mixing mechanism prompts the idea that the multichannel Kondo regime discussed in the previous section may be realized under certain conditions. Indeed the NFL two-channel ﬁxed point may be approached in the Δ -type setup [Fig. 3.14(f)] with isosceles TTQD [285]. To derive the dot Hamiltonian for this geometry it is worth to represent it as the Heisenberg exchange Hamiltonian involving spins in the sites l, c, r both for Hˆ d and Hˆ ex :

4.4 Kondo physics for small rings

189

Hˆ d = I Sc · (Sl + Sr ) + I (Sl · Sr ) Hˆ ex = J (Sl · sl + Sr · sr )

(4.162)

Then using the basis (4.155) and the projection operators (4.159) and repeating the operations described above one arrives at the following Hamiltonian valid in the vicinity of the critical point I = I : crit Hˆ ex = (Jb + Ja Tz )S · se + Jm (T + + T − )S · so + Δ±Tz

(4.163)

Here S = S+ + S− is the effective spin of orbitally degenerate spin doublet, Ja , Jb , Jm are SW exchange parameters expressed via the coupling constants of the original Anderson Hamiltonian. Thus we really come to the two-channel Kondo Hamiltonian where both even and odd spin modes of the lead electrons are involved in cotunneling process. Since the setup Fig. 3.14(f) is symmetric relative to left-right permutation of both leads and isosceles triangle, the global mirror symmetry is conserved, so that the even source-drain mode se is coupled with the longitudinal component of the pseudospin vector T , while the odd mode so is associated with the transversal components of this vector. Both channel anisotropy and effective ”Zeeman splitting” of the pseudospin states are inherent in the Hamiltonian (4.163), so the RG ﬂow diagram is quite complicated and the NFL point is apparently unattainable, but the model possesses very rich Kondo dynamics due to interplay between spin and orbital degrees of freedom (see also [445, 308]). The phase diagrams for complex quantum dots in a form of polygon where each dot in the vertex is occupied by one electron and attached to its own bath are quantitatively similar. The dot Hamiltonians have the form (4.154) (not only the nearest neighbors may be coupled: in quadruple quantum dot the Union Jack coupling scheme may be considered). Kondo screened, local moment and sometimes NFL domains form the phase diagrams constructed in the coordinates J, J where different coupling constants correspond to non-equivalent pairs of spins in the Hamiltonian (4.154). Some approach to classiﬁcation of these phase diagrams in terms of conformal ﬁeld theory is proposed in Ref. [105].

190

4 DYNAMICAL SYMMETRIES IN THE KONDO EFFECT

4.4.1 Kondo tunneling and Aharonov – Bohm interference Aharonov – Bohm (AB) effect known in quantum optics and quantum mechanics since late 40-es [6, 90] may be observed in tunneling through CQD in any ”which pass” geometry, including nanorings. If the magnetic ﬁeld has a component perpendicular to the plane of the ring, two paths have different chirality and as a result ”left” and ”right” components of electron moving through a CQD acquire electromagnetic phase difference between two paths through the device

φB =

$

A · dl = 2πΦ /Φ0

(4.164)

Here Φ is the magnetic ﬂux through the area between the paths and Φ0 = h/e is the magnetic ﬂux quantum. The phase difference in the ”meeting point“ of two trajectories in the drain electrode gives interference fringes in tunneling transport (oscillations with a ﬂux period of Φ0 ). Although the AB effect in nanoobjects is interesting per se, we focus our attention on the dynamical symmetry aspects of this phenomenon and speciﬁcally on the interplay between the AB interference and Kondo screening under strong Coulomb blockade. Typical conﬁgurations of CQD where the AB interference may be superimposed on the Kondo-resonance regime are those in Fig. 3.13(c), 3.14(e,f,g) and 3.15(c). Apparently, the simplest objects for observation of such effect are the ”which pass” conﬁgurations [Fig. 3.13(c) [179] and Fig. 3.14 [246] with one electron]. The Δ -like conﬁguration of Fig. 3.14(f) mimics the more complicated mesoscopic AB interferomenter fabricated in planar GaAs/GaAlAs heterostructures (see Fig. 4.40), provided the electron is located at the deep level in the side dot with strong Coulomb blockade and the dots in the source-drain channel provide ballistic channel for electron transport between the leads [52, 164]. In this case the Δ shaped device may be treated as a two-shoulder tunneling setup (Fig. 4.39, left panel). One may choose the gauge in such a way that the whole phase φB is ascribed to the electron passing through the ballistic shoulder. At zero magnetic ﬂux the interference picture is interpreted in terms of the Fano – Kondo effect described in Section 4.3.2, so that the Δ -shaped TQD in this limit is isomorphous to the T-shaped DQD, i.e. in the unitarity limit for the Kondo phase the antiresonance at the Fermi level arises in the tunneling transparency. The latter characteristics may be calculated by means of the Green function method [see Eq. (4.35) and ff ] or by the NRG technique at zero temperature. In the presence of magnetic ﬂux the interference of Kondo resonance with the phase shift π /2 at the

4.4 Kondo physics for small rings

191

d−shoulder

s

1111 0000 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111

Wd

Ws Φ W0 exp iφ B

11111 00000 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111

d

b−shoulder

Fig. 4.40 scheme of mesoscopic AB interferometer consisting of ballistic b-shoulder and tunneling d-shoulder, which includes a quantum dot.

Fermi level in the d-shoulder and the AB phase shift in the b-shoulder results in the change of the periodicity of the AB effect. Instead of canonical behavior of the AB interferometer with conductance oscillating as G ∼ A + B cos Φ , the frequency doubling of the AB oscillations takes place with the maximum of the conductance at φB = π /2. The physical reason of the period doubling is seen more distinctly in the simpliﬁed analytically solvable model of the ∇-shaped TTQD [432], where the leads are approximated by semi-inﬁnite one-dimensional tight binding chains (Fig. 4.39, right panel). Again, the AB phase is ascribed to the b-shoulder, so that the tunneling between the sites j = −1 and j = +1 is given by the coupling constant W0 eiφB , while the tunneling in the periodic chain is given by the nearest-neighbor matrix element wi,i±1 = w. Then the one-dimensional Hamiltonian of the left and right lead coupled with the b-shoulder may be diagonalized exactly. The problem in a new basis is described by the standard Anderson Hamiltonian Hˆ = Hˆ b + Hˆ d + Hˆ db , but the density of states is renormalized

ρ (ε , φB ) = ρ¯ 0 Here

ρ¯ 0 =

ε 1 + P(φB) . D0

ρ0 w2 2W0 w , P(φB ) = 2 cos φb . W02 + w2 W0 + w2

(4.165)

(4.166)

(the isosceles symmetry of the TTQD, Ws = Wd is adopted for simplicity). This renormalization is the source of Fano effect, which, in turn affects the Kondo tunneling through the d-shoulder.

192

4 DYNAMICAL SYMMETRIES IN THE KONDO EFFECT

The one-dimensional Kondo problem may be solved in the weak coupling regime T TK by means of the two-stage Jefferson – Haldane –Anderson RG approach (see Section 4.1). At the ﬁrst stage one ﬁnds the renormalized dot level

D0 εd ≈ εd + ρ¯ 0W 2 ln ¯ − 3P(φB) . (4.167) D This indicates that the dot energy level is tunable by the magnetic ﬂux Φ . This ﬁeld dependence is transposed in the SW exchange coupling constant and eventually in the Kondo temperature found as a solution of the RG equation for this constant:

3 TK (φB ) = TK0 exp − P(φB ) . (4.168) 2 Calculation of TK within the slave-boson mean ﬁeld theory [379] gives qualitatively the same result. The tunneling conductance is evaluated in the same way as in Eq. (4.25) G(T ) = G0

3π 2 1 − 2Γ cos φB + Γ 2 cos2 φB 2 16 ln [T /TK (φb )]

(4.169)

where Γ = 2wW0 /(w2 + W02 ). We see that in the high-temperature limit the ﬁeld dependence contains both Kondo corrections (ﬁrst term) and Fano corrections which includes the ﬁrst and the second harmonics of the AB oscillations. In the strong coupling limit T TK where the Fermi liquid approach is valid, the temperature expansion (4.24) is substituted for G = G0 (1 − Γ 2 cos2 φB + CT 2 ).

(4.170)

Here C unlike (4.24) cannot be presented using a single parameter TK , because potential scattering related to the b-shoulder cannot be ignored: it gives its own phase shift in the scattering amplitude and thus inﬂuences the interference picture. But in any case the period of AB oscillations is halved in agreement with the NRG results [164]. It is worth mentioning that the situation in real mesoscopic AB interferometers (see Fig. 4.40 for a simpliﬁed scheme) is quite complicated and various concomitant effects (see, e.g. [379]) hamper observation of this interference in a pure form. In nanoscopic AB interferometers the interplay between AB and Kondo effects is more intimately related to the dynamical symmetries inherent in CQD. Such in-

4.4 Kondo physics for small rings

193

terferometer may be realized, e.g., in two-terminal conﬁguration with vertical TQD as a tunneling structure [246]. This conﬁguration sketched in the right panel of Fig. 4.41 may be obtained from the three-channel two-terminal conﬁguration of equilateral TTQD realized in [12, 13] [Fig. 3.15(c)] by closing three dot-lead tunneling channels of six (see left panel of Fig. 4.41. External magnetic ﬁeld B applied in arbitrary direction create magnetic ﬂux Φ with two components Φ1 , Φ2 penetrating two closed loops in this device. Tuning the direction of B any of these ﬂuxes (e.g.

Φ2 ) may be turned to zero.

Fig. 4.41 Two-terminal three-channel triangular TQD. Left panel: vertical dot realization. Right panel: planar projection .

Evolution of the energy spectrum as a function of Φ1 shown in the upper panel of Fig. 4.34 now includes the avoided crossing effect (see Fig. 4.38) due to the loss of chiral symmetry. Hence, the ground state of TTQD is always orbitally nondegenerate. The gauge (4.164) is chosen is such a way that the tunneling integrals in the basic Anderson Hamiltonian are modiﬁed as W → W exp(iΦ1 /3), V1,2 → Vs exp[±i(Φ1 /6 + Φ2 /2)]. Then the SW exchange Hamiltonian reads H = Js S · ss + Jd S · sd + Jsd S · (ssd + sds ).

(4.171)

with ﬂux dependent coupling constants Js (Φ1 , Φ2 ), Jd (Φ1 , Φ2 ) and Jsd (Φ1 , Φ2 ). This dependence is crucial for the realization of the effect under consideration, so we write out the exchange constants explicitly. Since the SW procedure should be modiﬁed near the quasi degeneracy point Φ = Φ0 /2 (φB = π ), the vicinity of this

194

4 DYNAMICAL SYMMETRIES IN THE KONDO EFFECT

point should be considered separately. The magnetic ﬂux in the equations below is measured in units Φ0 /2π . ⎛ 4Vs2

1 + cos

Φ1 3

+ Φ2

⎞

1 ⎠, ε + Q − εF 2V 2 1 1 , Jd = d + 3 εF − ε ε + Q − εF 4VsVd 1 Φ1 Φ2 1 + Jsd = cos + 3 εF − ε ε + Q − εF 6 2

Js =

3

⎝

εF − ε

+

(4.172)

for 0 Φ1 < π , and

1 − sin Φ2 2 − < 0, ε + Q − εF ε + Q − εF 4V 2 1 1 + Jd = d 3 εF − ε ε + Q − εF 1 2VsVd 1 sin Φ2 Jsd = + 3 εF − ε ε + Q − εF V2 Js = s 3

(4.173)

for Φ1 = π . It is seen from these equations that at some values of magnetic ﬂuxes the constant Jsd turns to zero, which means that the source-drain tunneling is blocked due to the formation of ”dark states” in the TQD as a result of AB interference. To ﬁnd the inﬂuence of these dark states on the Kondo-resonance in the tunneling conductance, one should solve the system of RG equations for the Hamiltonian (4.171) 2 d js /d η = − js2 + jsd , 2 2 d jd /d η = − jd + jsd , d jsd /d η = − jsd ( js + jd ) . The solution of Eqs. (4.174) gives TK , ⎧ ⎫ ⎨ ⎬ 2 , TK = D0 exp − ⎩ j + j + ( j − j )2 + 4 j 2 ⎭ s s d d sd

(4.174)

(4.175)

and the conductance at T TK 2 jsd 3 G 1 . = 2 2 G0 4 ( js + jd ) ln (T /TK )

(4.176)

4.4 Kondo physics for small rings

195

Fig. 4.42 Interference of Kondo and Aharonov-Bohm effect in tunneling conductance G/G0 through three-channel two-terminal triangular TQD (Fig. 4.41). Left panel: Φ2 = 0; right panel: Φ2 = Φ1 (after [246]).

The conductance G(Φ1 , Φ2 ) (4.176) obeys the Byers – Yang theorem (periodicity in each phase) and the Onsager condition G(Φ1 , Φ2 ) = G(−Φ1 , −Φ2 ). As a numerical examples, we choose to display the conductance along two lines

Φ1 (Φ ), Φ2 (Φ ) in parameter space of phases, namely, G(Φ1 = Φ , Φ2 = 0) and G(Φ1 = Φ /2, Φ2 = Φ /2) (Fig. 4.42, left and right panels respectively). The Byers – Yang relation implies the respective periods of 2π and 4π in Φ . Experimentally, the magnetic ﬂux penetrates the whole sample as in the right panel of Fig. 4.41, and the ratio Φ1 /Φ2 is determined by the speciﬁc geometry, as was mentioned above. Strictly speaking, the conductance is not periodic in the magnetic ﬁeld unless Φ1 and Φ2 are commensurate. The shapes of the conductance curves presented here are distinct from those pertaining to a mesoscopic AB interferometer and termed as Fano – Kondo effect [164]. In the latter case the Fano interference effect results in a dip in G(Φ ), but does not cause a complete blockade of the source-drain tunneling. In the device of Fig. 4.41 the phase dependence is governed by interference effects on the level spectrum of the TTQD. The three dots share an electron in a coherent state strongly correlated with the lead electrons, and this coherent TTQD as a whole is a vital component of the AB interferometer. In the setup of Fig. 4.41 (right panel) the Kondo cotunneling vanishes identically on the curve Jsd (Φ1 , Φ2 ) = 0. The AB oscillations arise as a result of interference between the clockwise and anticlockwise ”effective rotations” of the TTQD in the tunneling through the {lc} and {rc} arms of the loop. The ”dark

196

4 DYNAMICAL SYMMETRIES IN THE KONDO EFFECT

state” responsible for the AB blockade of the Kondo tunneling is √ |Ψdark = (|l σ − |rσ )/ 2.

(4.177)

Similar role of dark states was revealed in a three-terminal geometry, Fig. 3.14(h) where two reservoirs are taken as the left and right source (LS and RS, respectively), and the third one plays part of the drain (D) [94]. In this paper the AB interference between two paths, namely {LS – TTQD – D} and {RS – TTQD – D} in the presence of the magnetic ﬂux is calculated. The Kondo effect is not taken into account, so the AB interference is seen in a pure form. The result for the conductance is practically the same. The dark state for a single electron which is not coupled with the central dot and the drain lead completely suppresses the tunnel current at Φ = Φ0 /2. This destructive AB interference abates and eventually disappears if the mirror symmetry of the isosceles triangle is violated.

Conﬁnement of countable number of electrons in complex quantum dots under strong Coulomb blockade revealed new facets of the Kondo paradigm in strongly correlated electron systems. The original SU(2) version of the effective Kondo model with odd occupation is now extended to the general SU(n) model not only in the limit n → ∞ where the mean ﬁeld approach is acceptable, but also at small n = 3, 4, where some new possibilities of transition to NFL behavior are discovered. In the models with even occupation real breakthrough was realized from the underscreened spin 1 SO(3) model to a novel family of systems with competing singlet and triplet states of the dot and great variety of SO(n) symmetries with n changing from 3 to 8 in a controllable way. Unceasing attempts to ﬁnd practical applications of controllable symmetries in CQD include various proposals to exploit spin current through CQD in spin ﬁlters and valves, spin-entangled ”dark states” in information storage devices, and so on. Undoubtedly, some of these proposals have a chance to be realized in future nanoelectronics and spintronics provided cheap and reproducible standard cells with CQD as tunneling elements will be fabricated by means of advanced methods of nanotechnology.

Chapter 5

DYNAMICAL SYMMETRIES IN MOLECULAR ELECTRONICS

Although the analogy between complex quantum dots and molecules as tunneling elements in the single-electron transport devices was recognized even before ﬁrst CQD were fabricated and investigated, real breakthrough in this ﬁeld took place after 2000, when both the Coulomb staircase in the current-voltage characteristics and the Kondo-type ZBA anomalies have been found in various molecular systems. The general description of individual molecules and molecular complexes in tunneling devices is given in Section 3.6. Basing on this description of molecular complexes as elements of nanostructures, we concentrate now on the various manifestations of many-particle correlations in speciﬁc molecular environments.

5.1 Kondo effect in molecular environment The wording ”environment” implies electronic and vibrational degrees of freedom inherent in molecular complexes which may assist or suppress Kondo tunneling in the spin channel. Besides, magnetic ions with unﬁlled shells possess their own spectrum of magnetic excitations due to exchange and spin-orbit interaction in the 3d or 4f shell of transition metal or rare-earth metal ions caged in the molecular complex. This internal multiplet structure also may contribute to dynamical symmetries relevant for the formation of the Kondo-resonance tunneling channel. A topical review of various cross-effects in single-molecule electronics may be found in Ref. [14]. We open this section with a brief discussion of the role of ”architecture” of molecular complex in the single-electron tunneling and in many-body effects which accompany this tunneling. Following the general picture of single-electron transistors including molecular complexes as tunneling elements (Section 3.6), we may K. Kikoin et al., Dynamical Symmetries for Nanostructures: Implicit Symmetries in Single-Electron Transport Through Real and Artificial Molecules, DOI 10.1007/978-3-211-99724-6_5, © 2012 Springer-Verlag/Wien

197

198

5 DYNAMICAL SYMMETRIES IN MOLECULAR ELECTRONICS

distinguish several types of such devices. As a rule molecular complex deposited on an etched metallic wire in a break junction geometry contains a cell (cage) where the magnetic moment is located. In this conﬁguration schematically sketched in the upper panel of Fig. 5.1(a) the tunneling problem is mapped on the standard Kondo problem in accordance with the SW procedure described in Section IV.1. If the Coulomb blockade in the molecular cage is strong enough to provide the single electron transport, like in endofullerenes and cerocene [Figs. 3.20 and 3.22], then the Kondo effect should be treated in terms of T-shape DQD [Fig. 3.13(b)] with localized spin in the side dot.

Fig. 5.1 Molecular complexes as elements of single electron molecular transistors. (a) Transition metal molecular complexes with one and two magnetic ions between two edges of broken quantum wire; (b) Molecular magnet in a contact with two metallic leads; (c) molecular complex adsorbed on a metallic substrate in a tunneling contact with a tip of scanning tunneling microscope. Tunneling channels tip/magnetic ions and tip/substrate are shown by solid and dashed lines, respectively. See further explanation in Section 5.1.3.

In some cases localized moments are located at the ends of linear complex molecule (see, e.g., [316]). Then the molecular bridge between two edge spins plays part of the media for indirect exchange [Fig. 5.1(b)], and the tunneling processes

5.1 Kondo effect in molecular environment

199

should be treated in terms of two-site Kondo model or serial DQD geometry. In organic complexes containing transition metal ions the localized spins form a quasi regular structures supported by organic ”scaffolds” [Figs. 3.22 and 3.24]. In this case the dot spins are described by the Heisenberg Hamiltonian with additional terms due to magnetic anisotropy (3.71) or (4.154) characteristic for these complexes. When deriving the scaling equation for the Kondo problem, one should take into account the complex structure of spin multiplets described by these Hamiltonians. Fig. 5.1(b) schematically illustrates the tunneling setup with TMOC. We will return to these conﬁgurations in Section V.2.

5.1.1 Chiral symmetry of orbitals and Kondo tunneling Electronic states in artiﬁcial molecules (complex quantum dots) fabricated in planar heterostructures as a rule have no deﬁnite symmetry unless the conﬁning potential imposes its own angular symmetry to the dot states, like in vertical quantum dots. Thus, the main source of extra orbital degrees of freedoms transforming the simple SU(2) or SO(3) groups into a semi-simple SU(n) or SO(m) group with n > 2 and m > 3 is the discrete symmetry of CQD exempliﬁed in Figs. 3.13 and 3.14. The situation is radically different in molecular complexes because the molecular orbitals are always classiﬁed in accordance with the discrete point group of a given molecule. Even in the simplest case of fullerenes where there is no speciﬁc source of localized spin except spatially quantized orbitals of carbonic cage, the HOMO and LUMO states involved in Kondo effect may possess orbital degeneracy (Section 3.6.1). More exquisite is the case of single wall nanotubes. Quantum dots prepared from short fragments of SWNT inherit double orbital degeneracy of 2D graphene sheets (Section 3.6.2 and Fig. 3.21). As a result of this degeneracy the system of Coulomb windows and resonances consists of fourfold patterns with occupations N = 1, 2, 3, 4 corresponding to subsequent ﬁlling of the two orbitals with opposite chirality by two electrons with opposite spin projections [15, 33, 183, 272]. From the level scheme shown in the upper panel of Fig. 5.2 one can see that the SU(4) Kondo effect should be realized in SWNT due to orbital degeneracy of electron states (see Section 4.3.1). Transitions between the ground state and excited states involved in Kondo screening mechanism are indicated by arrows. Occupation

200

5 DYNAMICAL SYMMETRIES IN MOLECULAR ELECTRONICS

Fig. 5.2 Upper panel: Nanotube between two electrodes as a quantum dot with discrete energy spectrum. Middle panel: Energy level scheme and inter-level transitions in nanotube quantum dots with occupation number N = 1, 2, 3. Lower panel: Energy level splitting in Zeeman ﬁeld .

N = 2 corresponds to the case of two-channel screening of S=1. Complete ”shell” N = 4 is a spin and orbital singlet without Kondo dynamics. The ﬁngerprint of this type of Kondo effect is its evolution in external magnetic ﬁeld [272]. Due to the opposite chirality of two orbital states the diamagnetic Larmor shift of the corresponding electron levels has opposite sign for clockwise and anticlockwise twisted orbitals. This shift together with the Zeeman splitting (which is noticeably weaker) removes spin and orbital degeneracy of the zero ﬁeld states in accordance with the scheme presented in the lower panel of Fig. 5.2. From the symmetry point of view such splitting means that at odd occupation the original SU(4) symmetry of the Kondo tunneling in zero ﬁeld which gives a ZBA bridge in the corresponding Coulomb window should be split into four FBA lines in the charge sector N = 1 in accordance with possible spin ﬂips and pseudospin ﬂips. In the sector N = 3 only three lines survive due to complete spin ﬁlling of one of the two orbital subsets. In the sector N = 2 diamagnetic splitting overcomes the Hund’s rule, so that the ground state of the system transforms into spin singlet at ﬁnite mag-

5.1 Kondo effect in molecular environment

201

netic ﬁeld. Then only the pseudospin Kondo effect is possible and two FBA arise due to pseudospin Larmor splitting. Just this picture was observed experimentally [272]. Additional veriﬁcation of this model was given by comparison of Kondo tunneling in zero magnetic ﬁeld with NRG calculations for the SU(4) Kondo effect [15]. In accordance with the general theory of the Kondo effect universal temperature behavior of the conductance in the SU(4) model in the strong coupling limit T TK slightly differs from the standard SU(2) law (4.26) reﬂecting different universality class of the former model. This difference was detected in comparison of calculated and measured curves G(T ). In this remarkable example speciﬁc contribution of molecular degrees of freedom in the Kondo tunneling in the presence of non-trivial dynamical symmetry is related to broken chiral symmetry of molecular orbitals in the axial magnetic ﬁeld.

5.1.2 Kondo effect in the presence of Thomas-Rashba precession Another manifestation of this type of symmetry is the interplay between the Kondo effect and spin-orbit interaction in its speciﬁc form characteristic for nanoobjects. This is the so called Rashba coupling characteristic for planar object in anisotropic conﬁnement potential characterized by the unit vector n normal to the plane [53, 348]. Physical conditions favorable for the appearance of Rashba term in the effective cotunneling Hamiltonian may be realized, in particular, in a system ’metallic substrate – absorbed molecule – nanotip of STM’ [330]. In this geometry the noticeable gradient of electric ﬁeld may exist in the substrate with spin-orbital interaction or in absorption potential retaining molecular complex on the surface. Rashba coupling is a speciﬁc manifestation of the fundamental Thomas effect of spin precession in the magnetic component of the electromagnetic ﬁeld due to relativistic spin-orbit interaction [399]. This precession is strongly enhanced in semiconductors, and in particular in 2DEG in semiconductor heterostructures [53]. Thomas precession is a kinematic effect resulting in a relativistic correction to the magnetic moment of a particle related to the angular velocity of its orbital motion. Interaction of spin with magnetic component of the electromagnetic ﬁeld e Bem = − ∇V (r) c

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5 DYNAMICAL SYMMETRIES IN MOLECULAR ELECTRONICS

results in the appearance of additional ”Thomas – Rashba” (TR) term in the effective Hamiltonian of a system ’adsorbate plus substrate’ HTR = αd nd · (S × pd ) + αl ∑ nl · (σ × p)

(5.1)

p

Here nd and nl are unit vectors in the direction of the gradient of the electric potential V (r) for an adsorbed molecule and a surface of the substrate, respectively;

αd and αl are the coupling constants obtained after averaging the TR coupling over the orbitals of the adsorbed molecule and the band states in the substrate. If the spin degrees of freedom of substrate may be represented by the spin sl of the local environment of absorbed molecule, then the second term in HTR may be written in the same form as the ﬁrst one, namely as αl nl · (sl × pl ). The vectors nd and nl are not necessarily parallel and the pseudoscalar coupling constants αd and αl are certainly different. Unlike the TR Hamiltonian for the two-dimensional electron gas [53, 348] where p = h¯ k is deﬁned in the whole Brillouin zone, in case of an adsorbed molecule momentum operator should be deﬁned in some local basis of the adsorbate and the substrate molecular orbitals which are subject to the potential V (r) in the vicinity of the adsorbed molecule. Then the effective spin Hamiltonian Hˆ s for the lead-dot device in an external magnetic ﬁeld B entering the Zeeman Hamiltonian Hˆ Z has the form Hˆ s = Hˆ TR + Hˆ Z + Hˆ cot =

(5.2)

nd · (S × wd ) + nl · (sl × wl ) + hd · S + hl · sl + JS · sl Here hi = gi μB B, wi = αi pi . It is seen from Eq. (5.2) that the spin precession described by HTR results in rotation of the spin axes established by the Zeeman term HZ , but the rotation angles are different for the dot and lead subsystems, S = T(Θd , Φd )S, sl = T(Θl , Φl )sl

(5.3)

It follows from (5.3) that the cotunneling part of the spin Hamiltonian (5.2) after rotation operation acquires the form (Ωd ) · s l (Ωl ) Hcot = JS

(5.4)

Here Ωd(l) = {Θd(l) , Φd(l) } are the Euler angles describing rotation of spin axes relative to the framework established by the external magnetic ﬁeld in the absence of TR precession. This means that the uniﬁed spin coordinate system for the adsorbed

5.1 Kondo effect in molecular environment

203

molecule and the metallic substrate may be established only at zero magnetic ﬁeld h = 0. Similar type of exchange between two spins with canted magnetic axes arises in the indirect RKKY interaction between two localized spins immersed in the twodimensional electron gas in the presence of Rashba coupling given by the last term in the Hamiltonian (5.1). In this case the twisting effect is explicitly contained in the RKKY exchange, so that the corresponding canting angle is proportional to the distance between the two magnetic ions [174]. The Hamiltonian (5.4) reminds one of the versions of Dzyaloshinskii – Moriya exchange Hamiltonian derived for the low-symmetry lattices with spin anisotropy, where the neighboring spins are localized in cells with different crystalline symmetry of the local environment [288]. Indeed, the Hamiltonian (5.4) may be reduced to a familiar Dzyaloshinskii – Moriya form with a mixed vector product in the limits of week magnetic ﬁelds hi wi and strong magnetic ﬁelds hi wi . To show this let us consider the simplest case when both TR vectors are parallel to the z-axis but the coupling constants are different in magnitude, ni = (0, 0, 1), wi = (wix , wiy , 0). Then the dot Hamiltonian Hˆ d = Hˆ d Z + Hˆ d TR = hdz Sz + (hdx + wdy )Sx + (hdy − wdx )Sy . is transformed to new spin axes by means of the matrix ⎛ ⎞ cos Θd cos Φd − cos Θd sin Φd sin Θd sin Φd cos Φd 0 ⎠ T(Φd , Θd ) = ⎝ − sin Θd cos Φd sin Θd sin Φd cos Θd

(5.5)

(5.6)

The Euler angles are deﬁned as tan Θd =

wdy + hdx |wd | , tan Φd = . hdz wdx − hdy

(5.7)

This means that the quantities w2d⊥ = w2dx + w2dy and h2d⊥ = h2dx + h2dy deﬁne the modulus of the planar component of the effective magnetic ﬁeld Δ⊥2 = w2d⊥ + h2d⊥. Similar transformation for the Hamiltonian Hl Z + Hl TR gives for the Euler angles (Φl , Θl ) the same equations (5.7) with wl , hl substituted for wd , hd . It is seen from Eq. (5.7) that in the absence of magnetic ﬁeld, h = 0, the Euler angles Θd = Θl = π /2, i.e. the quantization axes zd and zl coincide, although the reference frames for both subsystems are turned around the y axis [Fig. 5.3(a)] due to the TR effect, the only source of magnetic ﬁeld in this case. This means that the TR

204

5 DYNAMICAL SYMMETRIES IN MOLECULAR ELECTRONICS

z"

z

φ

S’ S

y y’

Θ Φ z’z" x

m

y"

x"

x"

x’

y"

Fig. 5.3 Left panel: Rotation of spin axes in accordance with Eq. (5.6); (x, y, z) - initial reference frame, x , y , z - reference frame after rotation Θ , x , y , z - reference frame after rotation Φ . Right panel: Rotation S → S resulting in transformation (5.6) .

precession is irrelevant for Kondo effect in accordance with general argumentation proposed in Ref. [282]. Since only the angle between the quantization axes in the two systems is relevant, we consider in further discussion the simpliﬁed model where the TR precession is signiﬁcant only for one subsystem, say, for the adsorbed molecule. In the limiting case hd wi the quantization axis slightly deviates from its direction z x. Turning the reference frame at a small angle φ is equivalent to turning the spin S in the old frame at the same angle φ , so that S = S + φ (m × S),

(5.8)

[see Fig. 5.3(b)]. Here m is the unit vector in the direction of quantization axis (in the original frame it is mx ). Then, substituting (5.8) in (5.4) we get Hex = JS · σ + j(S × σ )

(5.9)

where j = J φ m is the TR induced anisotropic component of the exchange coupling. Thus the TR correction to the exchange Hamiltonian indeed reminds the Dzyaloshinskii – Moria interaction. The same transformation may be realized in the opposite case hd wi . In this case one may refer to the limit hd → ∞ where the quantization axis is z. Then at

5.1 Kondo effect in molecular environment

205

ﬁnite hd the effective exchange Hamiltonian has the same form (5.9) but the unit vector m has only z-component. In both limits we deal with the Kondo effect in the presence of magnetic anisotropy. In the limit of strong magnetic ﬁeld the fully anisotropic Kondo Hamiltonian is 1 Hˆ = Hˆ d + Hˆ ex = hz Sz + wd S+ + w∗d S− 2 1 + − + + + J σlz Sz J− sl S + J+s− l S 2

(5.10)

with J± = J(1 ± iφ ),

(5.11)

and φ ≈ |wd |/h. Thus the main effect of TR precession is induced magnetic anisotropy of the Kondo tunneling. In the limit of weak magnetic ﬁeld Hˆ d has the same form as in (5.10), but the anisotropy of Hˆ ex manifests itself in a different way due to different direction of the axes in spin space: iφ J + − + Hˆ ex = J(S · sl ) + Sz (s− l − sl ) + (S − S )slz 2

(5.12)

where φ ≈ h/|wd |. This type of magnetic anisotropy is relevant for the Kondo effect. The RG scaling analysis shows that the difference J+ − J− characterizing the precession angle φ grows together with the constant J and distorts the ﬂow trajectories, but does not affect the inﬁnite ﬁxed point. The TR component of the effective magnetic ﬁeld also renormalizes together with the precession angle D0 δ wd ∼ | j|2 ln . |wd | T

(5.13)

In the limit of strong ﬁeld h wd this enhancement acquires the form of ”dynamical” contribution to the planar magnetic ﬁeld. This ”random” ﬁeld reminds the effect of exchange anisotropy induced by an edge spin coupled to an open spin-one-half antiferromagnetic Heisenberg chain [114]. The Kondo-induced component of the planar ﬁeld is weak at T TK , |δ w⊥ |/TK ∼ jΔ ⊥ . However, it generates its own energy scale T ∗ = TK e−1/ j

(5.14)

where the precession induced magnetic ﬁeld becomes comparable with the static magnetic ﬁeld.

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5 DYNAMICAL SYMMETRIES IN MOLECULAR ELECTRONICS

ky EF kx k

Fig. 5.4 Left panel: Renormalization of the conduction band Ekσ due to the Rashba interaction; Right panel: Two sheets of Fermi surface with opposite spiralities.

Rashba term in the two-dimensional leads also modiﬁes the tunneling matrix elements between the lead and the dot because the TR precession results in a splitting of the Fermi surface in accordance with ”spirality” (see Fig. 5.4). In particular, tunneling with given spirality admits spin ﬂip dot-lead matrix elements W↑↓ . Additional phase due to mismatch between the quantization axes (5.2) inﬂuences the interference fringe in the AB - Kondo interferometers [24, 389, 408].

5.1.3 Scanning tunneling spectroscopy via Kondo impurities Tunnel transport through atomic and molecular complexes adsorbed on metallic surface in tunneling contact with the nanotip of STM [Fig. 5.1(c)] is a complicated phenomenon because in this case only part of the tunneling channels involve localized spins of the adsorbed magnetic ions. Since direct tunneling contact between STM and the surface of substrate is unavoidable, Fano effect always accompany Kondo tunneling in this case. Due to its single atom spatial resolution, the scanning tunneling microscope became a powerful tool for studies of the surface microstructure [56]. Since the characteristic curvature Rt of STM tip is comparable with the size of molecular complexes adsorbed on the metallic surface or suspended between the edges of broken metallic wires, an STM may be used as an electrode in electric circuits containing these complexes and thus exploited in the studies of many-particle physics of nanoobjects.

