INTEGRABLE QUANTUM FIELD THEORIES AND THEIR APPLICATIONS Proceedings of the APCTP Winter School
ed b,
tiangrim Ahn haiho Rim r u Sasaki
INTEGRABLE QUANTUM FIELD THEORIES AND THEIR APPLICATIONS
INTEGRABLE QUANTUM FIELD THEORIES AND THEIR APPLICATIONS Proceedings of the APCTP Winter School Cheju Island, Korea 28 February - 4 March 2000
Edited by
Changrim Ahn Ewha Womans University, Korea
Chaiho Rim Chonbuk National University, Korea
Ryu Sasaki Kyoto University, Japan
V f e World Scientific V I
New Jersey 'London •Singapore Singapore ••Hong Kong
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V
Preface The APCTP Winter School on Integrable Quantum Field Theories and Applications was held in Cheju Island, KOREA, 28 February - 4 March 2000. The School was to promote the quantum field theories of integrable models and to propagate the recent developments of the non-perturbative results and new techniques to various background scholars. The research of integrable models had been limited to a few groups of scholars around the world and been mostly considered as a branch of mathematical physics of their own, not much related to the real physics world. This view, however, turns out to be too narrow. The non-perturbative approach has recently opened a new horizon to strongly correlated systems in low dimensional physics and provided invaluable tools otherwise nowhere to analyze analytically. This trend will continue as non-perturbative quantum phenomena are enhanced in low dimensional physics and the importance of the research will be emphasized in physics community. During the School, lectures were made on the fundamentals and elementary techniques of the integrable field theories and also on their applications to low dimensional physics system. New results of recent years are also presented in seminars. Through the lectures and seminars the School was arranged to increase the opportunity of communication among participant scholars ranging from researchers of many years to graduate students, and from high energy physicists to condensed matter physicists. In addition, the natural atmosphere of the Cheju Island was an invaluable asset to the informal discussion, being known for her beautiful landscape and mild temperature during winter season. Organizers of this winter school thank for lecturers and participants from all over the world and especially for Asia Pacific Center for Theoretical Physics (APCTP) for her generous financial support as well as for the time consuming secretarial works. Please notice that the contributions to this proceedings are arranged in alphabetic order of the first authors.
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Contents Preface
v
Applications of Reflection Amplitudes in Toda-Type Theories C. Ahn, C. Kim and C. Rim
1
Lax Pairs and Involutive Hamiltonians for C# and BCJV Ruijsenaars-Schneider Models Kai Chen, B.-Y. Hou and W.-L. Yang
35
Fateev's Models and Their Applications D. Controzzi and A. M. Tsvelik
55
The ODE/IM Correspondence P. Dorey, C. Dunning and R. Tateo
74
Integrable Sigma Models P. Fendley
108
Lorentz Lattice Gases and Spin Chains M. J. Martins
179
Quantum Calogero-Moser Models for Any Root System R. Sasaki
195
Quasi-Particles in Conformal Field Theories for Fractional Quantum Hall Systems K. Schoutens and R. A. J. van Elburg Towards Form Factors in Finite Volume F. A. Smirnov Static and Dynamic Properties of Trapped Bose-Einstein Condensates T. Tsurumi, H. Morise and M. Wadati
241
243
269
viii
Integrability of the Calogero Model: Conserved Quantities, the Classical General Solution and the Quantum Orthogonal Basis H. Ujino, A. Nishino and M. Wadati Conformal Boundary Conditions J.-B. Zuber
289
318
1
APPLICATIONS OF R E F L E C T I O N A M P L I T U D E S IN T O D A - T Y P E THEORIES CHANGRIM AHN Department
of Physics, Ewha Seoul 120-750,
Womans Korea
Department
of Physics, Seoul National Seoul 151-742
University,
CHANJU KIM University,
CHAIHO RIM Department
of Physics, Chonbuk Chonju 561-756,
National Korea
University,
Reflection amplitudes are defined as two-point functions of certain class of conformal field theories where primary fields are given by vertex operators with real couplings. Among these, we consider (Super-)Liouville theory and simply and nonsimply laced Toda theories. In this review we explain how to compute the scaling functions of effective central charge for the models perturbed by some primary fields which maintains integrability. This new derivation of the scaling functions are compared with the results from conventional TBA approach and confirms our approach along with other non-perturbative results such as exact expressions of the on-shell masses in terms of the parameters in the action, exact free energies. Another important application of the reflection amplitudes is a computation of onepoint functions for the integrable models. Introducing functional relations between the one-point functions in terms of the reflection amplitudes, we obtain explicit expressions for simply-laced and non-simply-laced affine Toda theories. These nonperturbative results are confirmed numerically by comparing the free energies from the scaling functions with exact expressions we obtain from the one-point functions.
1
Introduction
There is a large class of 2D quantum field theories (QFTs) which can be considered as perturbed conformal field theories (CFTs) 1. These theories are completely defined if one specifies its CFT data and the relevant operator which plays the role of perturbation. The CFT data contain explicit information about ultraviolet (UV) asymptotics of the field theory while its long distance property is the subject of analysis. If a perturbed CFT contains only massive particles, it is equivalent to the relativistic scattering theory and is completely defined by specifying the S-matrix. Contrary to CFT data the S-matrix data exibit some information about long distance properties of the theory in an explicit way, while the UV asymptotics have to be derived.
2
A link between these two kinds of data would provide a good view point for understanding the general structure of 2D QFTs. In general this problem does not look tractable. Whereas the CFT data can be specified in a relatively simple way, the general S-matrix is very complicated object even in 2D. However, there exists an important class of 2D QFTs (integrable theories) where scattering theory is factorized and S-matrix can be described in great details. In this case one can apply the nonperturbative methods based on the Smatrix data. One of these methods is thermodynamic Bethe ansatz (TBA) 2 3 ' . It gives the possibility to calculate the ground state energy E(R) (or effective central charge ceff(i?)) for the system on the circle of size R. At small R the UV asymptotics of cesf(R) can be compared with that following from the CFT data. Usually the UV asymptotics for the effective central charge can be derived from the conformal perturbation theory. In this case the corrections to ceff(0) = CCFT have a form of series in R1 where 7 is defined by the dimension of perturbing operator. However, there is an important class of QFTs where the UV asymptotics of ceg(R) is mainly determined by the zero-mode dynamics (see for example 4'?-?>?). In this case the UV corrections to CCFT have the form of series in inverse powers of log(l/i?). This UV expansion is also encoded in CFT data 6 . The simplest integrable QFT with the logarithmic expansion for the effective central charge is the sinh-Gordon (ShG) model, which is an integrable deformation of Liouville conformal field theory (LFT). It was shown in paper 6 that the crucial role in the description of the zero-mode dynamics in the ShG model is played by the "reflection amplitude" of the LFT, which determine the asymptotics of the ground state wave function in this theory. (The reflection amplitudes in CFT define the linear transformations between different exponential fields, corresponding to the same primary field of chiral algebra.) In this paper, we want to show that these zero-modes dynamics are quite general features of 2D QFTs with exponential interactions. Imposing the integrability, we will consider the Bullough-Dodd model which is another integrable perturbation of the LFT, supersymmetric ShG models 7 , and affine Toda field theories (ATFTs) associated with both simply laced and non-simply laced Lie algebras 8 ' ? . Each model shows quite unique properties in the Smatrix data; some has non-diagonal S-matrix, some has many particles with different masses, and so on. In this review of our recent works 7'8>9>10; we want to describe UV asymptotic behaviours of these 'Toda-type' models, i.e. a class of the integrable QFTs with exponential interactions, in unifying way in terms of the zero-mode dynamics. The perturbing term in the model re-
3
stricts the zero-mode dynamics to a box of size / ~ log(l/i?) in the auxiliary space with dimension equal to the number of independent zero-modes. This leads to the quantization condition for the momentum P conjugated to the zero-modes and the solution P{R) determines all logarithmic terms in the UV asymptotics of the effective central charge ceff(.R). Although it may seem quite model-dependent, it turns out that the basic dynamics in common is the reflection of a zero-mode off the interacting potential of the LFT. In all cases the results agree perfectly with TBA results based on the 5-matrix data. The remarkable feature is that effective central charge calculated from the CFT data with subtracted bulk free energy term (like in TBA approach) gives a good agreement with the TBA results even outside the UV region (at R ~ 0(1)). This "empirical" fact still needs the explanation. Finally the reflection amplitudes can be used n ' 1 2 > 1 3 to find the exact onepoint functions of this class of integrable models. One needs only symmetric properties of the given Lagrangian and analytic properties of the one-point function. Explicit calculation is given for various models 6 '->-' 13 '---'-'-. In the section 2, we introduce the reflection amplitudes for the CFTs we are interested in. These amplitudes are interpreted as quantum mechanical reflections off the potential wall of the zero-modes dynamics in sect.3. In sect.4 we analyze the off-critical integrable models. Due to the integrable perturbations, the zero-modes are confined in the potential well and the conjugate momenta are quantized. Using this, we calculate the UV asymptotics for the effective central charges. In sect.5 we compare this asymptotics with numerical solutions of TBA equations. We derive the exact one-point functions and free energies in sect.6. 2
Normalization Factors and Reflection Amplitudes
In this section, we introduce the reflection amplitudes for the Toda-type CFTs with background charges. 2.1
Toda-type CFTs
We start with non-affine Toda theories (NATTs) whose actions are given by
A = jd2x i - ( ^ ) 2 + ^ Mi e" e -^
(1)
where e^, i = 1 , . . . , r are the simple roots of the Lie algebra G of rank r. For simply laced algebras, the /Vs are all the same as \i. Non-simply laced ATFTs have standard simple roots with e? = 2 and nonstandard simple roots with
4
e 2 = £ 2 (^ 2). We choose the corresponding parameters /i; as \i (for standard roots) and \J! (for nonstandard ones) respectively". The simplest case is the A\ NATT, or the LFT with an action A=fd2x
l.(d^)2
+
fie^f
(2)
With appropriate background charges, these Toda-type QFTs are the CFTs. To describe the generator of conformal symmetry we introduce the complex coordinates z = X\ + 1x2 and ~z = X\ — 1x2 and vector:
ct = bp + \p\
PV = ^ E « V '
P= ^ £ ° ' <*>0
(3)
a>0 v
where the sum in definition of Weyl vector p (p ) runs over all positive roots a (co-roots a v ) of G. The holomorphic stress-energy tensor T{z) = -\{dz
(4)
ensures the local conformal invariance of the NATT with the central charge c = r + 12Q 2 . For the simply laced algebras, these expressions are simplified due to Q = Qp with Q = b + 1/fo that the central charge is given by c = r (l + h(h + l)Q2) (h is Coxeter number). Another model we are interested in is the N = 1 supersymmetric LFT with an action A = [d2x^(da4>)2 J
- ^-(jdij
07T
Z7T
+ i>8^) + inb2ipipeb* + ^ A e 2 6 * .
(5)
2
The super-LFT with the background charge Q = b+l/b has the central charge cSL = 3/2(1 + 2Q 2 ). 2.2
Reflection
Amplitudes
NATTs Besides the conformal invariance the NATT possesses extended symmetry generated by W(G)-algebra. The full chiral W/(G)-algebra contains r holomorphic fields Wj(z) (W2(z) = T(z)) with spins j which follows the exponents of Lie algebra G. The primary fields $„, of W(G) algebra are classified by r a
W e choose the convention that the length squared of the long roots are four for C). two for the other untwisted algebras.
and
5
eigenvalues Wj, j = 1 , . . . , r of the operator Wj j0 (the zeroth Fourier component of the current Wj(z)): WJi0$w = Wj$w,
Wjtn$w
= 0,
n > 0.