5.1 Kondo effect in molecular environment

207

The tunneling spectra essentially depend on the exact position of the tip relative to that of the molecular complex [330]. First, since there is no direct contact between the STM nanotip and the adsorbent, the condition Wt Wd is fulﬁlled in all practical situations[(see Fig. 5.1(c)]. Second, the center of the nanotip as a rule is settled not exactly above the symmetry center of the molecular complex, so the tunneling contact between the molecule and the tip reduces the molecular point symmetry. In particular, the tunneling matrix elements between the tip and the magnetic ions depend on ion position on the surface and the tip conﬁguration, Wti (Rti ) = wti (Z, R )e−κ Z where (Z, R ) are normal and parallel component of the vector Rti connecting the nanotip approximated by the sphere of the radius Rt and the ion i on the surface of the substrate, the parameter κ characterizes the exponential damping of the surface electron wavefunction in the normal direction. A simple model which allows parametrization of this matrix element is offered in Ref. [330]. Third, there exists a direct tunneling channel between the tip and the substrate characterized by the matrix element Wta . This channel is the source of the Fano effect in the tunneling spectra [330, 403]. The Fano lineshape in the complex conﬁgurations of Fig. 5.1(c) has a more complicated form than fσ (ω ) in Eqs. (4.101). Since the perturbation of the substrate density is characterized by the length ∼ Rt , one should use in equation (4.96) for the T-matrix the retarded Green function g(r, r ; ω ) g(r, r ; ω ) = ∑ k

ψk (r)ψk∗ (r ) ω − εk + iη

(5.15)

instead of the local function g0 (0, 0, ; ω ) used in Eqs. (4.101). Here the wave functions in the numerator and the energy levels in the denominator describe the unperturbed quasi two-dimensional, metallic substrate. Then the two quantities related to the real and imaginary parts of the Green function are deﬁned as

Λ (R, ω ) = eκ Z

Re g0 (R, 0; ω ) πρs (ω )

(5.16)

γ (R, ω ) = −eκ Z

Im g0 (R, 0; ω ) . πρs(ω )

(5.17)

and

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5 DYNAMICAL SYMMETRIES IN MOLECULAR ELECTRONICS

where ρs (ω ) is the density of states of substrate. These two functions carry the information about both the spatial extent of the surface electron perturbation at arbitrary R on the surface near the tip and also the spatial resolution of the tip. Then the Fano lineshape factor acquires the form [330, 377]: f (ω ) =

q2 − γ 2 + 2εγ q 1 + ε2

(5.18)

instead of (4.101). The lineshape factor (5.18) carries information not only on the modiﬁcation of the spectral density due to interference of the resonance and continuous states but also on the spatial structure of the nanoobject via the factor

γ (R, ω ). Turning to the practical realizations of correlated tunneling involving STM and molecules adsorbed on metallic surface, one should mention that involvement of several orbital states related to the substrate electrons is an implicit property of such setups. Even in the simplest case of single Co atom on a noble metal substrate the orbital degrees of freedom are involved in Kondo tunneling because the orbital degeneracy of 3d-electrons in transition metals on the surface is only partially lifted. So the Kondo effect in this case should be discussed in terms of SU(2m) symmetry, where m is the number of relevant orbitals [443]. One more relevant manifestation of the involvement of substrate states in Kondo tunneling is the magnetic anisotropy of the individual magnetic ions . This anisotropy arises when the magnetic atoms with spin S > 1/2 are incorporated into a polar covalent bonding with the substrate. This effect was discovered in the studies of magnetic ﬁeld dependent STM spectra of Mn, Fe and Co atoms on a copper substrate covered by a monolayer of Cu2 N [162, 317]. The spin excitations in an anisotropic environment are described by the Hamiltonian

E2 (5.19) Hˆ dani = DSz2 + [(S+ )2 + (S−)2 ] + h · S 2 As was noticed in Section 2.3, the dynamical symmetry of this Hamiltonian is SU(3). In the case S = 1 the generators of this group are the vector T (9.39) and the tensor Qˆ (9.40) [see also (2.57)]. In the case S = 3/2, S = 2 and S = 5/2 one should address the irreducible representations of the groups SU(4), SU(5) and SU(6), respectively. Positive and negative values of the parameter D mean hard axis and easy axis anisotropy, respectively.

5.1 Kondo effect in molecular environment

209

The sign of D is relevant for Kondo scattering. In the case D > 0 the ground state of magnetic ion with S = 3/2 is a doublet |S, Sz = |3/2, ±1/2 and the magnetic scattering with δ Sz = ±1 described by the ladder operators S± is possible. On the contrary in the easy axis case the ground state doublet is |3/2, ±3/2 and the Kondo processes are suppressed. The same is valid for integer spin: the ground state doublet for S = 2 is |2, ±2 for negative D. Transverse magnetic anisotropy, of course, intermixes the pure spin states, but the general trend survives provided E2 D. In particular, D is negative for Mn (S=5/2) and Fe (S=2) and the ratios E2 /D are -0.18 and -0.2, respectively [162]. The STM spectra demonstrate no sign of Kondo peak for these two impurities in accordance with the above arguments. In contrast to these two systems with easy axis magnetic anisotropy, Co impurities on the same substrate demonstrate the hard-axis anisotropy with D > 0. The Kondo peak is observed in the STM spectra, and its behavior in magnetic ﬁeld directed normally and in parallel to the substrate plane conﬁrms the presence of magnetic anisotropy [317]. It should be noted that the net spin of transition metal adsorbents is signiﬁcantly less than the nominal integer or half-integer values. These nominal values are obtained only provided the covalent transfer of spin density to the nearest orbitals of the adsorbate is properly taken into account [162]. Ti impurity with spin 1/2 shows the conventional Kondo behavior without any sign of magnetic anisotropy. This fact, apparently means that the binding of Ti ions to the substrate changes its electron conﬁguration from d 2 to d 1 due to strong ligand ﬁeld [317]. In any case these experiments give strong evidence for the intimate connection between magnetic anisotropy and covalent bonding between transition metal ions and the substrate. Theoretical calculation of Kondo tunneling in the presence of easy-axis magnetic anisotropy were performed in a framework of the NRG method [446]. These calculations reproduce the experimental STM spectra for Fe ion on the Cu2 N/Cu(100) surface [162]. Besides, the authors offer one more mechanism of magnetic ﬁeld induced Kondo effect. They have noticed that the kinematic suppression of the Kondo effect due to selection rule δ Sz = ±1 at D < 0 may be lifted of an external in-plane magnetic ﬁeld is applied in the x direction. This ﬁeld induces crossing of the two lowest levels and thus allows Kondo processes. The resonance magnetic ﬁeld at which the level crossing occurs is given by the equation h¯ x =

2B2 (|D| + B2).

(5.20)

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5 DYNAMICAL SYMMETRIES IN MOLECULAR ELECTRONICS

This effect is, however, extremely sensitive to misalignment of magnetic ﬁeld in z direction. 1◦ misalignment is enough to quench the Kondo effect for the anisotropy parameters characteristic for Fe:Cu2 N/Cu(100). Transition metal atoms may form quasi molecular trimers both linear and triangular on noble metal surfaces [182]. The problem of Kondo tunneling through such a complex should be described in terms of the triangular TQD model in a charge sector N = 3 (see Section 4.4). Apparently, the ground state of the triangular complex Cr3 studied in [182] is an orbital singlet, so the Hamiltonian Hˆ d for this complex has the Heisenberg-like form (4.154). This means that the geometrical spin frustration is essential for the Kondo effect. It was noted in Refs. [175, 252] that these frustrations may result in the NFL Kondo regime.

Fig. 5.5 Triangular complex Cr3 on the Au(111) surface.

111 000 0000 1111 000 111 0000 1111 000 111 0000 1111 000 111 0000 1111 000 111 0000 1111 0000 1111 0000 1111 000 111 000 111 0000 1111 0000 1111 0000 1111 000 111 0000 1111 0000 1111 000 111 0000 1111 0000 1111 000 111 0000 1111 0000 1111 000 111 0000 1111 0000 1111 000 111 000 111 000 111 0000 1111 000 111 000 111 000 111 0000 1111 000 111 000 111 000 111 0000 1111 000 111 000 111 000 111 0000 1111 000 111 000 111 000 111 0000 1111 000 000111 000 111 0000111 1111 000 111 0000 1111 000 111 0000 1111 0000 1111 000 111 0000 1111 0000 1111 000 111 0000 1111 0000 1111 000 111 0000 1111 0000 1111 000 111 0000 1111 000 111 0000 1111 000 111 0000 1111 000 111 0000 1111 000 111 0000 1111 000 111 0000 1111

Having in mind the above mentioned considerations about the essential role of local surrounding of the adsorbed complex, one should pay special attention to the crystalline structure of the substrate layer. In particular the equilateral Cr3 complex is stable on the Au(111) surface possessing the hexagonal symmetry (Fig. 5.5). In this case both the trimer and its local surrounding in the substrate possess the same point symmetry C3v , so that the description of the subsystem substrate + nanoobject may be mapped on that used for the three-terminal TTQD conﬁguration [Fig. 3.14(h)]. Then the effective Kugel – Khomskii exchange Hamiltonian (3.34) describes the magnetic interaction in this subsystem. If only the ground state spin 1/2 of Cr3 is retained in Hˆ d , the vector f in this Hamiltonian contains the generators λˆ of the SU(6) group corresponding to two projections of spin and three ”helicity”

5.2 Kondo effect in molecular magnets

211

indices of surface electrons obeying C3v point symmetry [see Eq. (3.52)]. This coupling provides either one or two tunneling channels depending on the sign of the crystal ﬁeld splitting for the surface states. The contact with STM nanotip makes the problem quite uncertain. First, one may speak about the triangular symmetry of the whole system only provided the nanotip of the STM is located exactly above the center of the Cr3 trimer. Even in this case there is no reason to suppose that three equivalent channels will be formed between the nearly spherical tip of the STM and the complex. More realistic is the assumption about the single fully symmetric state of the conduction electrons in a tip with A1 symmetry. The Fano effect cannot be excluded in these setups. To summarize the state of the art in the theory of tunneling via triangular Cr3 complex, one should say that the anticipations of NFL effects are reasonable (cf. the study of TTQD [285] in Section 4.3.5), and some results obtained in simpliﬁed models are available, a convincing theoretical description of this complex system is still absent. Complex molecular objects including magnetic ions may also be fabricated on metallic surfaces. Fabrication in this context means that the transition metal atoms and complex molecules can be induced to form rigid chemical bonds through STM manipulation. In particular, it was demonstrated [421] that individual V atoms may form stable complexes with π -electron acceptor tetracyanoethylene (TCNE). Complexes V(TCNE)x and Vx (TCNE) with x = 1, 2 on the Ag(100) surface were formed and Kondo peaks of single V site and two V site origin were studied by means of STM. Another example of such STM manipulation is the synthesis of complexes M(DCA)2 on insulating NaCl ﬁlm, where M=Fe, Ni and DCA is abbreviation for a ligand 9,10-dicyanoanthracene [263].

5.2 Kondo effect in molecular magnets In Section 5.1.3 we dealt with a situation where the covalent bonding between the magnetic ion and the ligands belonging to one of the leads (substrate) induces magnetic anisotropy, which in turn may affect electron cotunneling through magnetic ion in the Kondo regime. In molecular magnets described in Section 3.6.3 this anisotropy plays a decisive role in formation of Kondo resonances in tunneling spectra [142, 257, 355, 420]. Molecular magnets (MM) are typically transition metal-organic complexes made from magnetic ions of iron group held together by scaffold ligands which medi-

212

5 DYNAMICAL SYMMETRIES IN MOLECULAR ELECTRONICS

ate exchange interaction, spin-orbit interaction and hence magnetic anisotropy. These complexes are essentially non-spherical objects [Figs. 3.22 - 3.24) and the anisotropy of the total spin S of the MM is inherited from its constituents. The spin S is formed by all spins of the constituent magnetic ions which are coupled with exchange interaction described by the Hamiltonian (4.154). In case of ferromagnetic sign of the exchange constants I jl , the total spin is the arithmetic sum of the constituent spins, which may be large enough to make classical description of spin evolution a satisfactory approximation. If the ferromagnetic and antiferromagnetic coupling compete in the Hamiltonian (4.154), the spin in the ground state may be less than the above sum, 1 < S 10. In any case we consider below only the ground state of the MM and neglect all excitations with the energy cost ∼ I . We neglect also the orbital degrees of freedom related to nonzero angular momentum of the magnetic ions and concentrate on their spin degrees of freedom. This limitation is not too strict, because the interatomic exchange constants I are usually of the order of tens or hundreds of Kelvin , and the excitation spectrum at ε I is rich enough to leave a room for realization of various dynamical symmetries in the millikelvin temperature interval characterizing the spin and electron tunneling in MM. The Hamiltonian of the complete system reads: Hˆ = Hˆ dani + Hˆ b + Hˆ ex , where the second and third terms have usual SW form, and the ﬁrst term is a generalization of the Hamiltonian (5.19) introduced in the preceding section, 3

1 Hˆ dani = −DSz2 + ∑ E2n [(S+ )2n + (S−)2n ] + h · S. 2 n=1

(5.21)

We consider a molecule with easy z-axis and hard x-axis anisotropy. The dominant uni-axial anisotropy D favors the spin to align with the z-axis but maintains a continuous rotation SU(n) symmetry with respect to this axis so that the states |S, M remain the eigenstates of Hˆ dani (M is the eigenvalue of Sz ). The ground state is the doublet |S, ±S, and the Kondo scattering Hˆ ex with δ M = 0, ±1 is blocked, as was brieﬂy mentioned in Section 5.1.3. Now we have to discuss the role of the easy plane perturbation ∼ E2n more attentively than it was done in the previous section. This perturbation reduces the symmetry to that of the discrete group C2n of ﬁnite rotations relative to the easy axis. Since the operators S± do not commute with Sz , these rotations induce ”spin tunneling” between the eigenstates of Sz . In the classical limit of inﬁnite spin these processes indeed can be interpreted as tunneling through potential barrier generated by the anisotropy. There are two equivalent paths (clockwise and anticlockwise)

5.2 Kondo effect in molecular magnets

213

S=3/2

S=5/2

S=7/2

1 0 0 1 0 1 1 0 0 1 11 00 00 11

u

d

11 00 00 11 00 11

1 0 0 1 0 1 1 0 0 1

11 00 00 11 00 11 11 00 00 11

u

d

00 11 11 00 00 11

u

d

Fig. 5.6 Construction of states |u and |d (5.22) for spin multiplets S = 3/2, 5/2, 7/2.

for spin reversal passing and the two paths interfere with involvement of a Berry phase (see [123] and references therein). In the quantum case of ﬁnite spin the spin reversal consists of several ﬁnite steps (Fig. 5.6). The easy plane terms in the Hamiltonian Hˆ dani admix to any state |Sz other states |Sz ± (2n)kk=1,2... . In case of half-integer spin the corresponding linear combinations form two disjoint sets of states |u = {|S, +M ± (2n)k}k=0,1,2... , |d = {|S, −M ± (2n)k}k=0,1,2... .

(5.22)

[124, 420]. Hence the in-plane perturbation E2n does not connect the opposite basis states |S, ±S. Let us consider in more details the case of low-symmetry anisotropy E2 = 0, E4 = E6 = 0 illustrated by Fig. 5.6. Looking at this ﬁgure, we see that there are only two states |u, |d which form the basis set for Hˆ dani in this case. The Kondo scattering term can change Sz by one and thus connect these two disjoint states. This means that the Kondo effect assisted by quantum tunneling of magnetization is possible in case of easy-z axis magnetic anisotropy. To describe this peculiar Kondo effect one may resort to the standard trick with the pseudospin operator T , which in this case is built from two states (5.22). This operator may be constructed for half-integer spins in accordance with the kinematic scheme of Fig. 5.6, T + = |ud|, T − = |du|, Tz = (|uu| − |dd|)/2.

(5.23)

The effective exchange Hamiltonian in these terms is fully anisotropic like in the case of the Kondo – Rashba effect (Section 5.1.2). This Hamiltonian is obtained by

214

5 DYNAMICAL SYMMETRIES IN MOLECULAR ELECTRONICS

means of projecting the original Kondo Hamiltonian JS · s onto the set (5.22) by means of the corresponding Hubbard operators: Hˆ ex = J

∑

νν =u,d

X νν ν |S|ν s =

∑

α =x,y,z

Jα T α sα .

(5.24)

Here the effective exchange parameters explicitly depend on the constants D, E2 via the expansion coefﬁcients in the wave functions |u, |d, Jz = 2Ju|Sz |u, Jx,y = Ju|S+ ± S−|d.

(5.25)

The Hamiltonian (5.24) represents a class of Kondo Hamiltonians with interlaced spin and pseudospin degrees of freedom. In the SU(6) Hamiltonian (4.150) both the dot and reservoir are characterized by spin and pseudospin (orbital) degrees of freedom. The corresponding variables are separated on the level of the Hamiltonian and interfere only in the formation of the Kondo cloud. In the two-channel KugelKhomskii Hamiltonian (4.163) the anisotropy of the orbital degrees of freedom is different for the dot and the reservoir (even and odd lead modes are coupled with the longitudinal and transversal components of the dot pseudospin, respectively). In the Hamiltonian (5.24) the pseudospin representing the dot is coupled with the spin representing the lead electrons. One may say that appearance of pseudospin in the magnetically anisotropic system is the manifestation of hidden Z2 symmetry related to the two discrete paths from the northern pole to the southern pole of distorted Bloch sphere. The low-energy ﬂow diagram is obtained from the solution of the system of scaling equations of the RG group dJα /d η = −ρ0 Jβ Jγ

(5.26)

where α , β , γ are cyclic permutations of x, y, z. The scaling curves should be drawn in a three-dimensional phase space. Inversion of any pair of Jα , Jβ leaves the scaling equations invariant, whereas inverting a single one reverses the ﬂow. All scaling trajectories ﬂow to the usual strong coupling limit except those in planes of uni-axial symmetry, |Jα | = |Jβ | < |Jγ | with Jα Jβ Jγ < 0. In the latter case one has a ferromagnetic ﬁxed line which is unstable with respect to inﬁnitesimal perturbations perpendicular to it. In the vicinity of this line TK is strongly suppressed. Integration of the scaling equations gives the following equation for TK obtained for |Jz | |J |x |Jy | with |Jz | = |Jy |:

5.2 Kondo effect in molecular magnets

215

⎞ 2 2 | jy | jz − jx ⎠ ¯ = − 1 . ln(TK /D) cs−1 ⎝ 2 2 j2 − j2 | j2 − j2 | jz − jy ⎛

z

y

z

(5.27)

y

Here cs(p|q)−1 is the inverse of the elliptic integral cs(p|q). Here jα = ρ0 Jα . In the limit E2 D Eq. (5.27 ) reduces to

1 1 D TK ∝ D¯ exp − 1 − ln . 2j 2S E2

(5.28)

The Kondo temperature may be deduced also from the NRG level ﬂow [355]. The results of numerical calculations for three values of half-integer spin are shown

TK/D

in Fig. 5.7. We see that the Kondo effect is blocked for small enough in-plane anisotropy E2 in accordance with the arguments presented in Section 5.1.3.

E2/D' Fig. 5.7 Kondo temperature as a function of the magnetic anisotropy parameter E2 (5.21) for molecular magnets with different half-integer S = 3/2, 5/2, 7/2; D = 0.005 D, after [355].

If the dominant easy plane perturbation has higher symmetry, E4 E2 the picture is more complicated. In this case there are four disjoint subsets instead of two in the previous cases. Besides these sets are not necessarily matched with the Kondo scattering term to satisfy the kinematic conditions for Kondo scattering in the ground state. In the general case of dominant term E2n the conditions where the Kondo

216

5 DYNAMICAL SYMMETRIES IN MOLECULAR ELECTRONICS

effect can occur may be formulated as 2S − 1 = integer. 2n

(5.29)

In case of n = 1 this condition is fulﬁlled for S = 5/2 and S = 7/2, in case of n = 2, the ratio is integer for S = 5/2 but not for S = 7/2. In accordance with this criterion, Kondo tunneling through MM with integer S is completely suppressed. The weak coupling RG approach is valid only provided TK Δa where Δa ∼ (2S − 1)D is the gap separating the Kramers-degenerate ground state from spin excitations of the MM shown in Fig. 5.6. Only under this condition the general dynamical symmetry of the group Z2 × C2n is reduced to that of a simple two-level system. If TK is comparable in magnitude with the whole interval of spin excitations in MM, more general projection procedure than that used in (5.24) should be elaborated [420]. In this case the general form of the Hamiltonian Hˆ ex [middle term in Eq. (5.24)] is still valid but the index ν includes not only the quantum number u, d but also the indices enumerating disjoined subsets. Therefore transition to pseudospin representation is impossible. This case is direct manifestation of dynamical symmetry characterizing the supermultiplet of MM spectrum, because there is no selection rules for the interlevel transitions and the Kondo temperature loses its universality, TK = TK (J, Δ a ), i.e. it depends on the energy splitting like in the models with SO(n) symmetries discussed in Chapter 4. The Kondo effect in MM is sensitive to external magnetic ﬁeld. We mentioned already in Section 5.1.3 the magnetic ﬁeld induced Kondo tunneling in STM spectra of absorbed magnetic ions. Similar effect of reentrant Kondo effect is possible also in MM. Magnetic ﬁeld modulation effects in the Kondo conductance induced by h = hz in MM are quite multifarious due to multiple anticrossings in the excitation spectra induced by the Zeeman splitting (see Ref. [420] for further details). The ﬁeld with transverse in-plane components intermixes the sets |u, |d and thus gives additional splitting of the ground state (contributes to the g factor). This contribution is relevant for the Kondo effect [142, 257]. One may say that the spin ﬂip u ↔ d induced by Kondo scattering inherits essential features of spin tunneling S ↔ −S in the continuous model of classical spin S via clockwise and anticlockwise paths. One may also say that the states |u and |d accumulate the Berry phases, so that the sensitivity of Kondo effect to the direction and magnitude of the transversal magnetic ﬁeld is in some sense manifestation of interference of the two states with different Berry phases in a spin space with discrete symmetry.

5.3 Phonon assisted tunneling

217

5.3 Phonon assisted tunneling Probably, the main difference between artiﬁcial molecules (multivalley quantum dots) and natural molecules is the fact that the natural nanoobjects cannot be found in the quiescent state. Being attached to the edges of metallic wires or adsorbed on the metallic substrate, molecules oscillate even in the absence of tunneling current. Electron cotunneling through a molecule means injection of extra electron and/or hole in the molecular orbital, which may affect the vibrational subsystem of a molecule. The perturbation results in change of vibrational state due to emission or absorption of molecular phonons. Even if electron tunneling may be considered as a slow adiabatic process which occurs without changing the phonon quantum numbers, the mechanical motion of the molecule as a whole or distortion of the chemical bonds between the constituent ions may affect the parameters of the tunneling device, namely the relative positions of the electron levels and the tunneling matrix elements. Since the tunneling matrix element enter the effective exchange parameters, the many particle phenomena like the Kondo effect are also sensitive to adiabatic motion of the complex molecule. Vibrational motion is also relevant to dynamical symmetries of molecular nanoobjects. As was noted in Section 2.5, the spectrum of harmonic oscillator is characterized by its own dynamical symmetry SO(2, 1). Besides, the dependence of electronic levels on the distortion of the molecular bonds means the change of the relative disposition of these levels and sometimes induces level crossing. Such changes straightforwardly inﬂuence the dynamical symmetry in the corresponding energy scale. Finally, the phonon assisted interlevel transitions in the non-adiabatic regime pave the way to additional dynamical symmetries which are absent in the equilibrium regime. All these changes in dynamical symmetries eventually turn into effects observable in tunneling experiments. Description of adiabatic and phonon assisted single electron tunneling in molecular complexes is based on the conventional Born-Oppenheimer scheme described in Section 3.6.4. In accordance with this scheme the molecular distortions are included in the Anderson Hamiltonian (3.72). After second quantization of the normal vibrational modes this Hamiltonian is transformed into the electron-phonon interaction Hamiltonian. In the minimal Anderson – Holstein type model (3.74) the adiabatic terms included in Hˆ d acquire a form of energy levels depending parametrically on the conﬁguration coordinates Rα , where only one normal mode is retained as EΛ (R), and one vibration mode with the frequency Ω0 is taken into account.

218

5 DYNAMICAL SYMMETRIES IN MOLECULAR ELECTRONICS

Fig. 5.8 Simpliﬁed model of a molecule with magnetic ion vibrating between the two edges of the metallic wire. Wavy lines symbolize periodic change of tunneling rates due to molecular vibrations.

Before turning to speciﬁc manifestations of dynamical symmetries in phonon assisted tunneling let us brieﬂy survey the basic features of Kondo effect in vibrating environment. We consider the simplest model of a molecule carrying localized spin in a symmetric tunneling conﬁguration (Fig. 5.8). Wavy connection lines symbolize tunneling channels in the presence of vibration mode. In the most interesting cases the latter may be fully symmetric breathing mode (expanding and contracting molecular shell) or dipole mode (periodic displacement of the center of gravity to the left and to the right from its equilibrium position in the half-distance between two leads). The simplest effect related to the vibration motion of a molecule is an adiabatic modulation of the tunneling rates ΓL and ΓR due to, say, dipole displacements between the left and right electrodes. In analogy with the Debye – Waller effect for neutron scattering in vibrating lattice, the Kondo temperature becomes a function of the mean square displacement of the oscillating ion [208]. The modulation of the tunneling parameters may be incorporated in the effective exchange Hamiltonian by means of time-dependent SW transformation, where the time t parametrically enters the parameters of the Anderson Hamiltonian [192]. One may neglect small time-dependent shift of the energy levels in the denominator of the exchange constant (3.12) and retain the time argument only in the tunneling matrix elements parametrized as (0)

WL,R (t) = WL,R exp[±x(t)/l0 ]

(5.30)

where x(t) is the time-dependent displacement and l0 is the electronic tunneling length.

5.3 Phonon assisted tunneling

219

Then the second-order perturbation theory gives us the time-dependent exchange Hamiltonian Hˆ = Hˆ b +

1 ∑ Jαα (t) σ σ σ S + 4 δσ σ c†kσ ,α ck σ ,α kασ ,k α σ

(5.31)

where α = L, R and Jα ,α (t) =

Γα (t)Γα (t)/(πρ0 Ed ).

In the adiabatic approximation the RG equations for the Hamiltonian (5.31) can be solved in the same manner as those for equilibrium [192]. As a result, the Kondo temperature acquires periodic time dependence: πU ¯ exp − TK (t) = D(t) . (5.32) 8Γ0 cosh[2x(t)/l0 ] Since the amplitude of mechanical oscillations is small in comparison with l0 , the time averaged Kondo temperature can be written as TK = TK0 exp(−2W ),

(5.33)

with the Debye – Waller-like exponent W = −π Ux2 (t)/(8Γ0 l02 ),

(5.34)

giving rise to an enhancement of the static Kondo temperature. This counterintuitive exponentially large Debye-Waller factor results from the strong asymmetry in the tunneling rate at the turning points of the trajectory x(t). The next easily predictable effect is the appearance of vibrational sidebands in resonance Kondo tunneling [318]. We study this essentially non-adiabatic effect within the Anderson – Holstein model, where the multiphonon processes are included in the tunneling term by means of the Lang – Firsov canonical transformation + Ω 0 b† b Hˆ = Hˆ d + Hˆ b + Hˆ db

(5.35)

[cf. Eqs. (3.74) (3.75)]. Here the polaron shift is included in the molecular levels in Hˆ as d

EΛ (N ) = EΛ (N ) − N λ 2 Ω0 .

(5.36)

220

5 DYNAMICAL SYMMETRIES IN MOLECULAR ELECTRONICS

In case of strong polaron effect instead of the parameter W in the matrix S performing the SW transformation (3.10) the integral [318, 373] Wα =1,2 = W

∞

dte−i(Δkα −i0+ )t e−A(t)

(5.37)

0

should be used in the ﬁrst and second terms in the right hand side of Eq. (3.10), where

Δk1 = εk − εd , Δk2 = εk − εd − U and

A(t) = λ e−iΩ0t b − eiΩ0t b† .

Then the SW canonical transformation gives the Hamiltonian Hˆ ex (3.11) with modiﬁed exchange Hamiltonian which includes the multiphonon transitions in the denominator and the corresponding Franck – Condon weighting factors fmn = n|e−A(0)|m in the numerator:

∞ 1 1 n n 2 Jk k = W ∑ fmn fmn + Δk 1 + (m − n)Ω 0 Δk1 + (m − n)Ω0 m=0 − W2

∞

∑

fn m fnm

m=0

1 1 + Δk 2 + (m − n)Ω0 Δk2 + (m − n)Ω0

(5.38)

Due to the smallness of the ratio Ω0 /U the sum J n n may be represented as an asymptotic power series with leading terms J n n ∝ J(2λ Ω0 /U)|n−n | . Then the zerophonon coupling constant is simply

J

00

2λ Ω 0 ≈ J 1+ U

2 .

(5.39)

Comparing this result with the phonon line-shape factor in the resonance tunneling rate Γ (3.76), we see that the polaron-narrowing effect measured by the Huang – Rhys factor is absent in the phonon-assisted cotunneling processes. This is because the amplitude J 00 includes only virtual multiphonon excitations, whereas the phonon-dressed electronic transitions include real phonon excitations and thus strong overlapping effect for displaced phonon wave functions. Unlike the case of conventional polaron narrowing, there is a slight enhancement of the indirect exchange parameter and thereby of the Kondo temperature due to vibron induced reduction of the electron levels εd and εd + U.

5.3 Phonon assisted tunneling

221

More signiﬁcant is the contribution of phonon emission/absorption processes in the T-matrix deﬁned in Eq. (4.35) at ﬁnite energy and temperature. To third order in J, its imaginary part which enters the tunnel current [281] is 3πρ0 (1 − e−Ω0 /T )(1 + e−ω /T ) 16 × ∑ J mn J ln e−nΩ0 /T {1 − nF [ω + (n − l)Ω 0]}]θ (D − |ω + (n − l)Ω0|) Im Tk k (ω , T ) = − lmn

× δml + ρ0 J ml ln

D D + ln , ω + (n − m)Ω 0 ω + (m − l)Ω0

(5.40)

Here the shorthand notation ln |D/x| = ln |max{x, T }| is used, nF is the Fermi distribution function. The Kondo logarithms are calculated at ﬁnite source-drain bias eV so that the chemical potentials are taken as μs = 0, μd = −eV . The logarithmic singularities at voltages corresponding to multiples of the oscillator frequency reﬂect the onset of the Kondo-effect assisted by coherent vibron-exchange (Kondo sidebands). This effect may be observed as ﬁnite bias kinks in the tunnel conductance which arise on the low-energy and high-energy sides of the main Kondo ZBA. Other examples of Kondo singularities which appear at ﬁnite bias [218] will be considered in Chapter 7. √ As was pointed out above, only the even lead electron mode c†ek = (c†sk + c†dk )/ 2 enters the tunneling Hamiltonian. In case of the phonon assisted processes a new √ situation may arise, where the odd electron wave c†ok = (c†sk − c†dk )/ 2 is coupled with some vibration mode Ωc (e.g. the center of mass motion mode), = λo Ωc ∑[(b† + b)dσ† cokσ + H.c.] Hdb

(5.41)

kσ

which thus survives in the Kondo Hamiltonian [34] (only single-phonon processes are retained in Hdb for the sake of simplicity). This means that the phonon-assisted processes support the two-channel Kondo tunneling regime, albeit under condition

of strong channel anisotropy. Usually the even channel dominates, but in the case of S = 1 both channels are relevant for the conductance in the unitarity limit [see Eq.(4.63) and discussion below]. Anyhow the phonon-assisted processes should be taken into account in realizations of the two-channel scenarios in molecular complexes [34, 305]. In the examples considered above the phonon transitions accompany the conventional SU(2) Kondo effect, and the observable phenomena in some sense reproduce

222

5 DYNAMICAL SYMMETRIES IN MOLECULAR ELECTRONICS

the effects known in conventional spectroscopy (Debye – Waller effect) and optics (Stokes/anti-Stokes satellites in absorption spectra). Now we turn to many-particle effects possible only due to non-trivial dynamical symmetry of molecular complex. This is the phonon-induced Kondo tunneling [207, 208], which may be realized in TMOC with even occupation number in the case when the ground state is a spin singlet. Such situation arises due to the crystal/ligand ﬁeld splitting of 3d levels of magnetic ion in the potential of a molecular cage. As a result of this splitting the highest occupied electron level in 3d shell may be the doubly occupied nondegenerate level. For instance, for a distorted tetrahedral symmetry of the ligand ﬁeld we focus on the conﬁguration d 2 (e2 ) with two e-states split by δ due to this distortion. In case of distorted cubic symmetry, the same S/T multiplet arises for the conﬁguration d 8 (t 6 e2 ). Fig. 5.9 illustrates the conﬁguration of the tunneling structure (a) and states of the TMOC (b): it is seen that in the case of a weak intrashell exchange I < δ the ground state is singlet S and the lowest excitation is the S/T transition. The energy difference ΔT S ≡ ET − ES = δ − I is assumed to be larger than the Kondo temperature TK . The total spectrum consists from three singlets and one triplet and belongs to the SU(6) dynamical symmetry group [see Table (2.58)]. We are interested here only in the last stage of the RG procedure where the two singlet excited states S and S with the energies 2δ and δ , respectively, are integrated out and only the lowest excitation Δ T S competes with the phonon mode Ω0 . The latter vibronic excitation is ascribed to some deformation mode characterized by the √ conﬁgurational coordinate Q = (b† + b)/ 2 [Fig. 5.9(a)]. The effective SW Hamiltonian including the electron-vibron interaction rewritten in terms of the SO(4) group generators reads 1 He f f = Hb + ΔT S S 2 + JS (Q)S · s + JR(Q)R · s + Ω0 b† b. 2

(5.42)

The Q dependence of the exchange integrals arises via the corresponding dependence of the tunneling matrix elements W (Q) Fig. [5.9(a)] in the tunneling Hamiltonian Hˆ db + Hˆ . Here the ﬁrst term is the equilibrium part of the Hamiltonian db

corresponding to Q = Q0 , and only the linear terms in Q − Q0 are retained in the second term like in Eq. (5.41). This approximation allows one to take into account both single-phonon and two-phonon processes in the exchange vertices. Logarithmic singularities responsible for the phonon-induced Kondo tunneling arise in the vertex JR . It is convenient to illustrate the dynamics of the relevant processes by means of schematically depicted phonon contributions to this vertex (Fig. 5.10). The ground and excited levels in these schemes are marked by the indices S

5.3 Phonon assisted tunneling

223

Fig. 5.9 (a) Transition metal molecular complex represented as a vibrating molecular cage with magnetic ion inside. Tunneling transitions are accompanied by excitation of vibration modes. The excitation energy may be transferred to the d-shell of the TM ion. (b) Possible occupation of energy levels in e2 subshell of TM ion. (see text for further explanation) . Fig. 5.10 Two types of phonon assisted Kondo cotunneling processes. (a) Virtual phonon absorption initiates a S/T transition, Kondo processes take place in the intermediate triplet states, and the phonon is emitted at the end. (b) Every spin-ﬂip act in the intermediate triplet state is accompanied by a two-phonon process. Points and crosses denote spin-ﬂip transitions, wavy lines stand for phonon emission/absorption.

T S

t1

ti

tn

(a)

tn

t1

tf

ti

(b)

tf

and T . At some moment t1 the molecule absorbs a virtual phonon with frequency Ω0 comparable with ΔT S and excites triplet state. Then two possibilities should be taken into account. (a) Phononless Kondo processes with spin ﬂips symbolized by crosses take place in an intermediate triplet state and the phonon is emitted in the end at t = tn . (b) Every spin ﬂip process in the intermediate triplet state between the moments t1 and tn is accompanied by a two-phonon emission/absorption act. To compare the contribution of these two mechanisms of the Kondo screening, one has to collect the corresponding series in the vertices JS , JR dressed by the spin and phonon excitations. This can be done by means of the diagrammatic tech-

224

5 DYNAMICAL SYMMETRIES IN MOLECULAR ELECTRONICS

J

S

(a)

S

T

jS

jR

S,R

(d)

T

(b)

S

S

T

T

T

(c)

(e)

T

T

S

Fig. 5.11 (a) Bare exchange vertices JS,R ; (b) Single phonon correction jR to the vertex JR ; (c) Two-phonon correction jS to the vertex JS ; (d,e) Renormalized vertices γ1,2 corresponding to the processes illustrated by Fig. 5.10(a), (b), respectively.

nique in spin-fermion representation. In accordance with the mapping procedure described in Section 9.3, two vectors S and R1 from the set (2.46) generating the so(4) algebra may be represented in accordance with Eqs. (9.74) by means of four fermionic operators f 1† , f0† , f1¯† , g†1 where the operators fi† represent three projections of the spin 1 and the operator g†1 is related to the singlet state. The bare vertices JS,R are shown in Fig. 5.11(a), where the solid and dashed lines stand for spin fermion

and conduction electron operators, respectively. Single phonon correction to JR denoted as jR is shown in Fig. 5.11(b) (wavy line means emitted or absorbed phonon), and two-phonon correction to J S denoted as jS is shown in Fig. 5.11(c). It should be noted that the Kondo singularities arise in the scattering amplitude for the singlet ground state. Diagrams responsible for these singularities are shown in Fig. 5.11(d,e). The irst of these diagrams corresponds to the single phonon energy scheme of Fig. 5.10(a) and the second one corresponds to the multiple phonon emission/apbsorption process shown in Fig. 5.10(b). Here the phononless multiple electron scattering processes included in the ﬁlled square in the diagram (d) give the renormalized vertex JS . The corresponding scattering amplitude γS1 is ⎤ ⎡ D ln ⎥ ⎢ max[T, |Δ T S − Ω0 |] ⎥ (5.43) γS1 ∼ ( jR )2 ρ ⎢ ⎦; ⎣ D 1 − JS Aρ ln max[T, |Δ T S − Ω0 |] here A ∼ 1 is a constant determined by spin algebra. The Kondo temperature extracted from this equation reads

5.3 Phonon assisted tunneling

225

1 (1) TK ∼ D exp − Aρ JS

(5.44)

The dashed oval in diagram (e) corresponds to multiple scattering processes in triplet channel shown in Fig. 5.10(b). These processes renormalize the vertex jS . Summation of the corresponding diagrams generates the vertex γS2 , ⎡ ⎤ D ln ⎢ ⎥ max[T, |ΔT S − Ω0 |] ⎥ γS2 ∼ ( jR )2 ρ ⎢ ⎣ ⎦ D 1 − jS A ρ ln max[T, |ΔT S − Ω0 |]

(5.45)

The Kondo temperature characterizing this channel depends on the phonon-assisted exchange constant jS 1 (2) (1) TK ∼ D exp − (5.46) TK A ρ jS One concludes from these calculations that the single-phonon processes are sufﬁcient to compensate the energy of the S/T splitting and induce resonance tunneling through the TMOC provided a local vibration mode with appropriate frequency satisfying the condition (1)

|Ω S − Δ T S | TK

(5.47)

exists in the cage. As a result the zero bias anomaly (ZBA) arises in the tunneling conductance in spite of the zero spin ground state of the molecular complex. One can expect in this case a signiﬁcant enhancement of the tunnel conductance already (1) (1) at T > TK according to the law G/G0 ∼ ln−2 (T /TK ) (4.25). It should be emphasized that like in the general consideration of modulation of the exchange coupling by polaronic processes (5.38) these processes need no real phonon excitations and no non-equilibrium occupation of the excited vibration levels is required. The phonon overlap integral enters only the prefactor ( jR )2 in Eq. (5.43). This mechanism implies ﬁne tuning |Ω0 − ΔT S | < TK , which is a severe restriction. To make the situation more ﬂexible, we address here the case of strong electron-phonon interaction and apply magnetic ﬁeld h as an additional tuning instrument. The main ideas are illustrated in Fig. 5.12. The Zeeman splitting may be used for tuning the resonance condition. If the condition ΔT S − Ω0 − h < TK is satisﬁed [Fig. 5.12(a)], then the states |S, |T 1 form the effective SU(2) vector operator P1 with components (4.48) which acts effectively as a spin 1/2 operator and enters

226

5 DYNAMICAL SYMMETRIES IN MOLECULAR ELECTRONICS

Fig. 5.12 (a) Single phonon connects a singlet with spin 1 projection of triplet. (b) n-phonon processes connect a singlet with spin 1 projection of triplet (n¯hΩ < Δ ). (c) (n + m)-phonon processes connecting a singlet with spin 1¯ projection of triplet ((n + 1)¯hΩ > Δ ).

the SW Hamiltonian in accordance with the arguments offered in Ref. [341]. In this case one deals with phonon and magnetic ﬁeld induced Kondo tunneling. If the resonance condition is fulﬁlled for n-phonon processes,

ΔT S − h − n¯hΩ0 TK ,

(5.48)

then the Kondo tunneling is assisted by virtual excitation/absorption of n-phonon ”cloud” [Fig. 5.12(b)], so that the vertex jR is weighted with Pekarian distribution [phonon lineshape function, cf. (3.76)]

γh ∼ ∑ e−S n

Sn γS1 (ΔT S − nΩ0 − h) n!