(6)
The exponential fields Va(x) = e(Q+«)-v(*)
(7)
are spinless conformal primary fields with dimensions A(a) = W2(a) = (Q 2 — a 2 ) / 2 . The fields V^ are also primary with respect to all chiral algebra W(G) with the eigenvalues Wj depending on a. The functions Wj(a), which define the representation of H / (G)-algebra possess the symmetry with respect to the Weyl group W of Lie algebra G 1 6 '', i.e. Wj(sa) = Wj(a); for any s € W. It means that the fields Vsa for different s G W are reflection images of each other and are related by the linear transformation: Va(x) = Rs(a)VBa(x)
(8)
where R§{a) is the "reflection amplitude". This function is an important object in CFT and plays a crucial role in the calculation of the one-point functions in perturbed CFT 19 . To calculate the function Rg(a) for simply laced NATTs, we introduce the fields $w: $w(x)=N-1(a)Va(x)
(9)
where normalization factor N(a) is chosen in the way that field $w satisfies the conformal normalization condition <<Mz)<M2/)> = i — ^ (io) 1^ — 2/1 The normalized fields $w are invariant under reflection transformations and hence;
«•<«> = I S , -
<">
For the calculation of the normalization factor N(a), we can use the integral representation for the correlation functions of the W(G)-invariant CFT. (See 16 for details.) We note that operators Qi defined as Qi=H
ld2xebeiV^
(12)
6
commute with all of the elements of W(G)-a\gebr& and can be used as screening operators for the calculation of the correlation functions in the NATT. If parameters a satisfy the condition r
2Q + 2a + ^kiel
=0
(13)
8=1
with non-negative integer ki, we obtain form Eqs.(9) and (10) the following expression for the function N(a) in terms of Coulomb integrals 16 : N2(a) = M 4 A ( Va(x)Va(0) f [ | V ) \
i=i
(14)
I
where the expectation value in Eq.(14) is taken over the Fock vacuum of massless fields cp with the correlation functions (<Pa{x)ifib(y)) - ~^ab log \x - y\2. The normalization integral can be calculated and the result has the form: N2(a) = X
(7r^(b2))-2piQ+a)/b r ( l + Qa/b)T(l + Qab)T(l + a a / 6 ) r ( l + aab) n ^oT(l-Qa/b)r(l-Qab)r(l-aa/b)T(l-aab)
(
'
in terms of the scalar products Qa = Q • a ,
aa = a • a
(16)
where the product runs over all positive roots of Lie algebra G. We accept Eq.(15) as the proper analytical continuation of the function N2(a) for all a. It gives us the following expression for the reflection amplitude Rs(a): R s{a)
~
(17)
= mia) = I T
where a b
2 pa/ b Aa = (Ttiil{bl.2\\P )) '
I J r ( ! - aa/b)T(l
- aab).
(18)
For non-simply laced NATTs, the expression is generalized to Aa = n ^ 7 ( e ^ 2 / 2 ) r V - a / 6 J ] r ( ! - a<*v/b)T(l - aab), a>0
(19)
7
here aa = a • a, aav = a • a v and vectors w,v are the co-weights of G, satisfying the condition u)V • ej = 5ij. The reflection relation Eq.(8) can be written in more symmetric form as: AaVa(x)
= ASaVSa(x),
s e W.
(20)
In following we will be interested in the values of functions Aa for imaginary a = iP. We denote as V(P,x) = V,p(x) and A(P) = Aip. Using these objects we can construct the combination which is invariant under the Weyl reflections:
* P = J2 A(sP)V(sP).
(21)
sew From the above expression, we can find A(stP) A(P)
=
SL(et,P)
- f ^ . 7 ( e ? ^ / 2 ) 1 - g , - y / > r ( 1 + * P • e * m i + iP ' er/b) (22) - 1 ^ , 7 ^ , /2)] . (22) r ( 1 _ . p e _ f t ) r ( 1 _ . p _ ^/b) From Eq.(22), we see that the functional form of SL is independent of the algebra under consideration. Considering the simplest case {A\ Toda), we identify that SL is the reflection amplitude of LFT R(iP) = SL(V^P) where
c i P) - u,„(h2s\ -ip/» r (i + iPb) r (i + iP/b) SL(P) - fri7(& ))
T(l-iPb)T(l-iP/bY
(23)
super-LFT Due to the superconformal symmetry, the primary fields of the super-LFT can be devided into two sectors. The (NS) primary fields are given by Va(z,z,9,9)=4>a(z,z)
+
eipa(z,z)+9tpa(z,z)-ee4>a(z,z)
with dimensions Aa =
-a(Q-a),
and the (R) fields by a^Vn where CT<€) i s the 'twist field' with dimension 1/16 so that the dimension of the (R) fields are Aa =
16 + 2 a ( < 5 ~
a)
-
The reflection amplitudes of the super-LFT defined from the structure constants have been derived from the structure constants in 20 - 21 . The reflection amplitudes for the (NS) fields are
-(^r/))"
2 fcP
SNS(P)
--
and for the (R) fields SR(P)
3
=
f-Kfl
*mr :
'
r{l + iPb)T(l
+
iP\
T{l-iPb)T(l~
r ( i + iPb) T(\ + iPb)Y{\-
(24)
(25)
Reflections of Zero-Modes
In this section we introduce the wave functional interpretation of the primary fields of the LFT using the zero-mode following 6 and show how to generalize it to NATTs by considering the wave functionals $[ip(x)] whose asymptotic behaviours are described by the wave functions of the zero-modes in higher dimensions. 3.1
LFT
Consider LFT on a cylinder of circumference 2ir with the cartesian coordinates X\, X2 where x2 along the cylinder is defined as the imaginary time and X\ ~ x\ + 2ir is the space coordinate. The Hamiltonian acting in the space of states A of LFT H = - ^ + Lo + Lo
(26)
generates translations along the time x2The conformal structure can be formulated in terms of the "zero-mode" of the Liouville field <j)(x) defined by 0o = ^ % O r ) f ^
(27)
As 4>Q —> — oo in the configuration space, one can neglect the exponential interaction term in the LFT action so that one can expand
{x) as a free massless field (z = x\ + ix2) 4>{x) = <j>o-Hz-z)
+ Y,
(Ve™Z
+
Ve~m") '
(28)
9 where the m o m e n t u m conjugate to the zero-mode 4>0 and oscillators satisfy V
=
% 0 ~ o ^ ~ '
Tft [am,0.n] = -W&m+m [am,an]
l acpo
2
TTX = —Sm+n.
(29)
2
T h e Virasoro generators can be written in terms of these modes. T h e space of states is now represented as A0 = £ 2 ( - o o <4>Q < o o ) ® T
(30)
where £2 is the two-dimensional phase space spanned by 0 —> —00 asymptotic limit. In particular, the wave functional for the primary state vp corresponds t o * » P [ 0 ( S I ) ] = ( e i P 0 ° + S(P)e-iP*°)
|0)
as (h0 -»• - 0 0
(31)
where S(P) is the reflection coefficient of the asymptotic wave functional. One can check t h a t the wave functional of asymptotic form Eq.(31) has correct conformal dimension by acting LQ. T h e coefficient S(P) should be the reflection amplitude SL(P) introduced earlier since the wave functional ^y_P for the primary state V-p is SL(—P)^VP along with SL(P)SL(-P) = 1.
(32)
In this framework, one can check the validity of the reflection amplitude by taking semiclassical limit b —> 0 and using duality. Since P is of the order of 0(b), one can neglect the oscillators and keep only the zero-mode
+ 27r/ie ^ ^ .
(33)
T h e exact wave function of • —00 is given by Eq.(31) with
It is straightforward t o check this result is consistent with the non-perturbative reflection amplitude Eq.(23) perturbatively.
10 3.2
NATTs
The zero-modes of the fields — oo for alii), the fields can be expanded in terms of free field operators a n 0-V(z-z)
+ J2 (—einz + ^e-™)
,
(36)
where V = —iVVo is the conjugate momentum of ip0. In this region any state of the NATT can be decomposed into a direct product of two parts, namely, a wave function of the zero-modes and a state in Fock space generated by the operators a„. In particular, the wave functional corresponding to the primary state Eq.(21) can be expressed as a direct product of a wave function of the zero-modes
(37)
where the wave function ^p(• 0 where one can neglect the operators a n in Eq.(36) even for significant values of the parameters /ZJ . The full quantum effect can be implemented simply by introducing the exact reflection amplitudes which take into account also non-zero-mode contributions 6 . The resulting Schrodinger equation is given by
"
12
V^+j^e^o
tfp^oHiJotfp^o)
(38)
with the ground state energy E0 = - - + P2.
(39)
Here the momentum P is any continuous real vector. The effective central charge can be obtained from Eq.(39) where P 2 takes the minimal possible
11
value for the perturbed theory. Since only asymptotic form of the wave function matters, we derive the reflection amplitudes of the ATFTs in the way that we need only the LFT result. In the Hi -> 0 limit which will be of our interest, the potential vanishes almost everywhere except for the values of
ip e J--Wo
~eiP-*'o+SL(ei)P)ei*p-vo)
(40)
where S; denotes the Weyl reflection by the simple root e; and Pi the component of P along a direction. Sx,(ej,P) is defined in Eq.(22). Since the wave function interpretation makes sense only in the semiclassical limit, it is the b —• 0 limit of Eq.(23) which can be obtained from the solution of the Schrodinger equation for the LFT. We can see from Eq.(40) that the momentum of the reflected wave by the i-th wall is given by the Weyl reflection §i acting on the incoming momentum. If we consider the reflections from all the potential walls, the wave function in the asymptotic region is a superposition of the plane waves reflected by potential walls in different ways. The momenta of these waves form the orbit of the Weyl group W of the Lie algebra G; *P(
A(sP)eiip-*°.
- £
(41)
sew
This is indeed the wave function representation of the primary field (21) in the asymptotic region. It follows from Eq.(40) that the amplitudes A(P) satisfy the relations ^ = ^ (
e j
, P ) ,
(42)
which is the same as Eq.(23). Equation (42) is solved by r
A(P) = n [ w ( e ^ 2 / 2 ) r * V - p / " J ] T(l - iPab)T(l
- iPav/b),
(43)
12
and we recover the result (19) calculated in the previous section. 3.3
super-LFT
The super-LFT is a super-CFT which satisfies the usual super-Virasoro algebra. The space of states for the super-LFT can be expressed by A0 = £ 2 ( - o o < <po < oo,^o) ®F
(44)
where the fermionic zero-mode appears only for the (R) sector and T is the Fock space of bosonic and fermionic oscillators. The appearance of bosonic and fermionic zero-modes in Eq.(44) is well-known from the super-CFT results. In the (NS) sector, there is no fermionic zero-mode since the fermion field satisfies the anti-periodic boundary condition while it appears in the (R) sector with perodic one. The zero-modes appear in the super-Virasora generator LQ and So of the (R) sector in such a way that LQ contains the square of the conjugate momentum and So acts non-trivially only on the twist field. The primary state vp can be also expressed by a wave functional i$Vp[(f)(xi)] whose asymptotic form is given similarly as Eq.(40). The amplitude S(P) is either SNS(P) or SR(P) depending on the sector so that the wave functional ^V_P is given by S(—P)^vp. One can also check the validity of this expression by taking the classical limit of b —>• 0. Since P is small of order of 0(b), one can neglect the oscillator part in Eq.(44) and study only the dynamics of zero-modes. In the (NS) sector, only bosonic zero-mode appears so that the Hamiltonian becomes ff
™=-!-(&)'+"v*v"''
which is essentially the same as that of the LFT, hence the reflection amplitude becomes SNS(P)
(^VfPT(l
= - (TJ
+ iP/b)
T{1_iP/by
On the other hand, in the (R) sector, additional fermionic zero-mode is introduced in the hamiltonian by 22 H* = - i ^ - \
+ T r V V e 2 ^ + 27ri A i&Vo^e i *°.
Since the fermionic zero-mode satisfies {^o,^o} = 0,
^ 0 = ^ 0 = 2'
13
we can represent it by 1
-
1
-
i
TT/X&V^S
= (H+ £ V
and the Hamiltonian becomes
H* = V2+ TrVVe26^ The solution of H+ can be obtained as
g>+(0o) = fy^[Ki/2-iP/b{xHK1/2+iP/b(x)]
\
x =
w
^
0
where Kv(x) is the modified Bessel function. By taking the asymptotic limit 0o - • - c o , one can find the non-vanishing component is given by $ ~ eiP0° +
SR{P)e~iP4'0
with
c (p,(^Y^pn\
+ iPlb)
- \T)
Vi - iP/b) ra-iP/bY
SR{P)
These are consistent with the exact result Eq.(25) in the b —> 0 limit. 4
Quantization Conditions and Scaling Functions
In this section we derive the scaling functions for the various Toda-type models on a cylinder with circumference R. In the deep UV region R -» 0, the wave functional interpretation introduced in the previous section is used to obtain the quantization condition for the zero-mode momenta and the vacuum energies. 4-1
ShG model
We start by reviewing the analysis of 6 for the ShG model or A\ ATFT defined first on a circle of circumference R with periodic boundary condition. By rescaling the size to 2ir, one can write the action as /•27T
/
i
/ p\
2+26
(45) 2
2
dx2j dx! ^ ( ^ ) + / W | H (e^+e" ^) where fi ~ [mass]2"1"26 is the dimensional coupling constant with b the coupling constant.