(5.49)

Here S = ν /¯hΩ0 is the Huang – Rhys factor and ν = λ 2 /¯hΩ0 is the polaron shift. Kondo resonance picks out from this distribution those n which satisfy the condition (5.48). If the Pekarian distribution is wide enough to envelop the whole spin triplet [Fig. 5.12(c)], then the phonon-induced transitions restore the full SO(4) symmetry of spin multiplet in Kondo tunneling. Since phonon-induced Kondo tunneling takes place in the excited triplet state of the molecule, one should estimate the contribution of decoherence and dephasing effects which tend to smear the Kondo resonance. We will discuss this problem in Chapter 7 in a general context of the Kondo effect in non-equilibrium conditions. There is much in common between the electron tunneling through oscillating molecule and the shuttling mechanism, where the motion of nanoobject is induced electromechanically [143, 376]. This analogy is especially close for nanotubes

5.3 Phonon assisted tunneling

227

which possess collective vibration modes characterized by huge mean-square displacements comparable with their radius. In Chapter 8 we will continue the discussion of dynamical symmetries which may be revealed in tunneling through periodically moving nanoobjects.

5.3.1 Two-electron tunneling at strong electron-phonon coupling One more bright manifestation of dynamical symmetries in the electron transport in molecular nanodevices is the two-electron tunneling in presence of the strong polaron effect where formation of bipolaronic states is possible. Such bipolaron formation is known in the quantum chemistry as a disproportionation effect, where in the course of chemical reaction instead of one product in a given oxidation state two products appear with higher and lower oxidation numbers. Similar effect is known in the physics of amorphous semiconductors where an electron hops from one partner in a cluster to another one under the condition where the Coulomb energy loss is compensated by the gain in the elastic energy. This is the negative U scenario [20, 373, 396], which may be realized also in the Anderson model for electron tunneling in molecular complexes [9, 63, 64, 227, 228, 256]. The origin of the negative U effect is the phonon mediated attractive interaction between the electrons in the molecular complex, which results in the renormalization of the Coulomb energy. In the case of N = 1, the latter is given by the equation U = U − 2λ 2 Ω 0

(5.50)

This result arises in the Anderson-Holstein model (3.74), (3.75), (5.35) after the Lang – Firsov transformation which transforms the electron-phonon interaction into the polaron dressing exponent for the electron tunneling rate, the polaron shift of discrete electron levels and the phonon mediated electron-electron interaction. The latter renormalizes the Hubbard interaction term in the Anderson Hamiltonian in accordance with Eq. (5.50). In case of strong enough electron-phonon coupling λ , the condition U < 0 is satisﬁed, which means that the effective interaction between the electrons in the molecular dot is attraction. The interaction (5.50) should be included in the term Hˆ d , so that the Hubbard parabolas for the energy spectrum are reversed relative to the usual shape shown in Fig. 3.2. These “turned over” diagrams are shown in Fig. 5.13

228

5 DYNAMICAL SYMMETRIES IN MOLECULAR ELECTRONICS

E1

E

E1 U,V

W,Y U,V

E0

W,Y

E2

E0

Z

0

1

0

2

E10

ε

11 00 00 11 00 11 00 11 00 11 00 11 00 11

s

1

2

E12

E12 F 11 00 00 E20 /2 11

E2

Z

11 00 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11

d

ε

00 F 11 00 11

E10

11 E 20 /2 00 00 11 00 11 00 11 00 11 00 11 00 11 00 11

11 00 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11

Fig. 5.13 Upper panel: inverted Hubbard parabolas for the negative U Hubbard atom for the cases of empty and doubly occupied shells. The interlevel transitions are described by the operators generating the SU(4) dynamical group. Lower panel: single-electron levels corresponding to the transitions shown by the arrows in the upper panel (see text for further explanation).

Like in the positive U case, the transitions between the levels in the Hubbard supermultiplet are described by the operators (2.67) generating the SU(4) dynamical symmetry group. Let us consider the conﬁgurations, where the singlet states |Λ = |0, |2 are nearly degenerate, and the spin doublet |Λ = | ↑, | ↓ is an excited virtual state in the cotunneling processes. The two conﬁgurations presented in Fig. 5.13 correspond to the empty and completely ﬁlled two-electron shell of the Hubbard atom. They are connected by the particle-hole symmetry transformation, so it is enough to discuss one of them. We will show below that the negative U Anderson model may be formally mapped on the positive U model, by means of the multistage RG method, which generalizes the Jefferson – Haldane – Anderson procedure [19, 148, 185] described in Section 4.1. In the latter case after freezing out the high-energy excitations E10 and/or E 12 corresponding to injection of a hole or of an electron to the singly occupied quantum dot at the Jefferson – Haldane stage of the renormalization, one arrives at the Anderson stage of Kondo screening of the spin excitations in the sec-

5.3 Phonon assisted tunneling

229

tor N = 1 described by the vector operator T. In the negative U model the spin excitations are exponentially suppressed from the very beginning. After freesing out the charge excitations E10 and E12 generated by the operators U, V, W, Y we are left with the two-particle charge excitations E20 generated by the operator Z. Since the ﬁrst (Jefferson – Haldane) stage of the RG procedure is realized exactly in the same way as in the positive U Anderson model, we refer the reader to Section 4.1 for its description and concentrate on the second stage, where the SU(4) dynamical symmetry group is reduced to its SU(2) subgroup represented by the triad Z. These operators act in the subspace

Φ¯ 2 = (0, 2)

(5.51)

[cf. (2.68)]. The effective SW Hamiltonian in this subspace reads J⊥ + − Hˆ cotun = N Z B + Z− B+ + NJ Zz Bz , 2

(5.52)

Z+ = X 20 , Z− = X 02 , Zz = X 22 − X 00

(5.53)

where

[see Eq. (2.67)], B+ = N −1 ∑ c†k↑ c†k ↓ , B− = N −1 ∑ ck↓ ck ↑ , kk

kk

Bz = N −1 ∑ c†k↑ ck ↑ − ck ↓ c†k↓ = N −1 ∑ ∑ c†kσ ck σ − 1 kk

(5.54)

kk σ

The operators (5.54) form a matrix B in the space (5.51) with the components obeying the su(2) commutation relations [B+ , B− ] = Bz , [Bz , B± ] = ± 2B ±

(5.55)

The transversal part of the Hamiltonian (5.52) describes the tunneling of singlet electron pairs between the leads and the molecule, whereas its longitudinal part stems from the band electron scattering on the charge ﬂuctuations. Thus the Hamiltonian of two-electron tunneling is formally mapped onto the anisotropic Kondo Hamiltonian [64, 373, 396]. The origin of this anisotropy is the polaron dressing of tunneling matrix elements [64]. This dressing is different for the two-electron cotunneling and the electron scattering coupling parameters (5.38). In the strong electron-phonon coupling limit, (λ /Ω0 )2 = S 1,

230

5 DYNAMICAL SYMMETRIES IN MOLECULAR ELECTRONICS

J⊥ 2 = 2|0 ∼ e−2(λ /Ω0) . J

(5.56)

The eventual source of this anisotropy is the overlap between the phonon wave functions for a molecule in the charge states N = 0 and N = 2, i.e. the Huang – Rhys factor S. In a framework of the Anderson RG scalng procedure this means that the

J_|

J || 0(2)

0(2)

0

2 Z +B−

Z z Bz

_

2

0 Z +B−

2

0 _ Z B+

Z

0

0

2 Z +B

Z −B+

2 _ Z +B

0

_

0 Z zBz

2 Z z Bz

2_ Z +B

Fig. 5.14 RG diagrams in the space Φ¯ 2 = (0, 2). Upper panel: Bare vertices Jperp for the twoelectron tunneling and Jparallel for the charge scattering. Lower panel: diagrams for second-order renormalization of these vertices. Solid lines stand for conduction electron states, dashed lines denote the charge states of the molecule.

renormalization diagrams for the two models are the same, namely the ﬁrst two diagrams of Fig. 4.25 are mapped onto those shown in Fig. 5.14. The mapping procedure implies the substitution S → Z, s → B. The scaling equations, which follow from these equations are the same as for the conventional anisotropic Kondo model [19], namely d j

2 = − j⊥ dη d j⊥ = − j⊥ j

dη

(5.57)

( ji = ρ0 Ji ) In the case of strong anisotropy (5.56) solution of this system gives for the Kondo temperature the following equation [64]

5.3 Phonon assisted tunneling

TK ∼

231

j⊥ j

1/ j

πΩ0 ∼ D¯ exp − 2Γ

λ Ω0

4 .

(5.58)

The last equation in (5.58) is valid in the limit of strong electron-phonon coupling. Generally, the polaron narrowing of the tunneling rate results in a noticeable decrease of TK in comparison with its value for conventional Kondo effect. In spite of the formal similarity between the effective Hamiltonians for the single electron cotunneling and the electron pair cotunneling, the background physics is different. In the positive U Anderson model the tunneling in the middle of the Coulomb window arises exclusively due to the many-body Abrikosov – Suhl resonance. In the negative U model the resonance conditions for the two-electron tunneling arise at E02 = 0 irrelative to the many body particle-hole screening mechanism, so that the ZBA in tunneling conductance exists already at T TK , as well as FBA [227, 228, 256]. The many-body Kondo-like screening at low T only sharpen these anomalies. The ﬁnite difference E02 = 0 in the negative U model is equivalent to the ﬁnite magnetic ﬁeld in the positive U model: it results in the appearance of two split FBA peaks in the tunneling conductance. Having in mind all these differences, one may state that the multistage RG procedure reveals the hierarchy of reduced dynamical symmetries SU(4) → SU(3) → SU(2) in the Anderson model both with the Hubbard repulsion for odd occupation and with the Hubbard attraction for even occupation.

Chapter 6

DYNAMICAL SYMMETRIES AND SPECTROSCOPY OF QUANTUM DOTS

Hitherto we discussed the dynamical symmetries in nanoobjects which are realized in tunneling experiments in external electric and magnetic ﬁeld or induced by interaction with oscillator bath of elastic deformations of the constituent atoms (molecular vibrations). Irradiation of tunneling devices with periodic electromagnetic ﬁeld in optical and microwave frequency range also may activate transitions between the energy levels of nanoparticle. There is much in common between photon and phonon ﬁelds from the point of view of interaction with few electron quantum systems because being quantizes, both obey the Bose statistics without conservation of number of excitations. Excitations with a given polarization possess SO(2, 1) symmetry of 1D quantum oscillator with the basis functions (2.81). There is no insistent necessity to quantize the light in the majority of applications in photon assisted transport in nanostructures. Instead interaction with light may be treated as a response of a nanostructure to periodic in time electromagnetic ﬁeld. In this case the Floquet theorem is valid and one may use the quasienergy approach described in Section 2.7. Photon-assisted tunneling is essentially non-equilibrium problem. Photoexcited states in a nanoobject have ﬁnite life time and the excitation energy may be dissipated and redistributed between various subsystems of a tunneling device. Leaving discussion of relaxation and dephasing mechanisms for Chapter 7, we consider here the stationary regime of tunneling through quantum dots irradiated with electromagnetic ﬁeld in microwave and optical diapason. Microwave radiation may be absorbed by GaAs/AlGaAs planar dots, whereas the semiconductor self-assembled InAs/GaAs dots are active in the visible part of optical spectra. Electromagnetic ﬁeld may excite both the lead and the dot states. General survey of photon-assisted tunneling in quantum dots may be found in the review [329]. K. Kikoin et al., Dynamical Symmetries for Nanostructures: Implicit Symmetries in Single-Electron Transport Through Real and Artificial Molecules, DOI 10.1007/978-3-211-99724-6_6, © 2012 Springer-Verlag/Wien

233

234

6 DYNAMICAL SYMMETRIES AND SPECTROSCOPY OF QUANTUM DOTS

Various photon-assisted phenomena in single-electron tunneling in quantum dots may be described in terms of photon lineshape function similar to phonon lineshape function which appeared in tunneling and cotunneling amplitudes discussed above [see Eqs. (3.76) and (5.49), respectively]. Like in the phonon-assisted tunnel transport, photon side bands arise in the Coulomb staircase current-voltage characteristics. Physically these sidebands correspond to photoionization of a quantum dot due to reactions like d n + h¯ ω = d n+1 + h, where h denotes a hole in the metallic lead and h¯ ω is the energy of the absorbed photon. Photons also may assist interdot tunneling in DQD devices in the serial geometry. Such processes may be observed both at eV → 0 and at ﬁnite source-drain bias. Study of these processes adds nothing new to our understanding of the role of dynamical symmetries in single electron tunneling, and we refer to the current literature [231, 329] for further details. Here we concentrate on two types of manyparticle phenomena, namely on the photon-assisted/photon-induced Kondo cotunneling and on the exciton spectroscopy of self-assembled quantum dots. The deep analogy between the Kondo effect and the Dicke model of the superradiance [10, 77] should be also mentioned. This model describes the resonance interaction of radiation with an atom represented as a two-level system. The Dicke Hamiltonian may be treated as an analog of the Fano – Anderson Hamiltonian, where the bath B is the bosonic reservoir. In some cases (see, e.g., [255]) this “spinboson” problem may be mapped onto the anisotropic Kondo model .

6.1 Kondo effect in the presence of electromagnetic ﬁeld The study of photon involvement of light absorption/emission in Kondo tunneling begins with appropriate modiﬁcation of the Anderson Hamiltonian (3.2) describing this tunneling. Time-dependent terms describing interaction with the electromagnetic ﬁeld may appear in Hˆ b or in Hˆ d or in both these terms. Hˆ b = Hˆ b + ∑ Fb (t)c†lkσ clkσ k

Hˆ d = Hˆ d + ∑ Fd (t)dσ† dσ

(6.1)

σ

Here Fb (t) and Fd (t) are the amplitudes of the ac electromagnetic ﬁeld acting on the lead and dot, respectively. In any case this dependence may be transferred to the tunneling term by means of an appropriate canonical transformation [50, 140, 192].

6.1 Kondo effect in the presence of electromagnetic ﬁeld

235

Since we deal with time-dependent problem, this transformation should be applied to the operator L = H − i¯h

∂ , ∂t

(6.2)

so that Hˆ = Q−1 L Q 1 i t † † Q(t) = exp − dt ∑ Fb (t)clkσ clkσ + ∑ Fd (t)dσ dσ h¯ −∞ σ k

(6.3)

After this transformation the Anderson Hamiltonian has the form ˆ = Hˆ b + Hˆ b + Hˆ db H(t) (t), (t) = ∑ ∑(Wlj (t)d †jσ clkσ + h.c.) Hˆ db

(6.4)

jk

with Wlj (t)

i t = Wl j exp − dt [Fb (t) − Fd (t)] h¯ −∞

(6.5)

The SW transformation (3.10) for the time-dependent Anderson Hamiltonian implies the same type of generalization, −U ˆ (t) = eU H(t)e . Hˆ eff

(6.6)

Now the operator U should be found from the condition i¯h

∂U + Hˆ db (t) + [U , (Hˆ b + Hˆ d )] = 0. ∂t

(6.7)

This general scheme may be applied to any speciﬁc model of tunneling structure interacting with ac electromagnetic ﬁeld. Having in mind laser irradiation effects, we consider below the interaction (6.1) with monochromatic light characterized by a frequency ω0 , Fi (t) = Fi cos ω0 t. Then the time-dependent SW transformation gen erates the effective exchange vertex of the same type as the phonon assisted Jkn kn in Eq. (5.38), Jk k (t) = W 2

∞

∑

ei(m−n)ω0t Fm Fn

m,n=−∞

− W2

∞

∑

m,n=−∞

ei(m−n)ω0t Fm Fn

1 1 + Δk 1 + mω0 Δk1 + nω0 1 1 + Δk 2 + mω0 Δk2 + nω0

(6.8)

236

6 DYNAMICAL SYMMETRIES AND SPECTROSCOPY OF QUANTUM DOTS

Here the factors Fkn should be found from the solution of Eq. (6.7). In particular, if only the photon absorption/emission processes in the leads Fb (t) = Fb cos ω0t are taken into account, these factors are the Bessel functions Fn = Jn (W /ω0 ). In order to calculate the photon lineshape function for Kondo tunneling, one has to use the Keldysh technique for non-equilibrium Green functions in combination with fermionization/bosonisation procedures which allow one to calculate the conductance at T TK where the perturbation results in the appearance of photon sidebands which accompany the ZBA in close analogy with the phonon sidebands discussed in Section 5.3. When the ac electromagnetic ﬁeld is slow enough, h¯ ω0 εd ,U − εd , one can solve Eq. (6.7) in the adiabatic approximation, i.e. neglect the time derivative term [192]. In this limit one comes to the standard SW solution where the exchange integrals parametrically depend on time: 1 1 2 (6.9) − JkF ,kF (t) ∼ |W (t)| ΔkF 1 (t) ΔkF 2 (t) Here the time variable enters both the tunneling elements (6.5) in the numerator and the energy differences in the denominators in the form Fb (t) − Fd (t). The Kondo temperature also adiabatically oscillates with time: πεd ¯ TK (t) ≈ D exp − Γ

(6.10)

with εd and Γ parametrically depend on time. Photon assisted processes may also induce spin decoherence effects [192] which will be discussed in Chapter 7. Like in the case of electron-phonon interaction, one may think not only about photon-assisted Kondo tunneling but also about photon-induced Kondo effect. Several mechanisms of dynamically induced many-particle resonances have been proposed in the current literature. All these mechanisms are based on the idea that the Kondo-related effects develop only in a photoexcited state of quantum dot. In order to reveal the role of dynamical symmetries in the photon absorption/emission in tunneling structures, we start, like in many previous cases with a quantum dot with even occupation and singlet ground state and study its spectral properties under laser irradiation [202, 210]. To grasp the idea of the proposed

6.1 Kondo effect in the presence of electromagnetic ﬁeld

237

c v

Wv

Wc c

c

v

v

E c

Wv

T

E

Wc

v

G

Teh

The

E

Fig. 6.1 Left panel: Subset of states involved in the formation of midgap Kondo exciton in a semiconductor quantum dot with highest occupied states εv and lowest empty state εc divided by the gap Δcv . Transition from the ground state |G to the excitonic state |E occurs via intermediate states |Teh and |The in 2nd order in tunneling Wc ,Wv . Right panel: Diagram representing excitonic self energy in case where only the intermediate states |Theσ σ are involved .

mechanism let us consider a semiconductor quantum dot with the structure of lowlying states characteristic for such a system (Fig. 6.1, left panel). Here two dot levels in the ground state correspond to the highest occupied discrete state at the top of the valence band (v) of the dot and the lowest unoccupied discrete state at the bottom of conduction band (c) separated by the energy gap Δ cv from the set of occupied states. It is evident that there is no room for Kondo effect in the ground state of such system (the state |G in Fig. 6.1). However, the cotunneling between the lead (l) and the dot (d) at ﬁnite energy comparable with Δ cv may create a singlet exciton (right state |E) via intermediate states with odd occupation (two middle states |Teh , |The ). Multiple creation of electron-hole pairs in the excited states is the source of Kondo FBA in the tunneling spectrum of this system. The set of states represented in Fig. 6.1 is characterized by the SO(8) dynamical symmetry [(see Table (2.58)]. It consists of two singlets |G, |E (ground state σ σ = |e h , |T σ σ =|h e (extra electron and exciton) and two triplets |Teh dσ lσ dσ lσ he in the dot and hole in the lead and v.v.). Photon absorption/emission processes in a quantum dot are described by the time-dependent component of the Anderson Hamiltonian ]indexAnderson model! – time-dependent Hd = D ∑ dc†σ dvσ e−iω0t + H.c. σ

(6.11)

238

6 DYNAMICAL SYMMETRIES AND SPECTROSCOPY OF QUANTUM DOTS

εF

εc

εv

Fig. 6.2 Quantum dot formed in a heterostructure n−CdSe/ZnTe/n−CdSe.

n−CdSe

ZnTe

n− CdSe

where D is the matrix element of the dipole operator responsible for creation and annihilation of photons. It is convenient to introduce the Hubbard operator B† = |EG| which is the same as dc†σ dvσ in the subspace under consideration. The spectral properties of the system are described by the poles of the retarded exciton Green function Gee (ω ) = −i

dteiω t θ (t)[B(t)B† (0)]

(6.12)

where θ (t) is a step function and the brackets stand for the ground state of the system. In the closed subspace shown in Fig. 6.1 this Green function can be calculated exactly, but the main effect may be demonstrated in a minimal model with strongly localized hole hd σ where only the electron channel (upper of two intermediate states) should be taken into account [202]. Such situation may be realized, e.g., in a quantum dot fabricated in the II-VI semiconductor heterostructure CdSe/ZnTe/CdSe (Fig. 6.2). In this structure the level εv falls in the forbidden gap of the lead formed by n-type CdSe, so that it should be treated as a localized deep level εd below the band continuum, and only the states ed σ in the conduction band are involved in cotunneling processes. Then the exciton spectrum is given by the poles of Gee (ω ) deﬁned from the equation

ω − Δcv − Σee (ω ) = 0

(6.13)

where the self energy Σee (ω ) calculated in 2nd order of perturbation theory is given by the diagram shown in the right panel of Fig. 6.1. Here the broken lines denote excitonic states, and the loop is formed by the conduction electron (dashed line) and single electron state (solid line)) in the dot states shown in the middle column of the left panel,

6.1 Kondo effect in the presence of electromagnetic ﬁeld

Σee (ω ) = |Wc |2 ∑ k

1 − nF (εk ) ω − εc + εk

239

(6.14)

The real part of this self energy at ω ∼ Δ v may be estimated as ReΣee (ω ) ≈

Γc max{ω − Δv , T } ln 2π D

(6.15)

where Δv = εF − εv . Due to this logarithmic singularity the spectral density of excitonic states determined by the Green function Gee has not only the usual exciton pole at ω ∼ Δ cv but also a midgap peak at at ω ∼ Δ v . Formally this peak turns into a pole at some temperature T ∗ ≈ De−2πΔ c /Γc

(6.16)

(Δc = εc − εF ) but perturbation theory is valid only at T > T ∗ . The higher order terms and ﬁrst of all the damping effects due to the ﬁnite life time of any optical excitations should be taken into account at T → T ∗ . Although the calculation procedure reminds that for the conventional Kondo effect, the analogy between this midgap singularity and the Abrikosov – Suhl resonance is dubious. In fact the optical transitions do not imply the spin-ﬂip processes, so the genuine Kondo screening does not take place in this case. The common feature of the two effects is the ”orthogonality catastrophe” [18, 303]. It is more correct to interpret this result as a manifestation of shake-up effect well known in atomic spectroscopy, i.e. the ﬁnal-state interaction in the presence of many-particle reconstruction of the initial state of the system. The midgap exciton may be observed as a satellite in photoluminescence spectra of optically excited quantum dots. In principle, genuine Kondo effect induced by light irradiation is also possible, provided cotunneling with spin ﬂips could be sustained in the semiconductor quantum dot. Such Kondo effect is a manifestation of the SO(8) symmetry of the full supermultiplet of Fig. 6.2 or the SO(5) symmetry of the supermultiplet with one intermediate channel. One should remember, however, that the ﬁnite life time is detrimental for the Kondo effect in excited states and its role should be carefully checked. A somewhat related effect is predicted in nonlinear response of a Kondo center [375]. This effect arises as a response to the pump-probe excitation, where a strong pumping ﬁeld with frequency ω2 excites the dot and the optical absorption of weak probing ﬁeld with frequency ω1 is measured. If the ground state energy level is deep enough, εF − εd ω2 , pumping with such frequency is ineffective. However, the

240

6 DYNAMICAL SYMMETRIES AND SPECTROSCOPY OF QUANTUM DOTS

shake-up effect in the photoexcited state creates a logarithmic singularity and the absorption peak around ω2 arises. One more type of dynamically induced Kondo effect is proposed in Ref. [116]. If the ground state of a Kondo impurity corresponds to spin S fully screened with n channels (n = 2S), then the photoemission of electron from this impurity transforms the problem into that for overscreened spin with n − 2S = 1, whereas the inverse photoemission induces underscreened Kondo effect with n − 2S = −1. It should be noted that observation of this effect in quantum dots is not an easy task due to intangibility of multichannel Kondo regime in these devices (see Section 4.3.5).

6.2 Excitonic spectroscopy of quantum dots Quantum dots in self-assembled InAs/GaAs heterostructures possess rich dynamical symmetries discussed in Section 3.4. These are the symmetries of Fock – Darwin ”cylindrical atoms” (Section 3.3) supplemented with speciﬁc symmetries of electron-hole pairs (excitons). The structure of exciton spectra illustrated in Fig. 3.10(b) is well documented theoretically [42, 153, 349, 431], and the corresponding exciton lines are observed in high excitation photoluminescence spectra [349]. Due to interaction with the 2D electrons in the wetting layer on the interface, the self-assembled quantum dots acquire many features typical for the nanoobjects coupled with the bath. In particular, due to injection of electrons from the wetting layer (subsystem B) to the semiconductor InAs quantum dot (subsystem S ) not only Bose-like excitonic states but also Fermi-like electron-exciton complexes are observed in self-assembled nanostructures [417]. In this section we concentrate on the many-body effects in the optical spectra of these structures [146, 147]. The energy level scheme used in the theoretical description of InAs quantum dot is presented in Fig. 6.3(a). In the ground state the highest discrete electron level in the valence band of the QD is occupied, and the levels in the conduction band are empty. In the excitation photoluminescence experiment the system is irradiated with high energy light. Photon absorption results in the formation of electron hole pairs, which then relax by means of radiationless transitions to the lowest energy state with a hole in the top valence band level (marked s) and an electron with opposite spin at the level Es near the bottom of conduction band. Recombination of this pair has been detected in the luminescence spectrum.

6.2 Excitonic spectroscopy of quantum dots

241

Fig. 6.3 Energy level scheme and negatively charged exciton formation in InAs/GaAs selfassembled quantum dot. Negatively charged exciton with three electrons and one hole is formed under light irradiation. Electrons are bound in the potential well separated by the tunnel barrier from the wetting layer formed by 2D electrons in the interface. Recombination of electron-hole pairs is observed in the photoluminescence spectra (after [147]).

Occupation of the wetting layer which is separated from the quantum dot by a potential barrier is regulated by means of the gate voltage. Two-dimensional electrons in this layer are characterized by the Fermi energy εF . By application of the bias voltage to the wetting layer one can inject an electron through the potential barrier from the Fermi sea to the conduction band levels of the quantum dot. As a result not only neutral Fock – Darwin excitons Ex0 but also charged excitons appear in the quantum dot. The negatively charged exciton Exn− contains one hole and n + 1 electron. These composite electron-hole excitations are stabilized both by the electron-hole Coulomb attraction and by the conﬁning potential of the QD. They give additional peaks in the excitation photoluminescence spectra. Excitons with n up to 3 have been observed in InAs/GaAs quantum dots [107, 108, 417]. We are interested here in ﬁnal state interaction effects due to coupling between the excitons in InAs quantum dot and the wetting layer. These effects qualiﬁed as shake-up processes result in additional peaks in the excitation spectra in the same sense as in the previous section. The shake-up effects should be described in a framework of the Anderson model adjusted to this special case of composite excitations in a semiconductor quantum dot. This means that the terms Hˆ d and Hˆ db in the Anderson Hamiltonian (3.2) should be written in terms of excitonic states Hˆ d = ∑ Eλ X λ λ + ∑ EΛ X ΛΛ λ

Λ

(6.17)

242

6 DYNAMICAL SYMMETRIES AND SPECTROSCOPY OF QUANTUM DOTS

Hˆ db = ∑ ∑ W Λ λ X Λ λ ckσ + H.c. . Λ λ kσ

− Here the states |λ and |Λ denote the excitons |Ex− n and |Exn+1 involving elec-

trons in s, p, d... shells of the QD conduction states at given εF . Speciﬁcally, the excitonic states Ex2− and Ex3− are considered, which means that several electrons from the wetting layer may be captured in the s, p conduction levels of quantum dot. The s-shell in the conduction well of the QD is ﬁlled with two electrons, and the full spin of Ex2− may be either 0 or 1 depending on the mutual orientation of hole and p-electron spins. Although the angular momentum of the hole is 3/2, we are interested here only in its spin component, while the orbital part of the wave function is involved in the photon-assisted electron-hole recombination matrix element. The partners in the radiation recombination process are s-electrons from the bottom conduction level and the top valence hole (Fig. 6.3). In the absence of tunneling W one may expect three lines in the photolumines2− cence spectra in correspondence with three exited states |Ex3− , |Ex2− S and |ExT . Either the Ex3− or the Ex2− lines should be seen depending on the value of εF . Interaction in the excited state results in admixture of the states |φk = |Ex2− T , ek to the state |Ex3− . The mechanism of this admixture is illustrated by the diagram for the excitonic Green function similar to that shown in the right panel of Fig. 6.1. In this case the wavy line stands for the exciton |Ex2− T , and the loop contains the exciton |Ex3− and the electron ckσ from the wetting layer. This admixture may be also presented in a form of the intermediate state vector [405] 2− |i = A0 |Ex3− T + ∑ Ak |ExT , ek

(6.18)

k>

Here the sum includes only the empty states with k > kF , A0 is the normalization factor and the coefﬁcients Ak are determined by the above mentioned diagram. Following the scheme used in calculation of the Green function (6.12), we ﬁnd that the exciton |Ex− 2T has a satellite shifted to the energy εF − δ , where π |E p − εF | . (6.19) δ ≈ (D − εF ) exp − Γp Here D is the electron bandwidth in the wetting layer, E p is the energy of the pelectron corrected for the intradot Coulomb and exchange interaction, Γp is its tunneling width. This satellite is another example of the shake-up effect, but in this case the shake-up dressing takes place in the intermediate state of the excitation lumines-

6.2 Excitonic spectroscopy of quantum dots

243

]mbox

X2- exciton

b)

final states

X3- exciton Fig. 6.4 VI.4. Kondo effect in excitonic spectra of InAs/GaAs self-assembled quantum dots. (a) Initial Kondo state for recombination of X2− exciton. (b) Final Kondo state for recombination of X3− exciton (after [147]).

cence experiment. In other words one may treat the electron-hole cloud dressing the exciton |Ex2− T as an avalanche of Auger pairs in the wetting layer which accompanies the formation of the excitonic complex in the quantum dot. This type of shake-up effect is closer to the genuine Kondo effect, because the cloud formation may be accompanied by spin ﬂip processes. The procedure illustrated by the diagram of Fig. 6.1 and Eq. (6.18) is in fact a sort of variational procedure used in the poor-man’s explanation of the Kondo effect [158, 405]. More strict treatment of this effect is also possible. Similar shake-up effect takes place in the ﬁnal state after electron-hole recombination in the excitonic state |Ex3− (right panel of Fig. 6.4(b) (see Ref. [418]). Hybridization with the electrons in the wetting layer is detected also in the simplest excitonic complex Ex1− with two electrons and one hole in the quantum well (trion). This hybridization results in the noticeable blueshift of the photoluminescence and the asymmetry of the lineshape [71] due to the ﬁnal state interaction. One more type of excitonic complexes implying the orthogonality catastrophe mechanism is a Mahan exciton, i.e. a Coulomb-bound state between the hole in the localized state and the Fermi sea of the electrons in the wetting layer. [156, 268, 269, 303]. This type of excitons has also been detected in the photoluminescence spectra

244

6 DYNAMICAL SYMMETRIES AND SPECTROSCOPY OF QUANTUM DOTS

of self-assembled InAs/GaAs quantum dots [221]. In this speciﬁc case the Mahan exciton is realized as a Fermi-edge singularity due to the ﬁnal state interaction. Recombination of electron-hole pair means sudden change of the charge state of QD which induces multiple creation of electron-hole pairs in the vicinity of the Fermi level. Similarly to the Kondo effect, the mechanism of Fermi edge singularity of the optical line has two ingredients: the orthogonality catastrophe due to sudden change of scattering potential and the infrared divergence of the continuous electron-hole spectrum in the vicinity of the Fermi level. The difference is that in the Kondo effect these many-particle effects are realized in the spin sector, while in the Fermi edge problem the process develops in the charge sector of the excitation spectrum (detailed discussion of this difference may be found in [303, 304, 357]. Due to this difference the near-threshold behavior of the optical absorption/emission spectral density obeys the power law, I(ω ) ∼ (ω − ωth )−α . The exponent α < 1 is related to the scattering phase shift in the ﬁnal state or to the difference δ n in the occupation of the dot in the ﬁnal and the initial state of the recombination process,

α = δn−

(δ n)2 . 2

(6.20)

Due to the ﬁnite life time of excitations, this singularity transforms into an asymmetric sharp peak near the fundamental photoluminescence band.