14
We are interested in the ground state energy E(R) or, more conveniently, the finite-size effective central charge Ceft{R) = -—E(R)
(46)
7T
in the ultraviolet limit R —• 0. Since we are interested in the ground-state energy, only the zero-mode contribution counts. So the corresponding effective central charge at R —> 0 is determined mainly by P ceS(R) = 1 - 24P 2 + 0(R)
(47)
up to power corrections in R. For the ground state energy, one can consider only the zero-mode dynamics where the wave functional of o is confined in the potential well due to the ShG interaction term. The ShG potential introduces a quantization condition for the momentum P which depends on the finite size R. As R —>• 0, in particular, the wave functional is confined in the potential well where the potential vanishes in the most of the region and becomes nontrivial only at 2b(f>0 ~ ±ln/u(i?/27r) 2+26 near the left and right edges. Near these edges of the potential well, the potential becomes that of the LFT and the wave functional will be reflected with the reflection amplitude of the LFT introduced earlier. Therefore, the quantization condition is given by (R/2n)-4y/*iPQ
Sl(P)
= 1.
(48)
In terms of the reflection phase 6L (P) defined by SL(P) = -ei5^p\
(49)
the ground state momentum is qunatized as «5i(P) = 7r + 2 ^ P Q l n ^ .
(50)
Z7T
Thus determined quantized momentum will give the scaling function ceg(R) in the UV region by Eq.(47). To see this explicitly, one can expand the reflection phase in the odd powers of P, SL(P)
= S1(b)P + S3(b)P3 + S5(b)P5 + ...
(51)
where the coefficients can be obtained from the reflection amplitude Eq. (23) as follows: SJb) = hnb2 o
r ( 2 w ) r ( 1 + 2HW) -2Qln — —r^
+7E
(52)
15
s3(b)^-am3 h(b)
+ b'
(53)
:((5)(b5+b-5)
(54)
with Euler constant 7E- Now solving Eq.(51) iteratively, we get CeS(R)
= l+C^ + ^
C +
± + ..
(55)
where I = Si{b) d
-2V2Qln(R/2n) 2
= -6TT ,
C2 = 12TT%(&),
6
c3 = 12ir 65(b).
(56) (57)
The Gamma functions appear in <$i due to the relation between the mass of the physical particle and the coupling constant /x in the action 18 71"/Lt
m
li-b2) 4.2
VTT
1 \2 + 2b2
r
b2 i + 2 +2b2
2+26 2
(58)
ATFTs
The action of the ATFT is given by A
I
2
dzx ^(d^)
+ J2^ebe'*+fi0ebe°*
(59)
t=l
The additional potential term in the ATFT Lagrangian corresponding to the zeroth root eo introduces new potential wall in that direction (see Fig.l as a simplest example, the Ai ATFT). With this addition, the region of (p0 made of the non-afRne Toda potential walls (Weyl chamber) is now closed and the momentum of the wave function should be quantized depending on the size of the enclosed region. This quantized momentum defines the scaling function ceff in the UV region by Eq.(39). The quantization condition can be derived as follows. For simplicity let us first consider simply laced cases. Also, we assume for the moment that the circumference of the cylinder is 2TT. Consider the path C of a wave which starts with momentum P and comes back (after a series of reflections by other walls) to the zeroth potential wall with momentum P ' . It will then be reflected by the zeroth wall. Fig.1(b) illustrates a multiple reflection in the two-dimensional potential. To satisfy the self-consistency condition, the momentum P ' after the last reflection by the zeroth wall should be equal to the incoming momentum P so that SQP' = P . Furthermore, since the zeroth
16
e
o/
e
i
r
//
7P
(a)
(b)
Figure 1. (a) Potential walls in A2 afRne Toda theory, (b) A wave with momentum P near the zeroth wall comes back to the same wall with momentum P' after a series of reflections.
wall is again Liouville-type, the momenta P ' = SoP of the incident wave and P of the reflected wave should satisfy Eq.(42) which leads to A(S0P) A(P)
=
SL(e0,P).
(60)
On the other hand, since so is given by a product of the Weyl reflections corresponding to simple roots, each representing the reflection experienced by the wave along the path C, the left hand side of Eq.(60) can be obtained from Eq.(43). Therefore, Eq.(60) gives a nontrivial quantization condition for the momentum P . This condition can be generalized using the same arguments for other potential walls instead of the zeroth one. Then we obtain A(S0aP) A{SP)
SL(e0,sP).
where s is an arbitrary Weyl group element.
(61)
17
Using Eq.(18) we can write (61) 2^\ iP-sv/b
(^l(b2))
n
.a>0
G{sP-e0) G(-SP-eo)
G(sP • s0a) G(sP-a)
(62)
1
where v = s0p - p + e 0 = - ( e 0 • p)e0 + e 0 = he0 and we define a function G(P) = T (1 - iP/b) T (1 -
iPb).
The T-function factors in Eq.(62) can be further simplified. First, consider the action of s 0 on a positive root a : SQCC = a — eo(eo -a), which is either a or a + eo if a 7^ —eo since —eo is the maximal root. In the first case, the factor G(sP -s0a) in Eq.(62) is cancelled out by the same factor in the denominator, while, in the second case, there is no cancellation since a + eo is a negative root. Finally, s 0 e 0 = —e0 and the corresponding factor G(SP • e 0 ) appears twice in Eq.(61). Using the property e 0 • a = 0 or 1 for a > 0 ( a ^ — e 0 ) and eo • eo = 2, we can simplify Eq.(62) as
n
(7r/i 7 (fo ))
G(-P-sa)
(63)
= 1
G(P • set)
a>0
For the non-simply laced ATFTs, we obtain the condition for P by inserting Eqs.(43) and (22) into Eq.(61). After some transformations as above, it can be written in the form: -iiP-se^/6
2
n
n(^7(e?& /2)) ' U=o where we define
n LS(«>-P) g(q,P)
i,
(64)
a>0
G(a, P) = r ( l - iPab)T(l
-
iPav/b).
Now we consider the system defined on a cylinder with the circumference R. When we scale back the size from R to 2ir, the parameters //, in the action (59) rescales as / R \ 2+(>2^ f*i ->• Mi U ~ ) •
( 65 )
Then, Eq.(64) for the lowest energy state reduces to
LP = 2TTP-
^2aS{a,P), a>0
(66)
18
where
L = -hh + b2hw)\n— - i l n 0
Z7T
0
n(w(e?&72)r
(67)
Li=o
and 6 ( a , P ) = —ilog
T(l + iPQb)T(l + iPav/b) T(l-iPab)T(l-iPav/b)
(68)
This is the quantization condition for the momentum P in the UV region i?->0. The ground state energy with the circumference R is given by E(R) = -
with
6R
ceff = r - 12P 2
(69)
where P satisfies Eq.(66). In the UV region we can solve Eq.(66) perturbatively by expanding S(a, P) in powers of Pa, 6{a, P) = (Ji(a, b)Pa + 63(a, 6)P 3 + 65(a, b)P^
(70)
where the coefficients Si(a., b) and 6s(a, b), s = 3, 5 are:
Sx(a,b) = -2 7 B (6+ ^ ) M<*,b) = {-)** • fcW (V + ( ^ Using the relations: T,a>o(a)a(a)b we obtain that:
= ^V(5af>>
IP = 2TTP - J2 <M", 6)aP„ " ^ a>0
and
(71) E a > o ( a ) a ( a V ) 6 = Wa6,
<M", 6)aP* - • • • ,
a>0
with / = L - 2jE(bhv
+ h/b) = L - L0
(72)
The above equation can be solved iteratively in powers of \/l. Inserting the solution into Eq.(69), we find: Ceff
=
r
- r(h + l)/i v (?fj
+ ^C(3)[C 4 (G V )6 3 + C4(G)/fo3]
- ^ C ( 5 ) [ C 6 ( G V ) 6 5 + C 6 (G)/6 5 ] ( y )
+ 0(r»),
(-^ (73)
19 where the coefficients C(G) are defined as: C 4 (G v ) = ^ p l ,
C4(G) = £ p Q / £ v ,
C6(G^=J2p6a-
C6(G) = Y,P~Pl*, a>0
(74)
a>0
For simply laced algebras, these coefficients have the values:
™ ^nV-l)(2n2-3), -i-n2(n2-l)(n2-2)(3n2-5), 168 {l
3 ">\ = —(16n - 45n 2 + 27n + 8)n(n - l)(2ra - 1), 30 v 1 C6(£»{11)) = — (48n5 - 213n4 + 262n3 + 6n 2 - lOln - 32)n(n - l)(2n - 1 42
j(D For the non-simply laced algebras £?„ and C«( i ) , we can express the results through these values. Namely, we find:
CiicP)
= CiiD^),
For exceptional algebras G2
Ci(W) and F\
- jI C Ci{G22 ( l )) ^— ^ 44(D. ^ 4Vh
= Ci(D™n),
(i = 4,6). (75)
, we obtain:
392,
C6(G<1)) = |C 6 (L'i 1 ) ) = 7386,
C4(Gi 1 ) v ) = 1)V C 6 (G< ) = '2
C 4 (F 4 (1) ) = \C^E{^)
= 27378,
C4(F4^V)
C 6 (F 4 (1) ) = ICeiE™)
= 2203578,
C.(F 4 W V )
^ , 199516 243 ' 22815 4052763
(76)
We note that above equations relating coefficients Ci (G) for different Lie algebras follow from the similar exact relations between the ground state energies e(G) of quantum affine Toda chains associated with these Lie algebras. These exact relations are valid if the parameters /i, / / for non-simply laced Lie algebras and corresponding parameter \xsi for simply laced ones satisfy the condition: nh-z(2\i'H2)z = [ihsV where z = ^f1.
20
4-3
the BD model
The BD model is an integrable field theory associated with A\ ' affine Toda theory and can be regarded as an integrable perturbation of the LFT 19 . The action is given on a circle of circumference 2ir with periodic boundary condition; \.BD
= / dx2 /
s(**) , + "(s) , + "-"
dxi
n \ 2+62/2
+ /M^J
e-
(77)
This model possesses asymmetrical exponential potential terms compared with the ShG model. In the UV limit, the exponential potential becomes negligibly small except in the region where 0o goes to ±oo. This means that the BD model is again effectively described by the LFT. It is the quantization condition that makes the difference from the ShG model, due to the asymmetry of the potential well in the left and right edges. The conjugate momentum P is now quantized by the condition ,
R
x -4iP(Q+Q')
[~)
SL(P)S'L(P)
= l,
(78)
where S'L(P) is obtained by substituting 6 —> 6/2 for SL(P) given in Eq.(23) and Q = b + l/b,
Q' = 6/2 + 2/6.
(79)
Using the phase shifts defined as -ei5^p\
SL(P) =
S'L(P) ==
-ei6 AP)
the quantization condition becomes 6(P) = TT + 4QP\n^-
(80)
where 6(P) = \(SL(P)
+ 8'L(P)),
Q = \{Q + Q').
The relation between P and R in Eq.(80) gives the scaling function ceff as a continuous function of R, Eq.(73), with Q replaced by Q and J's with S's
21
defined by power series expansion of the phase shift in P 5(P) = 61P + 63P3 + 55P5 + ... S1
^lnj-2{Q
=
x <
In
+ Q') 2/i'Trr ( l + £ )
UnT (1 + b2) 2
r(-& )
+ 7E
r(-£)
53 = 3C(3) ( b3 + ~ b\ 33 5 _^ 3 2 \ Si5 = - ^ C ( 5 ) ( 6 6 X + ^5 l
(81)
6 J
Then, the central charge is given by C\
/ „s
C-r
C->
(82)
where l = d
(83)
51(b)-4Qla(R/2w)
= -24TT
2
c2 = 48TT 4 4(6)
c 3 = 4 8 * % (6). 4-4
the SShG model
Now we consider an integrable model obtained as a perturbation of the superLFT, the SShG model. By rescaling the size to 2ir, one can express the action of the SShG model by A. SShG
/ d%2 I
dx\
- ^ ( 3 « 0 ) 2 - - ^ W + lW)
R \ 1+b2 ( B \ 2+2b2 2 2 + 2inb ( — J i>i> cosh(&) + -Kfi b I — \ 2
[cosh(26(A) - 1] (84)
In the UV limit, the exponential potential becomes negligible except in the region where o goes to ±oo. This means that the SShG model is effectively described by the super-LFT as R —> 0. From the ground state energy for the
22
primary state labelled by P , the effective central charge can be obtained by CeffCR) = ^-12P2+0{R) 2
= -12P
(NS)
+ 0{R)
(R).