Chapter 7

DYNAMICAL SYMMETRIES AND NON-EQUILIBRIUM ELECTRON TRANSPORT

The concept of dynamical symmetry used in this book is realized in the Fock space of eigenstates of many-body systems under the tacit assumption that the spectral density completely determines the properties of the system S + B. Time-dependent external ﬁeld discussed in the previous chapter did not undermine this concept, because only perturbation periodic in time which induces resonance transitions between these states has been considered. These transitions are easily incorporated in the general scheme. In principle, working with periodic ﬁelds one may resort to the concept of quasienergy spectrum (Section 2.7) and thus extend the Fock space of the problem from dimension M to dimension M + 1. Some aspects of this approach are discussed in the review [329]. The main change in the excitation spectrum is the appearance of photon satellites. In terms of dynamical symmetries these satellites arise due to additional SO(2, 1) symmetry which is added to the basic SO(n) or SU(n) symmetry in a form of direct product. It is timely to remind that we apply the machinery of dynamical symmetry group generators to the calculation of tunneling spectra, i.e. to the problem of quantum transport. Till only the zero bias anomalies in tunnel conductance are considered, we are still on the ﬁrm ground of the dynamical symmetries of supermultiplets. It should be noted, however, that the photon and phonon satellites may be observed as ﬁnite bias anomalies in tunnel conductance, and in any case FBA should be included in the general scheme of dynamical symmetries in single electron tunneling along with ZBA. Electron transport at ﬁnite bias is essentially non-equilibrium phenomenon, so before turning to the problem of Kondo-like anomalies out of equilibrium in quantum dots with non-trivial dynamical symmetries, one should consider the properties of the conventional SU(2) quantum dot in nonequilibrium environment. K. Kikoin et al., Dynamical Symmetries for Nanostructures: Implicit Symmetries in Single-Electron Transport Through Real and Artificial Molecules, DOI 10.1007/978-3-211-99724-6_7, © 2012 Springer-Verlag/Wien

245

246 7 DYNAMICAL SYMMETRIES AND NON-EQUILIBRIUM ELECTRON TRANSPORT

Having in mind future applications of the theory to the Kondo effect, we should consider ﬁnite but not too large dc bias voltage, where the effect of electric potential applied to the leads may be expressed via electrochemical potential μs = μ + eV ,

μd = μ . In the case of strong Coulomb blockade the single-electron tunneling regime is realized, so the current through the dot is weak enough and there is no noticeable nonequilibrium component in the distribution of band electrons in a form of deviation from the Fermi function. However the level occupation in a few electron QD may alter noticeably in the presence of stationary current, provided eV δ ε (see Fig. 7.1), because all electrons within the energy interval μs > ε > μs − eV

(7.1)

may participate in inelastic cotunneling and thus repopulate discrete levels in the dot denoted by dashed lines in Fig. 7.1. This repopulation is detrimental for Kondo tunneling in planar and vertical QD provided eV TK . However one may consider the Kondo effect under the latter condition in the tunneling barrier following the original model of quantum barrier with strongly correlated paramagnetic defect [17, 27]. In this case the nanoobject possesses only the levels εi and εi + Q for singly and doubly occupied states. Then the problem of Kondo tunneling may be treated perturbatively (e.g., with the help of RG approach) under restrictions TK eV (D, Q)

(7.2)

(see [321, 356] and references therein). The SU(2) Kondo effect loses its universality at ﬁnite bias because electrons in the energy window (7.1) contribute to the low-energy properties so that additional ultraviolet cutoff parameter eV arises along ¯ with D. Diagrammatic representation of the RG equations is still the same as in Fig. 4.25, but the coupling parameter J in the single-loop scaling equation (4.17) acquires frequency dependence J(ω ) due to explicit appearance of a cutoff parameter containing eV in the ﬂow equations. The solution becomes more complicated (especially at ﬁnite magnetic ﬁeld B) but the net result of this weak coupling theory is the appearance of logarithmic singularities in the renormalized J(ω ) at ω ∼ μs , μd . This result is intuitively clear from the right panel of Fig. 7.1. Respectively, the ZBA in the conductance disappears, and the zero bias conductance is determined by the equation G(T ) = G0

3π 2 . 16 ln (eV /TK ) 2

(7.3)

7 DYNAMICAL SYMMETRIES AND NON-EQUILIBRIUM ELECTRON TRANSPORT 247

ε

εi + Q μs εi

μd J(ε)

Fig. 7.1 Left panel: Energy level scheme for quantum dot at ﬁnite bias eV = μs − μd . Empty discrete levels in a quantum dot involved in non-equilibrium tunneling are marked by dashed lines. Right panel: Zero bias and ﬁnite bias anomaly in the Kondo tunneling current.

which appears instead of Eq. (4.25) due to the ﬁnite bias related cutoff (see also [192]). The source-drain bias also substitutes for temperature in the paramagnetic response of the quantum dot, so that the static paramagnetic susceptibility χ behaves as χ ∼ 1/V at V T TK instead of the high temperature Curie law χ ∼ 1/T which is valid at V = 0 [321, 356]. These perturbative results correlate with earlier studies of Anderson-Kondo model out of equilibrium using more reﬁned Keldysh diagrammatic formalism for calculation of non-equilibrium Green function (see, e.g., [281, 282]. In this method [199, 347] the matrix of Green functions 0 GA G(t) = GR F

(7.4)

is deﬁned on the ”Keldysh contour”, which starts and ends at t = −∞ and envelops the whole time axis. The retarded and advanced Green functions GR,A together with the function F = G< + G> carries information on the non-equilibrium occupation of the energy levels, G< (t) = ia† (0)a(t), G> (t) = −ia(0)a† (t), Here a, a† are Fermi operators describing excitations in the Anderson-Kondo model. The components of the matrix (7.4) should be found from the solution of the matrix Dyson equation which generalizes Eq. (4.35) for the case of QD in tunneling contact with the Fermi bath out of equilibrium.

248 7 DYNAMICAL SYMMETRIES AND NON-EQUILIBRIUM ELECTRON TRANSPORT

The current through a symmetric QD device is determined by the following equation: J=

ie h¯

d ε [ fs (ε ) − fd (ε )]Im Tr{Γ GRdot (ε )}

(7.5)

valid at ﬁnite bias. Here Γ = ΓsΓd /(Γs + Γd ), GRdot is the retarded Green function of the dot electrons and fs,d are the electron distribution functions in the leads. Calculation of nonequilibrium density of states by means of the approximate slave-boson representation indeed show splitting of a single Abrikosov – Suhl resonance at the Fermi level into two suppressed peaks, one at each chemical potential μs , μd . The main mechanisms of this suppression are related to dephasing and decoherence effects in the nonequilibrium state. The regular investigation of these effects and in particular the study of their temperature behavior in a multistage Kondo screening regime is a very complicated problem which is still waiting a solution.

7.1 Dynamically induced ﬁnite bias anomalies in tunneling spectra Based on the arguments formulated in the preceding section, one may conclude that in a strong bias regime eV TK all ﬁne effects generated due to the interplay between the relevant states within a dynamical symmetry multiplet are suppressed by strong bias. The mechanism of erosion is rather crude and invincible: this is the nonequilibrium repopulation of the states within the supermultiplet in the process of cotunneling charge transfer through a QD. The equilibration of level population induced by the tunnel current is effective at least till the characteristic energies E ∼ eV at which the Kondo singularties are developed essentially exceed the characteristic energy interval Δ E ∼ TK at which the inﬂuence of the dynamical symmetry on the Kondo screening is noticeable. The situation changes radically at small bias eV ∼ ΔE ∼ TK where the repopulation effect is weak and the main source of erosion is dephasing of excitations at ﬁnite temperature. It is clear that the simpler is the structure of the supermultiplet, the more robust is the dynamical symmetry against equilibration of level occupation. So the ﬁrst candidates for FBA in the conductance related to the dynamical symmetries are quantum dots with even number of electrons possessing the SO(4) symmetry of singlet/triplet spin states. Indeed such effect has been predicted theoretically for DQD in the T-shape geometry [218] and then observed experimentally [319] in nanotube quantum dots.

7.1 Dynamically induced ﬁnite bias anomalies in tunneling spectra

249

μs Fig. 7.2 Energy level scheme for quantum dot with N = 2 at ﬁnite bias eV = μs − μd . Cotunneling processes responsible for the resonance Kondo tunneling are shown by dashed lines.

E

T

μd

ES

Let us consider the model of singlet/triplet states in the non-equilibrium conﬁguration at ﬁnite bias (Fig. 7.2). This model is an extension of the minimal model presented in Fig. 7.1 for the case of double electron occupation. Various realizations of this model in equilibrium conditions have been discussed in previous chapters of this book. To demonstrate the mechanism of FBA let us start with one of the simplest realizations of this SO(4) model, i.e., with DQD in the T-shape geometry [Fig. 3.13(b)] where the orbital degrees of freedom are absent. The ground state of the DQD with N = 2 is the spin singlet, and the triplet state is separated from the ground state by the exchange gap Δ T S . As was shown in Section 4.2, in equilibrium conditions eV → 0 the Kondo screening is effective at energy/temperature E > ΔT S where the full SO(4) symmetry is realized, but at E → 0 the singlet-triplet transitions are quenched and the Abrikosov – Suhl resonance is absent in the ground state (see Fig. 4.7 and discussion around). Accordingly, there is no ZBA in the tunneling conductance. Fig. 7.2 illustrates the mechanism of appearance of the FBA at eV ≈ ΔT S . Under these ”resonance” conditions conduction electrons accelerated in the electric voltage applied to the source may induce inelastic cotunneling processes in which the exchange energy ΔT S is compensated by the kinetic energy of the injected electron and the Kondo singularities arise at ﬁnite energy ε ≈ eV . The Kondo channel opens at |eV − Δ T S | < TK . To describe the mechanism of this FBA in terms of the renormalized Green functions, we consider the effective SO(4) Hamiltonian at ﬁnite bias μs − μd = ΔT S . We write it as Hˆ = Hˆ b + Hˆ d + Hˆ ex T TS Hˆ ex = ∑ Jαα S + Jαα R sαα α =s,d

(7.6)

250 7 DYNAMICAL SYMMETRIES AND NON-EQUILIBRIUM ELECTRON TRANSPORT

Here some notations are changed in comparison with Eq. (4.51) and an explicit index α denoting the states in the leads is introduced. Like in the case of phonon induced Kondo effect, the resonance cotunneling is determined by the nondiagonal component JT S of effective SW exchange. Thus the FBA in this model is realized exclusively due to the non-trivial dynamical symmetry of the QD. We are interested in renormalization of the exchange vertices in the weak coupling perturbative regime T TK . To ﬁnd the relevant diagrams of perturbation series we use the spin-fermion representation for the generators of the SO(4) group. The fermionization algorithm for generators of the SO(n) groups is described in Section 9.3 [see Eq. (9.74)]. In the speciﬁc case of n = 4 this fermionization is realized by means of four creation operators and their conjugates: S+ = +

R =

√ √

2( f0† f1¯ + f1† f0 ), S− = 2( f1† g − g† f1¯ ),

√

2( f1¯† f0 + f0† f1 ), Sz = f1† f1 − f1¯† f1¯ ,

√ R = 2(g† f1 − f1¯† g), Rz = −( f0† g + g† f0 ).

(7.7)

−

The Casimir constraint S2 + R2 = 3 in spin fermion representation transforms to the local constraint Nˆ ≡ ∑ fλ† fλ + g†g = 1. (7.8) λ =1,0,1¯

Having in mind the weakness of repopulation effects in S/T multiplet, we use the quasiequilibrium perturbation theory (cf. similar approach to the description of decoherence rate in Refs. [281, 282]). In order to develop the perturbative approach we introduce the temperature Green functions for spin fermions in the dot, Gλ (τ ) = Tτ fλ (τ ) fλ† (0), Gg = Tτ gλ (τ )g†λ (0)

(7.9)

and for electrons in the leads, Gα ,kσ (τ ) = Tτ cα kσ (τ )c†α kσ (0),

(7.10)

where τ is imaginary time changing in the interval 0 ≤ τ ≤ β ≡ T −1 and Tτ is the operator of chronological ordering in this interval [106]. To take into account the kinematic constraint (7.8) we use the trick introduced for SU(2) spin fermions by V. Popov and S. Fedotov [337] and generalized for the SO(n) groups in Ref. [214]. In the Popov – Fedotov representation the Hamiltonian Hˆ + iπ Nˆ f /3β is used in calculation of partition function. This transformation means that the purely imaginary Lagrange factor controlling the number of spin

7.1 Dynamically induced ﬁnite bias anomalies in tunneling spectra

251

fermions is introduced in the statistical ensemble. Then the nonphysical states are automatically excluded from the temperature averages with appropriately modiﬁed partition function (see Section 9.3 for more details). The Fourier transformation of the Green functions (7.9) and (7.10) in imaginary time brings them to the form Gλ (ωm ) = (iωm − ET )−1 , Gg (ωm ) = (iωm − ES )−1 , Gα ,kσ (εn ) = (iεn − εα k + μα )−1 .

(7.11)

with discrete frequencies determined as εn = 2π T (n + 1/2), iωm = 2π T (n + 1/3). Then the standard Feynman diagrammatic rules [106] may be applied to calculation of perturbation series. The retarded and advanced Green functions as well as the renormalized vertices are obtained by means of analytical continuation from the imaginary axis to the real ω axis.

S a)

11 00 00 00 11 11 00 11 00 00 11 11 00 11 00 11 00 11 00 11 11 00 00 11 00 00 11 11

L S d) L

T

S S11 00 b)

11 00 00 11

00 11 11 00 000 11 1 1 0 00 11 00 11 00 11

11 00 00 11 00 11

R L 00 11 11 00 00 11 00 11 11 00 00 11 00 11

T11 00 11 00 00 11 00 11 11 00 00 11

L

S S

11 00 00 11 00 11 00 11 00 11 00 11 00 11

L T

11 00 11 00 11 00 00 11

L

11 00 00 11 00 11 00 11 000 111 000 111 000 111 000 111

c)

R R 11 00 00 11 00 11 00 11 00 11 00 11 00 11

S

T e)

R

L

T 00 11 11 00 00 11

11 00 00 11 00 11

11 00 00 11 00 11

000 111 111 000 000 111

L

S

L 00R 11 00 11 00 11

T

000 111 000 00111 11 000 111 11 00 000 111 00 11 00 11 00 11 00 11 00 11 00 11 11 00 00 11 00 00 11 11 00 11 00 11

11 00 00 11 00 11

T

L ...

11 00 00 11 00 11 00 11

L

S

SS (a), leading (b),(d) and next to leading logarithmic (c),(e) corrections. Fig. 7.3 Bare vertex JLR Solid lines denote electrons in the leads, dashed lines stand for spin-fermions in the dot.

SS and the ﬁrst leading and next to leading parquet diagrams The bare vertex JLR containing logarithmic singularities are shown in Fig. 7.3. The ﬁrst of these correc-

tions, namely the diagram (b) is (2b)

ST T S ΓLR (ω ) = JLL JLR ∑

ST T S ∼ JLL JLR ν ln

k

f (εks − eV ) ω − εks + μs + ΔT S D¯

max{ω , (eV − ΔTS ), T }

.

(7.12)

252 7 DYNAMICAL SYMMETRIES AND NON-EQUILIBRIUM ELECTRON TRANSPORT

We see that under the condition |eV − Δ T S | max[eV, Δ T S ] this correction does not depend on eV and becomes quasielastic. Unlike the diagram (b) its parquet ”counterpart” term (c) behaves as (2c)

(2b)

ST T S ΓLR (ω ) ∼ JLL JLR ν ln (D/(eV + ΔT S )) ΓLR (ω )

and can be neglected in the vicinity of the FBA. Similarly diagram (d) should be retained and diagram (e) may be neglected in the leading logarithmic approximation in the third order in Jαα .

T

00 11 11 00 00 11

111 000 000 111 000 111 00 11 00 11 11 00 00 11

T11 00 11 00 11 00 00 11 00 11 00 11 T 000 111 000 111

S

00 11 11 00 00 11 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00 11 00 11 00 11 00 11

000 111

000 111 111 000 000 111

11 00 11 00

S

111 000 000 111 000 111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 00 11 00 11 00 11 00 11

11 00 11 00

T 11 00 11 00

T11 00 11 00

11 00 00 11

T 111 000 000 111 000 111

11 00 11 00

00 11 00 11

00 11 11 00 00 11

S

00 11 00 11 00 11 11 00 11111 00000 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00 11 00 11 00 11

Fig. 7.4 Diagrams contributing to the RG equations. Hatched boxes and circles stand for triplettriplet and single-triplet vertices, respectively.

Thus, the Kondo singularity is restored under non-equilibrium conditions where the electrons in the left lead acquire additional energy in the external electric ﬁeld, which compensates the energy loss Δ T S in a singlet-triplet excitation. The leading sequence of most divergent diagrams degenerates in this case from a parquet to a ladder series. Instead of summing these series one may use the scaling RG approach in a single loop approximation (see Fig. 7.4). The system of ﬂow equations for LL and LR exchange vertices reads ST T dJLL dJLL T 2 ST T = −ρ0 (JLL ) , = −ρ0 JLL JLL , dη dη T ST dJLR dJLR T T ST T = −ρ0 JLL JLR , = −ρ0 JLL JLR , dη dη S dJLR ST T S = −ρ0 JLL JLR . dη

The solution of the system (7.13) is:

(7.13)

7.1 Dynamically induced ﬁnite bias anomalies in tunneling spectra

253

J0ST J0T , JαST,α = , ¯ ¯ ) ln(D/T ) 1 − ρ0J0T ln(D/T

JαT ,α =

1 − ρ0J0T

¯ ) 3 ln(D/T S . = J0S − ρ0 (J0ST )2 JLR T ¯ ) 4 1 − ρ0J0 ln(D/T

(7.14)

One should note that the Kondo temperature is determined only by the triplettriplet processes in spite of the fact that the ground state is a singlet. Its value is TK = D exp[−1/(ρ0J0T )]. This temperature is noticeably smaller than the equilibrium temperature TK0 = D exp[−1/2(ρ0 J0T )], which emerges in tunneling through triplet 2 ¯ The reason of this difference is /D. channel in the ground state, namely, TK ≈ TK0 the reduction of the usual parquet equations for TK to the ladder series. ST 2 | is a universal function of two The differential conductance G(eV, T )/G0 ∼ |JLR parameters T /TK and V /TK :

G/G0 ∼ ln−2 (max[(eV − ΔT S ), T ]/TK )

(7.15)

Its behavior as a function of bias and temperature is shown in Fig. 7.5. We ﬁnd that in contrast to the standard Kondo related anomaly in the tunneling conductance (Figs.4.1, 4.5), which emerges as ZBA and gradually splits at large enough bias (see preceding section), the dynamically induced Kondo peak appears as FBA at some threshold bias eVth ≈ Δ T S . This peak is asymmetric owing to its threshold character and to dephasing effects in excited triplet state.

G/G0

G/G0

0.1

T/TK eV/TK

0 5

10

eV/TK

Fig. 7.5 Kondo tunneling conductance as a function of dc-bias eV /TK and temperature T /TK .

254 7 DYNAMICAL SYMMETRIES AND NON-EQUILIBRIUM ELECTRON TRANSPORT

Dephasing in the Kondo effect usually arises due to inelastic spin relaxation [192, 356]. It results in low-energy cutoff in the singular electron – spin-fermion loops in the Kondo-related diagrams (Fig. 4.25). Mathematically, this means that the retarded spin-fermion Green function Gλ (ω ) acquires a self energy part Σλ (ω ) which determines the spin relaxation time τr deﬁned as h¯ /τr = −2ImΣλ (ω ). In contrast to the standard situation where relaxation takes place in the ground state of the QD, in dynamically induced Kondo effect the spin triplet appears only as a virtual state in accordance with the diagrams of Fig. 7.3, and, hence, the nonequilibrium processes are not so destructive. The leading diagrams for the damping h¯ /τr in a triplet state are presented in Fig. 7.6(a-d). The diagrams (e) and (f) describe the threshold processes resulting in the asymmetry of the FBA.

L(R)

L(R) T

a)

11 00 00 11 00 11 00 11 00 11 00 11 00 S11 00 11 00 11

T

T

b)

c) e)

11 00 00 11 00 11 00 11 00 11

S111 000S

T

000 111 000 111 11 00 00 11 00 11 00 11 00 11 00 11 00 T11 00 11 00 11

T

R(L)

L(R) T

111 000 000 111 000 111 00 11 00 11 00 11 000 111 S 000 111 000 111

11 00 00 11 00 11 00 11 00 11

T

T

d) f)

T

T

111 000 000 111 000 111 000 111 000 111 000 111

111 000 000 111 000 111 000 111 000 111 000 S111 S111 000 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 T111 S 000 000 111 000 111

T

T

Fig. 7.6 Leading diagram describing spin relaxation in excited triplet state.

The two leading diagrams (a) and (b) describe the damping of triplet excitation due to its inelastic relaxation to the ground singlet state. The relaxation rate associated with these diagrams near the resonance is h¯ /τrT S ∼ (J T S /D)2 max{eV, ω , TK } ≈ (eV )3 /D

(7.16)

(here the small difference between eV ≈ J T = ΔT S and J T S is ignored). Thus the T → S relaxation effect does not contain the logarithmic enhancement factor in the lowest order. Higher order corrections to this damping given by the diagrams (c),(d) are estimated in [214]. These corrections arise only beyond the main logarithmic approximation, so one may ignore them when formulating the condition for the existence of the anomalous Kondo peak at a ﬁnite bias

7.1 Dynamically induced ﬁnite bias anomalies in tunneling spectra

ΔT S (ΔT S /D)2 TK ΔT S .

255

(7.17)

Since ΔT S /D ∼ eV /D 1, the condition (7.17) is satisﬁed in a broad range of parameters. The repopulation of the triplet state as a function of eV and temperature T is controlled by the real part of the self energy Σλ which may change the energy gap between the triplet excitation and the singlet ground state. The modiﬁed exchange splitting Δ ∗ is given by the solution of equation

Δ ∗ − ΔTS = ReΣλ (Δ ∗ , eV, T ).

(7.18)

The right hand side of Eq. (7.18) given by the diagrams (a) and (b) in Fig. 7.6 is J D ReΣλ (ω , eV, T ) ∼ ω ln . (7.19) D max[ω , eV, T] Estimating it at ω ∼ Δ ∗ we ﬁnd that Δ ∗ (eV ) − Δ T S ΔT S and the repopulation of the triplet state given by the factor P(eV ) = exp[−Δ ∗ (eV )/T ] is exponentially small at eV ≈ Δ T S . In agreement with the arguments presented in the beginning of this chapter, the effects of repopulation become important only at eV ΔT S . In that case the quasiequilibrium approach is not applicable and one should resort to the Keldysh formalism. This regime is deﬁnitely not realized in conditions considered above. Dynamically induced Kondo anomalies have been observed experimentally in the same SWNT quantum dots which provided many examples of dynamical symmetry related Kondo tunneling phenomena [319]. The FBA shown in Fig. 7.7 arises exactly at eV = ±ΔT S . The conductance anomaly at positive bias is seen distinctly. It is asymmetric in energy and its temperature dependence shown in the inset follows the pattern of Kondo effect. The asymmetry of peaks relative to the change of the bias polarity, eV → −eV is apparently connected with asymmetry of conﬁning potential (Fig. 5.2) and the tunneling rates, ΓL = ΓR . The theory of dynamically induced Kondo effect in the QD with SO(4) symmetry formulated above needs some generalization in order to apply it to the nanotube QD because the orbital degeneracy related to two possible spiralities of electrons in SWNT (Fig. 3.21) should be taken into account. In the experimental study of the Coulomb window diagram for the tunneling conductance G(eV,Vg ) the parameters

256 7 DYNAMICAL SYMMETRIES AND NON-EQUILIBRIUM ELECTRON TRANSPORT

Fig. 7.7 Finite bias anomalies of non-equilibrium Kondo-type tunneling conductance in SWNT quantum dot with even occupation at different temperatures. Inset: Temperature dependence of the FBA peak (open circles) in comparison with ZBA in the neighboring Coulomb window with odd occupation, after [319].

Fig. 7.8 Left panel: electron levels in SWNT quantum dot. δs is the mismatch in the energies of the states with opposite spiralities, δ ε is the interlevel spacing. Right panel: Ground state and two lowest excitations in SWNT quantum dot with N = 2.

δε δs

E S’ ET ES

of the discrete spectrum have been restored. It turned out that the mismatch in the energies of states with opposite spiralities is δs = 1.5 meV, the interlevel spacing in SWNT is δ ε = 4.6 meV, and the intradot exchange J is really small, J ≈ 0.05δs . In this case the low energy states of SWNT with even occupation form SO(5) multiplet (Fig. 7.8). In the ground state |S two electrons with opposite spin projections occupy the same level, in the spin triplet state |T λ and the excited spin singlet state |S two electrons have opposite spiralities. The energies of these states are

7.1 Dynamically induced ﬁnite bias anomalies in tunneling spectra

257

ES = 2εd , ET = 2εd + δs − J, ES = 2εd + δs .

(7.20)

Thus the effective spin Hamiltonian for the RG procedure should be taken in the form (4.76), and the scaling equations should generalize the system (4.77) for the case of ﬁnite bias. Two more important parameters have been extracted from the experimental data. This is the Kondo temperature TK ≈ 0.4 K, and the spin relaxation rate γr = h¯ /τr ≈ 350 mK. Reduced value of the Kondo temperature TK = 1.0 K obtained from tunneling spectra in the adjacent Coulomb window with odd occupation (see inset in Fig. 7.7 correlates well with the trend discussed above. Small damping also corresponds to our anticipation of a moderate contribution of spin relaxation in dynamically induced Kondo effect. Since the condition TK ΔT S is satisﬁed in this experimental setup, the ﬁnite bias RG theory [356] was used and the RG equations have been solved numerically [319]. The resulting FBA anomaly in the tunneling conductance is generated by the operators S and R1 in the Hamiltonian (4.76). Besides, additional very weak peaks in the frequency dependent vertices are generated by the operator R2 mixing the excited states |S and |T λ . The shape of FBA is close to that obtained within a quasiequilibrium approach (Fig. 7.5) and it ﬁts the experimental data satisfactorily. The threshold ﬁnite bias anomalies observed in SWNT with double occupation at ﬁnite magnetic ﬁeld [272] have similar shape and the same nature. The corresponding energy levels are shown in the middle column of Fig. 5.2(b).

εd +U Fig. 7.9 Two-electron cotunneling in a dot at ﬁnite bias eV = μs − μd . In the initial state there is an empty dot, in the ﬁnal state there are two electrons with opposite spins in the dot and two holes near the Fermi level in the source electrode.

μs εd μd

Not only the spin degrees of freedom may be activated dynamically at ﬁnite bias. Inelastic cotunneling may also allow a pair tunneling in the single electron regime, where the charging energy of empty dot is compensated by the energy gain

258 7 DYNAMICAL SYMMETRIES AND NON-EQUILIBRIUM ELECTRON TRANSPORT

of such cotunneling [256] (see Fig. 7.9).

One may say that in this process the

full SU(4) dynamical symmetry of a ”Hubbard atom” [see Section 5.3.1 and Fig. 5.13] including two-electron transition operators X 02 , X 20 from the triad Z of group generators is involved.

7.2 Dephasing and decoherence in quantum tunneling When considering quantum tunneling under nonequilibrium conditions we used the terms ”decoherence” and ”dephasing” in order to denote ﬁnite lifetime effects in many-body cotunneling processes, but did not care about distinction between these two notions and about the strict meaning of these terms. In this section these phenomena will be at the center of our attention. We are especially interested in the mechanisms of dephasing and decoherence intimately related to dynamical symmetries of CQD. Before turning to speciﬁc models where dephasing and decoherence play decisive part in many-particle tunneling, it is expedient to turn to basic deﬁnitions of these phenomena in quantum mechanical systems. All these processes may be discussed in a general context of the theory of decoherence in quantum objects which are put in a contact with external quantum system [438]. One should discriminate between (i) decoherence by measurement, (ii) environmental decoherence and (iii) decoherence due to appearance of ”superselection rules”. The ﬁrst type of decoherence arises in a contact between the pure quantum mechanical ensemble and the probe instrument. This contact transforms the measurable object into a mixed quantum ensemble described by the density matrix, and it is impossible to deﬁne a ”complete” system of quantum measurements which makes the ﬁnal state of the object predictable. In the second case the density matrix includes the states of environment (bath) which may be both of classical and quantum nature, and again the state of a quantum object becomes uncertain. Being interested in manifestations of dynamical symmetries in the ground state of a nanoobject, we turn to the third type of decoherence introduced in the seminal ”WWW” paper [425] (see the reviews [370, 427] for further development). Decoherence due to superselection rules in accordance with the WWW theory arises when two or more nearly degenerate states with different quantum numbers become superimposed in the ground state due to intrinsic or extrinsic symmetry breaking. Wick, Wightman and Wigner put forward this idea in order to interpret the baryonic charge.

Their approach eventually paved the way to the inclusion

7.2 Dephasing and decoherence in quantum tunneling

259

of proton and neutron belonging to the same baryonic family but differing by the charge quantum number in the baryonic octet containing also six hyperons. Quarks triads have been shown to be the source of intrinsic SU(3) symmetry broken due to electroweak interaction. case. The idea of superselection is intimately related to the Eight-fold Way of Ne’eman and Gell-Mann [127] which opened the studies of dynamical symmetries (see Section 2.4.1). Baryons with spin 1/2 but different charge turned out to be included into a SU(3) supermultiplet formed by two doublets, singlet and triplet. This octet is classiﬁed by means of hypercharge and isospin quantum numbers (see, e.g., [92]). Nanoobjects are described by non-relativistic quantum mechanics, where the laws of spin and charge conservation are strict. However, the symmetry of complex quantum dot S is violated due to interaction with the bath B. We have seen above that SO(n) and SU(n) dynamical symmetries of various nanoobjects include transitions between states with different spin and charge states induced by this interaction (see the diagrams of Fig. 2.4). Following the WWW ideas, we deﬁne the decoherence as a state of nanoobject S where its spin state is indeterminate in the ground state due to dynamical symmetry effects induced by interaction with the bath B. In the context of Kondo effect it is clear that such decoherence prevents formation of coherent Kondo-singlet state with fully screened local spin of the object S . Dephasing is a more conventional phenomenon. In accordance with classiﬁcation used in [438], dephasing arises in the scattering amplitude at ﬁnite energy and/or temperature so that any wave ϕn in a coherent superposition of quantum states transforms into a scattered wave Sin ϕn , where Sin is a scattering amplitude. Among the great variety of dephasing mechanisms we choose those related to the dynamical symmetries of the CQD. In all cases decoherence and dephasing arise as a result of transitions between the eigenstates of the Hamiltonian Hˆ d due to time-dependent interaction with environment (reservoir) Hˆ b , which does not respect the symmetry of Hˆ d . Before turning to our main object that is complex quantum dot, let us illustrate the idea of spin dephasing and decoherence due to time dependent interaction with the bath in the conventional spin 1/2 QD with odd occupation. We return to the model of QD in the presence of ac electromagnetic ﬁeld [192] described in Section 6.1. Let this ﬁeld be applied only to the dot so that the dot Hamiltonian Hˆ d (t) (6.1) reads

260 7 DYNAMICAL SYMMETRIES AND NON-EQUILIBRIUM ELECTRON TRANSPORT

Hˆ d = Hˆ d + ud cos ω t ∑ dσ† dσ .

(7.21)

σ

Performing the time dependent SW transformation in accordance with the prescriptions of Eqs. (6.4), (6.7) and (6.8), we retain in the latter equation only the ﬁrst nonvanishing term in the expansion of a diagonal component of the cotunneling exchange integral in the small parameter ξ = ud /εd (the limiting case Q → ∞ is considered): (7.22) Jαα (t) = J0 [1 + ξ cos ω t], where J0 is the time independent part of the indirect SW exchange parameter. This means that only single photon processes are taken into account. Then the term

∑ Jαα (t)S · sα α

in the exchange integral serves as a source of spin dephasing and decoherence. The electromagnetic ﬁeld is assumed to be too weak to induce photoionization of the dot, but it can create electron-hole pairs with energy h¯ ω and opposite spins, which accompany spin reversal in the dot. Creation of such a pair is nothing but the inelastic spin ﬂip scattering (see Fig. 7.10). Applying the Fermi golden rule, the corresponding lifetime τr is estimated as

h¯ 1 Γs + Γd 2 2 γr = = ξ . h¯ ω τr 8π εd

(7.23)

hω εd

εd Fig. 7.10 Light induced inelastic spin-ﬂip cotunneling from the initial state (i) with creation of electron-hole pair in the ﬁnal state ( f ).

(i)

(f)

If the damping γr is much larger than the Kondo temperature TK , one may use the time-dependent RG procedure in the adiabatic approximation where t is taken into account parametrically [cf. Eq. (6.9)]. The transformation must be stopped when the

7.2 Dephasing and decoherence in quantum tunneling

261

current bandwidth D is reduced to the values of the order of the energy h¯ ω . Then recalculation of the damping rate with logarithmically enhanced exchange integral gives

h¯ 3π 1 δ TK 2 = h¯ ω . (7.24) τr 32 [ln(¯hω /TK )]4 Tk Here

δ TK εd =ξ . TK Γs + Γd

Both TK and δ TK vary adiabatically with time via the time dependent level position

εd (t) = εd (1 + ξ cos ω t). The inﬂuence of these processes on the high-temperature tunneling conductance can be evaluated as G 3π 2 1 = . (7.25) G0 16 [ln(¯h/τ TK )]2 Comparing this equation with the high-temperature formula (4.25) we note that the scattering rate substitutes for temperature in the logarithmic enhancement factor due to the infrared cutoff of Kondo singularity. In accordance with the above classiﬁcation, this effect may be interpreted as a result of dephasing processes due to optically induced spin ﬂip scattering at ﬁnite energy and temperature. In the opposite limit γr TK the spin-ﬂip scattering does not prevent crossover from the weak coupling Kondo regime to the strong coupling limit. However, radiation-induced spin ﬂip processes do not allow the system to reach the unitarity limit. The characteristic spin ﬂip time τr should be compared with the time of singlet-to-triplet ﬂuctuations above the Kondo ground state, h¯ /TK . One may estimate the fraction of time which the system spends in the coherent singlet state as 1 − a¯h/τr TK where a ∼ 1. Then the resulting time-averaged ZBA in tunneling conductance of the QD at T → 0 is G h¯ = 1−a . G0 τr TK

(7.26)

This radiation-induced S/T ﬂuctuations of spin state of the QD at T → 0 can be qualiﬁed as a decoherence effect preventing formation of the coherent ground state of Kondo singlet in quantum dot coupled with the Fermi bath. In the above example both dephasing and decoherence arose due to the same irradiation ﬁeld at different values of the ﬁeld parameters. Now we turn to CQD with dynamical symmetries SO(5) and SU(4) [217, 219] where the difference between dephasing and decoherence is more profound although the source of both phenomena is the same time-dependent field Fd (t) in the dot Hamiltonian (6.1). In the

262 7 DYNAMICAL SYMMETRIES AND NON-EQUILIBRIUM ELECTRON TRANSPORT

model under consideration [217, 219] this ﬁeld is applied to the side dot in the Tshaped DQD (Fig. 3.13(b) or in the equilateral TQD [Fig. 3.14(h)]. In this model the source of the random potential is ﬂuctuatig of gate voltage Vg (t) applied to one of the dots (Fig. 7.11)

3 2

1

1

Vg(t)

Vg(t)

(a)

2

(b)

Fig. 7.11 Time-dependent gate voltage Vg (t) applied to two-terminal DQD (a) and three-terminal TQD in a magnetic ﬁeld perpendicular to the plane of triangle.

Following the symmetry approach used in this book, one should ﬁrst diagonalize the dot Hamiltonian Hˆ d , i.e. express it via diagonal operators X ΛΛ and then

rewrite all perturbations in terms of the full set X ΛΛ combined in the generators of appropriate dynamical symmetry group. In order to realize this scheme for the Hamiltonian (6.1), we return to the original form (3.27) of the multidot Hamiltonian for conﬁgurations shown in Fig. 7.11: Hˆ d = ∑ Hˆ 0j + j

j= j

∑ Hˆ 0j j + Hˆ 1(t),

(7.27)

jj

where Hˆ 0j = ε j n j + Q j n2j , Hˆ 0j j = −V ∑ d †jσ d j σ σ

Hˆ 1 (t) = [vg (0) + vg (t)]n1 ,

(7.28)

where the gate voltage contains both a static component vg (0) and a time-dependent perturbation vg (t). The potential wells in the CQD are enumerated by the index j in

7.2 Dephasing and decoherence in quantum tunneling

263

accordance with Fig. 7.11, and by convention j = 1 is reserved for the well coupled to the gate. Let us consider a general situation, where the gate potential applied to a multivalley complex QD contains both coherent and stochastic components vg (t) = v˜g (t) + δ vg (t).

(7.29)

Here v˜g (t) is the coherent (deterministic) contribution and δ vg (t) is the stochastic noise component which is deﬁned by its moments

δ vg (t) = 0

(7.30)

δ vg (t)δ vg (t ) = v2 f (t − t ) The overline stands for the ensemble average, the characteristic function f (t − t ) will be speciﬁed below. Unlike the previous example of photon-assisted spin dephasing and decoherence [192], here we consider an adiabatic situation with slowly varying vg (t), so the time-dependent perturbation is of purely charge nature. We will see below that this charge-related input signal may be converted in the Kondo-type response of spin origin. This conversion is possible exclusively due to implicit dynamical symmetries of the CQD. The time-dependent perturbation Hˆ 1 (t) may be transformed into time-dependent inter-dot hopping by means of canonical transformation (6.3) with the exponent U(t) = exp[iφ1 (t)n1 ],

(7.31)

where the phase φ1 (t) is given by

φ1 (t) =

1 h¯

t

dt vg (t ).