(85)
For the (NS) sector, P corresponding to the ground state is determined again by the quantization condition coming from the super-LFT reflection amplitudes: SNS(P)=n
+ 2QP\n^,
(86)
where (5NS (-P) is the phase factor of (NS) reflection amplitudes. This quantization condition can be solved iteratively by expanding <$NS(-P) in powers of P, SNS (P) = 6?SP + 6*SP3 + <55NSP5 + •••
(87)
*,NS = |<(3) ((-» + £
The (R) sector shows very different behaviour from the (NS). The physical meaning becomes clear if one considers the P —• 0 limit where SR(P) —>• 1 comparing with SNS(P) ->• — 1. While for the (NS) sector * p ~ 2iP<j>0 so that the quantum number n should be 1 as in Eq.(86), the wave functional for the (R) sector becomes constant corresponding to n = 0. Therefore, the quantization condition becomes SR(P) = 2QPIn ~ .
(88)
Z7T
Obvious solution is P = 0 so that ceS(R) = 0 + O(R).
(89)
In the & —»• 0 limit, one can verify this from the (R) sector zero-mode dynamics of the SShG model which is governed by the Hamiltonain H^ = - ( -— I +
4TT 2 /X 2 & 2
s i n h 2 b0 cosh b0.
(90)
23
This is a typical supersymmetric quantum mechanics problem and in general there exists a zero-energy ground-state 23 if the supersymmetry is not broken. Explicitly, the wavefunction of the state is found to be / „ — 2TTH cosh b<j>o \
*o(o)=(^
J-
0
(91)
This state is normalizable and its energy is exactly zero. Thus at least in 6 —» 0 limit, ceff is exactly zero regardless of r without any power correction. 5
Comparison with the T B A results
A standard approach to study the scaling behaviour of integrable QFTs is to solve the TBA equations. In this section we compute the scaling functions in the UV region from the TBA equations and compare them with the results in the previous section based on the reflection amplitudes. 5.1
TBAs of ATFTs and BD model
The TBA equations for the ATFTs are given by (i = 1, • • •, r) C
(TBA) eff
(R) = J 2 ^
1
/ c o s h 9 l o g ( l + e - ^ ' f i > ) dQ.
(92)
where m^'s are particles masses and functions Ci(6,R) (i = l , - - - , r ) satisfy the system of r coupled integral equations: TTHRcosh9 = 6i(8, R) + Y,
f V>ij{0 ~ e')
lo
S 0 + e- £ ' ( e '' f l ) ) ^ ,
(93)
with the kernels ifiij, equal to the logarithmic derivatives of the 5-matrices Sij(6) Of ATFTS 24,25,26,27^
Equation (93) becomes the TBA equation of BD model when r = 1 and the kernel is given by b2 f{d) = *2/ 3 (0) + *-B/ 3 (0) + *(B-2)/ 3 (0).
With
B =
1 +
.
where ,
...
4sin7rxcosh# cos 2TTX - cosh 29
,_,
24
The function E(TBA"> (R) defined from the TBA equations differs from the ground state energy E(R) of the system on the circle of size R by the bulk term: E^TBA\R) = E(R) - fR, where / is a specific bulk free energy 3 . To compare the same functions we should subtract this term from the function E(R) defined by Eq.(73) i.e.
&BA,(*)=c£A)(*) + —/(G).
05)
7T
The specific bulk free energy f(G) can be calculated by Bethe Ansatz method. For ATFTs 28,29,9,10^ m2sin(7i7/i)
-/^ /(G)
m S[n{i /H) f
l
^
flr) J { )
^
= 8Sin(,B/h)Sin(!(l-B)/h)
:
,^^
° =A D E
'
„
G = B?\C?\ J
T
8sin(7rS/H) sin(7r(l - B)/H) '
m 2 cos(7r(l/3-l/g))
_
S6neS
T
'
_
16cos(jr/6)sin(7rB/ff)sin(7r(l-B)/H)'
(1) u
'
(1)
2 > ^4 , l»°;
where
^-rr^'
ff
-TT62-
(97)
For the BD model 19 , f = —r • 08) 16-v/3sin(7rB/6)sin(7r(2-B)/6 The contribution of bulk term / ( G ) becomes quite essential at R ~ 0(1). The TBA equations (93) are solved numerically for various algebras. The effective central charge Cgff (R) is then computed from Eq.(92) for many different values of parameter TnR. After taking into account the bulk term, the numerical solution for Cgff (R) is fitted with the expansions (73) in l/l considering the coeffiecients as fitting parameters. To compare the numerical TBA results with analytical ones from reflection amplitudes, we need to know the exact relations between parameters Hi of the action and masses of particles m,. This is because TBA equations are derived from S-matrix data while the method of reflection amplitudes deals with the paramters of the action directly. These "mass-//" relations 18>29>9 are given in the Appendix. With the help of them, we can express ceff(R) obtained in the previous section purely in terms of particles masses m^. For
25
example, the function L(R) of ATFTs defined in Eq.(67) becomes -T(h + b
b2hv)\n ? M G ) r
(i_£)r(1 + |
-ln(6 2 "(C 2 /2) z ).
(99) Similarly we can rewrite fi and / / in (81) in terms of the particle mass m as, 6 b2 S1 = ^ln--2(Q
+ Q') In
-
r-/,^
x
2V5r(|)
- + 7E
(100)
Comparing the results, we found that, up to the order l// 7 , the numerical TBA results are in excellent agreement with the analytic results given in previous sections. (For the details of the comparison, see 8 and 9 .) To see the agreement more concretely, we plot the functions ceff (R) and ce({ (R) for non-simply laced ATFTs setting B — 0.5. The first function is computed numerically from TBA equations. The second one is calculated using Eqs.(66) and (69), based on the reflection amplitudes, with taking into account the bulk free energy term according to Eq.(95). Figure 2 shows that, for all models, the two curves are almost identical without essential difference in the graphs even at R ~ (9(1). This good agreement outside the UV region looks not to be accidental. However, at present, we have no satisfactory explanation of this interesting phenomena in ATFTs. 5.2
TBA of SShG model
Finally we briefly describe the TBA analysis of SShG model (see 7 for details). There are two sectors in SShG model and the corresponding TBA equations are different from each other. They can be written as 7 . ceff(r) = ^
fcosh9\n(l±e-ei^de
(101)
where the pseudo-energies are the solution of the equations, —-(0-6l')ln[l±e-£2(o'>], rift'
/
|%(0-0')m[l±e-^'>].
(102)
In the above equations, the plus (minus) sign corresponds to the NS (R) sector and the kernel is given by tp(B) = - * B ( 0 )
with B = ^ 5 1 ,
26
0.01
0.001
0.1
1
R
o
i
0.001
• —
• —
0.01
0.1
•
•
' • • '
1
R
Figure 2. (a) Plot of c eff for A 2 , A 3 , A 4 , D4, A 6 , E6 ATFTs and BD model at B = 0.5. (b) Plot of c<™ A ) for C(21}, C f ) , B ^ 1 ) , G21} and F 4 ( 1 ) ATFTs at B = 0.5. As an example, we also display c ^ A ' for C2
calculated without taking into account the bulk term.
The difference between this function and c^ff according to Eq.(95).
gives the bulk free energy of C2
ATFT
27
where $ B ( # ) is defined in Eq.(94). In the case of NS sector, we can perform the numerical analysis following the method described before and again we have an excellent agreement between the results of TBA and reflection amplitudes. On the other hand, we find that the R sector TBA equation is trivial, giving vanishing ceff(R). This is consistent with the existence of the supersymmetric zero mode as explained in Sec. 4.3. 6
Vacuum expectation value of exponential operator
The reflection amplitude, being the quantity derived from CFT, plays a crucial role in the calculation of one-point functions in perturbed CFT. In this section, we will demonstrate how powerful this method is by constructing explicitly the one-point funtion of ATFTs. One point function G(a) we are considering is defined as the vacuum expectaion value of the vertex operator Va{x) = exp(a • tp(x)) of the ATFTs (59) : G(a) = (Va(x)) = (exp(a •
(103)
We start with the one point function of the simplest case, sinh-Gordon model, A[1] ATFT 11,? and Bullough-Dodd model, A^K This approach is generalized to ATFTs 13-?.?>?. This same approach was also applied to twoparameter family of integrable models 15 . 6.1
One-point function of sinh-Gordon model and Bullough-Dodd model
The sinh-Gordon model (45) has the one component field, which can be considered as the perturbed LFT. The operator VsG(x) = exp(%/2a0(:r)) satisfies the "reflection relation" exp(V2a(x)) = RsG(a) exp(V2(Q - a)(x)).
(104)
We put a —> \[2a in the vertex operator for convenience. RsG(a) is given in terms of Liouville reflection amplitude, SL(V^P) in Eq. (23). The vacuum expectation value GSG (a) of the sinh-Gordon model is considered to satisfy the same reflection relation, GsG(a) = RsG(a)GsG(Q
- a)
(105)
In addition, since the sinh-Gordon model is invariant under the parity transformation <{> —> —0, the one-point function is even function of a. GsG(-a)
= GsG(a).
(106)
This symmetric consideration determines GsG(a) up to a periodic function.
28
The one-point function of the sinh-Gordon model is given as the minimal solution of Eqs. (105, 106). GsG(a) = ( - 7TM7(1 + b2))
_ ? T
^ x exp J j (2a2e~2t - T{a, tj) (107)
with
T{a,t) =
sinh 2 (2abt) L2sinh(i) sinh(6 2 i) sinh((l + b2)t)
(108)
The result is checked by various consideration including the classical equation, free energy of the theory and perturbative calculation n . For the Bullough-Dodd model (77), the one-point function GBD(%) = (exp(2a(f)(x))) is similarly found 13 . (Here the factor 2 is included in the exponential again for convenience. This has to be related with the normalization of the action.) GBD{O)
= RBD{O)GBD{Q
-
(109)
a)
where RBD — SL(2P). However, the parity symmetry is no longer the symmetry of BD. Instead, we have GBD{O)
= R'BD{a)GBD(Q'
- a).
(110)
where Q' is given in Eq. (79) and R'BD(a) = S'L(2P) as in Eq. (78). From this one obtains the one-point function of the BD model as
GBD{O)
x exp
I
™r(I+5jfc ) r ( e w )
=
2*V5r(|) 2
+°° dt / (
sinh((2 + 6 )i)*(i,a) sinh (3(2 + b2)t) sinh(2<) sinh (b2t)
+ 2a2e~2t
(111)
where tf (*, a) = sinh (2abt) [sinh ((4 + b2 + 2ab)t) - sinh ((2 + 262 - 2ab)t) + sinh ((2 + b2 + 2ab)t) - sinh ((2 + b2 - 2ab)t) - sinh ((2 - b2 + 2ab)t)' and m is given in Appendix.
29 6.2
One-point function in simply-laced ATFTs
Simply laced ATFTs can be considered as the obvious generalization of the sinh-Gordon model. The complexity comes from the fact that more than two operators are of the same conformal dimension and satisfy the reflection relation 14,? . The Weyl reflection of Lie algebra space takes over the simple parity reflection in sinh-Gordon model. We summarize here the general rule for the one-point function to satisfy the requirements: • [Rl] Analytic property: G(a) is meromorphic in a. • [R2] Normalization: G{a = 0) = 1. • [R3] Reflection Relation from (8): G(Q + iP) = Rs(a)G{Q
+ isP)
• [R4] Symmetry of the system: G(ra) = G(a). r is the symmetry operation of the action. The minimal solution satisfying these requirements is given in G ( a ) = ( - 7 r M 7 ( l + ^ ) ) _ 5 ? ^ T * exp f j (a2e~2t
14>
- ^(o,t))
' (112)
with ,
s
r sinh((6 2 + l k ) 1 Lsinh(^) smh(b2t) sinh((l + b2)ht) J
,
,
,
,
and
I(a, *) = J Z [ s i n h ( Q • abt)sinh
(( a • ab ~ 2a.Q6 + h(l + b2)t) 1 . (114)
«>0
One may simply prove that [Rl] and [R2] are satisfied. Requirement [R3] is checked by confirming the result for the Weyl reflection Si with respect to any simple root e^ : G(Q + iPi) = SL{eil,P)
G{Q + isiP)
where Pi = P • e^. Here are two ingredients to be checked, a2 and which under the reflection, result in (Q + i P ) 2 - (Q + isiP)2 = 2i(b + i ) P i = 2iQPi o
(115) I(a,t),
(116)
30
and I(a,t)
= / ( Q + iP) - 7(Q + iSiP)
(117)
2
2
= - (cosh[2biPit + h(l + b )t] - cosh[-26iP^ + h{\ + b )i\) = smh{2biPit) sinh/i(l + b2)t. Combining this two reflection property, G(a) satisfies the Eq. (115). The symmetry operation r in the the requirement [R4] is the Dinkyn diagram symmetry of ADE series. Obviously a2 is invariant under r. One needs to prove that I{a,t) is invariant. Let us rearrange I(a,t) in Eq. (113) as I(a,t)
= -[J{a,t)-J(ai=0,t)]
(118)
where J(a, t) = ^2
cosh
i2b(a -a-Q-a)t
+ h(l + b2)t)\,.
(119)
a>0
The problem reduces to prove J(a, t) invariant under r. Under r, if a positive root goes to another positive root, this only reshuffles the terms in J(a,t). However, there are cases where a postive root goes to a negative root, —/3 = TOL. Then apart from the reshuffling, J(a,t) contains the term, cosh (2ba • (-/3)f - 26Q • at + h{\ + b2)t)
(120)
This is the case when r changes a simple root aT to zeroth root eo : raT = eo. Noting that in ADE series, any postive root can contain at most one such a root a r , and the root satisfies an identity, p- {ex - ret) = p • (aT - e 0 ) = h.
(121)
From this one can find a unique positive root a satisfying 2foQ • a - h(l + b2) = - 2 6 Q • 0 + h{l + b2) and therefore, J(a,t) 6.3
(122)
is invariant under r.
One-point function in nonsimply-laced
ATFTs
One-point function of non-simply laced ATFTs can be obtained in the same way as follows 9 '': G{a) = K(a) x exp f — ( a 2 e " 2 t - ^ ( a , *))
(123)
31
^here ,9i
,2
(a.*)(<*.ab~2a.Qb
M
ifW^K-Mll^))"
+ HQb)
W
]
(124)
a>0
and ^
r sinh(a.ata) sinh ((a.ab - 2a.Q6 + HQb)i)
^o
L
smh
W
sinh(^-Lt) s i n h ( ( % ^ + l)t) sinh(tfQfa)
where Qb=l + b2, B = ^ and H = h(l - B) + hyB. It is simple to show that G(a) in Eq. (123) satisfies the requirements [Rl] and [R2]. For the requirements of [R3], we can proceed same as in the simply laced case for a2 and T parts. Only the factor K(a) needs care. The exponent of the factor K La(a)
= (a-a)(a-a-2a.Q
+ HQ)
(125)
is rewritten as L Q (a) = ( a - a - ( a - Q - ^ ) )
2
- ( a - Q - ^ )
2
.
(126)
Under the Weyl reflection, one has La(Q + zP) - La(Q + i§iP, a ) = i P • ( a - hot)
(127)
Now the Weyl reflection changes positive roots to another positive root with the same length except the simple root on, SiOii = —on, we have the ration of K(a)'s as K(Q + iP) = ( " W ( 1 + ^ ) ) " ^ KiQ + iStP)
(128)
which proves the requirement [R3]. Finally, if the Dynkin diagram symmetry operation r changes a positive root to another positive root, this will reshuffle the multiplication factors in G(a). On the other hand, When TOL becomes a negative root —/? = TOL, one may apply the similar argument of the simply laced case and find G(a) is invariant. Here one needs the identities, (a + 0)• pv = h and (a+(3)-p — hy. Of course, the symmetry operation requirements [R1]-[R4] do not guarantee that G(oc) is the correct one-point function. The solution we have is the minimal solution which is meromorphic function of a . In fact, one can show
32
that this one-point function G(a) does give the correct bulk free energy in Eq. (96) 9 . The one-point function of the BD model introduced earlier can be reproduced since it is the dual theory of nonsimply-laced ATFTs, A\ '. Furthermore, one may check that the result is consistent with the perturbative calculation 10 . Acknowledgement We thank P. Baseilhac and V. Fateev for fruitful collaborations and F. Smirnov and Al. Zamolodchikov for valuable discussions. This work is supported in part by MOST 98-N6-01-01-A-05 (CA), and KOSEF 1999-2-112001-5(CA,CR). Appendix: Mass-/i relations In the BD model, the parameters fi and fi' in the action are related to thte mass of on-shell particle by 19 2 r
^r(i)
6+3i> 2
b2)
UTTT (1 +
r(-f)
r(-62)
( 1 + 6+36*) r (e+lfc*)
2/i'7rr(l + g ) (129)
For SShG model, m-fi relation is given in 1 + b2 2^7V 2
15
sin
TT62_
(130)
The spectrum of simply laced ATFTs consists of r particles with the masses rrii (i = 1 , . . . , r) given by rxii = mvi
(131)
where m
=
^?m?
(132)
i=\
and v\ are the eigenvalues of the mass matrix: r
Mab = X>*(e')a(e*)6 + M a M 6
(133)
33
The parameter m characterizing the spectrum of physical particles can be related with the parameter /j, in the action using Bethe ansatz method 18,29 and the result is /
i
-7T/i7(l + 6 2 ) =
\
/
,.2
\n2(1+(>2)
(134)
2r(l//i)
where l/2h
HG)=
n<"
(135)
In the case of non-simply laced ATFTs, the exact mass ratios are different from the classical ones and get quantum corrections 2 6 ' ? . Using the notation (97), the spectrum of ATFTs are expressed in terms of one mass parameter m as: Mr = m,
Ma = 2msm{-Ka/H),
Ma = 2m sm(ira/H),
a = 1, 2 , . . . , r — 1
a = 1,2,..., r
G.( i )
Mi = m,
M 2 = 2mcos(7r(l/3 - l/ff))
?M
Mi = m,
M 2 = 2mcos(7r(l/3 - l/ff)),
M 3 = 2m COS(TT(1/6 - 1/H)),
M 4 = 2M 2 cos(ir/H).
(136)
Again, m can be written as a function of the parameter in as 9 , 2
2
n [ - w ( l + e 6 /2)r =
^r(I^)r(l+|
2i/(l+6 2
i=0
(137) where &(G) is a function depending on the algebra: n-2/H
o^B/H
r(i/H)' r(2/3) (i). _ r(2/3) fc(G<x)) = 2 r ( i / 2 ) r ( i / 6 + i/H)' HFn 2T(l/2)T(l/6+l/H)
r(i//f)'
References 1. A. B. Zamolodchikov, Int. J. Mod. Phys. A4 (1989) 4235. 2. C. N. Yang and C. P. Yang, J. Math. Phys., 10 (1969) 1115. 3. Al. B. Zamolodchikov, Nucl. Phys. B342 (1990) 695.
(138)
34
4. Al. B. Zamolodchikov, "Resonance factorized scattering and roaming trajectories", prepreint ENS-LPS-335 (1991). 5. V. A. Fateev, E. Onofri, and Al. B. Zamolodchikov, Nucl. Phys. B406 (1993) 521. 6. A. B. Zamolodchikov and Al. B. Zamolodchikov, Nucl. Phys. B477 (1996) 577. 7. C. Ahn, C. Kim, and C. Rim, Nucl. Phys. B556 (1999) 505. 8. C. Ahn, V. Fateev, C. Kim, C. Rim and B. Yang, Nucl. Phys. B565 (2000) 611. 9. C. Ahn, P. Baseilhac, V. Fateev, C. Kim, and C. Rim, Phys. Lett. B481 (2000) 114. 10. C. Ahn, P. Baseilhac, C. Kim, and C. Rim, in preparation. 11. S. Lukyanov and A. Zamolodchikov, Nucl. Phys. B493 (1997) 571. 12. V. Fateev, S. Lukyanov, A. Zamolodchikov and Al. Zamolodchikov, Phys. Lett. B406 (1997) 83. 13. V. Fateev, S. Lukyanov, A. Zamolodchikov and Al. Zamolodchikov, Nucl. Phys. B516 (1998) 562. 14. V. Fateev, Mod. Phys. Lett. A15 (2000) 259. 15. P. Baseilhac and V. A. Fateev, Nucl. Phys. B532 (1998) 567; Al. B. Zamolodchikov, in private communications. 16. V. Fateev and S. Lukyanov, Sov. Sci. Rev. A212 (Physics) (1990) 212. 17. P. Bouwknegt and K. Schoutens, "W-symmetry" Singapore, World Scientific (1995). 18. Al. B. Zamolodchikov, Int. J. Mod. Phys. A10 (1995) 1125. 19. V. Fateev, S. Lukyanov, A. Zamolodchikov, and Al. Zamolodchikov, Phys. Lett. B406 (1997) 83. 20. R. C. Rashkov and M. Stanishkov, Phys. Lett. B380 (1996) 49. 21. R. H. Poghossian, Nucl.Phys. B496 (1997) 451. 22. T. Curtright and G. Ghandour, Phys. Lett. B136 (1984) 50. 23. E. Witten, Nucl. Phys. B185 (1981) 513. 24. A. Arinstein, V. Fateev, and A. Zamolodchikov, Phys. Lett. B87 (1979) 389. 25. H. W. Braden, E. Corrigan, P. E. Dorey, and R. Sasaki, Nucl. Phys. B338 (1990) 689. 26. G. W. Delius, M. T. Grisaru, S. Penati, and D. Zanon, Phys. Lett. B256 (1991) 164; C. Destri, H. de Vega, and V. Fateev, Phys. Lett. B256 (1991) 173. 27. E. Corrigan, P. E. Dorey and R. Sasaki, Nucl. Phys. B408 (1993) 579. 28. C. Destri and H. de Vega, Nucl. Phys. B358 (1991) 251. 29. V. A. Fateev, Phys. Lett. B324 (1994) 45.
35
LAX PAIRS A N D INVOLUTIVE HAMILTONIANS FOR CN A N D BCN R U I J S E N A A R S - S C H N E I D E R MODELS KAI CHEN, BO-YU HOU AND WEN-LI YANG Institute of Modern Physics, Northwest University Xian 710069, China E-mail: [email protected] We study the elliptic Cn and BCn Ruijsenaars-Schneider models. The Lax pairs for these models are constructed and the involutive Hamiltonians are also given. Taking nonrelativistic limit, we also get the Lax pairs of the corresponding CalogeroMoser systems.