(7.32)

It is expedient to introduce ”even” and ”odd” hopping operators (±)

T1 j = ∑[d1†σ d jσ ± d †jσ d1σ ]

(7.33)

σ

To lowest orders in Vg , the time dependent part of the dot Hamiltonian acquires the form 1 (−) (+) (7.34) δ Hˆ d (t) = −V ∑ iφ1 (t)T1 j + φ12 (t)T1 j 2 j

264 7 DYNAMICAL SYMMETRIES AND NON-EQUILIBRIUM ELECTRON TRANSPORT

where φ1 (t) is deﬁned in Eq. (7.32) and

φ12 (t) =

v2 h¯ 2

t

dt

t

dt f (t − t ).

(7.35)

Thus, we obtain the effective time-dependent dot Hamiltonian dot = Hˆ (0) + δ Hˆ coh (t) + δ Hˆ stoch (t), H dot

(7.36)

where the time-dependent perturbation contains a coherent component [the ﬁrst term in (7.34)] and a stochastic one [the second term in (7.34)]. The coherent perturbation may be easily taken into account in the adiabatic approximation, whereas the second term in (7.36) results in the stochastization of the quantum state of the complex QD. The coherent part of δ Hˆ d (t) results in conversion of charge input signal to the Kondo screening, and its stochastic component brings an end to this screening due to dephasing and decoherence processes. We leave the discussion of coherent component of the time-dependent potential till Chapter VIII, were the adiabatic time-dependent potential of the type (7.34) will be considered in a context of tunneling through moving quantum dot, and concentrate on the incoherent component.

ε 1+ Q

ε1

ε 2+ Q Δ12

E S’ Δ12

ε 1+ ε 2 ε2

ΔS’S(t)

ET ES η

Fig. 7.12 Left panel: electron levels in a biased DQD with N = 2. Interdot transitions induced by the gate voltage Vg (t) are shown by the dashed line. Middle panel: two-electron states in a biased DQD. Right panel: time-dependent Jefferson-Haldane RG scaling trajectories EΛ (η ), where η = ln D0 /D is the scaling parameter.

The object under consideration is the T-shaped DQD in a biased conﬁguration [Fig. 3.13(b)] with the level scheme shown in the left panel of Fig. 7.12 and even occupation N = 2 with one electron per potential well. The general structure of the energy spectrum of the DQD is shown in the middle panel (see also Fig. 3.16). The asymmetry of the spectrum is controlled by the potential Vg (0) applied

7.2 Dephasing and decoherence in quantum tunneling

265

to the dot 1. It is enough for our purpose to retain only the lowest charge transfer exciton in this spectrum, so that the effective dynamical symmetry of the DQD is SO(5) (see the middle panel of Fig. 7.12). The level crossing ES /ET shown in the right panel arises due to the Jefferson-Haldane renormalization of these levels with different tunneling rates ΓT > ΓS [203] [see Eq. (4.93) in Section 4.3.2]. Thus the Kondo tunneling is realized in a system with low-lying S/T multiplet and remote state ES . In equilibrium one may forget about this high energy excitation, but in the presence of charge ﬂuctuations, it plays a key role in decoherence effect, as will be shown below. Now one may return to the diagonal representation (2.45) of the dot Hamiltonian where the time-dependent perturbation is explicitly built in the parameters EΛ . The leading (ﬁrst) term in the Hamiltonian (7.34) rewritten via the generators of √ (−) the SO(5) group contains the operator T12 = iA3 2 where A3 is the scalar operator from the set (2.46) of generators of the so(n) algebras which intermixes the two singlet states |S and |S . Thus, in accordance with our anticipations, the ﬁeld vg (t) affects only the charge degrees of freedom, and the time dependent potential affecting the DQD is

δ Hˆ d (t) = V φ1 (t)A3

(7.37)

The Hamiltonian Hˆ d (7.27) with this mixing term may be diagonalized and reduced to the canonical form (2.45) or (2.55) for DQD with the SO(5) symmetry:

Hˆ d = ET ∑ X T λ ,T λ + ES (t)X SS + ES (t)X S S =

(7.38)

λ

! 1 1 1 ET − [ES (t) + ES (t)] S 2 + ES (t)R12 + ES (t)R22 . 2 3 3

The levels ES and ES acquire an explicit time dependence due to mutual repulsion induced by the ﬂuctuating gate potential (7.37): ES (t) = ES − δstoch (t), ES (t) = ES + δstoch (t) where

δstoch (t) =

2V 2 2 φ (t) Δ S S 1

and

ΔS S = ES − ES .

(7.39)

266 7 DYNAMICAL SYMMETRIES AND NON-EQUILIBRIUM ELECTRON TRANSPORT

Thus the stochastic part of the dot Hamiltonian is

Hˆ d,stoch = δstoch (t)(X S S − X SS ) =

δstoch (t) 2 R2 − R21 . 3

(7.40)

The reason why ﬂuctuations in the charge sector affect spin excitations now clears up. The energy gap ΔS S between the two singlet states ﬂuctuates in time, but the center of gravity of this quasi doublet is ﬁxed so that the gap ΔT S ﬂuctuates as well. The Kondo effect is sensitive to the smallest energy gap ΔT S , because the higher singlet is quenched at low enough energies and the dynamical symmetry reduces from SO(5) to SO(4). In this reduced symmetry the system behaves as if the triplet/singlet gap ﬂuctuate with time [see Fig. 7.12(c), left panel]. In the reduced singlet/triplet subspace the exact constraint (2.49) imposed by the SO(5) symmetry transforms into the ﬂuctuating conﬁnement for the SO(4) group explicitly included in the effective Hamiltonian by means of the Lagrange multiplier, 1 1 ET S 2 + ES R1 2 − δstoch (t)R1 2 − μ (S 2 + R1 2 − 3). Hˆ d (t) = 2 3

(7.41)

Here the SO(4) symmetry conﬁnement controlled by the Lagrange multiplier μ is preserved only approximately. Thus the stochastic component of the dot Hamiltonian appears explicitly in the constraint. Further reduction of the energy scale is possible if TK Δ T S . Eventually one comes to the SO(3) Hamiltonian with ﬂuctuating conﬁnement 1 1 Hdot (t) = ET S 2 − μ (S 2 − 2) + δstoch (t)S 2 . 2 3

(7.42)

so that the ﬂuctuations of the charge transfer gap ΔS S are transformed into the ﬂuctuations of the spin constraint. Stochastic renormalization of the states |S and |S generates appropriate contribution the lead-dot tunneling in the corresponding channels, and as a consequences the stochastic component arises in the generalized SW Hamiltonian obtained as a result of the time dependent transformation (6.7): Hˆ ex (t) = J0T S · s + J T S (t)R1 · s + J T S (t)R2 · s.

(7.43)

(see [217, 219] for detailed derivation). Again, pure spin scattering is not affected by the charge perturbation, but the time-dependent spin-ﬂip transitions in the leads described by the two last terms in (7.43) arise due to the fact that the dynamical symmetry of the dot spin multiplet is activated by the time-dependent gate potential

7.2 Dephasing and decoherence in quantum tunneling

267

(7.37). The exchange parameters in the time dependent part of the Hamiltonian are 2V vg (t)φ1 (t) TS TS − φ12 (t) J (t) ≈ J0 1 + ΔSS εd = J0T S + js ϕ (t),

(7.44)

and similar term for J T S . We treat the stochastic ﬂuctuations as a classical ﬁeld and represent them by a single mode ϕ (t). Following the general deﬁnitions of dephasing and decoherence given in the beginning of this Section, one may claim that the dot Hamiltonian (7.38) is the source of decoherence in the many-particle Kondo state, while the dephasing effects induced by dynamical symmetry are related to the ”scattering” term (7.43) in the effective Hamiltonian. In accordance with our general pattern of treating the Kondo tunneling related dynamical symmetries in nanoobjects, we ﬁrst consider the high energy/temperature region T TK , where the perturbative approach is valid. We will use the same spinfermion diagrammatic technique as in Section 7.1 and include the ﬂuctuations ϕ (t) as a weak external ﬁeld characterized by the corresponding correlation functions. In case of SO(5) symmetry the set of spin fermions (9.73) contains three f -operators, two g-operators and their Hermitian conjugates. The generators S, R1 , R2 and A3 are represented by means of these fermions in accordance with Eqs. (9.74).

Τμ1

Τμ

Τμ2

Τμ1

(a)

S

Τμ2

(b)

Fig. 7.13 First Kondo exchange diagram (a) and noise-related correction to it (b).

The ﬁrst logarithmically singular vertex correction and its noise-related counterpart are shown in Fig. 7.13. Like before, the solid and dashed lines represent electron and spin-fermion propagators Gα ,kσ (t), Gλ (t), Gg [see Eq. (7.11) for their Fourier transforms]. The wavy line in the nonadiabatic correction (b) stands for the correlation function S(t − t ) = ϕ (t)ϕ (t ).

(7.45)

268 7 DYNAMICAL SYMMETRIES AND NON-EQUILIBRIUM ELECTRON TRANSPORT

In a weakly nonadiabatic regime the phase ﬂuctuations are presumed to be slow, S(t) = −ia2 e−γ |t|

(7.46)

with the amplitude a predetermined by the vertex corrections in Eq. (7.44). The decrement γ is small in comparison with the characteristic energies Δ ET and ΔST . The Fourier transform of this correlation function is S(ω ) =

2ia2 γ . ω2 + γ2

(7.47)

S Fig. 7.14 Noise-related contribution to the self energy of the spin-fermion propagator.

Τμ

Τμ

Dephasing induced by the T/S non-adiabatic ﬂuctuations results in the appearance of ﬁnite life time τr in the spin fermion propagator Gλ (ω ). This time is given by the imaginary part of the self energy shown in Fig. 7.14. The dephasing rate due to thermal triplet-singlet ﬂuctuation scattering is ⎧ ⎪ ΔST e−ΔST /T , T ΔST ⎪ ⎨ h¯ ∼ (a j2 )2 γdeph = ⎪ 2τr (ω → 0) T2 ⎪ ⎩ , ΔST < T < ΔET ΔST

(7.48)

We see that this temperature dependent contribution to the dephasing effect is frozen out at low temperatures because the singlet state responsible for dephasing is depopulated at T Δ T S . Inserting the spin fermion propagator with damping GT μ (ω ) = (ω + iγdeph )−1 ,

(7.49)

in the RG diagram [Fig. 4.25(b)], we conclude that this imaginary part transforms ¯ into the infrared cutoff of the Kondo singularity ∼ ln(D/max( ω , T, γdeph ). This means that there is no signiﬁcant effect of ﬂuctuations on the Kondo tunneling at high T ∼ ΔST provided γ2 TK , where γ2 ∼ T 2 /ΔST . At low T TK this type of dephasing slightly affects the temperature behavior of tunneling transparency in

7.2 Dephasing and decoherence in quantum tunneling

269

the Fermi liquid regime but does not prevent achievement of the unitarity limit at T → 0. Instead, in the low-energy limit the decoherence effects given by the last term in the Hamiltonian (7.42) become dominant. To investigate the inﬂuence of stochastic corrections to the constraint μ on the spin properties of the isolated dot, it is convenient to reformulate the Hamiltonian (7.42) in a fermionized form Hdot (t) =

∑

[ET /2 − μ (t)] fλ† fλ .

(7.50)

λ =0,±1

In this reduced SO(3) Hamiltonian only three fermions fλ† from the set (9.74) represent the spin degrees of freedom. The stochastic component of μ may be treated as a random potential in the time domain, which describes the ﬂuctuations of global fermionic constraint. The problem of propagation of spin fermions in a random time-dependent potential may be considered by means of the ”cross technique”[87] originally developed for the study of electron propagation in a ﬁeld of impurities randomly distributed in real space. In this technique correlated disorder induced by external noise is described by correlation functions D(t − t ) = h¯ 2 μ (t)μ (t ),

(7.51)

which connect different ”points” on the time axis. As was mentioned above, the ﬂuctuations are extremely slow in the nearly adiabatic limit. We use the ”anti-white noise” approximation where the frequency spectrum of noise ﬂuctuations is deltalike 2ζ 2 γ D(ω ) = lim 2 = 2πζ 2 δ (ω ). (7.52) γ →0 ω + γ 2 In this limit the averaged spin propagator describes the ensemble of states with chemical potential μ = const in a given state, but this constant is random in each realization. Similar problem of stochastization of the Landau-Zener (LZ) level crossing problem in two limiting cases of of anti-white noise (7.52) and white noise (frequency independent spectrum) was considered in Ref. [198]. We will return to this problem in Chapter 8. As was noted in [217, 219], the problem of decoherence of the spin state in a stochastically perturbed DQD in this limit can be mapped on the so-called Keldysh model [89, 200, 360] originally formulated for systems with δ -correlated in momentum space impurity scattering potential. The problem can be solved exactly and the decoherence time is controlled by the variance of the Gaussian correlation given by the parameter ζ 2 .

270 7 DYNAMICAL SYMMETRIES AND NON-EQUILIBRIUM ELECTRON TRANSPORT

The retarded spin-fermion propagator at T = 0 is deﬁned as GTRν (t − t ) = fν (t) fν† (t )R = −i[ fν (t) fν† (t )]+ .

= (a)

+ (b)

+ (c)

(7.53)

+ ... (d)

Fig. 7.15 Diagrams for the self energy of spin-fermion Green function in random time-dependent potential. Each circle corresponds to some moment on the time axis. Solid lines stand for bare spinfermion propagators g(t − t ), dashed lines denote noise-ﬂuctuation correlation function D(t − t ). (a) Full self energy with vertex corrections; (b)-(d) irreducible diagrams for the self energy of the scalar Keldysh model in the time domain. The ﬁrst non-vanishing vertex correction is represented by diagram (d).

The Keldysh model in time domain admits exact solution because all diagrams in a given order of the cross technique (see Fig. 7.15) for the Green function (7.53) are equivalent due to the delta-function character of the correlation function D(ω ) (7.52). As a result, the contribution of all diagrams in a given order n is completely determined by the combinatorial coefﬁcient An = (2n − 1)!! which gives the total number of diagrams corresponding to all possible pairwise connections of n vertices by wavy lines. Then the sum of the perturbation series for the retarded spin fermion Green function is ∞

(R)

Gλ (ε ) = g(ε ) 1 + ∑ An ζ 2n g2n (ε )

(7.54)

n=1

Here g(ε ) = (ε + iδ )−1 is the free spin-fermion propagator with ET /2 − μ0 = 0 taken as the reference energy. Summation of this series may be performed by means of the integral representation 1 (2n − 1)!! = √ 2π

∞ −∞

t 2n e−t

2 /2

dt.

(7.55)

Changing the order of summation and integration in accordance with Borel summation procedure, one obtains 1 G (R) (ε ) = √ ζ 2π

∞ −∞

2 /2ζ 2

e−z

dz ε − z + iδ

(7.56)

7.2 Dephasing and decoherence in quantum tunneling

271

As was mentioned, the solution (7.56) represents the set of spin states under a stochastically ﬂuctuating chemical potential averaged with a Gaussian exponent characterized by the variance ζ 2 . Remarkably, the spin-fermion Green function in this model has no poles, singularities or branch cuts. Alternatively, the same result can be obtained by means of the Ward identity connecting the self energy Σ (ε ) shown in Fig. 7.15(a) and the vertex Γ (triangle in the same ﬁgure) dG −1 (ε ) Γ (ε , ε ; 0) = . (7.57) dε Here and below the index λ is omitted, since the ﬂuctuations of constraint are related to the global U(1) symmetry. The self energy in the Keldysh model is easily calculated:

Σ (ε ) =

dω Γ (ε , ε − ω ; ω )G (ε − ω )D(ω ) = ζ 2 Γ (ε , ε ; 0)G (ε ) 2π

(7.58)

Using (7.57) and (7.58), we transform the Dyson equation for the spin-fermion propagator into an ordinary differential equation

ζ2

dG + εG − 1 = 0 dε

(7.59)

This equation is supplemented by the boundary condition G (ε → ∞) =

1 ε

(7.60)

(the universal ultraviolet asymptotic for the Green functions). The gaussian average (7.56) is a solution of Eq. (7.59). The net effect of this type of decoherence is stochastization of spin degrees of freedom. The measure of this stochastization is the Gaussian variance ζ 2 . The noise induced stochastization of the local spin competes with the Kondo screening, so that the magnitude of the parameter ζ should be compared with the Kondo temperature TK . One may expect that strong enough noise is detrimental for the Kondo effect. Indeed, direct calculation of the static magnetic response function χ (0 by means of the same Ward identity gives ∞ 2 1 yζ (7.61) χ (0) = √ dye−y /2 y tanh 2T 8πζ −∞ so that the Gaussian averaging is superimposed on the thermodynamic averaging. The asymptotic behavior of the static susceptibility χ (0) is

272 7 DYNAMICAL SYMMETRIES AND NON-EQUILIBRIUM ELECTRON TRANSPORT

χ (0) =

⎧ ⎨ CC /T, T ζ ⎩

(7.62) CK /ζ , ζ T

where CC and CK are constants. At high T the behavior of the dot is Curie-like with Curie constant CC modiﬁed by averaging. At low T the noise dispersion ζ plays the role of an effective temperature in the Keldysh model with the corresponding constant CK in the numerator. This effective temperature should be compared with the energy argument in the Kondo singularity at high T and eventually it appears in the Kondo logarithm as an infrared cutoff. Even if ζ TK the noise-induced stochastization of spin does not allow the system to reach the unitarity limit of the full Kondo screening at T → 0. We conclude from these calculations that the charge noise at the gate electrode is converted into spin stochastization in the DQD due to its implicit dynamical symmetry SO(5). In accordance with the above general classiﬁcation this type of disorder may be interpreted as decoherence in the ground state of Kondo system. It is interesting that due to the speciﬁc dynamical symmetry yet another type of magnetic response arises [219]. This response is described by the correlation function

λλ χST (t − t ) = −i[R1λ (t), R1λ (t )] → δλ λ χP (ω )

Calculation of this response function shows that the static S/T susceptibility which mimics the Curie law at very large temperatures T (Δ ST , ζ ), is suppressed exponentially ∼ exp(−ΔST /T ) at low temperatures T (Δ ST , ζ ) and has an intermediate asymptotic behaviour χP (0) ∼ 1/ζ ln(ζ /T ) if Δ ST T ζ while

χP (0) ∼ 1/ΔST when ζ T ΔST . Another type of decoherence due to dynamical symmetry involving orbital degrees of freedom [219] may be traced in the triangular TQD shown inFig. 7.11. The energy spectrum of the equilateral TQD is presented in Section 3.5.2 and the Kondo related tunneling in transversal magnetic ﬁeld is described in section 4.4. The basis functions of a singly occupied equilateral TQD in the three-terminal geometry are given by Eqs. (3.50) and (3.52), and the energy spectrum as a function of magnetic ﬁeld is shown in the upper panel of Fig. 4.34 [see also Eq. (4.146)]. As was pointed out in the discussion of magnetically driven Kondo tunneling through equilateral TQD, the dynamical symmetry SU(4) is involved in the Kondo effect at magnetic ﬂux Φ ∼ (n + 1/2)Φ0 , where the energy level crossing takes place. The gate voltage Vg (t) applied to one of the three dots violates the C3v point

7.2 Dephasing and decoherence in quantum tunneling

273

symmetry of the triangle and thus transforms level crossing into level anticrossing (Fig. 4.38). If Vg (t) has a noise component, the time dependent gap Δ± = E+ − E− is the source of decoherence in the Kondo tunneling regime. In order to calculate the contribution of δ Hdot (t) in the charge-spin conver(±) sion, one needs the matrix elements of the operators T1 j (7.33) in the basis |Γ (3.50) of the C3v point group, in particular those involved in the above level crossing/anticrossing in the sector Φ0 Φ 2Φ0 . The relevant matrix elements in the subspace |E± are

Δ (stoch) (t) = E+ |δ Hˆ stoch (t)|E−

(7.63)

so that only the second term δ Hˆ stoch ∼ φ12 (t) in the Hamiltonian (7.34) contributes to the ﬂuctuating off-diagonal matrix elements. These stochastic inter-level transitions are responsible for the ﬂuctuation induced avoided crossing E+ −E− (dashed lines in Fig. 4.38). One may write the corresponding part of the Hamiltonian in terms of the pseudospin operator T (3.55), as (stoch)

Hdstoch = Δ±

∗(stoch)

(t)T − + Δ±

(t)T +

(7.64)

This term should be added to the effective Kondo Hamiltonian of TQD (3.57). Thus in this case the stochastic ﬂuctuations of gate voltage induce random pseudospin scattering, and the Keldysh approximation of anti-white noise correlations may be used for this type of random potential. However, unlike the DQD case, the scattering has a vector character, so that the fermionized Hamiltonian (7.64) has the form stoch (t) = ρ (t)g†↓ g↑ + ρ ∗ (t)g†↑ g↓ Hdot

(7.65)

where ρ (t) is a random scattering potential which stems from (7.64), g↑ and g↓ are ”pseudospin-fermions” for the vector T . Only the transversal component of pseudospin scattering is involved in the stochastic perturbation. Then, following the pattern of the scalar model (7.52), we introduce the correlation function C(t − t ) = ρ (t)ρ ∗ (t ) and its Fourier transform proportional to the Gaussian variance ξ , 4ξ 2 γ = 4πξ 2δ (ω ) γ →0 ω 2 + γ 2

C(ω ) = lim

(7.66)

The perturbation theory series for the Fourier transform of the retarded Green’s function for the pseudospin operators FσR (t − t ) = gσ (t)g†σ (t )R

(7.67)

274 7 DYNAMICAL SYMMETRIES AND NON-EQUILIBRIUM ELECTRON TRANSPORT

[cf. Eq. (7.53)] has the same form as in the ”scalar” Keldysh model, [200, 360], namely,

∞ √ F (ε ) = f (ε ) + ∑ Bn ( 2ξ )2n f 2n+1 (ε ).

(7.68)

n=1

Here f (ε ) = (ε + iδ )−1 is the free pseudospin-fermion propagator, Bn = n! is the total number of 2n-th order diagrams, which is again a combinatorial coefﬁcient. This coefﬁcient differs from that in the expansion (7.54) because only the diagrams with the pseudospin operators ordered as . . . T + T − T + T − . . . survive in the self energy. Thus we come to a two-color Edwards’ technique (Fig. 7.16, left panel).

Fig. 7.16 Two-color diagrammatic representation for the planar vector Keldysh model. Left panel: the ﬁrst non-vanishing vertex correction to the self energy of the transversal spin-fermion propagator. Right panel: the ﬁrst non-vanishing vertex diagram (see text for further details).

To generalize the above Borel summation procedure for this ”planar vector model” we use the integral representation n! =

∞ 0

dzzn e−z .

The result of summation is the two-dimensional Gaussian average 1 F (ε ) = 2

2 2 +∞ +∞ dye−y2 /2ζ 2 dxe−x /2ζ

−∞

√ ζ 2π 1

ε−

−∞

√ ζ 2π 1

, + x2 + y2 + iδ ε + x2 + y2 + iδ

(7.69)

which is a natural generalization of the one-dimensional Gaussian averaging (7.56). Only the modulus of the random ﬁeld r = x2 + y2 is averaged with the Rayleigh distribution function PR (ε ) = (ε /ζ 2 ) exp(−ε 2 /2ζ 2 ),

(7.70)

7.2 Dephasing and decoherence in quantum tunneling

275

whereas the angular variable remains irrelevant due to the in-plane isotropy of the system. Like in the scalar model, the averaged pseudospin-fermion Green’s function has no singularities. A similar version of the cross technique in a real space arises in electron systems in the domain of long-range Gaussian ﬂuctuations near the charge density wave (CDW) instability, although the physical mechanism is radically different (alternating incoming and outgoing Umklapp ﬂuctuations of CDW order parameter in 1D [360, 361] and 2D [38, 241] systems). To generalize the differential equation (7.59) for the two-dimensional Keldysh model, we calculate explicitly the derivative dF (ε )/d ε using the same method of summation of the series in the r.h.s. of Eq. (7.68) as the one which was used for the calculation of the integral representation (7.69) for the Green’s function. This calculation gives the following result:

ξ

2 dF (ε )

dε

ξ2 = 1 − ε F (ε ) 1 − 2 . ε

(7.71)

To emphasize the internal unity of the two models, we rewrite Eqs. (7.59) and (7.71) in a uniﬁed way:

ε G (ε ) − 1 = ζ 2 G 2 (ε )

d −1 G (ε ) dε

1 d ε F (ε ) − 1 = ξ 2 F 2 (ε ) ε F −1 (ε ) ε dε

(7.72)

Then, appealing to Eq. (7.57), we deﬁne the vertex Γ (ε , ε , 0) for the vector model (see Fig. 7.16, right panel) as

Γ=

1 d ε F −1 (ε ) . ε dε

(7.73)

It is worth noting that the differential operator on the r. h. s. of Eq. (7.73) is nothing but divε in polar coordinates. This form reﬂects the effective two-dimensionality of Gaussian averaging in the planar Keldysh model, which has been noticed already in Eq. (7.69). The stochastization of pseudospin manifests itself in the transformation of the response function χ⊥ (t) = T − (t)T + (0)R . Using the Ward identity for the pseudospin Green function, one ﬁnds the static pseudospin susceptibility 1 ∞ 2 −y2 /2 yξ (7.74) χ⊥ (0) = y dye tanh ξ 0 2T

276 7 DYNAMICAL SYMMETRIES AND NON-EQUILIBRIUM ELECTRON TRANSPORT

and its asymptotic

χ⊥ (0) ∼

1/T, T ξ , 1/ξ , ξ T.

(7.75)

Thus the pseudospin in the vector model loses its local characteristics in the same way as the spin in the scalar model. Accordingly, stochastization affects the Kondo tunneling. We conclude from the above results, that the case of SU(4) symmetry supported by the interplay between spin and orbital degrees of freedom in a TQD differs radically from the case of SO(4) symmetry involving only the spin variables. In the latter case the external charge noise results in the stochastization of spin degrees of freedom, so that the DQD ”loses” its spin moment at low enough energy/temperature. In the former case the spin 1/2 is robust, and only the orbital (pseudospin) degrees of freedom are affected by the charge noise. Pseudospin stochastization means that the logarithmic divergences in the corresponding Kondo loops are subject to a cutoff similar to that given by Eq. (7.62). As a result, only the pseudospin fermion - electron loops determine the Kondo screening at low T . Thus a noise induced SU(4) → SU(2) crossover takes place in a TQD threaded by magnetic ﬂux.

7.2.1 Vector Keldysh model in the time domain The Keldysh models in time domain have potential applications in other ﬁelds of nanophysics, e.g., in the studies of ultracold gases. Having this in mind, we conclude this chapter by a general formulation of 3D vector Keldysh model for an ensemble of two-level systems in a stochastic external ﬁeld [216]. The Hamiltonian of an isolated asymmetric double-well trap has the form Hˆ DW =

∑

[ε j n j + Un j (n j − 1)] − Δ 0 (c†l cr + H.c.).

(7.76)

j=1,2

Here n j = c†j c j is the particle occupation number, ε j is the discrete energy level in the valley j, U is the interaction (repulsion) parameter for two particles in the same valley. The condition U Δ 0 is usually assumed. We consider spinless particles, having in mind that the theory can be applied both to interacting bosons and fermions (electrons) with frozen spin degrees of freedom. In case of singly occupied trap HDW acquires the form (see Section 2.3)

7.2 Dephasing and decoherence in quantum tunneling

277

(1) Hˆ DW = −δ0 σz − Δ 0σx − μ0 (N − 1).

(7.77)

in pseudospin space. Here the asymmetry parameter δ0 = εr − εl plays part of an effective ”magnetic” ﬁeld, the chemical potential μ0 controls the occupation N of the trap. Both asymmetry and tunnel transparency parameters of the double-well trap are assumed to ﬂuctuate in time and induce stochastic ﬁeld with longitudinal and transversal components Hˆ stoch (t) = h (t)σz + h⊥(t)σ + + h∗⊥(t)σ − .

(7.78)

Like in previous examples, the ﬂuctuation ﬁelds are introduced as random ﬂuctuations of double well trap parameters, namely as time dependent ﬁelds determined by their moments h (t) = 0,

h (t)h (t + τ ) = D(τ ),

h⊥ (t) = 0,

h⊥ (t)h∗⊥ (t + τ ) = C(τ ).

(7.79)

Here the overline stands for the ensemble average. Thus we have reduced the original model to the effective spin Hamiltonian in a vector magnetic ﬁeld with random time-dependent components. The problem can be reformulated as a study of propagation of excitations along the time axis in the presence of time-dependent random vector potential h(t)σ . We use the same Keldysh – Efros conjecture (7.52), (7.66) for the longitudinal and transversal correlation functions D(ω ) and C(ω ). To demonstrate the key features of the vector model we commence with stochastization of isolated symmetric double well trap with impenetrable barrier (δ0 = 0, Δ0 = 0), where the only source of dynamics is the Hamiltonian (7.78). We solve this problem by means of path integral formalism [336]. The Lagrangian and the corresponding action are deﬁned on the Keldysh contour K [199], L (t) =

∑

ˆ c¯ j i∂t c j − H,

SK =

j=1,2

K

L (t)dt.

(7.80)

Here c¯ j , c j are Grassmann variables describing particles. The time-dependent gauge transformation c j (t) → c j (t)eiϕ j (t) , converts the ﬂuctuation of the well depth to the ﬂuctuation of the phase of the tunnel matrix element under the choice

ϕ j (t) =

t −∞

h j (t )dt .

(7.81)

278 7 DYNAMICAL SYMMETRIES AND NON-EQUILIBRIUM ELECTRON TRANSPORT

We therefore identify the longitudinal and transverse noise with phase ﬂuctuations of the barrier transparency and ﬂuctuations of the modulus of the tunnel matrix element, respectively and unify them in the path integral description. The ensemble averaging ...noise =

dhρ Pl (hρ )

d Δ ρ∗ d Δρ Ptr (Δ ρ∗ , Δρ )...

is done with the help of probability distribution functions for longitudinal and transverse ﬂuctuations

1 Pl = √ exp − dtdt hρ (t)D−1 (t − t )hρ (t ) K ζ 2π 1 ∗ −1 exp − dtdt Δ (t)C (t − t ) Δ (t ) . Ptr = ρ ρ 2πξ 2 K

The Green functions can be calculated by means of the generating functional corresponding to the Keldysh action SK in a standard way [336]. In the ”inﬁnite memory” limit (7.52), (7.66) the Green function in the symmetric double well potential is easily expressed as Gv (ε ) =

1 2 ζ ξ (2π )3/2 2 /2ξ 2

dw∗ dwe−|w|

∞ −∞

dze−z

2 /2ζ 2

ε ±z (ε + iη )2 − z2 − |w|2

(7.82)

Noticing that the Green functions do not depend on the well index j and performing the integration over angles in spherical coordinate system we get 1 ∞ ρ2 d ρρ exp − 2 Gv (ε ) = 2ξ 0 2ξ 2 2 Erf ρ ξ2ξ−2 ζζ2 1 1 + (7.83) ε − ρ + iη ε + ρ + iη ξ2 − ζ2 Here Erf(z) is the error function. The three-dimensional Gaussian averaging in (7.82) reﬂects the vector character of the random ﬁeld distributed on an ellipsoid. Characteristic size of semi-axes is deﬁned by the variances of the longitudinal and transverse noise. This ellipsoid transforms into a Bloch sphere for pseudospin if ζ = ξ . Two extreme cases of ”planar” limit ξ → 0 of easy plane anisotropy and ”’scalar” limit ξ → 0 of easy axis

7.2 Dephasing and decoherence in quantum tunneling

279

anisotropy represented by the averages (7.56) and (7.69) may be obtained from Eq. (7.83) straightforwardly. The perturbation series for the full vector Green function Gv which generalizes (7.54) and (7.68) may be restored from the integral (7.83) by means of expanding its integrand in powers of g(ε )m = (ε + iη )−m . Using the series expansion of the error function 2 ∞ (−1)n z2n+1 Erf(z) = √ ∑ , π n=0 n!(2n + 1) and the integral representation (7.55), we transform (7.83) into ∞

Gv (ε ) = 2 ∑

∞

ξ 2m+1

∑ Cmn α 2n ζ 2n+1 g2m+1 (ε ).

(7.84)

n=0 m=0

Here α =

(ξ 2 − ζ 2 )/2, and Cmn =

[2(m + n) + 1]!! n!(2n + 1)

is the combinatorial coefﬁcient in the 2m-th order of the ”three-color” Edwards’ diagrammatic expansion for the full vector Keldysh model combining diagrams of Figs. 7.15 and 7.16. The differential equations (7.72) may be generalized for isotropic threedimensional vector Keldysh model, ξ = ζ = λ , describing rotation of pseudospin on the Bloch sphere:

ε Gv (ε ) − 1 = λ 2 Gv (ε )

1 d 2 −1 ε G ( ε ) , v ε 2 dε

(7.85)

and the corresponding Ward identity is

Γv =

1 d 2 −1 ε Gv (ε ) . ε 2 dε

(7.86)

The most instructive property of the Keldysh model for asymmetric double-well trap is the robustness of avoided crossing effect against stochastization of tunneling ﬂuctuations. The corresponding dip in the density of states of stochastisized double well trap is shown in Figs. 7.17 and 7.18 for scalar and planar Keldysh models with ﬁnite static asymmetry δ0 of the trap. The density of states (DoS) is given by the imaginary parts of the Green functions (7.56), (7.69) for scalar, planar and isotropic vector models, respectively:

280 7 DYNAMICAL SYMMETRIES AND NON-EQUILIBRIUM ELECTRON TRANSPORT

DoSs

0.2 0.5 2.0

a) -0.5 0

0.5

Fig. 7.17 Density of states in a stochastisized double trap with scalar Keldysh noise.

DoSv

0.5 1.0 1.5

b) -1

0

1

Fig. 7.18 Density of states in stochastisized double trap with planar vector Keldysh noise.

2 ε + δ02 1 εδ0 νs (ε ) = √ exp − cosh , 2ζ 2 ζ2 ζ 2π 1 ε2 ν p (ε ) = 2 |ε | exp − 2 . 2ξ 2ξ

ν p (ε ) =

λ3

1 √

ε2 ε exp − 2 . 2λ 2π 2

(7.87)

(7.88)

(7.89)

In the scalar model νs (ε ) is a superposition of two Gaussians centered around ε1 and ε2 , respectively. In the planar model ν p (ε ) is represented by a single Gaussian with a dip ”burnt” around zero energy. In full anisotropic vector model with ﬁnite

7.2 Dephasing and decoherence in quantum tunneling

281

δ0 the DoS has more complicated form, but the dip behavior is given by the same exponent as in (7.89) with pre-exponential factor ∼ ε 2 . Real-time manipulation with occupation of ultracold atoms in optical lattices formed by double-well traps [55] provides the possibility of experimental observation of anti-white noise effects in nanoobjects. Of course, this noise is a concomitant effect which may accompany adiabatic repopulation of the wells due to slow variation of the optical lattice parameters. In the next chapter the combination of coherent and incoherent motion of nanoobjects will be studied in a more general context of tunneling in moving few-electron systems with dynamical symmetries changing in time.

Chapter 8

TUNNELING THROUGH MOVING NANOOBJECTS

Universally recognized parallels between real molecules and artiﬁcial complex quantum dots embrace electronic and spin properties of these few-electron systems included in tunneling electrical circuits. This analogy is not complete because the artiﬁcial nanoobjects do not possess vibrational eigenmodes which are integral part of excitation spectrum of natural molecular complexes. However, modern nanotechnologies are able to provide artiﬁcial atoms and molecules with mechanical degrees of freedom. As was mentioned in Section 3.6.4, mechanical motion of nanoisland may be induced by external electromotive or magnetomotive forces. For this sake the nanoisland may be mounted as an electromechanical pendulum formed by a gold clapper [97], silicon nanopillar [367, 368] or suspended silicon nitride string [229]. Nanostring itself [404] as well as a suspended carbon nanotube [172, 362, 376, 383] may be used as a nanomechanical oscillator. Practical implication of moving nanoislands in fabrication of single-electron tunneling transistors, current standards, atomic force microscopy, etc encourages experimentalists to reﬁne the electrical and nano-mechanical components of tunneling circuits in order to make the tunneling current more controllable and the parameters of devices more reproducible. Nanoisland oscillating between two ”banks” (metallic electrodes) may catch up electrons from one electrode and eject them at another one, i.e. play part of a shuttle, which transports charge between two banks [143]. If the radius of the nanoisland is small enough, strong Coulomb blockade promotes one-by-one electron shuttling in NEMS-SET (nano-electro-mechanical shuttling single-electron-tunneling) regime. Besides, the spin-ﬂip cotunneling mechanism (Fig. 3.4) paves the way to spin shuttling in the Kondo regime [220]. Three basic conﬁgurations which may be used in electron shuttling by means of single and double quantum dots are presented in Fig.8.1. We deﬁne these conﬁguraK. Kikoin et al., Dynamical Symmetries for Nanostructures: Implicit Symmetries in Single-Electron Transport Through Real and Artificial Molecules, DOI 10.1007/978-3-211-99724-6_8, © 2012 Springer-Verlag/Wien

283

284

(a)

8 TUNNELING THROUGH MOVING NANOOBJECTS

(b)

(c)

Fig. 8.1 Basic conﬁgurations of single electron suttle (a): QD oscillating between two metallic electrodes; (b) DQD oscillating as a pendulum; (c) DQD oscillating as a turnstile. Two extreme positions and tunneling contacts of the moving island are shown by solid and dashed line, immovable island is shown as a ﬁlled circle.

tions as ”shuttle” (a), ”pendulum” (b) and ”turnstile” (c). Shuttling regime implies that the lead-dot tunneling is possible only when the shuttle ”moors” to the bank, and dwells near it for a certain time τd , while the tunneling contact is broken during the transportation time T /2 − τd where T /2 is a half period of the shuttling cycle. In various experimental situations the contact may vary from sudden to adiabatic. In case of sudden contact the perturbation of the system due to switching on the tunneling channel results in various shake effects, including multielectron ionization of the dot, provided the electric ﬁeld inducing mechanical motion is strong enough. This regime is practically beyond the reach of reliable theoretical description. In the opposite ”smooth” regime the contact occurs adiabatically without inelastic excitations of the internal degrees of freedom of the shuttle and the electrodes. All intermediate regimes are also realizable. It is evident that the dynamical symmetries, which are in the center of our interest, manifest themselves in the adiabatic or nearly adiabatic regime, and we will consider below only the limit of smooth single or double electron shuttling (tunneling or cotunneling). Besides, from the point of view of time-dependent dynamical symmetries the complete interruption of the tunneling contact is not a decisive point, and all principal effects related to the time evolution of the dot spectrum may be observed if the tunneling rate changes from strong to exponentially weak coupling during the shuttling period but does not breaks completely. Having this in mind, we consider here the DQD and TQD in side geometries with moving side dot [Fig.8.2(a,b)]. These geometries have been exploited above in the study of dephasing and decoherence in Kondo tunneling (Section 7.2), where the stochastic motion of the side dot have been considered. Here we will study the slow periodic motion regime and its inﬂuence on the Kondo effect.

8 TUNNELING THROUGH MOVING NANOOBJECTS

285

x

(a)

(b)

Fig. 8.2 DQD (a) and TQD (b) with moving side dot. Perpendicular magnetic ﬁeld may be applied in conﬁguration (b) Notations are the same as in Fig. 8.1.

It is clear that in the adiabatic or nearly adiabatic regime the motion of quantum dot may be described via coordinate-dependent parameters of the basic Anderson Hamiltonian (3.2), namely

ε j → ε j (x), Wl j → Wl j (x),

(8.1)

where x(t) is a time-dependent coordinate of the moving dot. The simplest effect related to cyclic motion of a dot between two banks is the analog of Debye – Waller effect considered in Section V.3 in connection with tunneling through oscillating molecules. This effect exists in all geometries shown in Fig. 8.1. If the tunnel coupling Wl j (x) is not broken at any x, the averaging of the dot position over the oscillation period results in enhancement of the static Kondo temperature (5.33) proportional to the mean-square displacement of the dot from its static position [220]. If the shuttle-bank contact exists only when the shuttle approaches one of the banks, then the matrix elements Wl j are enhanced exponentially in comparison with the static equidistant position of the shuttle between the two electrodes, but this enhancement is reduced by the factor ∼ τd /T . This Debye – Waller like factor is universal and independent of the dynamical symmetry of a nanoobject. We know from previous analysis that the simplest case where dynamical symmetry is involved in tunneling mechanisms is a doubly occupied quantum dot. Below we conﬁne ourselves with this case and study various manifestations of SO(4), SU(4) and SU(3) symmetries in the physics of moving quantum dots, although the relevant mathematical technique is well elaborated only for the simple SU(2) and SO(3) groups.

286

8 TUNNELING THROUGH MOVING NANOOBJECTS

8.1 Conversion of coherent charge input into the Kondo response The interest in charge-spin conversion effects is spurred by challenging prospects of spintronics. Most of the mechanisms of such conversion are related to the interconnection between electrical and spin current due to spin-orbit interaction [85], which results in spin accumulation near the sample edges. Such an accumulation in threeand two-dimensional electron gas in elemental and III-V semiconductors may result in a spin-Hall effect [290, 388]. It was argued also that the Rashba-type spin-orbit interaction in a quantum dot assists pure spin current by modulation of the voltages applied to the leads in a three-terminal device [266]. In all these propositions the possibilities of conversion of charge current into spin current were discussed. Noise-induced dephasing and decoherence effects in Kondo tunneling considered in Section 7.2 where the crossovers SO(5) ↔ SO(3) and SU(4) ↔ SU(2) in the side T-shape and Δ -shape geometries have been studied, may be qualiﬁed as generation of spin response to charge input. In moving quantum dots with nontrivial dynamical symmetries similar effects take place in the coherent regime. In this section we will use these two model systems for demonstration of conversion of coherent charge input given by slow periodic oscillation of the side dot into the coherent Kondo response [217, 219, 220]. Like in the above case, dynamical symmetries ﬂuctuating in time are responsible for conversion effect. We start with the T-shape DQD described by the Hamiltonian (7.28). The induced motion of the side dot in Fig.8.2 may result in renormalization (8.1) of the model parameters. It was shown in Section 7.2, that renormalization of the dot level

εd may be converted in renormalization of the interdot tunneling W1 j by means of the canonical transformation (7.31) (like above, the moving dot is labeled by the index ’1’), so we take into account only the time-dependent term in the tunneling integral δ Hd (t) = −υg cos Ω t ∑(T1−j d1†σ d jσ + H.c.) (8.2) 1j

[cf. Eqs. (7.33) and (7.34)]. To reveal generic properties of the object we consider here the coherent monochromatic modulation of the tunneling amplitude with frequency Ω . The mechanism of conversion is the same as in the case of charge noise-induces decoherence of spin singlet ground state, namely the time-dependent modulation of the states ES (t) = ES − δcoch (t),

8.1 Conversion of coherent charge input into the Kondo response

ES (t) = ES + δcoch (t)

287

(8.3)

with δcoch (t) = υg2 cos2 Ω t/Δ S S in the Hamiltonian (7.38). Following the above pattern, we reduce the actual energy interval to E Δ S T , so that the higher singlet exciton is frozen out and the dynamical symmetry is reduced from SO(5) to SO(4). Then the adiabatic time-dependent SW procedure [219] transforms the original Hamiltonian into HSW = Hd (t) + Hex(t), 1 ET S2 + ES (t)R21 + Q(Nˆ − 2)2 , Hdot (t) = 2 S (t)]R1 · s Hex (t) = J0T S · s + [J0ST + Jad

(8.4)

The time dependence of the exchange parameter J ST , that is directly related to the variation of the energy gap Δ ST (t), transforms into the corresponding time dependence of the Kondo temperature. The behavior of TK in case of SO(4) symmetry has been discussed in detail in Section 4.2 [see Eqs. (4.51) – (4.56)]. The time dependent tunneling conductance in a weak coupling regime at T TK can be extracted from Eqs. (4.25) and (4.55): G(t) ∼ {ln(T /TK0 − ρ ln[TK0 /Δ ST (t)]}−2 . G0

(8.5)

Several examples of numerical solution of parametrically time-dependent equations for TK (t) and G(t) are presented in Fig. 8.3. It is seen from these curves that the input sinusoidal potential ∼ cos Ω t transforms into periodic oscillations of the Kondo-related ZBA in G(t). The sinusoid is reproduced with a slight distortion when the gap Δ ST (t) remains positive under temporal perturbation (row 1 in Fig. 8.3). Additional minima impart the ”Kremlin wall” shape to the periodic curve G(t) when the sign of Δ ST (t) changes under the periodic perturbation (rows 2,3). The same regime with larger amplitude υg may result in complete suppression of Kondo tunneling due to the periodic triplet-singlet crossover (row 4). Finally, if the system remains completely in the singlet sector Δ ST (t) < 0 near the crossover point, the charge perturbation results in a pulsed Kondo output signal (row 5). In the case of Δ -shaped TQD in perpendicular magnetic ﬁeld [Fig.8.2(b)] the motion of the side dot affects the spectrum near the level crossing points corresponding to magnetic ﬂux Φ = (2n + 1)Φ0 /2 (Fig. 4.38). above that only the terms ∼

(+) T1 j

It was noticed

contribute to the off-diagonal matrix elements

288

8 TUNNELING THROUGH MOVING NANOOBJECTS

1

1

1

b

a 0

2

c

0

3

4

5

Fig. 8.3 Left column: Adiabatic temporal variation of Kondo tunneling characteristics in DQD with moving side dot. Left panel: TK as a function of DST . Middle column: time-dependent TK (t) corresponding to the evolution of TK (ΔST ) in the left column. The intervals over which the evolution is followed are shown by straight lines in the left column. Right column: Evolution of ZBA in the tunneling conductance .

δ Δ± = E+ |δ Hˆ d (t)|E− . This means that we may repeat the procedure used for derivation of the Hamiltonian (7.36) for the case of coherent perturbation given by the time dependent modulation of the level position ε1 (t) = ε1 + v˜g (t). This perturbation is converted in the interdot tunneling term V (+) δ Hˆ d (t) = − φ2 (t) ∑ T1 j 2 j by means of the canonical transformation (7.31). Here 1 φ2 (t) = 2 h¯

t

dt v˜g (t )

t

dt v˜g (t ).

(8.6)

8.1 Conversion of coherent charge input into the Kondo response

289

Referring to the solution of the Kondo problem in the static case (Section 4.4) with effective SW Hamiltonian (4.150), we note that this Hamiltonian near the crossing point E+ /E− (Fig. 4.38) reduces to

Hcotun =

∑

Γ Γ =E±

JΓ Γ SΓ Γ sΓ Γ + Jo T · τ .

(8.7)

Thus, near the orbital degeneracy point the symmetry crossover SU(2) → SU(4) → SU(2) takes place, and the Kondo temperature has sharp peaks at the degeneracy points (Fig. 4.34). The adiabatic time modulation of the energy gap Δ± (t) = [E+ (Φ ) − E− (Φ )]2 + V 2 φ22 (t) (8.8) around these points results in corresponding modulations of TK (t) and G(t).

Fig. 8.4 Left panel: Time dependent TK corresponding to the evolution of Δ± in TQD. Inset: TK as a function of Δ± . The intervals of this evolution are shown by straight lines. Right panel: Time dependent ZBA in the conductance in accordance with the evolution of TK .

Two types of adiabatic temporal oscillations TK (t) are presented in the left panel of Fig.8.4. If Δ± (t) oscillates between zero and some maximum due to ﬂuctuations of the gate voltage, TK (t) reaches its maximum at Ω t = 2π n. The tunneling conductance G(t) shown in the right panel evolves with the same periodicity. If Δ± (t) oscillates symmetrically around zero value, the period is halved, and G(t) reaches minima at Ω t = π n. In the general case when Δ ± varies between −Δa and +Δ b , the oscillations of G(t) are periodic but not monochromatic. Experimentally one may turn from one regime to another by changing the magnetic ﬁeld (shifting the value of Φ ) in the vicinity of the point Φ = 3Φ0 /2.

290

8 TUNNELING THROUGH MOVING NANOOBJECTS

We see that the situation with the SU(2) → SU(4) → SU(2) crossover is close to the case of the SO(4) → SO(5) → U(1) crossover discussed in the previous section from the point of view of the conversion of the charge signal into the Kondo response. However, due to the fact that the orbital degrees of freedom are involved in the formation of an effective exchange, the perturbation vg (t) directly affects the Kondo tunneling in the Δ -like geometry via the orbital degrees of freedom by means of time-dependent lowering of the point symmetry of the triangle induced by the gate voltage.

8.1.1 Single-electron shuttling The theory of electron shuttling in the presence of strong interaction effects (Coulomb blockade, Kondo screening, etc) is not worked out as yet, and we conﬁne ourselves by some introductory remarks and formulation of speciﬁc features of this problem in comparison with static single-electron tunneling.

W

Fig. 8.5 Periodic change of the tunneling amplitude for a nanoisland oscillating between two leads

0

T/2

T

As was mentioned already, in a shuttling regime the tunneling amplitude W (t) is a periodic function of time (Fig.8.5). From the mathematical point of view there is no principal difference between complete breaking and exponential weakening of W at t ∼ T /2, because the Fourier components Wn of periodic perturbation are relevant for tunneling current: W (t) =

∞

∑ Wn eiΩnt

(8.9)

n=0

so that the shape of the signal W (t) is reﬂected in the form of its Fourier transforms. The really signiﬁcant factor is the steepness of peaks centered around t = (2n + 1)T /4. The steeper are the slopes of these peaks, the stronger are non-adiabatic

8.1 Conversion of coherent charge input into the Kondo response

291

effects. If, e.g., the signal W (t) has Gaussian shape, then its Fourier transforms Wn are Gaussians as well, and the parameter with controls the non-adiabaticity is r = td /T (the larger is r the more adiabatic is the single-electron tunneling).

W

Fig. 8.6 Coordinate dependence of the tunneling amplitude for a nanoisland oscillating between two leads.

− d/2

0

d/2

In the adiabatic approximation the contact is smooth and a single electron tunneling act is not accompanied by inelastic excitations in the leads or in the shuttle. In this case the main effect of shuttling motion is the above mentioned Debye-Wallerlike enhancement of the Kondo temperature (5.33). The mean square displacement (5.34) may be huge in this case due to the speciﬁc shape of the time and coordinate dependence of the tunneling amplitude, because the maximum of the tunneling potential is shifted towards the banks (Fig.8.6). More reﬁned effects arise due to the dynamical symmetries implicit in the pendulum-like and the turnstile like shuttles (Fig.8.1). In case of even occupation of these shuttles this is the SO(4) symmetry of the singlet/triplet multiplet. Temporal variation of the tunneling parameters of interdot and dot-lead coupling, V (x,t) and W (x,t), respectively, results in corresponding oscillations of the effective SW exchange JSW (t) and hence, it causes an oscillating conductance according to the conversion mechanism discussed in Section 8.1 (see Fig.8.3 for possible shape of such oscillations). This effect may be interpreted as a manifestation of ”Kondo shuttling”, where the spin-ﬂip screening cloud is transmitted by the shuttle from one metallic bank to another [220]. In the general case the shuttling takes place in the non-adiabatic regime, and the higher terms in the Fourier series (8.9) should be taken into account. Here we are in a situation close to that discussed in Section 6.1, but the time-dependent tunneling integral Wl j in the Anderson Hamiltonian (6.4) has the form (8.9). Then the timedependent SW transformation (6.6), (6.7) should be performed with the matrix U

292

8 TUNNELING THROUGH MOVING NANOOBJECTS

containing the integrals Bk, l j (t) = −i namely, U =

∑

kσ ,l j

t

dt ei(εkl −ε j )(t −t)Vl j (t )

" # Bk, l j (t)X Λ λ ckl σ − H.c.

(8.10)

(8.11)

Here the indices Λ , λ are related to the electron eigenstates in the shuttle corresponding to two adjacent charge sectors N , N − 1. Then, insertion of the Fourier expansion (8.9) in the integral, leads to equation similar to (6.8). Eventually the satellites at ﬁnite bias eV ∼ mΩ arise in the tunneling conductance of a quantum shuttle. Detailed shape of the tunneling spectra depends on the type of the shuttle. Alternatively, the shuttling effect may be approached by means of Floquet representation. In this approach (see Section 2.7) the time dependent shuttling problem is turned into a time-independent one at the expense of extending the effective Hilbert space to an inﬁnite number of states within a quasienergy formalism. The corresponding secular equation (2.142) gives an inﬁnite number of equivalent solutions. In other words, any solution of the Floquet problem with quasienergy fα can be rewritten as fα + mΩ in close analogy with Bloch states in periodic potential, where the equivalent levels in the extended Brillouin zone are connected with reciprocal lattice vectors G. To summarize the principal features of few-electron shuttling and the role of dynamical symmetry in this transport mechanism, one should say that from the theoretical point of view tunneling through periodically moving nanoobject (quantum dot or nanoisland) is akin to tunneling transport in ac electromagnetic ﬁeld. Sideband resonances which accompany the fundamental resonance lines in the tunneling transparency may be seen as additional FBA in the tunneling spectra. If the shuttle possesses complex spectrum characterized by dynamical symmetries, conversion of electromechanical perturbation into the Kondo response is possible, and the manyparticle effects such as transport of ”Kondo cloud” from one bank to another (Kondo shuttling) arise provided the non-adiabatic effects in shuttling are not too strong.

8.2 Time-dependent Landau-Zener effect The idea of time-dependent dynamical symmetry is based on the adiabatical treatment of the dot level position ε (x,t) and the tunneling parameters W (x,t) in the

8.2 Time-dependent Landau-Zener effect

293

Anderson Hamiltonian [192]. We have seen in the preceding section that the main effect of dynamical symmetry in Kondo tunneling is the time-dependent level crossing characterized by some energy gap ΔΓ Γ which directly affects the nonuniversal

behavior of the Kondo temperature TK (ΔΓ Γ ) and the respective ZBA in the tunneling conductance. This problem is intimately related with the time-dependent Landau-Zener (LZ) tunneling effect, where level crossing may or may not result in transition of the system from one quantum state to another (adiabatic or diabatic evolution in time). The original theory of LZ effect [249, 441] (see also [270, 386]) deals with a twolevel system, where the time-dependent level crossing parameter δ12 = E1 − E2 is described by the linear law.

δ12 (t) = α (t − t0 ).

(8.12)

E1 Fig. 8.7 Bose particle in an asymmetric two-well trap.

E2

δ 12

The two levels cross at t = t0 . One may refer to a two-well trap (Fig.8.7) occupied by a single Bose particle as a physical realization of this model (see also Section 2.3). If one assumes that the level positions E1,2 are changed under external perturbation in accordance with the law (8.12) without change of the barrier transparency, the effective tunneling Hamiltonian written in pseudospin terms via Pauli matrices τ has the form (8.13) Hˆ tun = α t τz + gτx . In this simplest case one deals with time-dependent SU(2) symmetry (Fig. 8.8, left panel). In general case the LZ effect may be realized in a trap with n wells each containing a single level. These levels may intercross in various fashion (see papers [334, 381] for introductory discussion of time-dependent LZ effect and the book [293] for a general survey of the LZ problem in multilevel systems. In the general case the basic dynamical symmetry of this model is SU(n), provided the interlevel transitions take place in accordance with Fig. 8.9. The sequential level crossings

294

8 TUNNELING THROUGH MOVING NANOOBJECTS

1

2

E

E

2

1

t

1 1

2 2

2 2

1 1

t

Fig. 8.8 Left panel: Landau-Zener level crossing for spinless particle in a two-well trap. Right panel: Landau-Zener level crossing for fermion with spin 1/2 in external magnetic ﬁeld.

are described in terms of SU(2) subgroups of SU(n) with corresponding pairwise permutations of level indices.

E

t Fig. 8.9 Multiple level crossing resulting in time dependent dynamical symmetry SU(n).

In many physical realizations of multilevel LZ scheme some interlevel transitions are forbidden due to selection rules, so that the dynamical symmetry is determined by certain combinations of avoided and allowed level crossings. The simplest example of such combination is a spin/pseudosin LZ problem with basic SU(4) symmetry. If the tunneling particle is an electron or an atom in an optical trap with half-integer spin obeying Fermi statistics, then the basic symmetry of a two-level system is SU(4). However, since the electron tunnels through the interwell barrier without spin ﬂip, the basic matrix representation of this group reduces to (2.44) and the time-dependent dynamical symmetry is SU(2) × SU(2) (Fig.8.8, right panel). In the presence of spin-orbit interaction of, say, Rashba-type,

8.2 Time-dependent Landau-Zener effect

295

tunneling with spin ﬂips is allowed and the full SU(4) symmetry (2.43) is restored. Returning back to nanosystems, one should note that the time-dependent LZ effect with involvement of dynamical symmetries has been realized in complex quantum dots. An example of such realization may be found in Ref. [112]. In this experimental study LZ transitions take place in a DQD with N = 2. In this experiment repopulation of the two wells of the planar GaAs/GaAlAs DQD is controlled by time-dependent gate voltage vg (t), so that the systems transforms from the spin singlet state (1,1) (one electron in each well) into the spin singlet state (0,2). Excited triplet state in conﬁguration (1,1) is also involved in LZ level crossing in external magnetic ﬁeld, because of Zeeman splitting inducing level crossing ES , ET 1 [341]. As a result the full level crossing picture acquires the form shown in Fig. 8.10. Looking at this ﬁgure, one notes that the LZ transition takes place between the states |T 1(1,1) and |S+ , where |S+ = sin θ (t)|S(1,1) + cos θ (t)|S(0,2) .

(8.14)

E

E Fig. 8.10 Energy level crossing diagram for DQD with N = 2 in an external magnetic ﬁeld as a function of external parameter ε which controls the repopulation of the two wells forming the quantum dot from the triplet state (1,1),T ± , T 0 to the singlet states (0, 2), S and (1, 1), S (after [112]).

296

8 TUNNELING THROUGH MOVING NANOOBJECTS

The S/T transition in this system occurs due to hyperﬁne interaction with the nuclear spins. Thus we see that in the process of time evolution of the full SO(5) spectrum (two singlets and one triplet, see table (2.58) only the subgroup SU(2) of the dynamical symmetry group is realized in the LZ effect in the presence of Zeeman splitting in accordance with the mechanism offered in Ref. [341]. As a result the operator P1 (4.48) is involved in the LZ Hamiltonian, Hˆ tun = β P1x ,

(8.15)

where β = Bxnuc,L − Bxnuc,R is the difference of nuclear magnetic ﬁelds in the left and right wells of the DQD [112], P1x = (P1+ + P1−)/2. The LZ Hamiltonian (8.15) is linear in terms of the generators of the SU(2) group. This property can be used for exact evaluation of the dynamics of timedependent LZ tunneling in terms of dynamical correlation functions. For this sake one should ﬁnd the quantum mechanical time evolution operator U(t,t0 ) = |ψ (t)ψ (t0 )|, where |ψ (t) is the state vector of a system. This operator can be constructed, e.g., by means of the Wei – Norman [422] procedure. Below we describe this method using an example of the SU(2) group. Let us ﬁnd the time-evolution operator for the system with linear realization of the SU(2) symmetry described by the generic Hamiltonian ˆ = Θ (t) · S H(t)

(8.16)

where all 3 generators of the SU(2) group are present. The equation for the time ˆ evolution operator for the time dependent Hamiltonian H(t) i

dU(t,t0 ) ˆ = H(t)U(t,t 0) dt

(8.17)

with initial condition U(t0 ,t0 ) = 1 can be represented by a ﬁnite product of n exponential operators [422]. The index n is a dimension of the Lie algebra generated by ˆ H(t). In our case n = 3. The solution of (8.17) can be parametrized as − z + ˆ U(t,t 0 ) = exp{ f (t)S } exp{h(t)S } exp{g(t)S }

(8.18)

where S± = Sx ± iSy and Sz are three generators of the SU(2) group. Subsituting (8.18) in (8.16) and (8.17), one ﬁnds that the functions f (t), h(t) and g(t) satisfy the

8.2 Time-dependent Landau-Zener effect

297

system of differential equations [73, 120] ⎧ ⎨ i f˙ = 12 Θ + − Θ z f − 12 Θ − f 2 ih˙ = Θ z + Θ − f ⎩ ig˙ = 12 Θ − e−h

(8.19)

with initial condition f (t0 ) = h(t0 ) = g(t0 ) = 0. This system can be easily obtained ˆ ˆ Aˆ with the help of the Hausdorff expansion for the product of operators e−A Be [422]. The solution of the system of equations (8.19) depends on the solution of the ﬁrst of these equations, which is nothing but the Riccati differential equation. To solve the system, one has to introduce one more parametrization of (8.19) by three new functions κt± , ρt deﬁned as h(t) ˙ κt+ = − f (t), κt− = −ig(t)e ˙ , ρt = −ih(t).

(8.20)

Thus parametrization leads to the Kolokolov [232] representation of the timeevolution operator + − S

ˆ = e−κt U(t)

e

3 iSz 0t ρt1 dt1

t 3t + − −i t01 ρt2 dt2 exp iS κt1 e dt1

(8.21)

t0

The initial condition f (t0 ) = 0 transforms into κ + (t0 ) = 0. The initial conditions g(t0 ) = h(t0 ) = 0 are satisﬁed by construction of functions κt− and ρt . For the LZ problem the functions in Eqs. (8.19) are Θ x = 2Δ , Θ y = 0 and Θ z = α t, and initial conditions are usually deﬁned at t0 = −∞. Then the functions f (t), h(t) and g(t) are expressed in terms of parabolic cylinder (Weber) functions: f (t) =

αt i d ln pt − , h(t) = 2 ln pt , Δ dt 2Δ

g(t) = −iΔ

t dt −∞

pt2

and pt satisﬁes the Weber equation

1 z2 d2 p+ n− − p=0 dz2 2 4 with n =

iΔ 2 α

and z =

√ −iπ /4 α te .

Having this solution at hand, one may calculate the transition probability in a two level systems with states 1,2 (or spin ↑ and ↓) P1→2 (t) = |2|U(t)|1|2

(8.22)

Respectively, the probability of level crossing without tunneling is 1 − P1→2.

298

8 TUNNELING THROUGH MOVING NANOOBJECTS

T−

T−

T−

T−

T0

T0

S T0

S T0

S T+

S T+

T+

T+

t0 (a)

t

t0

t

(b)

Fig. 8.11 Energy level crossing diagram for DQD with N = 2 in a time dependent external magnetic ﬁeld h(t) = v(t − t0 ). (a) ΔT S > 0, (b) ΔT S = 0.

One may also imagine a situation, where the time-dependent parameter in the LZ problem is magnetic ﬁeld h(t) which changes sign at t = t0 . Depending on the sign and magnitude of Δ T S , various level-crossing schemes may be realized. From the point of view of dynamical symmetries the most interesting cases are those shown in Fig.8.11. Spin-ﬂip transitions are possible due to ﬂuctuations of the effective magnetic ﬁeld induced by the hyperﬁne interaction in QD. These ﬂuctuations result in spin ﬂips in the two wells of the DQD which cause S/T transitions. In case (a) ΔT S > 0 and only the transitions S ↔ T+ are involved. In case (b) with ΔT S ∼ 0 all four states of the SO(4) multiplet take part in the LZ transitions. Here the ”polar” states (0,2) and (2,0) are supposed to be suppressed by strong Coulomb blockade effect. Time-dependent LZ effect may be also observed in optical lattices with cold atoms captured in multiwell traps. The generic model for these systems is the same two-well trap (Fig. 8.7) occupied by two Bose particles. To make the notations symmetric let us put the reference energy in the middle between the left and right levels E1 and E2 and introduce the time-dependent potential (8.12) in such a way that the center of gravity of the TLS is ﬁxed. Then three states for occupation N = 2 in conﬁgurations (2,0), (1,1) and (0,2) are characterized by the energies E1 = δ12 (t) + U, E0 = 0, E2 = −δ12 (t) + U.

(8.23)

Here U is the hard-core repulsion energy for two bosons in the same well (the analog of Coulomb blockade and electron repulsion parameters in quantum dots and in the Hubbard model, respectively); δ12 (t) is deﬁned in Eq. (8.12). The time evolu-

8.2 Time-dependent Landau-Zener effect

299

tion of this spectrum is illustrated in Fig. 8.12. LZ transitions due to single particle tunneling between two wells are shown by dashed lines. At zero temperature two consecutive transitions (0, 2) → (1, 1) → (2, 0) occur at moments t∓ = t0 ∓ δ t.

(2,0)

(1,1) Fig. 8.12 Time-dependent level crossing for a doublewell trap with two spinless Bose particles (see text for further details).

(0,2)

t t− t0 t+

¯ of pseudospin 1, we inTreating the three states as three projections |1, |0, |1 troduce the operators Tz = X 11 − X 11 , T + = ¯¯

√

2(X 10 + X 01), T − = ¯

√ ¯ 2(X 01 + X 10 ),

Tz2 = X 11 + X 11 , (T + )2 = 2X 11, (T − )2 = 2X 11 ¯¯

¯

¯

(8.24)

which describe the spectrum and the interlevel transitions. Comparing these operators with the components of irreducible tensor (2.57), we see that the model of doubly occupied double-well trap may be mapped on the anisotropic spin 1 system with SU(3) symmetry. It is more convenient to use the Gell-Mann operators (2.56) for the diagonal terms and their combination from the triads (9.39) for the interlevel tunneling transitions. We derive these operators from Eqs. (2.56) reduced from SU(4) to SU(3). In the subspace ¯ 0) Φ¯ 3 = (1, 1,

(8.25)

1 X3 X8 + + √ , 3 2 2 3 X8 1 X3 + √ , = − 3 2 2 3 √ = 1 − 3X8 .

(8.26)

we have X 11 = ¯¯

X 11 X 00

300

8 TUNNELING THROUGH MOVING NANOOBJECTS

Then the effective LZ Hamiltonian acting in Φ¯ 3 reads 4U U · 1 + δ12(t) · Tz + · (Vz + Uz ) Hˆ aniso = 3 3 + g1 · (T+ + T−) + g2 · (V+ + V− ) + g3 · (U+ + U− ).

(8.27)

Here the deﬁnition (9.38) is used, 1 is the unit matrix in the space (8.25) U plays part of the axial pseudospin anisotropy parameter and δ12 (t) may be treated as effective time-dependent ”magnetic ﬁeld” splitting of the spectrum EΛ ; g1 , g2 , g3 are the tun¯ 1 ↔ 0, 1¯ ↔ 0, respectively. It is seen from neling constants for transitions 1 ↔ 1, (8.27) that the LZ tunneling transitions activate the dynamical SU(3) symmetry of a double well occupied by two bosons. The Wei-Norman approach described above, may be used for this system as well. One should represent the evolution operator U(t) including the generators of the SU(3) group in a parametrized form similar to (8.18) and ﬁnd the corresponding system of differential equations, which generalizes the system (8.19) of ordinary differential equations but includes vector Riccati equation. A multiwell level crossing system has been realized experimentally in optical traps for cold gases [55]. In this experiment the trap with occupation N = 4 was split into the two-well potential, and repopulation of the two wells was monitored by an adiabatic change of the shape of the double trap. The corresponding level crossing diagram shown in Fig. 8.13 demonstrates that the full dynamical symmetry of this system is SU(5). Calculation of transition probabilities (8.22) by means of the Wei-Norman method, of course, reproduces a direct solution of corresponding Schr¨odinger equation [441].

In the conventional LZ effect the measure of non-adiabatic transitions between crossing levels (the probability P1↔2 of tunneling from one diabatic state to another one is predetermined by the velocity 2α (8.12) in the Hamiltonian (8.13), π g2 . (8.28) P1↔2 = 1 − exp − α Depending on the magnitude of the parameter ν = 2π g2 /α one discriminates between the ”adiabatic” and ”sudden” regimes. In the adiabatic regime the transition P1↔2 (t) occurs monotonously with characteristic transient time τLZ,ad ∼ g/2α . In a sudden regime of fast level crossing the transition to the ﬁnal state |2 is accompanied by weak oscillations of P1↔2 (t) with characteristic oscillation time √ τLZ,sud ∼ α .

8.2 Time-dependent Landau-Zener effect

-3

-2

301

-1

0

1

2

3

Bias Fig. 8.13 Energy level crossing diagram for a double well optical trap with N = 4 Bose atoms as a function of the asymmetry parameter (potential bias) which controls the repopulation of the two wells (after [55]). Both energy and bias are given in units of U.

An external noise may modify the generic picture of LZ transitions, because in this regime the temporal characteristics of the noise signal should be compared with τLZ [198]. Such noise may involve ﬂuctuations of both the level positions and the tunnel barrier parameters. We are interested mainly in the limit of slow noise having in mind establishing a connection with dynamical symmetry related dephasing and decoherence effects discussed in Section 7.2.

Fig. 8.14 Landau-Zener transitions in a two-level system with time-dependent interlevel distance ε21 (t) = α t in the presence of transverse ﬂuctuations.

2

1

1

2 0

t

Let us consider the Landau-Zener effect in the presence of slow transverse noise which induces the inter-level transitions on the background of variation of the interlevel distance that is linear in time (Fig. 8.14). The Hamiltonian of this problem is

302

8 TUNNELING THROUGH MOVING NANOOBJECTS

H(t) = α t σz + ∑ fi (t)σi , i = x, y

(8.29)

i

The noise correlation function is given by fi (t) = 0, fi (t) f j (t ) = J 2 δi j exp{−γ |t − t |}

(8.30)

[cf. Eq. (7.46)]. In order to calculate the transition probability one has to solve the density matrix equation i

dρ (t) = [H(t), ρ (t)] dt

(8.31)

which after some algebra is transformed to

ρ˙ (t) = −4

t −∞

cos[α (t 2 − t12 )] f+ (t) f− (t1 )ρ (t1 )dt1 .

(8.32)

where ρ = ρ11 − ρ22, and the indices 1,2 denote two crossing levels. The solution of this equation is to be averaged over all possible realizations of the two-level system (ensemble average). The result of this averaging is different for the two limits of fast and slow noise. If the noise is fast, τnoise ∼ 1/γ τLZ , one can average equation (8.32) directly [333] and decouple the product f+ (t) f− (t1 )ρ (t1 ) → f+ (t) f− (t1 )ρ (t1 ). Solving Eq. (8.31) in this case, one eventually comes to a master equation for the average ρ which gives the conventional LZ equation for the transition probability in which the quantity g2 should be replaced by the equal time noise correlation function f+ (t) f− (t). This decoupling procedure is not applicable in the limit of slow noise, when γ → 0, so that the density matrix equation cannot be reduced to the master equation. Instead, one has to solve the equation (8.32) and then perform ensemble average. In order to solve this equation, we ﬁrst integrate both sides of it over time, implementing the condition ρ (−∞) = 1. As a result, the integral equation for the density matrix acquires the form

ρ (t) = 1 − 4

t −∞

dt1

t1 −∞

cos[α (t12 − t22 )] f+ (t1 ) f− (t2 )ρ (t2 )dt2 .