1
Introduction
Ruijsenaars-Schneider(RS) and Calogero-Moser(CM) models as integrable many-body models recently have attracted remarkable attention and have been extensively studied. They describe one-dimensional n-particle system with pairwise interaction. Their importance lies in various fields ranging from lattice models in statistics physics 1 ' 2 , the field theory to gauge theory 3 ' 4 , e.g. to the Seiberg-Witten theory 5 et al. Recently, the Lax pairs for the CM models in various root system have been given by Olshanetsky et al 6 , Inozemtsev7, Bordner et al 8 , 9 ' 1 0 ' 1 1 , D'Hoker et al 12 and Hurtubise et al with more general algebra-geometric construction 13 with or without spectral parameter respectively, while the commutative operators for RS model based on various type Lie algebra given by Komori 14 - 15 ; Diejen 16 ' 17 and Hasegawa 1,18 et al. An interesting result is that in Refs. 19 ' 20 , the authors show that for the A^v-i-type RS and CM models exist the same non-dynamical r-matrix structure compared with the usual dynamical ones. On the other hand, similar to Hasegawa's result that AJV-I RS model is related to the Zn Sklyanin algebra, they also reveal that corresponding CM model's integrability can be depicted by SZJV Gaudin algebra 21 . As for the C„ type RS model, commuting difference operators acting on the space of functions on the Ci type weight space have been constructed by Hasegawa et al in 18 . Extending that work, the diagonalization of elliptic difference system of that type has been studied by Kikuchi in 22 . Despite of the fact that the Lax pairs for CM models have been proposed for general Lie algebra even for all of the finite reflection groups 11 , however, the Lax integrability of RS model are not clear except only for AJV-I -type 2 3 ' 2 ' 2 4 ' 2 5 ' 2 6 , 2 7 , i.e. the Lax pairs for the RS models other than A^-i -type have not yet been
36
obtained. Extending the work of Ref. 28 , the main purpose of the present paper is to provide the Lax pairs for the Cn and BCn Ruijsenaars-Schneider(iJS) models with the elliptic interaction potentials. The key technique we used is Dirac's method on the system imposed by some constraints. We shall give the explicit forms of Lax pairs for these systems. It is turned out that the Cn and BCn RS systems can be obtain by Hamiltonian reduction of A2n-\ and A^n ones. We calculate involutive Hamiltonians, which can be generated characteristic polynomials of the Lax matrix. In particular, taking their non-relativistic limit, we shall recover the systems of corresponding CM types. The paper is organized as follows. The basic materials about AM-\ RS model are reviewed in Section 2, where we propose a Lax pair associating with Hamiltonian which has a reflection symmetry with respect to the particles in the origin. This includes construction of Lax pair for A^-i RS system together with its whose symmetry analysis etc al. The main results are showed in Section 3 and 4. We shall propose the explicit form for the Lax pairs for the Cn and BCn RS models by reducing AN-\ RS models. The characteristic polynomials, which gives the complete sets of involutive constant motions for these systems, will be given. Section 5 is to show the various limits of the elliptic case: the trigonometric, hyperbolic and rational cases. In Section 6, we derive the nonrelativistic limits of these systems. The last section is brief summary and some discussions. 2
Ajv-i-type
Ruijsenaars-Schneider model
As a relativistic-invariant generalization of the Ajv-i-type nonrelativistic Calogero-Moser model, the Af^-i-type Ruijsenaars-Schneider systems are completely integrable whose integrability are first showed by Ruijsenaars 23 ' 29 . The Lax pairs for this model have been constructed in Refs. 23,2,24,25,26,27 Recent progress have showed that the compactification of higher dimension SUSY Yang-Mills theory and Seiberg-Witten theory can be described by this model 5 . Instanton correction of prepotential associated with s/2 RS system have been calculated in Ref. 30 . 2.1
The Lax operator for AN-I RS model
Let us briefly give the basics of this model. In terms of the canonical variables Pi, Xi(i,j = 1 , . . . , N) enjoying in the canonical Poisson bracket {Pi,Pj} - {xi,Xj} = 0,
{xitPj} = Sij,
(1)
37
we give firstly the Hamiltonian of ^4jv-i RS system N
(2) 1=1
where a{x — 7)
f(x) : = a(x)
g(x) : = /(x)| 7 _ > _ 7 ,
•Eifc • — Xj
Xk 5
and 7 denotes the coupling constant. Here,
X
J ] ( l - J ) exp wer\{o}
1 .X . 9
(3)
H - ( —)
where T = 2wiZ + 2W3Z is the corresponding period lattice. Defining a third dependent quasiperiod 2w2 = — 2wi — 2w3, one can have a{x + 2ujk) = -<j(x)e2^x+u"<\
Q{x + 2ojk) = C(x) + 2Vk,
k = 1,2,3,
where
C(*)
a'(x)
p^) = -C(x),
and % = ((wfc) satisfy ^ w3 - 773 wi = ^ . Notice that in Ref. 2 3 Ruijsenaars used another "gauge" of the momenta such that two are connected by the following canonical transformation: Xi—> Xi,
1 N f(Xij) + -/nJJ
pi—^Pi
2'" f£ g{xijY
(4)
The Lax matrix for this model has the form (for the general elliptic case) W,N
N x - ^ <&(xi — Xj + 7 , A) *j=i
.
. , ,_,
(5)
$ ( 7 , A)
where ( x A :=
-' ^
g(i)g(A)'
bj : =
n
- ^ ~
x
^'
(^J)*'=^J*-
38
It is shown in Ref. 26 that the Lax operator satisfies the quadratic fundamental Poisson bracket {Li,L 2 } — L\ L2 a% - a2 L\ L2 + L2 «i L\ - L\ s2 L2, where L\ = LAN_1 ® Id,
(6)
L2 = Id® LAN_1 and the four matrices read as
a,\ = a + w, Si = s — w, a2 = a + s — s* — w, s2 = s* + w.
(7)
The forms of a, s, w are N
a(A, n) = -C(A - /i) ]T] Ekk <8> Ekk - ^
$(XJ - xk,X-
n)Ejk ® £fcj-,
TV
s(X) = C(A) ^
£ t t ®Ekk + Y^ *(
w = ^2 C(xk - Xj)Ekk Ejj.
(8)
k+j
The * symbol means N
r* = n>n with II = ^
Ekj JS,-*.
Noticing that rm-i 1 h
=
i
ah + \)a(X + (N-lh) a{X)a(\ + Nj)
E
$(xi - Xj - 7,A + N'y)
.
.,<
tij—*•
b'j • = ~[[g{xj -xk),
(9)
one can get the characteristic polynomials of LAN_1 and L^1
det(L(\)-v.id) j=o = J2^,xr3(-v)N-j4h
t
x g(
tm 7) '
(10)
39
where(i?5fc)j4„_1 = (H„)AN_1
(^u_,=
= 1 and
ex
E JC{1,...,JV}
(Hi-)^.! = E Jc(i
II /(**-**). (12)
p(E^) \jgjr
/
>6J
ex
pE-^
N}
\j€J V
\J\ = i
I '
n
J
ff^--1*)-
(i3)
'£
*6{l....,iV}\J
Define (B,)A»-,
= {H+)AN_,
+ (HfU^,
(14)
from the fundamental Poisson bracket Eq.(6), we can verify that {(Hi)A„-1,(Hj)A„-1}
= {(Hf)Afl_1,(H?)AN_1} e,e' = ±,
= 0,
i,j = l,...,N.
(15)
In particular, the Hamiltonian Eq.(2) can be rewritten as TV
HAN_X
=H,
= (H+)AN^
+ (HrUs-i
= J2(eP,bi
+ e Plb
~ 'j)
It should be remarked the set of integrals of motion Eq.(14) have a reflection symmetry which is the key property for the later reduction to Cn and BCn cases, i.e. if we set
Pi <—> -Pi,
then the Hamiltonians flows (Hi)AN_1 metry.
Xi <—> -Xi,
(17)
are invariant with respect to this sym-
40
2.2
The construction of Lax pair for the AN~\ RS model
As for the A^-i RS model, a generalized Lax pair has been given in Refs. 23,2,24,25,26,27 g u t ^jjgj-g j s a c o m m o n character that the time-evolution of the Lax matrix LAN_X is associated with the Hamiltonian (H+)AN_1We will see in the next section that the Lax pair can't reduce from that kind of forms directly. Instead, we give a new Lax pair which the evolution of LAN_1 are associated with the Hamiltonian HAN_1 LAN.X
where
MAN_X
= {LAN^,HAN_X}
= [MAN.^LAN.,],
(18)
can be constructed with the help of (r,s) matrices as follows
*,,., = r,((s, -M)(1.(W) - ^ +°ffff++(y' ,h)LW-))). (19)
The explicit expression of entries for MAN_1 is Mi:j = ^(xij,X)ePjbj
- <S>(Xij,\ +
N^e-P'b'j,
MH = (C(A) + C(7))ep'6i - (C(A + 7) - C W K " ^
+ J2 ((Cfc; + 7) - Cixi^y'bj
+ {
X) {Xii x+N r)e tb * 7(?x ' ~' d*(7,A)* >
3
(20)
Hamiltonian reduction of C„ and BCn RS models from ^4/v-i-type ones
Let us first mention some results about the integrability of Hamiltonian (2). In Ref. 29 Ruijsenaars demonstrated that the symplectic structure of Cn and BCn type RS systems can be proved integrable by embedding their phase space to a submanifold of A2n-i and A2n type RS ones respectively, while in Refs. 16,17 and 15 , Diejen and Komori, respectively, gave a series of commuting difference operators which led to their quantum integrability. However, there are not any results about their Lax representations so far, i.e. the explicit forms of the Lax matrixes L, associated with a M (respectively) which ensure their Lax integrability, haven't been proposed up to now except for the special case of C2 28 . In this section, we concentrate our treatment to the exhibition of the explicit forms for general Cn and BCn RS systems. Therefore, some
41
previous results, as well as new results, could now be obtained in a more straightforward manner by using the Lax pairs. For the convenience of analysis of symmetry, let us first give vector representation of AN- i Lie algebra. Introducing an N dimensional orthonormal basis of RN ej-ek=$j,k,
j , k = l,...,N.
(21)
Then the sets of roots and vector weights are: A={
e j
-e, :
A={ej:
j,k = l,...,N},
(22)
j = l,...,N}.
(23)
The dynamical variables are canonical coordinates {XJ} and their canonical conjugate momenta {pj} with the Poisson brackets of Eq.(l) . In general sense, we denote them by N dimensional vectors x and p,
x = (xi,...,xN)
e K , p = (pi,...,pN)
eM. ,
so that the scalar products of x and p with the roots a • x, p • ft, etc. can be defined. The Hamiltonian Eq.(2) can be rewritten as HAN-!
= ^2 I exp(n-p) M6A \
Y[
f(P-x)+exP(-V-p)
A3/3=^-i/
II AB0=-ft+i>
9(P-x)\, J
(24) Here, the condition A 9 ft = p. — v means that the summation is over roots ft such that for 3u £ A p-v
= ft £ A.
So does for A 9 ft = —/i + v. 3.1
Cn model
The set of Cn roots consists of two parts, long roots and short roots: AC„=ALUAS,
(25)
in which the roots are conveniently expressed in terms of an orthonormal basis of Rn: AL = {±2ej: As = {±ej±ek,:
j = l,...,n}, j , k = 1,... ,n}.
In the vector representation, vector weights A are
(26)
42 A
c„ ={ej,-ej
:
j = l,...,n}.
(27)
The Hamiltonian of Cn model is given by Hc
"
=
2 *52 (exP(M-p) x
H
II
f(P-x)+exp{-(j,-p)
gW-x)).
Ac„9/3=-M+"
(28)
/
From the above data, we notice that either for A^-i or Cn Lie algebra, any root a € A can be constructed in terms with vector weights as a = (i — v where p., v £ A. By simple comparison of representation between AN-\ or Cn, one can found that if replacing ej+n with — ej in the vector weights of A2n-i algebra, we can obtain the vector weights of Cn one. Also does for the corresponding roots. This hints us it is possible to get the Cn model by this kind of reduction. For A2n-i model let us set restrictions on the vector weights with ej+n + ej = 0,
for j = 1 , . . . , n,
which correspond to the following constraints on the phase space of type RS model with (_rj = ^6j_)_n -f- Ci) ' X — Xi + Xi-\-n
(29) A2n-i-
— U,
Gi+n = (ei+n + ei) -p = Pi +Pi+n = 0 , i-l,...,n,
(30)
31
Following Dirac's method , we can show {GUHA^-O,
for
Vte{l,...,2n},
(31)
i.e. HA2„-! is the first class Hamiltonian corresponding to the above constraints Eq. (30). Here the symbol ~ represents that, only after calculating the result of left side of the identity, could we use the conditions of constraints. It should be pointed out that the most necessary condition ensuring the Eq. (31) is the symmetry property Eq. (17) for the Hamiltonian Eq. (2). So that for arbitrary dynamical variable A, we have A = {A^HA^D
~{A,HA2n_,},
= {A^HA^}
-
{A^^Ar^Gj^A^}
i,j = l,...,2n,
(32)
43
where AlJ = {Gl,GJ}=2^_0Id™y
(33)
and the {,}D denote the Dirac bracket. By straightforward calculation, we have the nonzero Dirac brackets of {Xi,Pj}D
{xi,pj+n}D
= {xi+n,Pj+n}D
= {xi+n,pj}D
=^j.j,
(34)
= - 2&,:>'•
Using the above data together with the fact that HAN_1 is the first class Hamiltonian (see Eq. (31), we can directly obtain Lax representation of Cn RS model by imposing constraints Gk on Eq. (18)
{-k/tsn-i'-ft^n-iJ-D = {^yl 2n _ 1 ,i?>l2„_i}|Gfc,*=l,...,2n,
{LA2n_1,HA2_1}D
= [MA2„_„LA^_A\Gh,k=l,...,2n = {LCn,HCn},
= [MCn,LCJ,
(35) (36)
where Hcn
-
Lc„
—
2^2n-i|Gfc,fc=l,...,2n, LA2n-i\Gk,k=l,...,2n,
MCn =Mj42„_1|Gfc,fc=l,...,2n,
(37)
so that Lcn = {LCn ,HCJ = [MCn ,LCJ.
(38)
Nevertheless, the (H^~)AN_1 is not the first class Hamiltonian, so the Lax pair given by many authors previously can't reduce to Cn case directly by this way. 3.2
BCn model
The BCn root system consists of three parts, long, middle and short roots: ABC„ = A x , U A u A s ,
(39)
44
in which the roots are conveniently expressed in terms of an orthonormal basis of Rn: AL = {±2ej: A = {±ej±ek: As = {±ej:
j = l,...,n}, j,k = 1,... ,n}, j = l,...,n}.
(40)
In the vector representation, vector weights A can be ABCn={ej,-ej,0:
j = l,...,n}.
(41)
The Hamiltonian of BCn model is given by H
BC„
=2 ]C
x
[expiVP)
J]
II
f(0-x)+exp(-n-p)
9(fi-x)\.
ABc„9/3=-M+'/
(42)
/
By similar comparison of representations between AN_I or BCn, one can found that if replacing e J + n with — e,- and e2ra+iwith 0 in the vector weights of A-in Lie algebra, we can obtain the vector weights of BCn one. Also does for the corresponding roots. So by the same procedure as Cn model, it is expected to get the Lax representation of BCn model. For Ain model, we set restrictions on the vector weights with ej+n + ej = 0 ,
for j = 1 , . . . , n,
e2n+i = 0,
(43)
which correspond to the following constraints on the phase space of ^42n"type RS model with CTJ =z y€i-\~n "r 6jJ • X — Xj -r Xi±n
Gi+n = (ei+n + ei) -p = pi+ pi+n (*2n+l = e 2 n + l ' X = X2n+1
= 0,
G2n+2
= 0.
= C2n+1 • P = Pln+1
— U,
= 0, i =
l,...,n,
(44)
Similarly, we can show {Gi,HAaJ~0,
for
V » e { l , . . . , 2 n + l,2n + 2}.
(45)
45
i.e. HA2n is the first class Hamiltonian corresponding to the above constraints Eq. (44). So LBC„ and MBC„
LBCn
can
be constructed as follows
= LA2n\G'
fc=l,...)2n+2' k'
'
'
•
MBCn = M A a J G i i f c = l i . . 2 n + 2 ,
(46)
#BC„ = 2^2»|G*,*=l,-..,2n+2,
(47)
while HBC„ is
due to the similar derivation of Eq. (32-38). 4 4-1
Lax r e p r e s e n t a t i o n s a n d involutive H a m i l t o n i a n s for Cn a n d BC„ RS models Cn model
The Hamiltonian of Cn RS system is Eq.(28), so the canonical equations of motion are = ePibi-e-pibi,
xi = {xi,H} p. PJ
=
r„. # )
=
(48)
^TsePih.(L^2lL
_ /j[£l±fil\
U.,,//) 2. { e 6 , ( / ( ^ } 9{Xji) bi{1
f{2Xl)
& (
' %(2^)
/(x.+ X | ) )
g(xj+Xi) +
^f{xlj) +
f^{9(xij)
+
+
f^+xj)" g(xt+Xj))h
where
/ w = = ™ .•<„ = *£>. fc = / ( 2 H ) U(/(ar< - a;*)/(n + xk)),
(49)
46
b
i = 9{2xi) ~[[(9(xi - xk)g{xi + xk)).
(50)
The Lax matrix for Cn RS model can be written in the following form for the rational case (LcJ^-e
*bv
^ - ^
,
(51)
which is a 2n x 2n matrix whose indices are labelled by the vector weights, denoted by \i, v G Ac„, Mc„ can be written as MCn =D + Y,
(52)
where D denotes the diagonal part and Y denotes the off-diagonal part Y^=e"-pbv*(xltl„\) +e~>ipbl/$(x^,\ D^
+ N1),
(53)
= (C(A) + C ( 7 ) ) e " ^ - (C(A + 7) + E
C W ^ " ^
( ( C ( ^ + 7) - ax»»))e"vK
+ i % ^ ^ $ ( ^ > A + Ny)e-"-"b'v)
(54) (55)
and
/^-^
n
K=
A c „ 3/3=/*-"
*V =
II
9(13-x),
(56)
A c „9/3=M-i /
z,n/
:
={n-v)-x.
(57)
The Lc„, MQ„ satisfies the Lax equation LCn = {LCn ,HCn}
= [MCn, LCJ,
(58)
which equivalent to the equations of motion Eq.(48) and Eq.(49). The Hamiltonian Hcn can be rewritten as trace of Lc„
HCn = trLCn = \
Y. M6A C „
^'PK
+ e""-^).
(59)
47
The characteristic polynomial of the Lax matrix Lcn generates the involutive Hamiltonians
det(L(A) -
v
( C T ( A
• Id) = £
^1^
where (H0)cn = (#271)0, = 1 and (Hi)Cn
J 7 )
(-^)2"-J(^)c.
= (H2n-i)cn
{(Hi)c„,(Hj)cJ=0,
(60)
Poisson commute
i,j = l,...,n.
(61)
This can be deduced by verbose but straightforward calculation to verify that the (Hi)A2n_1,i = l,...2n is the first class Hamiltonian with respect to the constraints Eq.(30), using Eq.(15), (32) and the first formula of Eq.(37). The explicit form of (Hi)cn are {Hi)cn=
ex
Yl
P(Pej)^J;^cf/^,/-|J|.
/ = l , . . . , n , (62)
JC{1....,»},|J|
with PeJ = ]T] £jPj, jeJ
Fej, K = J J f2(£JXJ + £i'x3' ) I I f(£3X3 + Xk)f(eJxJ - xk) J[ UI
>P=
i,j'€J
i£J
i<3'
*£K
Y
ffajXj),
j£J
I I f(xJk)f(xJ+xk)9{xjk)g{xj+xk)l^
(p° ven )
jei'
I'CI
\r\=[P/2]kei\r
Here, [p/2] denotes the integer part of p/2. As an example, for C2 RS model, the independent Hamiltonian flows (i?i)c 2 a n d {H2)c2 generated by the Lax matrix Lc2 are 28 (Hi)c 2 = HC2 = e" 1 /(2a: 1 ) f(x12)f{Xl + x2) + e " P l g{2xx) g(x12)g{xi + x2) +ep*f(2x2)f(x2l)f(x2+Xl) +e-p*g(2x2) g(x2i)g(x2 (H2)C2
1+
= e' *V(2*i)
+Xl), 2
(f(x1+x2)) f(2x2)
(63)
48
+e-pi-™g(2x1)
(g(Xl +
x2))2g(2x2)
+e»i-r*f(2x1)(f(x12))2f(-2x2) +e^-^g(2x1)(g(x12))2g(-2x2) +2/(a;i2) 5(2:12) f(xi + x2)g(xi + x2). 4.2
(64)
BCn model
The Hamiltonian BCn model is expressed in Eq.(42), so the canonical equations of motion are ii = {xuH}=ep'bi-e-pibi,
(65)
+e-p>b'J(9-^4-9)Xi+Xih} 9\Xji)
g{Xj + Xi)
+ r,f'(2xi"> , y ^ / ' f o j ) f(2Xi) + ^ f(Xij)
_ Pih (ISEA l[ f(Xi)
+
C-Pib't9(xi) A
| 2g'(2xj)
g{xi)
, ,f'(xi)
-
6o(
g{2xi)
|
y^,g'(xij) ?-f g{xij)
+
, f'(xi + xi)^ f(Xi + xj) "
+
g'jxj+Xj) g{Xi + Xj)
, 9(XJ)
+
(66)
7^y ^ y > '
where
h = f(xi)f(2xi)
J J ( / ( x * - xk)f(xi
+ xk)),
n
b'i = g(xi)g(2xi)
Y[(g(xi - xk)g{xi + xk)),
n
bo = Y[f(xi)g(xi).
(67)
i=l
The Lax pair for BCn RS model can be constructed as the form of Eq.(51)-(55) where one should replace the matrices labels with [i,v G A-BC„, and roots with /? € &BCn •
49
The same as for the C„ model, the Hamiltonian HBC^ can also be rewritten as trace of the Lax matrix LBC„
HBcn=trLBCn=\
Yl
(e^b^+e-^).
(68)
M6ABC„
The characteristic polynomial of the Lax matrix LBcn Hamiltonians 2
det(L(A) -vld)= where (H0)Bcn commute
-
{ a {
£
f ' ^
J 1
\ - v r ^ ( H
= 1 and (Hi)Bcn
(H2„)BC„
{(Hi)BcT,,(Hj)BCn}
= 0,
generates the involutive
j ) B
c ,
= (H2n+i-i)BCn
Poisson
i,j = l,...,n.
(69)
This can be deduced similarly to C„ case to verify that the (Hi)A2n ,i — 1, ...2n is the first class Hamiltonian with respect to the constraints Eq.(44). The explicit form of {H{)Bcn are (Hi)BCn=
^
^viPeriFej-^cUjc^^,
I = l,...,n,
(70)
JC{l,...,n},|J|
Ei=±i,jeJ
ith PeJ = 5 Z £iPi> Fej-K = J J f{ejXj 3<j'
+erxr)
2£ x
J J f(ejXj +xk)f(ejxj
- xk)
keK
x n f( i i) n /(e^i). Ui,P=
Yl
II
fixik)f{xjJrxk)g{xjk)g{xj+xk)
\i'\=[P/2}kei\r x
f Ui€i\r f(xi)g(xi), lYli'ei' f(xi')9(xi>),
(p odd), (peven).
(7i)
50
5
Degenerate cases
Let us now consider its various special degenerate cases. As is well known, if one or both the periods of Weierstrass sigma function a(x) become infinite, there will occur three degenerate cases associated with trigonometric, hyperbolic and rational systems respectively. The degerate limits of the functions $(x, A) , a(x) and ((x) will give corresponding Lax pairs which include spectral parameter. Moreover, when the spectral parameter takes on certain limit, the Lax pairs without spectral parameter will be derived. 5.1
Trigonometric limit
The limit can be obtained by sending w3 to ioo with Wi = | , so that a\x) —> e 6
sinar,
((x) —> cot a; + -x, o
and the function $(x, A) = //jfwA reduce to $(x, A) -> (cot A - cot x)e*xu. By replacing the corresponding functions (x, A), a{x) and C,(x) to the form given above for the Lax pairs, we will get the corresponding spectral parameter dependent Lax pairs. For the simplicity, we notice that the exponential part of the above functions can be removed by applying suitable "gauge" transformation of the Lax matrix on which condition the functions can be valued as follows: cr{x) —>• sin a;, Q(x) —> cotx, $(ar, A) -> (cot A - cotx). As for the spectral parameter independent Lax pair, furthermore, we can take the limit A —> ioo, so the function $(i,A) ->
, sinx while the corresponding Lax matrix become to L,v = evnv
Sin 7 . . sm((/i — v) • x + 7)
(73)
51 which are exactly the same as the spectral parameter independent Lax matrix given in . 5.2
Hyperbolic limit
In this case, the periods can be choosen as by sending by sending Wi to ioc with u>3 = f, so following all the procedure in achieving the result of trigonometric case, we can find the hyperbolic Lax pairs by simple replacement of the functions appeared in trigonometric Lax pair as follows: sin a; —> sinhi, cos a; —> cosh a;, cot a: —> cotha;. The same as for the trigonometric case, we can get the Lax pairs with and without spectral parameter. 5.3
Rational limit
As far as the form of the Lax pair for the rational-type system is concerned, we can achieve it by making the following substitutions a(x) —> x,
C(x) -> I *(*,A)-± + ± for the spectral parameter dependent Lax pair, while furthermore, taking the limit A —> ioo, we can obtain the spectral parameter independent Lax pair. The explicit form of Lax matrix without spectral parameter is
£„, = e"-*b„-
7——-
(74)
(fj, - v) • x + 7
which completely coincide with the spectral parameter independent Lax matrix given in 32 . 6
Nonrelativistic limit to the Calogero-Moser s y s t e m
The Nonrelativistic limit can be achieved by rescaling pM i—> /3p^, 7 i—• /?7 while letting /? i—> 0, and making a canonical transformation
52
Pn^—>Pn+l
Y^
C(»7 -a;),
(75)
here pM = fi • p, such that L^Id
+ pLCM
+ 0(f32),
(76)
and + 2P2HCM + 0(P2).