(8.33)

This equation can be solved iteratively, assuming noise to be Gaussian. Like in the Keldysh model considered in Section 7.2, all pairings in each term of iteration series contribute to the integral on an equal footing, and calculation of the average value of the expanded integral (8.33) is reduced to the calculation of the combinatorial number of pairings at each iteration step in a vector Keldysh

8.2 Time-dependent Landau-Zener effect

303

model [216]. Therefore, the nth step iteration solution of a slow noise driven LZ problem differs from the same order solution of LZ problem without noise only by these combinatorial factors. In case of transverse XY-model (8.33) these factors are Bn = 2n n!. In the scalar model with “longitudinal” noise affecting the term ∼ σz in the Hamiltonian (8.29), these factors are An = n!!. Using this simplifying circumstances, one may ﬁrst solve iteratively the LZ problem without noise, and then append combinatorial factors in each order of the iteration procedure. The iterative solution reads

2 1 π J2 π J2 ρ (t) = 1 + 2 − F(t) + F(t) + . . . + − G(t, J 2 /α ) α 2! α

π J2 2 F(t) − G(t, J /α ) . (8.34) = 1 + 2 exp − α Here the function F(t) is deﬁned as F(t) =

1 2

2 2 1 1 , + C(t) + + S(t) 2 2

C(t) and S(t) are Fresnel integrals 2 t πt dt , C(t) = cos 2 0

S(t) =

t 0

(8.35)

π t 2 dt , sin 2

and G(t, J 2 /α ) includes all corrections to the exponential solution. Then the transition probability at a ﬁnite time is determined as 1 [1 − ρ (t)] 2 π J2 = 1 − exp − F(t) + G(t, J 2 /α ) α

PLZ (t) =

(8.36)

The correction function G(t, J 2 /α ) vanished at t → ∞, so that asymptotically one comes back to the conventional LZ formula (8.28), π J2 . PLZ = 1 − exp − α

(8.37)

In order to ﬁnd the correction function G(t, J 2 /α ) we use the exact expression for non-stochastic LZ probability PLZ [441], PLZ (t) ≡ |AD−n−1(−iz)|2

304

8 TUNNELING THROUGH MOVING NANOOBJECTS

where D−n−1 (−iz) is the Weber’s parabolic cylinder function, n = ig2 /α and A is a normalization factor. Then π J 2 F(t) , G(t, J 2 /α ) = PLZ (t) − 1 − exp − α It is convenient to rewrite PLZ (t) as π J2 PLZ (t) = 1 − exp − [F(t) + lnW (t)] α

(8.38)

(8.39)

with

2 α πJ 2 F(t) . lnW (t) = − 2 ln 1 − G(t, J /α ) exp πJ α

(8.40)

Then rewriting Eq. (8.39) in the form of perturbative expansion over the parameter π J 2 /α and attaching the combinatorial coefﬁcients, we in fact repeat the procedure described in Section 7.2.1 devoted to the Keldysh model and eventually come to similar result (at t → ∞ where all transient oscillations of LZ tunneling probability die): the probability PLZ in the presence of slow noise is averaged with a two-dimensional Gaussian distribution:

∞ ∞ 1 PLZ (x) PLZ (y) exp{−(x2 + y2 )/2J 2 }dxdy 2 2π J −∞ −∞ −1

2π J 2 = 1− 1+ , α

xy = Pnoise

(8.41)

where PLZ (Q) is the transition probability for the LZ transition with off-diagonal coupling constant given by Q = x, y, PLZ (Q) = 1 − exp{−π Q2/α }.

(8.42)

Similar averaging procedure for the scalar noise gives

∞ 1 PLZ (x) exp(−x2 /2J 2 )dx 2π J −∞

−1/2 2π J 2 , = 1− 1+ α

x = √ Pnoise

(8.43)

The difference between Eqs. (8.41) and (8.43) is a consequence of the effective two-dimensional character of noise ﬂuctuation spectrum in the former case and its one-dimensionality in the latter case. Dependence of the transition probability (8.43) on the amplitude of noise J is shown in Fig.8.15 (ν = 2π J 2 /α ).

8.2 Time-dependent Landau-Zener effect

305

Fig. 8.15 Landau-Zener transitions in a two-level system. Dependence of the transition propability on the amplitude of noise J and the asymmetry parameter Δ ; k = πΔ 2 /α ν = 2π J 2 /α .

This expansion method allows one to describe the transition probability |1 → |2 at ﬁnite time t. For example in case of scalar noise x (t) = 1 − Pnoise

1+

2π J 2 α

1 . F(t) + lnW (t)

(8.44)

In the limit t → ∞ F(∞) = 1 and lnW (∞) = 0 and we return to Eq. (8.43). Fig. 8.16 illustrates the noise induced time-dependent LZ effect in the sudden and adiabatic approximations. Up to this point we dealt with LZ transitions induced by the slow noise. This approach cam be generalized straightforwardly to the case when LZ transitions are induced by an external magnetic ﬁeld (in the case of spin systems) or by an effective “ﬁeld” associated with the ﬁnite transparency of the interwell barrier in the double-well potential in cold gases [see Eqs. (7.77),(7.78)]. To describe this effect one should take into account the non-zero mean value of the stochastic ﬁeld fi (t) in Eq. (8.29), fi (t) = Δ + fi (t) (i = x, y),

(8.45)

306

8 TUNNELING THROUGH MOVING NANOOBJECTS

Fig. 8.16 Scalar (a), (b) and vector (c), (d) noise induced time-dependent Landau-Zener transitions in the sudden and adiabatic approximations.

where the noise correlation function for fi is given by Eq. 8.30. Then, averaging over slow scalar noise ﬂuctuation, results in the following transition probability

x Pnoise = √

1 2π J

∞

[1 − exp{−π x2/α }]exp[−(x − Δ )2 /2J 2 ]dx 1 1 πΔ 2 1 . = 1− exp − 2 α 1 + 2π J 2 1 + 2παJ α −∞

(8.46)

In the limit α /2π J 2 1

x Pnoise

= 1−

α Δ2 exp − 2 . 2π J 2 2J

(8.47)

so that the argument of the exponent does not depend on the velocity. Similarly, the slow noise with constant magnetic ﬁeld in the xy-plane results in

8.2 Time-dependent Landau-Zener effect xy Pnoise = 1−

307

1 1 + 2παJ

2

1 2πΔ 2 , exp − α 1 + 2π J 2

(8.48)

α

and in the limit α /2π J 2 1 xy Pnoise = 1−

α Δ2 exp − . 2π J 2 J2

(8.49)

The theory of noise induced LZ effect may be extended to the multilevel LZ problems (Figs. 8.9, 8.11, 8.13), where more complicated patterns of transient oscillations in the tunneling probability of transition from initial to ﬁnal state of a nanosystem with nontrivial dynamical symmetry is expected.

Chapter 9

MATHEMATICAL INSTRUMENTATION

In this supplementary chapter the basic ingredients of the theory of matrix representations for SU(n) and SO(n) groups is collected and the main bosonization and fermionization procedures for the generators of these groups are described in brief.

9.1 SU (2) group for arbitrary spin The SU(2) group is a group of unimodular complex matrices of 2nd rank with unit determinant. These matrices U may be represented as a b U= −b∗ a∗

(9.1)

with |a|2 + |b|2 = 1. Here a and b are Cayley – Klein parameters or ﬁnite rotation matrices for an angular momentum 1/2 1/2

1/2

a = D1/2,1/2 (α , β , γ ), b = D1/2,−1/2 (α , β , γ ),

(9.2)

α , β , γ are three Euler angles parametrizing this rotation. The corresponding Lie algebra su(2) is given by Pauli matrices τ = (τ1 , τ2 , τ3 ), and the unit matrix is denoted as τ0 : 01 0 −i 1 0 10 τ1 = , τ2 = , τ3 = , τ0 = . (9.3) 10 i 0 0 −1 01 The Pauli matrices obey commutation and anticommutation relations [τ j , τk ] = 2iε jkl τl , {τ j , τk } = 2δ jk τ0 . K. Kikoin et al., Dynamical Symmetries for Nanostructures: Implicit Symmetries in Single-Electron Transport Through Real and Artificial Molecules, DOI 10.1007/978-3-211-99724-6_9, © 2012 Springer-Verlag/Wien

(9.4) 309

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9 MATHEMATICAL INSTRUMENTATION

The ladder operators τ ± = (τ1 ± iτ2 )/2 are the matrices 01 00 + − , τ = τ = 00 10

(9.5)

The commutation relations for this representation are [τ + , τ − ] = τ z , [τz , τ ± , ] = ±2τ ± .

(9.6)

SU(2) is a symmetry group of the Schr¨odinger equation describing an electron with two component spinor wave function ψˆ , which transforms at any rotation in R3 as

ψˆ = Uψˆ , ψˆ =

ψ1 ψ2

(9.7)

with indices 1,2 corresponding to up and down spin projections. The spin 1/2 matrices are deﬁned as σ j = 12 τ j Although the Lie algebra su(2) is isomorphous to so(3), the group SU(2) is a double cover for the group SO(3). The Lie group SU(2) describes the symmetry properties of a particle with arbitrary spin, both half-integer and integer. From the viewpoint of its symmetry the wave function of a particles with spin S = n/2 is equivalent to that of n particles with spin s = 1/2. To represent these wavefunctions, the symmetric tensor of n-th rank ψ λ ...μζ ...η realizing the irreducible representations of rotation group are introS+M

S−M

4 56 7 4 56 7 duced. Here the indices λ . . . μ and ζ . . . η stand for up and down spin projection, respectively. The wave function 8

ΨSM =

(2S)! ψ λ ...μζ ...η (S + M)!(S − M)!

(9.8)

describes the quantum state with total spin S and projection M. Then the operators Sz , S± = Sx ± Sy acting in the space (9.8) possess the commutation relations similar to those in Eq. (9.6) with appropriate normalization factors. The spin matrices for S = 1 and S = 3/2 are given below. S = 1: ⎛ ⎛ ⎛ ⎞ ⎞ ⎞ 010 0 −1 0 10 0 1 ⎝ i 1 0 1 ⎠ , Sy = √ ⎝ 1 0 −1 ⎠ , Sz = ⎝ 0 0 0 ⎠ ; Sx = √ 2 010 2 0 1 0 0 0 −1

(9.9)

9.2 Kinematical constraints for systems with SO(n) and SU(n) symmetries

311

S = 3/2: ⎛ ⎞ 0 1 0√ 0 √ ⎟ 3⎜ ⎜ 1 0√ 2/ 3 0 ⎟ , Sx = ⎝ 0 2/ 3 0 1 ⎠ 2 0 0 1 0

⎛ ⎞ 0 −1 0√ 0 √ ⎟ i 3⎜ ⎜ 1 0√ −2/ 3 0 ⎟ Sy = ⎝ 0 1⎠ 0 2/ 3 2 0 0 1 0 ⎛ ⎞ 30 0 0 ⎟ 1⎜ ⎜0 1 0 0 ⎟ Sz = (9.10) ⎝ 2 0 0 −1 0 ⎠ 0 0 0 −3

These matrices may be represented via Hubbard operators by means of the expansion (2.8). For examples the spin operators for S = 1 have the form Sz = (X 11 − X 11 ), S+ = ¯¯

√

2(X 10 + X 01 ), S− = ¯

√

¯

2(X 01 + X 10 ).

(9.11)

9.2 Kinematical constraints for systems with SO(n) and SU (n) symmetries 9.2.1 SO(4) group The group SO(4) of rotations on a 4D sphere is the ﬁrst semisimple group in the set of SO(n) groups which represents the dynamical symmetry of some quantummechanical objects. Its generators perform inﬁnitesimal rotations around the axis j or in the plane kl: −iMkl = xk

∂ ∂ − xl ( j, k, l = 1, 2, 3, 0). ∂ xl ∂ xk

(9.12)

Let us denote them as ⎛

⎞ ⎛ ⎞ 0 M12 M13 M10 0 L3 −L2 −K1 ⎜ 0 M23 M20 ⎟ ⎜ 0 L1 −K2 ⎟ ⎜ ⎟=⎜ ⎟ ⎝ 0 M30 ⎠ ⎝ 0 −K3 ⎠ 0 0

(9.13)

Three operators L j of angular momentum on a 3D sphere obey the algebra o(3) and generate a subgroup SO(3) ⊂ SO(4). Three more generators of SO(4) group are K j ≡ M0 j ( j = 1, 2, 3). Six operators L j , K j form the so(4) algebra

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9 MATHEMATICAL INSTRUMENTATION

[L j , Lk ] = iε jkl Ll , [K j , Kk ] = iεik j Ll , [K j , Lk ] = iε jkl Kl ,

(9.14)

[M jk , Mlm ] = −(δ jl Mkm + δkm M jl − δ jmMkl − δkl M jm ).

(9.15)

or

A linear transformation J1 j =

L j + Kj L j − Kj , J2 j = 2 2

(9.16)

converts (9.14) into another basis with algebra [J1 j , J1k ] = iε jkl J1l , [J2 j , J2k ] = iε jkl J2l , [J1 j , J2k ] = 0

(9.17)

The Casimir operators for the SO(4) group are L 2 + K 2 = 3, L · K = 0

(9.18)

J12 + J22 = 3/2, J12 − J22 = 0

(9.19)

or

This means that together with the usual kinematical constraint which stems from the trace completeness for the projection operators ∑λ Xλ λ = 1, additional constraint is imposed, which demands the orthogonality of the operators L and K or equality of the eigenstates j1 ( j1 + 1) = j2 ( j2 + 1) of the operators J12 and J22 . The group SO(4) is semisimple because the so(3) algebras spanned by the operators J1 and J2 are ideals of so(4). In other terms, there exists a local isomorphism SO(4) = SU(2) × SU(2),

(9.20)

Generally, the enveloping algebra so(4) is a direct sum of the two subalgebras, so(4) = so(3) + so(3). The coordinates xi on a 4D sphere with radius R are characterized by 3 angles x1 = R sin α sin ϑ cos ϕ , x2 = R sin α sin ϑ sin ϑ , x0 = R cos α

(9.21)

The rotation operators L and K in these spherical coordinates have the form ∂ ∂ ∂ + i cot ϑ , (9.22) , L3 = −i L± = L1 ± iL2 = e±ϕ ± ∂ϑ ∂ϕ ∂ϕ

1 ∂ ∂ ∂ K ± = K1 ± iK2 = e±iϕ cot α ∓ + i sin ϑ + i cos ϑ sin ϑ ∂ ϕ ∂ϑ ∂α

9.2 Kinematical constraints for systems with SO(n) and SU(n) symmetries

K3 = i cos ϑ

∂ ∂ − i sin ϑ cot α . ∂α ∂ϑ

313

(9.23)

The subalgebras so(3) for the operators J1,2 may be realized in spherical coordinates x1 = R cos β cos φ , x2 = R cos β sin φ , x3 = R sin β cos θ , x0 = R sin β sin θ where these operators have the form 1 ±i(φ +θ ) ∂ ∂ ∂ ± − i cot β + i tan β ∓ , J1 = e 2 ∂β ∂θ ∂φ ∂ ∂ i J1z = − + 2 ∂θ ∂φ 1 ∂ ∂ ∂ J2± = e±i(φ −θ ) ∓ + i cot β + i tan β , 2 ∂β ∂θ ∂φ ∂ ∂ i − J2z = 2 ∂θ ∂φ

(9.24)

(9.25)

(9.26)

9.2.2 Noncompact groups SO(p, n − p) The pseudorotation groups SO(p, n − p) are formed by transformations which leave the form x21 + x22 + . . . x2p − x2p+1 − . . . − x2n invariant. These Lie groups are called noncompact because the parameter space is unbounded. The simplest of these groups is SO(1, 1) sometimes denoted as SH(2): the group of 2D pseudorotations on a hyperbola which conserves invariant ρ 2 = x21 − x22 . Such transformations are used in various physical applications: superconductivity (Bogolyubov transformation), antiferromagnetism (Holstein-Primakoff transformations) etc. From the point of view of dynamical symmetries, the groups of pseudorotations with n 3 are of special interest. In particular, we consider the group SO(3, 1) of pseudorotations in a 4D space (x1 , x2 , x3 , x0 ) which leaves invariant the form r 2 − x20 , where r is a 3D vector with components x j , j = 1, 2, 3 (metrics with signature {+, +, +, −}). If the coordinate x0 = ct is chosen where t is time and c is the speed of light, then the SO(3, 1) group is called the Lorentz group describing the spacetime symmetry of our universe. The algebra so(3, 1) of group generators differs from that of so(4) due to different signature. The operators L j with j, k, l = 1, 2, 3 are the same as in Eq. (9.30), whereas

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9 MATHEMATICAL INSTRUMENTATION

the operators of pseudorotations K¯ j = M¯ o j are deﬁned as ∂ ∂ ¯ + xj ( j, k, l = 1, 2, 3). K j = i x0 ∂xj ∂ x0

(9.27)

Respectively, the commutation relations differ from those in Eq. (9.14) by the sign in the last equation [L j , Lk ] = iε jkl Ll , [K¯ j , K¯k ] = iεik j Ll , [K¯ j , Lk ] = −iε jkl K¯ l ,

(9.28)

and the Casimir operators are L 2 − K 2 = 3, L · K = 0

(9.29)

9.2.3 Groups of conformal transformations Groups of pseudorotations SO(p, n− p) are isomorphous to conformal groups which consist of transformations xj = a jk xk + b j x = λ x x x = 2 +d x 2 x

(9.30)

The ﬁrst of these transformations is the usual linear rotation-displacement operation in n-dimensional space, the second one is the scaling (dilation) transformation, and the third one is a special conformal transformation. The conformal group SO(4, 2) is related to the symmetry of the Maxwell equations in classical electrodynamics and the massless Klein – Gordon equation in quantum electrodynamics. The special conformal transformation in this group has a form xj =

x j + c j x2 1 + 2cx + c2x2

(9.31)

The generators of these transformations may be combined in 15 antisymmetric operators Lμν (μ , ν = 0, 1, 2, . . . , 5) which obey the standard commutation relations for o(p, n − p) algebras [Lμν , Lρσ ] = i(δμρ Lνσ + δνσ Lμρ − δμσ Lνρ − δνρ Lμσ ).

(9.32)

9.2 Kinematical constraints for systems with SO(n) and SU(n) symmetries

315

One may consider the conformal group SO(4, 2) as a group of pseudorotations in an effective 6D space with a signature g μ μ = 1 (μ = 1, . . . 4), g55 = g00 = −1. The 4D rotations form the subgroup SO(4) of this group. Another subgroup of SO(4, 2) is the de Sitter group SO(4, 1) with the subalgebra consisting of the operators L μν (μ , ν = 1, . . . 5). There are three Casimir operators for the group SO(4, 2):

∑ Lμν Lν μ = 6, ∑ μν

κλ μνρσ

εκλ μνρσ Lκλ Lμν Lρσ = 0,

∑

κλ μν

Lκλ Lλ μ Lμν Lνκ = −12.

(9.33) (see [310] for general discussion of Casimir invariants and irreducible tensor operators in Lie algebras).

9.2.4 From SU (2) to SU (n) The special unitary group SU(n) is a group of unimodular n × n matrices. The corresponding Lie algebra su(n) is formed by n2 − 1 generators, which obey the commutation relations [Oi j , Okl , ] = δ jk Oil − δli Ok j ,

(9.34)

where i, j, k, l = 1 . . . n. The simplest non-trivial group in this family is the SU(2) group described in Section 9.1 The next in order is the group SU(3). Its dimension is 32 − 1 = 8. Its generators g j = λ j /2 are given by eight Gell-Mann matrices, 3 × 3 analogs of Pauli matrices. ⎞ ⎞ ⎞ ⎛ ⎛ ⎛ 010 0 −i 0 1 0 0 λ1 = ⎝ 1 0 0 ⎠ , λ2 = ⎝ i 0 0 ⎠ , λ3 = ⎝ 0 −1 0 ⎠ , 000 0 0 0 0 0 0 ⎞ ⎞ ⎞ ⎛ ⎛ ⎛ 001 0 0 −i 000 λ4 = ⎝ 0 0 0 ⎠ , λ5 = ⎝ 0 0 0 ⎠ , λ6 = ⎝ 0 0 1 ⎠ , 100 i 0 0 010 ⎞ ⎛ ⎞ ⎛ 10 0 00 0 1 (9.35) λ7 = ⎝ 0 0 −i ⎠ , λ8 = √ ⎝ 0 1 0 ⎠ . 3 0 0 −2 0 i 0 The corresponding su(3) algebra is deﬁned by the commutation relations [g j , gk ] = i ∑ f jkl gl l

(9.36)

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9 MATHEMATICAL INSTRUMENTATION

with structure factors given by the expression f jkl =

1 Tr[[λ j , λk ], λl ]. 4i

(9.37)

For any 2D subspace of the 3D Fock space one may construct a subgroup SU(2) of the group SU(3). One of such subgroups is determined by the matri√ ces {λ1 , λ2 , λ3 }. Two other sets are {λ4 , λ5 , (λ3 + 3λ8 )/2} and {λ6 , λ7 , (−λ3 + √ 3λ8 )/2}). These subgroups may be extended to SU(2) × U(1) by adding an operator performing the corresponding 3D rotation. In case of subgroup {λ1 , λ2 , λ3 } this extension is accomplished by adding the matrix λ8 to this subset. The ladder operators for three SU(2) subgroups are T± = (λ1 ± iλ2 )/2, Tz = λ3

√ U± = (λ6 ± iλ7 )/2, Uz = (−λ3 + 3λ8 )/2 √ V± = (λ4 ± iλ5 )/2, Vz = (λ3 + 3λ8 )/2.

(9.38)

Their matrix form is ⎛

⎞ ⎞ ⎞ ⎛ ⎛ 010 000 1 0 0 T+ = ⎝ 0 0 0 ⎠ , T− = ⎝ 1 0 0 ⎠ , Tz = ⎝ 0 −1 0 ⎠ , 000 000 0 0 0 ⎞ ⎞ ⎞ ⎛ ⎛ ⎛ 000 000 00 0 U+ = ⎝ 0 0 1 ⎠ , U− = ⎝ 0 0 0 ⎠ , Uz = ⎝ 0 1 0 ⎠ , 000 010 0 0 −1 ⎞ ⎞ ⎞ ⎛ ⎛ ⎛ 001 000 10 0 V+ = ⎝ 0 0 0 ⎠ , V− = ⎝ 0 0 0 ⎠ , Vz = ⎝ 0 0 0 ⎠ . 000 100 0 0 −1 (9.39)

Each of three vectors T, U, V form the su(2) algebra given by the commutation relations (9.6). Commutation relations between operators belonging to different subsets can be derived from these basic deﬁnitions. Only eight matrices from this set should be chosen as the generators of the su(3) algebra operating in a 3D Fock space. Physical realizations of the SU(3) group are various dynamical symmetries of 3-level systems [the ﬁrst of these realizations were proposed for the (u, d, s) quark family (see [92, 127] for details). The dynamical symmetry of this supermultiplet may be revealed by means of irreducible tensor operators in several ways. First, one may combine the Gell-Mann matrices in two irreducible vectors T, V and two scalars U+ , U− . Second, one may represent them as an irreducible vector T and an

9.2 Kinematical constraints for systems with SO(n) and SU(n) symmetries

ˆ of the 2-nd rank with components irreducible tensor Q ⎞ ⎛ ⎛ 01 0 0 0 1 1 ˆ (−1) = (U− − T− ) = ⎝ −1 0 ˆ (+1) = (T+ − U+ ) = ⎝ 0 0 −1 ⎠ , Q Q 2 2 00 0 0 1 ⎞ ⎛ 1 0 0 ˆ (0) = 1 ⎝ 0 −2 0 ⎠ = 2 (Tz − Uz ). ˆ (+2) = V+ , Q(−2) = V− , Q Q 3 3 0 0 1

317

⎞ 0 0 ⎠, 0 (9.40)

Dynamical symmetries of the four-level multiplet are given by the group SU(4). 15 generators of this group obey the su(4) algebra. The Gell-Mann matrices of the 4th rank are:

⎞ ⎞ ⎞ ⎛ ⎛ 0100 0 −i 0 0 1 0 00 ⎜1 0 0 0⎟ ⎜ i 0 0 0⎟ ⎜ 0 −1 0 0 ⎟ ⎟ ⎟ ⎟ ⎜ ⎜ λ1 = ⎜ ⎝ 0 0 0 0 ⎠ , λ2 = ⎝ 0 0 0 0 ⎠ , λ3 = ⎝ 0 0 0 0 ⎠ , 0000 0 0 00 0 0 00 ⎞ ⎞ ⎞ ⎛ ⎛ ⎛ 0010 0 0 −i 0 0000 ⎜0 0 0 0⎟ ⎜0 0 0 0⎟ ⎜0 0 1 0⎟ ⎟ ⎟ ⎟ ⎜ ⎜ (9.41) λ4 = ⎜ ⎝ 1 0 0 0 ⎠ λ5 = ⎝ i 0 0 0 ⎠ , λ6 = ⎝ 0 1 0 0 ⎠ , 0000 00 0 0 0000 ⎞ ⎛ ⎞ ⎞ ⎛ ⎛ 10 0 0 00 0 0 0001 ⎟ ⎜ 0 0 −i 0 ⎟ ⎜0 0 0 0⎟ 1 ⎜ ⎜0 1 0 0⎟ ⎟ ⎟ ⎜ λ7 = ⎜ ⎝ 0 i 0 0 ⎠ , λ8 = √3 ⎝ 0 0 −2 0 ⎠ , λ9 = ⎝ 0 0 0 0 ⎠ , 00 0 0 00 0 0 1000 ⎞ ⎞ ⎞ ⎛ ⎛ ⎛ 0 0 0 −i 0000 000 0 ⎟ ⎟ ⎜0 0 0 0 ⎟ ⎜ ⎜ ⎟ , λ11 = ⎜ 0 0 0 1 ⎟ , λ12 = ⎜ 0 0 0 −i ⎟ , λ10 = ⎜ ⎝0 0 0 0 ⎠ ⎝0 0 0 0⎠ ⎝0 0 0 0 ⎠ i 00 0 0100 0 i 0 0 ⎞ ⎛ ⎞ ⎞ ⎛ ⎛ 100 0 0000 000 0 ⎟ ⎜ ⎟ ⎜0 0 0 0⎟ ⎜ ⎟ , λ14 = ⎜ 0 0 0 0 ⎟ , λ15 = √1 ⎜ 0 1 0 0 ⎟ , λ13 = ⎜ ⎝0 0 0 1⎠ ⎝ 0 0 0 −i ⎠ 6 ⎝0 0 1 0 ⎠ 0 0 0 −3 0010 00 i 0 ⎛

The ﬁrst eight matrices contain the familiar 3-rd rank Gell-Mann operators as submatrices, and the operators λ9 − λ15 generate transitions between the triplet and the fourth level. Three more SU(2) subgroups characterized by the vectors W, Y, Z together with those given by T, U, V (9.39) form six linearly dependent triads. The generators of these subgroups are

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9 MATHEMATICAL INSTRUMENTATION

1 1 1 4 W = (λ9 ± iλ10) , Wz = λ3 + √ λ8 + √ λ15 2 2 3 6 1 1 1 4 ± −λ3 + √ λ8 + √ λ15 Y = (λ11 ± iλ12) , Yz = 2 2 3 6 √ 1 1 Z± = (λ13 ± iλ14) , Zz = √ −λ8 + 2λ15 . 2 3 ±

(9.42)

The commutation relations for the irreducible operators for the SU(4) group interlace the states belonging to different triads. Namely, the operators O belonging to the same subgroup (triad) obey the standard SU(2) commutation relations [Oz , O± ] = ±2O± , [O+ , O− ] = Oz .

(9.43)

The rest non-zero commutation relations are [U± , V∓ ] = ±T∓, [U± , Vz ] = ∓U± , [Uz , V± ] = ±V ± , [V± , T∓ ] = ∓U± , [V± , Tz ] = ∓V± , [Vz , T± ] = ±T± [U± , T± ] = ∓V± , [U± , Tz ] = ±U± , [Uz , T± ] = ∓T± [W± , Y∓ ] = ±T± , [W± , Yz ] = ∓W± , [Yz , W± ] = ±W ± , [W± , Z∓ ] = ±V± , [W± , Zz ] = ∓W± , [Wz , Z± ] = ±Z± [Y± , Z∓ ] = ±U± , [Y± , Zz ] = ∓Y± , [Yz , Z± ] = ±Z± [W± , T∓ ] = ∓Y± , [W± , Tz ] = ∓W± , [Wz , T± ] = ±T± , [V± , W∓ ] = ∓Z∓ , [V± , Wz ] = ∓V± , [Vz , W± ] = ±W± , [V± , Z± ] = ±W± , [V± , Zz ] = ±V± , [Vz , Z± ] = ∓Z± , [U± , Z± ] = ±Y± , [U± , Zz ] = ±U± , [Uz , Z± ] = ∓Z± , [Y± , T± ] = ±W± , [Y± , Tz ] = ±Y± , [Yz , T± ] = ∓T± , [Y± , U∓ ] = ∓Z± , [Y± , Uz ] = ∓Y± , [Yz , U± ] = ±U± .

(9.44)

Among various physical applications of the SU(4) group the most important is the (spin, pseudospin) symmetry, where the effective Fock space is built by means of direct product of two spinors χ × φ one of which is the conventional spin 1/2 with components α = (↑, ↓) and another is a pseudospin representing the two-level system labeled as ρ = (l, r) in real space. The basis functions are deﬁned as Φ¯ = ↑ l ↓ l ↑ r ↓ r

(9.45)

9.2 Kinematical constraints for systems with SO(n) and SU(n) symmetries

319

The basis for matrix representation of SU(4) group is formed by the linear combination of Gell-Mann matrices (9.41). These 15 traceless block-diagonal matrices in the representation Φ | . . . |Φ¯ (9.45) may be constructed from two sets of Pauli matrices σ = {σ0 , σ + , σ − , σz } and τ = {τ0 , τ + , τ − , τz } in the following way

λˆ = σ ⊗ τ − σ0 ⊗ τ0

(9.46)

The commutation relations for these matrices may be derived from those for the Pauli matrices: [λαρ , , λα ρ ] = [σα τρ , σα τρ ] = [σα , σα ]τρ τρ − σα σα [τρ , τρ ].

(9.47)

Another realization of the SU(4) symmetry is a mixed Fock space with basis Φ¯ = ↑ ↓ 0 2

(9.48)

The ﬁrst two components of the 4-spinor correspond to the states |1 ↑, |1 ↓ in the charge sector N = 1 and the two remaining states |0, |2 correspond to the empty and the doubly occupied Hubbard cell introduced in Section 2.4. The original Hubbard operators X ΛΛ are represented via Gell-Mann matrices in the basis Φ | . . . |Φ¯ (9.48) in the following way: 1 1 X ↑0 = (λ4 + iλ5), X 0↑ = (λ4 − iλ5 ), 2 2 1 1 X ↓0 = (λ6 + iλ7), X 0↓ = (λ6 − iλ7 )/2, 2 2 1 1 X 2↑ = (λ9 − iλ10), X ↑2 = (λ9 + iλ10) 2 2 1 1 2↓ ↓2 X = (λ11 − iλ12 ), X = (λ11 + iλ12 ) 2 2 1 1 ↑↓ ↓↑ X = (λ1 + iλ2), X = (λ1 − iλ2 ) 2 2 1 1 20 02 X = (λ13 − iλ14 ), X = (λ13 + iλ14 ) 2 2 1 2 2 ↑↑ λ0 + 2λ3 + √ λ8 + √ λ15 , X = 4 3 6 1 2 2 λ0 − 2λ3 + √ λ8 + √ λ15 , X ↓↓ = 4 3 6 √ 4 1 1 2 X 22 = (λ0 − 6λ15 ), X 00 = 1 − √ λ8 + √ λ15 4 4 3 6

(9.49)

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9 MATHEMATICAL INSTRUMENTATION

Here λ0 is the unit matrix supplementing the set of Gell-Mann matrices. Three operators from the triad T± , Tz describe spin-ﬂip excitation in the homopolar subspace N = 1 of the Hubbard atom. Three operators from the last triad Z± , Zz in the set (9.42) may be used in the description of the excitonic sector N = {0, 2} in the Hubbard model. The six operators from the triads U and V intermix the states from the charge sectors N = 0 and N = 1, and the six operators W and Y do the same for the sectors N = 2, N = 1. In many physical applications the reduced Hubbard Hamiltonian with U → ∞ is exploited. In this case the doubly occupied state is completely suppressed. In the appropriately reduced Fock space possessing SU(3) symmetry Φ¯ = ↑ ↓ 0

(9.50)

1 1 X ↑0 = (λ4 + iλ5 ), X 0↑ = (λ4 − iλ5 ), 2 2 1 1 X ↓0 = (λ6 + iλ7 ), X 0↓ = (λ6 − iλ7 )/2, 2 2 1 1 X ↑↓ = (λ1 + iλ2 ), X ↓↑ = (λ1 − iλ2 ) 2 2 1 2 1 λ0 + λ 3 + √ λ8 , X ↑↑ = 2 3 3 2 1 1 ↓↓ X = λ0 − λ 3 + √ λ8 , 2 3 3 √ 00 X = λ 0 − 3λ 8 .

(9.51)

the system (9.49) transforms into

The general parametrization algorithm for the SU(n) groups is described in Refs. [161, 397, 398]. In the general case the structure factors for the commutation relations (9.36) may be calculated by means of the formula (9.37).

9.3 Bosonization and fermionization for arbitrary spins All the representations for generators of the Lie groups by means of sets of bosonic and fermionic operators are based on the commutation rules for bilinear combinations of boson and fermion creation and annihilation operators, b†k bl and fk† fl , respectively. Such bilinear products obey the following algebras:

9.3 Bosonization and fermionization for arbitrary spins

321

[b†k bl , b†m bn ] = b†k bn δlm − b†mbl δkn [ fk† fl , fm† fn ] = fk† fn δlm − fm† fl δkn

(9.52)

These commutation relations are in fact the same as the commutation relations for the Hubbard operators (2.7) of Bose type. Nearly all the generators of the dynamical groups SO(n) and SU(n) considered in this book are expanded in the Clebsh – Gordan series (2.8) for Bose-like Hubbard operators, so that the representation of these operators via conventional Fermi or Bose operators may serve as a helpful tool. One should remember, however, that such transformations implement additional constraint conditions imposed on the phase space in order to eliminate unphysical states which inevitably arise as a result of any mapping procedure. Bosonization and fermionization mapping operations for any matrix A have the forms A → Ab = b† Ab = ∑ b†k Akl bl kl

A → A f = f A f = ∑ fk† Akl fl †

(9.53)

kl

with the following commutation rules for bosonized and fermionized operators: [Ab , Bb ] = b† [A, B]b = b†Cb = Cb [A f , B f ] = f † [A, B] f = f †C f = C f

(9.54)

Thus the algebras are isomorphous for both mapping procedures, [A, B] = C ⇔ [A , B] = C ,

(9.55)

and the commutation relations for the irreducible tensor operators O introduced in Eq. (2.8) are conserved after this mapping as well. Various bosonization and fermionization procedures for generators of dynamical Lie groups are brieﬂy described below

9.3.1 Schwinger boson representation for the SU (2) group The Schwinger boson (SB) approximation for the SU(2) group represents the spin operator S via two sets of Bose operators for harmonic oscillator:

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9 MATHEMATICAL INSTRUMENTATION

S+ = a† b, S− = b† a, Sz = (a† a − b†b)/2,

(9.56)

or S = a† σ b. Then the eigenvector |S, M is mapped on the eigenvector of the twodimensional oscillator (a† )S+M (b† )S−M |S, M = |vac (S + M)! (S − M)!

(9.57)

The physical subspace {na , nb } for this representation is deﬁned by the constraint na + nb = 2S.

(9.58)

The Schwinger representation for the generators O j of the SU(3) group is realized by means of two independent triplets of harmonic oscillators aρ and bρ with ρ = 1, 2, 3: (9.59) O j = a† λ j a − b†λ¯ j b, j = 1, 2 . . . 8. Here λ j are the Gell-Mann matrices deﬁned in (9.35) and λ¯ j are transposes of

λ j . Similar construction exists for the SU(4) group. In this case the basis of representation includes three boson quadruplets which form the 12-dimensional space for 15 Gell-Mann matrices. Generalization of this procedure to the SU(n) group is straightforward but cumbersome [277].

9.3.2 Holstein – Primakoff boson representation for the SU (2) group This transformation of spin operator S goes beyond the above paradigm of bilinear representations (9.53): (9.60) S+ = 2S − b† b b, S− = b† 2S − b†b, Sz = S − b† b The physical constraint for this representation is 0 ≤ nb ≤ 2S.

(9.61)

The Holstein – Primakoff (HP) approximation is used for the description of magnon-like excitations. In the harmonic approximation valid for large S, the ladder operators in the representation (9.60) acquire the form

9.3 Bosonization and fermionization for arbitrary spins

√ √ S+ = b 2S, S− = b† 2S.

323

(9.62)

In order to describe magnon interaction, damping and decay, one should take into account the next terms in the expansion of square roots in Eq. (9.60). Comparing (9.56) and (9.60) we establish the correspondence between Sw and HP representations : bSB ↔ bHP , aSB ↔ 2S − nb,HP.

9.3.3 Dyson – Maleev representation for the SU (2) group The Dyson – Maleev (DM) representation maps real magnon states onto the space of ﬁctitious harmonic magnons β : √ √ 1 † + S = 2S 1 − β β β , S− = 2Sβ † , Sz = S − β † β . (9.63) 2S The equivalence between the DM and the HP representation is given by the following equations: −1/2 1/2 β †β β †β b† → β † 1 − , b → 1− β , β † β = b† b. 2S 2S

(9.64)

The DM representation is non-Hermitian, but it gives the same eigenvalues as the other bosonization representations at least at low energies.

9.3.4 Pomeranchuk – Abrikosov spin fermion representation for the SU (2) group The Pomeranchuck – Abrikosov (PA) representation uses bilinear combinations (9.53) of neutral fermions: S = S f μ† τ μν fν

(9.65)

where the indices μ , ν acquire 2S+1 values from −S to S. In the case S = 1/2 one needs two spin fermions, μ , ν =↑, ↓, so that 1 S+ = f↑† f↓ , S− = f↓† f↑ , Sz = ( f↑† f↑ − f↓† f↓ ). 2

(9.66)

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9 MATHEMATICAL INSTRUMENTATION

A constraint condition is necessary to exclude the spurious states |0, 0, |2, 2 with two and zero fermions and retain only the physical states |1, 0, |0, 1 corresponding to up and down spin projection. This constraint condition may be formulated as n↑ + n↓ = 1

(9.67)

In the case S = 1/2 the constraint (9.67) is satisﬁed automatically, because all three operators (9.66) give zero when acting on the spurious states |0, 0, |2, 2. One only has to take into account the change of normalization condition when averaging only over physical states, i.e. add factor 2m in a numerator of any average containing m spin operators. In the general case of arbitrary spin the constraint reads n f = ∑ f μ† f μ = 1

(9.68)

μ

(the spin has a deﬁnite projection). To take this kinematic constraint into account, one may add an extra term δ Hˆ to the model Hamiltonian:

δ Hˆ = λ (n f − 1),

(9.69)

i.e. introduce ﬁctitious chemical potential λ which plays part of a Lagrange factor in any procedure of minimization of the free energy with the partition function Z . As a result Z acquires additional exponent Z ∼ e−δ H/T = e−λ (n f −1)/T . ˆ

(9.70)

This exponent allows one to freeze out the unphysical states in the limit λ /T → ∞. The PA fermionization procedure is in fact nothing but the factorization of Hubbard operators which enter the spin matrices (see Section 9.1):

X SM,SM = |SMSM | → fM† fM .

(9.71)

For example, in the case S = 1 the expansion (9.11) transforms into Sz = ( f1† f1 − f1¯† f1¯ ), S+ =

√

2( f1† f0 + f0† f1¯ ), S− =

being fermionized by means of three spin fermions.

√ † 2( f0 f1 + f1¯† f0 ).

(9.72)

9.3 Bosonization and fermionization for arbitrary spins

325

9.3.5 Spin-fermion representations for the SO(n) groups The Pomeranchuk – Abrikosov fermionization procedure may be straightforwardly extended to supermultiplets described by SO(n) groups by means of the factorization (9.71) and its generalization to the other states from the corresponding multiplet. Using the bilinear representation A f (9.53) for the generators S, Ri , A j of SO(n) groups deﬁned in Eqs. (2.46), one immediately ﬁnds that one needs 2n spin-fermion operators and one constraint to map these generators onto an auxiliary fermionic Fock space. The spin-fermion representation of the SO(n) group (n = 4, 5, 6) is characterized by the operator n-vector q T = ( f1† , f0† f1¯† , g†1 , g†2 . . . g†n−3 )

(9.73)

Here the ﬁrst three operators realize the spin-fermion representation for the spin 1 operator S, and the remaining operators are related to the singlet states entering the corresponding supermultiplets in accordance with Table (2.58). The components g†i are enumerated in the same order as the singlet states in the basis. Then constructing

the group generators by means of matrix elements q T |X ΛΛ |q, we project (2.46) onto √ † √ 2( f0 f1¯ + f1† f0 ), S− = 2( f 1¯† f0 + f0† f1 ), Sz = f1† f1 − f1¯† f1¯ , √ † √ † † † † z R+ 2( f1 gi − g†i f1¯ ), R− i = i = 2(gi f 1 − f 1¯ gi ), Ri = −( f0 gi + gi f 0 ), S+ =

A j = i ∑ ε jkl g†k gl .

(9.74)

kl

The kinematic constraint is q T |q T = ∑ fi† fi + ∑ g†j g j = 1. i

(9.75)

j

The high symmetry groups SO(n) with n ≥ 7 include more than one spin triplet. In this case the separate set of fermions fα i should be included in q for each spin 1 vector Sα .

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9 MATHEMATICAL INSTRUMENTATION

9.3.6 Popov – Fedotov semi-fermion representation One more elimination procedure for unphysical states which arise when projecting the original spin states to fermionic ones is realized in the Popov –Fedotov (PF) representation. Like in the PA procedure (9.66), two fermions are used in this representation in the case S=1/2 1 S+ = a† b, S− = b† a, Sz = (a† a − b† b). 2

(9.76)

Instead of the Abrikosov’s trick (9.69), in the PF procedure an additional term with purely imaginary Lagrange factor is introduced

δ Hˆ =

iπ T nf , 2

(9.77)

where n f = a† a + b† b. In the partition function with this additional term Z = Tr exp[−(Hˆ + iπ T n f /2)/T ]

(9.78)

the states from the unphysical sector with n f = 0, 2 are authomatically annihilated due to the identity (−i)0 + (−i)2 = 0. The price of the PF trick is the ”intermediate” statistics of quasiparticles as poles of the thermodynamic (Matsubara) Green functions deﬁned at a discrete set of points on the imaginary axis. This set includes even frequencies ωn = 2π inT for bosons and odd frequencies ωn = 2π i(n + 1/2)T for fermions. As a result of the shift δ Hˆ in the Hamiltonian the Matsubara frequencies are deﬁned as ωn = 2π i(n + 1/4)T, i.e. intermediate between those for Fermi and Bose statistics. Such ”semi-fermionic/semi-bosonic” behavior reﬂects in some sense the dual nature of spin excitation which may be Fermi-like or Bose-like in different physical realizations of spin systems. Similar procedure for spin S = 1 demands three fermions S+ =

√ † √ 2(a c + c†b), S− = 2(b† c + c† a), Sz = a† a − b†b.

(9.79)

The additional term δ Hˆ has the form

δ Hˆ =

iπ T nf . 3

(9.80)

with where n f = a† a + b† b = c† c. It is easy to see that the states with occupation numbers n f = 0, 3 cancel each other, whereas the states with occupations n f = 1, 2

9.3 Bosonization and fermionization for arbitrary spins

327

are equivalent due to the particle-hole symmetry and thus can be taken into account on an equal footing by proper normalization of the partition function. The PF procedure may be generalized for higher spin states S > 1 and for SU(n) Lie groups [214].

9.3.7 Majorana fermionization Spin 1/2 operator S may be expressed as a bilinear combination of three real Majorana fermions ηα = ηα† with anticommutation relations {ηα , ηα } = δαα (α = x, y, z): i Sα = − εαβ γ ηβ ηγ . (9.81) 2 Majorana representation is helpful in situations, where the system possesses hidden Z2 symmetry, since the transformation η → −η keeps the commutation relations. The mixed Majorana – Dirac representation S+ = ηz f , S− = f † ηz , Sz = 1 − 2 f † f

(9.82)

(drone-fermion representation) contains one Majorana fermion ηz and two spinless Dirac fermions f , f † .

9.3.8 Mixed fermion-boson representations In the context for strongly interacting electron systems the symmetry group SU(4) is used for the description of dynamical symmetries in the Fock subspace involving three charge sectors with electron number N = 0, 1, 2 and spin sector which contains spinor algebra of single electron states with spin 1/2 in the sector N = 1. The state vector for this space has the form (9.48). In order to represent the generators of this group via commuting or anticommuting operators, one should resort to the mixed representations including both fermions and bosons. One of the possible representations of this type is the mixed spin-fermion/slave boson representation using Fermi-operators for spin variables and Bose operators for charge degrees of freedom. Hubbard operators X ΛΛ describing transitions be-

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tween the states constituting the multiplet (2.62) may be represented via four neutral spin fermionic operators fσ† , fσ and four charged bosonic operators h, h† ; e, e† (”holons” and ”doublons”): X σ 0 = fσ† h, X 0σ = fσ h† , X 2σ = fσ e† , X σ 2 = fσ† e,

X σ σ = fσ† fσ hh† = fσ† fσ ee† , X 00 = fσ fσ† h† h, X 22 = fσ fσ† e† e X 20 = fσ fσ† e† h, X 02 = fσ fσ† h† e.

(9.83)

This representation preserves the multiplication rules for the Hubbard operators and hence the commutation relations (2.7). All the operators affect both charge and spin degrees of freedom. The kinematic constraint imposed on the occupation numbers of fermions and bosons has quite complicated form in the general case. It stems from the constraint

∑ X ΛΛ = 1 Λ

. Alternative procedure implying charged spinless fermions for hole excitations and neutral spinful Schwinger bosons for spin excitations

X 0σ = f † b†σ , X σ σ = b†σ bσ , etc. is also available.

(9.84)

Chapter 10

CONCLUSIONS AND PROSPECTS

The realm of nanostructures was surveyed in this book with deﬁnite viewpoint. We looked at these systems from the point of view of dynamical symmetries. In such a framework any nanoobject is considered as a nanosize trap containing the few strongly interacting electrons or few atoms with complicated spectrum characterized by its own internal symmetry, which is violated due to interaction with a macroscopic bath and/or external electromagnetic ﬁeld. We have chosen for illustration of the main statements and clauses of our mathematical theory those systems for which the inﬂuence of dynamical symmetries on the experimentally measurable properties of nanoobjects is especially signiﬁcant. These are complex quantum dots and cold gases in optical traps. It was assumed in the theory presented in the book that a few-particle nanoobject S is characterized by deﬁnite individual and collective spin and orbital degrees of freedom, which are interlaced due to particle exchange and direct interaction with the bath B. The latter subsystem was treated mainly as a Fermi sea of conduction electrons itinerating in metallic electrodes (leads). In this basic conﬁguration the paradigmatic behavior of a nanosystem possessing dynamical symmetries may be summarized as follows. • The systems S + B usually are not integrable, and only the low-energy part of their excitation spectra may be found within a controllable accuracy. Respectively, the subsystem S may be characterized by a dynamical symmetry DS of low-energy supermultiplet only within some energy interval E . • Dynamical symmetries are activated due to interaction with the subsystem B, because this interaction does not conserve spin and charge invariants of the subsystem S . The interaction Hamiltonian as a rule may be written in terms of K. Kikoin et al., Dynamical Symmetries for Nanostructures: Implicit Symmetries in Single-Electron Transport Through Real and Artificial Molecules, DOI 10.1007/978-3-211-99724-6_10, © 2012 Springer-Verlag/Wien

329

330

10 CONCLUSIONS AND PROSPECTS

scalar, vector and mixed products of operators generating the dynamical symmetry groups. • Under the restriction of strong Coulomb blockade in quantum dots and strong Hubbard repulsion in optical traps the number of particles N in the subsystem S is ﬁxed, and the charge degrees of freedom are represented by the pseudospin operators characterizing the redistribution of particles within the local minima of a trap potential, which conﬁnes these particles and quantize their energy states. • Nanoobjects with even and odd number N belong to two different classes of dynamical symmetries. As a rule, the objects with even and odd N are characterized by the semisimple groups SO(n) with n > 3 and SU(n) with n > 2, respectively. This statement is, however, not completely strict. In some complex conﬁgurations odd N objects behave as systems with integer spin or pseudospin and v.v. Several examples of such disguise are described in Chapter 4. • One of the most important features of dynamical symmetry in nanoobjects is its dependence on the energy scale E . Reduction of this scale results in quenching of the highest states in the multiplet, and the dynamical symmetry DS modiﬁes appropriately. Due to this trend, the experimentally observable properties of a device S + B are predetermined by the characteristic energy scale ε¯ of the interaction between the complex nanoobject and the bath. • The most remarkable many-body effects which emerge in nanoobjects are related to the phenomenon of Anderson’s orthogonality catastrophe, which results in a complete reconstruction of the ground state and the low-energy excitations at ε < ε¯ in the system S + B relative to the states of this system in the limit of asymptotic ultraviolet freedom (large E ). • This class of many-body effects in complex quantum dots is represented by the Kondo-type resonance tunneling with ε¯ ∼ TK . The generic Kondo effect exists already in tunneling through quantum dots with the symmetry SU(2). The novel features which the dynamical symmetries SU(n) and SO(n) introduce in the Kondo effect are related to the sensitivity of these symmetries to the energy scale E . These are the multistage screening of the local spin and pseudospin degrees of freedom by the itinerant electrons and the loss of universality of the Kondo effect in the strong coupling low-energy limit. All these features may be observed experimentally in measurements of the zero bias anomalies in the tunnel conductance G as a function of temperature or energy,

10 CONCLUSIONS AND PROSPECTS

331

• Dynamical symmetries manifest themselves also in the nonequilibrium conditions and in the presence of time-dependent perturbations. In these conditions Kondo singularities and other many-body effects are observed as ﬁnite bias anomalies in G(ε ). Dynamical symmetries also affect the optical properties of nanoobjects Restricting ourselves within the aforesaid limited domain of nanophysics, we, however, did not exhaust all possibilities proposed by inventive theoreticians and provided by advanced nanotechnologies. The leads B may have their own nontrivial symmetries both in spin and charge sectors. In particular, the electrodes may be prepared from magnetic materials, e.g. from dilute magnetic semiconductor like (Ga,Mn)As. Then the electrons in the subsystem B are partially or completely spin polarized and their state is characterized by a nonzero magnetic moment. Partial magnetization of the leads does not suppress Kondo tunneling but provides additional possibilities for controlling the tunneling current. For example, the magnetization vectors ms and md in the source and drain electrodes are not necessarily parallel, and the tunneling current, including its Kondo component, depends on the angle between them. This effect together with an external magnetic ﬁeld opens the possibilities for constructing spin ﬁlters, spin valves and other elements of nanospintronics [230, 267]. Another challenging possibility is the use of superconductor leads, especially in combination with metallic ones. In this case one should expect various types of interplay between superconductor pairing and Kondo effect both in equilibrium and non-equilibrium regime, including Cooper pair cotunneling, Andreev reﬂection, etc [26, 31, 59, 74, 103, 177, 193, 258]. Of course, these phenomena exist already for the conventional SU(2) symmetries for spin 1/2, but involvement of dynamical symmetries should enrich the physical picture of tunneling between the leads with broken symmetries linked by means of quantum dots under strong Coulomb blockade. Another option is the use of Bose-type reservoirs B. The ensemble of Bose excitations in the bath may be formed by a system of harmonic oscillators of various sorts and by spin ﬂuctuations in magnetic reservoir. One may expect that dynamical symmetries inherent in a nanoobject S will enrich the theory of Bose-Fermi Kondo models developed during the recent decade (see, e.g., [84, 259]) One of the most promising potential applications of CQD is their use as material carriers for quantum qubits, entangled quantum states etc. Since the systems with

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10 CONCLUSIONS AND PROSPECTS

dynamical symmetries contain more than two quantum levels both of spin and charge nature, they render new possibilities for the construction of logical elements as well as for recording and reading information.

We did not touch in this book the problem of dynamical symmetries in nanowires, ladders, arrays, networks etc. One may formulate the conditions favorable for manifestation of these symmetries based on the knowledge accumulated in the studies of quantum dots. The network should have a geometrical structure where the dimers, triangles or squares in the elementary cell α with small intersite distances riαj are

organized in such periodic structures that the lattice period rαβ noticeably exceeds riαj . Several examples of such quasi 1D and 2D Hubbard lattice models are presented in Fig. 10.1. Multiwell optical traps organized in periodical optical lattices may become another realization of similar structures in Bose – Hubbard models.

rij r αβ Fig. 10.1 Decorated chains and two-dimensional lattices with elements possessing dynamical symmetries (see text for detailed explanation).

One may use the cluster marked by the bold lines as a subsystem S α and classify the electron states in such cluster in accordance with representations of the dynamical groups SU(n) or SO(n), depending on the odd or even number of electrons per unit cell. The clusters S α form periodic structures, and the electron hopping between the neighboring sites α , α may be treated as a perturbation under the above mentioned condition rαα riαj . This perturbation activates dynamical symmetries

10 CONCLUSIONS AND PROSPECTS

333

in the cluster S α in the same way as the cotunneling between a complex quantum dot and the leads activates the dynamical symmetries DS . One may anticipate that the excitation branches in such Hubbard models will be classiﬁed along with representations of the group DS like in other tight-binding models. Without any doubt the phase diagram of these arrays and lattices will be extraordinarily rich. Some preliminary studies in this direction are available [28, 29, 51, 205, 215], but serious investigation of this class of low-dimensional structures will become relevant only if and when the corresponding breakthrough in fabrication of such structures will take place. The physics of graphene and graphene-like systems is an encouraging example of such a breakthrough.

Index

Abrikosov – Suhl resonance, 115, 118, 149, 152, 248 accidental degeneracy, 5, 180, 182 addition/removal energies, 24, 53, 56, 62, 93 Aharonov – Bohm (AB) effect, 189, 190 – in quantum dots, 189, 190, 192, 193 – – in tunneling conductance, 190, 191, 194, 195 – mesoscopic AB interferometer, 190, 192, 195 Anderson model, 28, 51, 101, 108, 110, 111, 119–121, 150, 176 – Anderson – Holstein Hamiltonian, 217, 219, 221, 227 – excitonic representation, 241 – negative U, 227–229, 231, 258 – time-dependent, 234, 235, 259, 262, 285–287, 291–293 – two-orbital, 132–135, 156, 164 Berry phase, 213, 216 bipolaronic states, 227–229, 231 Borel summation, 270, 274, 279, 303 Born – Oppenheimer approximation, 101, 102, 104, 217 Bose/Fermi representations for group generators, 33, 42, 320, 321 – boson representations, 33 – – Dyson – Maleev representation, 323 – – for SO(2,1) group, 33 – – Holstein – Primakoff representation, 322, 323 – – Schwinger bosons, 44, 321, 322 – fermion representations, 42 – – drone fermions, 327 – – Majorana fermions, 42, 327 – – Popov-Fedotov fermions, 42, 250, 326

– – spin-fermions, 42, 133, 163, 224, 250, 267, 323–325 – mixed representations, 42 – – slave bosons, 42, 44, 116–118, 191, 327 – – slave fermions, 42, 44, 328 Byers – Yang theorem, 194 Casimir operators, 9, 12, 19, 20, 32, 61, 78, 146, 250, 312, 313 Cayley – Klein parameters, 309 charge-spin conversion in SET, 286, 287 – due to dynamical symmetries, 286, 287, 289 chiral symmetry, 179, 193, 200, 201 Clifford algebra, 37, 41 colored quarks, 2, 31, 258, 316 conﬁguration diagrams, 102, 104, 217, 222 conformal ﬁeld theory, 163, 169, 172, 179, 189 Coulomb blockade, 3, 7, 17, 24, 28, 52–57, 59, 60, 70, 73, 78, 89, 108, 136, 139, 149, 167, 169, 330 Coulomb diamonds, 54–56, 59, 95, 108 crystal and ligand ﬁeld effects, 121, 123, 212, 222 current-voltage characteristics, 3, 62 – Coulomb staircase, 54, 55, 95, 102, 136, 234 dark states, 193, 195, 196 de Broglie wavelength, 49, 50 Debye – Waller effect, 218, 219, 285 decoherence, 248, 258 – in Kondo effect, 259–261, 263, 264, 266–270, 272, 274, 275, 286 – related to dynamical symmetries, 258 dephasing, 248, 258

K. Kikoin et al., Dynamical Symmetries for Nanostructures: Implicit Symmetries in Single-Electron Transport Through Real and Artificial Molecules, DOI 10.1007/978-3-211-99724-6, © 2012 Springer-Verlag/Wien

335

336 – in Kondo effect, 253–255, 260, 261, 263, 264, 266–268, 286 – related to dynamical symmetries, 259 Dicke model, 234 dynamical symmetries, 1, 49, 55 – deﬁnition, 5, 329, 330 – in 1D and 2D nanosystems, 332, 333 – in Kondo effect, 107, 108, 136, 330 – in Landau-Zener effect, 293, 294, 296–298 – in non-equilibrium systems, 4, 21, 104, 245, 248–255, 257, 331 – noise induced, 4, 261, 263, 264, 266–270, 272, 274, 275 dynamical symmetry breaking, 1 – Anderson – Nambu mechanism, 1 – Higgs – Anderson mechanism, 1 – electroweak interaction, 31 dynamical symmetry groups, 1 – deﬁnition, 2 – point C2n group, 212, 216 – point C3v group, 82, 88, 180, 211 – point D2d group, 83, 85, 185 – SO(1,1) subgroup, 313 – SO(12) group, 89, 91, 183 – SO(2,1) group, 32, 34, 63, 104, 217, 233, 245 – SO(3) subgroup, 10–12, 19, 142, 183, 268–270, 272, 310 – SO(3,1) group, 9, 313, 314 – SO(4) group, 2, 9, 11, 12, 20, 21, 29, 40, 41, 50, 60, 63, 77, 79, 80, 88, 89, 91, 104, 125, 127–133, 142, 146, 149, 161, 173, 183, 222, 224, 249–255, 285, 287, 291, 292, 295, 311–313 – SO(4,1) group, 314 – SO(4,2) conformal group, 2, 13, 14, 41, 314 – SO(5) group, 20, 21, 29, 60, 78, 80, 87, 142–144, 239, 257, 261, 263, 264, 266–268 – SO(6) group, 19, 21, 60, 76, 126, 132 – SO(7) group, 21, 142–144 – SO(8) group, 21, 89, 139, 161, 177, 178, 239 – SO(9) group, 139 – SO(n) group, 3, 14, 15, 21, 23, 59, 91, 117, 160, 245, 325, 330 – SO(n,1) group, 2, 14 – SO(n,2) group, 14 – SO(p,n-p) group, 313, 314 – SU(10) group, 123 – SU(2) subgroup, 15, 17, 26, 34, 57, 64, 68, 75, 76, 85, 86, 108, 112, 113, 120, 121, 124, 127, 128, 131, 132, 138, 180, 216,

Index 228, 229, 231, 276, 295, 309, 310, 312, 316–319 – SU(3) group, 2, 10, 23, 27, 29, 31, 42, 43, 59, 111, 112, 116, 145, 208, 259, 285, 299–301, 315, 316, 320, 322 – SU(4) group, 17, 22, 27, 56, 58, 65, 73–75, 84–86, 91, 110, 118, 120, 123, 137, 138, 147, 182, 183, 185, 187, 201, 208, 227–229, 231, 258, 261, 263, 264, 272, 274, 275, 285, 289, 294, 317–319, 322 – SU(5) group, 208 – SU(6) group, 65, 85, 86, 91, 123, 180, 208, 210, 214, 222 – SU(n) group, 3, 15, 23, 73, 108, 117, 119, 121, 175, 196, 208, 212, 245, 293, 315, 320, 322, 330 eightfold way, 2, 29, 31, 258 electron shuttling, 4, 104 Euler angles, 8 evolution operator, 296, 297 exchange interaction, 17, 51, 59, 63 – double (Zener) exchange, 87 – Dzyaloshinskii – Moriya exchange, 203–205 – indirect exchange, 17, 28, 58, 59, 157, 171, 182, 193, 212, 220, 224–226, 235, 236, 250–253 – RKKY exchange, 99, 125, 126, 139 excitons in quantum dots, 18, 59, 61, 238–240 – charge transfer excitons, 18, 20, 28, 77–79, 87–89, 165, 177, 265 – exciton-electron complexes, 240–242 – in optical response, 239–244 – many-exciton states, 69, 243, 244 – singlet excitons, 59, 60, 69, 90, 139, 222, 257 – singlet-triplet excitons, 19, 20, 28, 59, 63, 77–79, 87–89, 98, 104, 125, 127, 128, 130–134, 139, 146, 148, 173, 177, 222, 257, 265, 268, 287 Fano effect, 149, 150, 207 – Fano – Cooper factor, 151, 208 – interplay with Kondo effect, 150–153, 159, 190, 191, 195 Floquet – Bloch theorem, 45, 47, 233, 292 Fock – Darwin model, 31, 33, 34, 61, 64, 135, 177 – energy levels, 32, 34, 39, 40, 61, 68 – two-band, 67, 69, 240, 241 – wave functions, 33 Friedel sum rule, 114, 134 frustration, 185, 187, 210

Index Gaussian distribution function, 270, 274, 275, 278, 279, 302–306 Gell-Mann matrices, 22 – of 3rd rank, 25, 29, 111, 116, 299–301, 315, 316, 320, 322 – of 4th rank, 22, 25, 26, 56, 110, 145, 317–319, 322 Glazman – Raikh rotation, 55, 74, 81, 84, 86, 164, 167 Green function, 118, 119, 150, 151, 159, 190, 207, 238, 242, 249, 250, 254, 255, 268–270, 272, 274, 275, 278, 279 – for nonequilibrium systems, 247 harmonic oscillator, 2, 32, 44, 49 hidden symmetry, 2, 12, 14, 41, 214 Huang – Rhys factor, 103, 220, 226, 229 Hubbard atom, 23, 27, 29, 31, 41, 43, 102, 227, 228, 319, 320 – energy levels, 24, 27, 42, 55, 56, 110, 265 – – time-dependent, 293, 299 Hubbard model, 7, 23, 44 – Hubbard chain, 136 Hubbard molecule, 72 – dimer, 72, 73 – – energy levels, 76–79 – – even occupation, 76–80 – – odd occupation, 73–75 Hubbard operators, 6, 7, 16, 17, 21, 24, 55, 75, 176, 178, 187, 214, 238, 241 – Bose-type, 7, 25, 42, 57, 116 – commutation relations, 6, 25, 26 – Fermi-type, 7, 25, 42, 55, 116 – relation to SO(n) group generators, 19, 21, 311, 325 – relation to SU(n) group generators, 17, 22, 25, 26, 56, 58, 110, 187, 258, 299, 311, 319, 320, 324, 328 Hubbard parabola, 53, 55, 56, 227, 228 Hund’s rules, 124, 200 hydrogen atom, 2, 3, 7, 10, 14, 20, 40, 49 – n-dimensional, 14, 41 hydrogen molecule, 18, 92 hyperspherical harmonics, 8 integrable systems, 2, 4, 5, 15, 50, 68, 163 – Bethe ansatz, 114 irreducible representations, 5–7, 321 – for SU(3) group, 29, 30 irreducible tensor operators, 7, 9, 10, 22, 23, 26, 317 Keldysh contour, 277 Kolokolov representation, 297

337 Kondo effect, 28, 50, 65, 69, 108 – in molecular complexes, 199, 208, 209, 211–216 – – even occupation, 199, 200, 229, 231 – – for pair tunneling, 229, 231 – – in presence of TR precession, 205 – – odd occupation, 199–201, 210 – – phonon assisted, 218–220 – – phonon induced, 222–226 – in nonequilibrium systems, 245–247, 260, 261 – – even occupation, 249–255, 257 – in SET, 92 – – even occupation, 126–128, 130–135, 138–140, 142–145, 148, 149, 160, 169, 171–173, 175, 177, 178, 183, 185, 196, 236–239 – – odd occupation, 109–113, 115, 117–119, 121, 123, 124, 137, 138, 145–148, 156–158, 165–167, 169, 176, 180, 182–185, 187, 189, 196 – – photon assisted, 236 – – photon induced, 236–240 – interplay with Aharonov – Bohm effect, 189–193, 206 – magnetic ﬁeld induced, 126–128, 130, 138, 145, 169, 171–173, 175, 180, 182, 183, 216, 225, 226, 287, 289 – multichannel, 162–167 – orbital, 90, 175, 177, 178, 201 – overscreened spin, 124, 162, 240 – spinons and holons, 116–118, 163 – three-channel, 153, 167, 181 – two-channel, 90, 132, 153, 167, 169, 171, 176–178, 188, 221 – – non-Fermi liquid regime, 154, 163–167, 169, 171–173, 175, 187, 189, 210 – – orbital anisotropy, 124, 125, 164–167 – two-site Kondo model, 124, 126, 139, 155 – underscreened spin, 124, 196, 240 Kugel – Khomskii Hamiltonian, 75, 84, 187, 210 Landau – Zener effect, 4, 99 – in n-level systems, 293, 294, 296–301, 307 – in presence of noise, 301–307 – in two-level systems, 293, 301–306 Landau levels, 39 Landauer formula, 151 Lang – Firsov canonical transformation, 102, 219, 227 Larmor (diamagnetic) shift, 63–65, 68, 69, 177, 185, 200

338 line-shape function, 103, 220, 226, 234–236 Luttinger liquid, 155 magnetic ﬂux, 180–185, 189–193, 287, 289 molecular complex, 3–5, 91, 96–101, 197 – in single-electron transistors, 198 – in STM spectroscopy, 201, 206–211 – molecular grid, 98, 99, 199 – rare-earth metal organic complexes (REMOC), 96, 101 – transition metal organic complexes (TMOC), 96, 101, 212–216, 222–226 molecular excitons, 94, 101 molecular magnets, 44, 99, 211–216 molecule – as quantum dot, 92, 95, 108, 128, 132, 197, 198 – carbon peapod, 96 – fullerene, 92–94, 100, 132, 199 – – dimetallofullerene, 94 – – endofullerene, 94 – in break junction geometry, 91, 93, 96 – in STM geometry, 91 – metallocene, 94, 96, 97 – nanotube, 92, 94, 95, 100, 108, 128, 199, 255 multiplets, 1, 2, 9, 15–18, 20, 21, 23–25, 29, 42, 56, 60, 73, 93, 125, 156, 157, 165, 177, 180, 257 – hadron multiplets, 1, 29 – – isospin and hypercharge, 29, 31, 258 nanoelectromechanical single-electron tunneling transistor (NEM-SET), 105, 283, 284, 290 – shuttling in Kondo regime, 283, 291, 292 noise, 263 – in quantum dots, 264–268 – in TLS, 276–280 – in ultracold gases, 280 – Keldysh model, 269 – – scalar, 270, 272, 279, 280, 304–306 – – vector, 272, 274, 275, 277–280, 302–306 Onsager relations, 194 optical traps, 300, 330 orthogonality catastrophe, 107, 330 – shake-up effects, 239, 241, 243, 244 paramagnetic susceptibility, 247, 272, 275 path integral method, 277–279 Pauli matrices, 16, 17, 75, 309, 310 polaron shift, 220, 226, 227

Index projection operator, 6, 9, 50 pseudospin operator, 17, 26, 75, 84, 90, 175, 178, 182, 183, 187, 189, 201, 213, 272, 274, 275, 293, 299, 318, 319, 330 quantum box, 175 quantum criticality, 3, 173, 175 quantum dot, 3, 5, 91, 330 – complex quantum dots, 4, 5, 70, 71, 91 – – double quantum dots (DQD), 70–73, 75–80, 125, 137, 138, 147–149, 154, 160, 167, 169, 171–173, 175, 249–255, 286, 287, 295–298 – – Fulde molecule, 28, 29, 60, 72, 94 – – triple quantum dots (TQD), 70, 71, 80, 81, 83, 85–90, 139, 140, 142–146, 152, 153, 155–158, 160, 165–167, 177, 189–193, 195, 289 – geometry, 71, 136 – – Δ -shape, 71, 83, 185, 187, 189, 287, 289 – – ∇-shape, 71, 83, 185, 187, 188, 190 – – cross, 71, 81, 155–158 – – parallel, 71, 74, 81, 159, 160, 165–167, 177, 189 – – ring-shaped, 179 – – serial, 71, 74, 81, 137–140, 142–146 – – T-shape, 71, 74, 81, 96, 147–149, 152–155, 167, 169, 171–173, 175, 249–255, 261, 263, 264, 266–268, 286 – – triangular, 71, 82, 83, 179, 180, 182–185, 187, 189–195, 210, 261, 263, 264, 272 – – V-shape, 71, 83, 84 – lateral(planar), 51, 52, 72, 108, 153 – – level spacing, 51, 66, 169 – self-assembled, 65, 67, 69, 238, 240–244 – – level spacing, 67 – – wetting layer, 67, 69, 240–244 – vertical (disk-like), 61–65, 71, 160, 176, 185, 192 quantum wire, 155 quasienergy levels, 46, 47, 292 quasienergy states, 45, 46, 233, 245, 292 Rayleigh distribution function, 274, 275 renormalization group (RG), 110–113, 115, 146 – Anderson procedure, 113, 122, 140, 228, 229, 231, 266 – – for nonequilibrium systems, 246 – ﬁxed point, 114, 124, 126, 163, 164, 166, 214

Index – ﬂow trajectories, 113, 142, 157, 161, 164, 166, 171, 214 – Jefferson – Haldane procedure, 110, 112, 121, 140, 146, 149, 157, 160, 177, 191, 228, 266 – multistage renormalization, 110, 112, 116, 121, 125, 130–134, 137, 138, 144, 148, 149, 154, 158, 178, 180, 191, 222, 228, 229, 231, 266, 330 – numerical RG, 114, 115, 119, 134, 176, 187, 190, 192, 201, 209, 215 – scaling equations, 113, 114, 129, 130, 133, 134, 137, 146, 158, 161–164, 166, 171, 174, 182, 193, 214, 231, 252, 253 – scaling invariants, 113, 140 Riccati equation, 297, 300 rigid rotator, 2, 7–9, 14, 20 – n-dimensional, 14 rotation operators, 11, 311–313 Runge – Lenz vector, 2, 3, 10, 11, 14, 20, 40 scattering phase shift, 114, 134, 159, 163, 190, 244 Schr¨odinger equation, 2, 5–7, 10, 15, 16, 20, 45–47, 301 Schrieffer – Wolff (SW) Hamiltonian, 58, 73, 75, 81, 83, 89, 93, 108, 148, 153, 156, 164, 166, 182, 187, 193, 220 – Coqblin – Schrieffer model, 120, 122 – Cornut – Coqblin model, 121, 122 – for negative U Anderson model, 228, 229 – in presence of dynamical symmetries, 61, 78, 84, 87, 88, 90, 91, 107, 128, 129, 133, 143, 144, 146, 158, 171, 178, 188, 222–226, 228, 229, 250, 266, 267, 287, 289, 291, 292 – in presence of TR precession, 202–205 – magnetic anisotropy, 171, 172, 208, 212–214, 229 – three-site exchange Hamiltonian, 185 – time-dependent, 219, 235, 236, 260, 266, 267, 287, 289, 291, 292 – two-site exchange Hamiltonian, 124, 125 selection rules, 2, 10, 209 single electron tunneling (SET), 3, 52, 54 – electron cotunneling, 57, 59, 73, 177 – – inelastic, 108, 109, 258 – – pair tunneling, 227–229, 231, 258 – – phonon assisted, 217, 218, 220, 222–226 – – photon assisted, 233–236, 260, 261 – in non-equilibrium systems, 236 – through molecules, 92, 197, 199 – – nanotubes, 95 – – phonon assisted, 92, 100, 102

339 spin ﬁlters and valves, 196 spin relaxation, 254, 255 spin-orbit interaction, 121, 123 spirality, 95, 96, 206, 255 strongly correlated electron systems (SCES), 7, 24, 26, 41, 54, 107 superalgebra, 6, 25, 35–37, 43 – Z(2) graded, 36, 37, 40 supersymmetry, 3, 14, 35–42 – in mesoscopic systems, 44 – supercharge operators, 35–38, 40–43 – supersymmetric Hamiltonian, 37, 43 – – energy spectrum, 38, 41 – – spinor representation, 38 Thomas – Rashba (TR) precession, 201 – TR Hamiltonian, 202–206 three-level systems, 23, 81, 316 – even occupation, 80, 145 – – energy levels, 80, 86, 88, 177, 183 – – wave functions, 86, 87 – odd occupation, 81, 157, 158 – – energy levels, 81, 82, 84, 85, 156, 165, 180 – – wave functions, 82, 146, 156, 165 transition matrix (T -matrix), 115, 119, 150, 221 trimers, 86, 210 tunneling conductance, 54, 164, 167, 169 – ﬁnite bias anomaly (FBA), 109, 201, 237, 245, 248–255, 257, 331 – unitarity limit, 115, 134, 137, 159, 163, 168, 190 – zero bias anomaly (ZBA), 108, 114, 115, 128, 130–132, 136, 152, 201, 225, 245, 287 – – in presence of magnetic ﬁeld, 172, 173, 175, 180, 190–194 two-level systems (TLS), 15, 47, 90, 175, 276 – even occupation, 17, 28, 76, 138 – – energy levels, 18, 28, 298–301 – – wave functions, 18 – for Bose particles, 293, 298–301 – odd occupation, 15–17, 73, 137, 293, 305, 306 – – energy levels, 16 – – wave functions, 16 Ward identities, 271, 275 Wei – Norman method, 296, 297 Wigner theorem, 5 Wigner-Eckart theorem, 9 Zeeman effect, 39, 62, 68, 90, 99, 126, 138, 144, 169, 177, 179, 184, 200, 296

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