H^N
(77)
where N = 2n for C„ model and N = 2n + 1 for B C n model. L C M can be expressed as LCM
=p-H
+ X,
(78)
where X^
= 7 $ ( x ^ , A)(l - JM„).
The Hamiltonian / ? C M of CM model can be given by
HCM = 2p2 ~ \ ^ 2
p Q
(
' *)
=
4 i r Z / 2 + c o n s *'
( 7Q )
( where const = ^ ' 7 p(A). All of the above results are identified with the results of Refs. 6.9>io,11,12 up to a suitable choice of coupling parameters. As for the various degenerate cases, one can follow the same procedure as for the RS model(please refer to Eq.(73)-(74)).
7
Summary and discussions
In this paper, we have proposed the Lax pairs for elliptic Cn and BCn RS models. The spectral parameter dependent and independent Lax pairs for the trigonometric, hyperbolic and rational systems are derived as the degenerate limits of the elliptic potential case. Involutive Hamiltonians are showed to be generated by the characteristic polynomial of the corresponding Lax matrix. In the nonrelativistic limit, the system leads to CM systems associated
53
with the root systems of Cn and BCn which are known previously. There are still many open problems, for example, it seems to be a challenging subject to carry out the Lax pairs with as many independent coupling constants as independent Weyl orbits in the set of roots, as done for the Calogero-Moser systems 6 ' 8 ' 9 ' 10 ' 11,12 . What is also interesting may generalize the results obtained in this paper to the systems associated with all of other Lie Algebras even to those associated with all the finite reflection groups 11 which including models based on the non-crystallographic root systems and those based on crystallographic root systems. Acknowledgement One of the authors K. Chen is grateful to professors K. J. Shi and L. Zhao for their encouragement. This work has been supported financially by the National Natural Science Foundation of China.
References 1. K. Hasegawa, Commun. Math. Phys. 187, 289 (1997). 2. F.W. Nijhoff, V.B. Kuznetsov, E.K. Sklyanin and O. Ragnisco, J. Phys. A: Math. Gen. 29, L333 (1996). 3. A. Gorsky, A. Marshakov, Phys. Lett. B 375, 127 (1996). 4. N. Nekrasov, Nucl. Phys. B 531, 323 (1998). 5. H.W. Braden, A. Marshakov, A. Mironov and A. Morozov, Nucl. Phys. B 558, 371 (1999). 6. M.A. Olshanetsky and A.M. Perelomov, Phys. Rep. 7 1 , 314 (1981). 7. VI. Inozemtsev, Lett. Math. Phys. 17 (1989) 11 8. A.J. Bordner, E. Corrigan and R. Sasaki, Prog. Theor. Phys. 100, 1107 (1998). 9. A.J. Bordner, R. Sasaki and K. Takasaki, Prog. Theor. Phys. 101, 487 (1999). 10. A.J. Bordner and R. Sasaki, Prog. Theor. Phys. 101, 799 (1999). 11. A.J. Bordner, E. Corrigan and R. Sasaki, Prog. Theor. Phys. 102, 499 (1999). 12. E. D'Hoker and D.H. Phong, Nucl. Phys. B 530, 537 (1998). 13. J.C. Hurtubise and E. Markman, e-print, "Calogero-Moser systems and Hitchin systems", e-print math/9912161. 14. Y. Komori, K. Hikami, J. Math. Phys. 39, 6175 (1998).
54
15. Y. Komori, "Theta Functions Associated with the Affine Root Systems and the Elliptic Ruijsenaars Operators," e-print math.QA/9910003. 16. J.F. van Diejen, J. Math. Phys. 35, 2983 (1994). 17. J.F. van Diejen, Compositio. Math. 95, 183 (1995). 18. K. Hasegawa, T. Ikeda, T. Kikuchi, J. Math. Phys. 40, 4549 (1999). 19. K. Chen, B.Y. Hou, W.-L. Yang and Y. Zhen, Chin. Phys. Lett. 16, 1 (1999); High energy physics and nuclear physics. Vol.23, No. 9, 854 (1999). 20. B.Y. Hou and W.-L. Yang, Commun. Theor. Phys. 33, 371 (2000); J. Math. Phys. 4 1 , 357 (2000). 21. B.Y. Hou and W.-L. Yang, Phys. Lett. A 261, 252 (1999); K. Chen, H. Fan, B.Y. Hou, K.J. Shi, W.-L. Yang and R. H. Yue, Prog. Theor. Phys. Suppl. 135, 149 (1999). 22. T. Kikuchi, "Diagonalization of the elliptic Macdonald-Ruijsenaars difference system of type C2," e-print math/9912114. 23. S.N.M. Ruijsenaars, Commun. Math. Phys. 110, 191 (1987). 24. M. Bruschi and F. Calogero, Commun. Math. Phys. 109, 481 (1987). 25. I. Krichever and A. Zabrodin, Usp. Math. Nauk, 50:6, 3 (1995). 26. Y.B. Suris, "Why are the rational and hyperbolic Ruijsenaars-Schneider hierarchies governed by the same R-operators as the Calogero-Moser ones?" e-print hep-th/9602160. 27. Y.B. Suris, Phys. Lett. A225, 253 (1997). 28. K. Chen, B.Y. Hou and W.-L. Yang, "The Lax pair for C 2 -type Ruijsenaars-Schneider model," e-print hep-th/0004006. 29. S.N.M. Ruijsenaars, Commun. Math. Phys. 115, 127 (1988). 30. Y. Ohta, "Instanton Correction of Prepotential in Ruijsenaars Model Associated with N=2 SU(2) Seiberg-Witten Theory", e-print hep-th/9909196. 31. Paul A.M. Dirac, Lectures on Quantum Physics(Yeshiva University, New York, 1964) 32. K. Chen, B.Y. Hou, W.-L. Yang, "Integrability of the Cn and BCn Ruijsenaars-Schneider models" e-print hep-th/0006004.
55
FATEEV'S MODELS A N D THEIR APPLICATIONS D.CONTROZZI AND A.M.TSVELIK Department of Physics, Theoretical Physics, 1 Keble Road, 0X1 3NP Oxford, UK. We present two families of integrable models recently introduced by Fateev and suggest some possible application to physical systems.
1 1.1
Fateev's models Introduction
Recently Fateev introduced various families of models x'2,3 that, beside being amusing examples of integrable field theories, may also have some interesting physical applications. In this paper we revise some aspects of two of these families and discuss some possible physical applications. The complete list of models can be found in 1'2. These theories exist in two representations -bosonic and fermionic - and the weak coupling limit of one representation corresponds to the strong coupling of the other one. The two families are characterized by the presence of one or two fermionic/bosonic fields respectively, coupled to Toda chains. We will refer to them as type I and type II models. After introducing the models, we report some of the mathematical results obtained in 4 . We will also present two possible applications to physical systems. A common ingredient in both these interpretations is the presence of a phononic background that interacts with some additional bosonic or fermionic degrees of freedom. We will discuss in some details one of this applications. According to this interpretation the Euclidean version of Fateev models is seen as Ginzburg-Landau free energy functionals describing thermal fluctuations in superconducting films interacting with an insulating substrate. The major result of this analysis is that the effective critical temperature increases with the thickness on the insulating stratum. Some of this results were presented before in 10 . The paper is organized as follow. In the first section we introduce Fateev's models and briefly describe some possible physical applications. In Sec.II we recall few essential results from the exact solution, and in the last section we will study the physical consequences of our interpretation of the models.
56
1.2
Type I models
The models of type I have a Lagrangian formulation in terms of a complex fermion field {ip,ip) and n scalar fields ip = (
Sjf" = J d2x [i^O^
+ ^ (^VO 2 - mtH*-^
(1)
+2U^f (fa ^ ) ^2 - ~^W2 L-2^ +2 g e W. +1 -0.) + e 2^ m 2
and can be considered as a MTM coupled to AfEne Toda Theories (ATT). Fateev showed that these models are integrable for g = P2/(4n + p2)
(2)
They posses a U(l) symmetry and the corresponding conserved charge is given by Q = / dxipJoi>
(3)
Thus in presence of external field (chemical potential) the Hamiltonian becomes H -> H -hQ
(4)
In absence of chemical potential the spectrum of the model consists of fundamental charged particles of mass M = M(m,P) and their neutral bound states with mass MQ = 2Msin(7rj/2A),(j = 1 , 2 , . . . n - 1 ) , X = n-
^
P2
=n-g.
+
(5)
The two-body S-matrix of fundamental particles is diagonal and is given by c /m_c c m _ TT Zn,++[V) - :>»,+-IITT V) - y
sinh
(A/ 2 + Ml^) _ M/2n)
ginh(e/2
U sinh(g/2 + in(k - g~ 1 )/2n) JUL ginh^/2 _ i7r(k - g-!)/2n)
(6) Fateev also showed the existence of a dual representation in terms of complex bosonic fields x — Xi+*X2 and (n — 1) scalar fields 4> = {4>i, ...,n-i):
S^=jd2x
"I 2 1 + ( 7 /2)2 X x
fl^x__^w2exp(_7^)+i(M2 2 '*•' " v '^'
(7)
2
n-2
^ - j < 2exp(-7i) + 2 ^ e x p 7 ( ( / . i -n_1)
57 where the coupling constant of the two models are connected by the duality relation 7 = 47r//3
(8)
As we have mentioned before, the weak coupling limit of the bosonic theory, 7 < < 1 (1 — > 1. T h e duality transformation relating fermionic to the bosonic r e p r e s e n t a t i o n ' s then £ ' ( / ) (/?, „ ) = SI{b) (47T//S, n - 1)
(9)
T h e theory (7) possesses £/(l)-symmetry associated to the charge (xdoX ~ Xdox)
Q
(10)
"2./ s/' [l + (7/2)2xx] d:E
For small n the action (7) requires some modifications. In particular for n = 1 it becomes: *
1.3
•
"
>
-
/
1 duXduX 2 1 + (7/2)2xX
d^x
m
-XX
(11)
Type II model
T h e models of type II are a direct generalization of (1,7). In the fermionic representation they can be described in terms of two complex fermionic fields tpi,'
+ ^ ( & 7 ^ . ) 2 ] + (d^f r
—rm/^V^e'
m 2(P
- m&tfue-^1
(12)
n-i exp(-2/3
T h e bosonic representation can be expressed in terms of two complex scalar fields XiiX2 and (n — 2) scalar fields 4> = (i,--,
d^XsdnXs
/<<•• \ZT - 1 + (7/2) X Xs 2
-[XiXi exp(-7i) + X2X2 exp(70 n _ 2 )]+
s
n(d,
m 272
2 exp(-71) + 2 ^2 exp yifc - 4>i+1) + 2 exp(7„_ i=l
(13)
In this case the duality transformation is given by £ " ( / ) (/3, n) = <S / / ( 6 ) (4TT//3, n - 2)
(14)
58
Again for small n it has to be modified and becomes
s (b)
" = h2x\\tilttt* + (7/2) X X
-^feixi + x,x2+fi^mx^)\ 2
2
s
s
(15) It possesses U ( l ) x U ( l ) symmetry and the corresponding conserved charges are given by n
(A
J
i
i
[A
(XsdoXs ~ XsdoXs)
Qs = / d ^ w s = - - y d , [ 1
+
(7/2)2xs_s]
,1C,
(i6)
Thus one can introduce two chemical potentials and modify the Hamiltonian H -»• H - /nQi - /i 2 Q 2
(17)
The U ( l ) x U ( l ) symmetry of the Hamiltonian reflects in the symmetry of the particle multiplets. In this case, in absence of chemical potentials, the spectrum of the model consists of fundamental particles carrying quantum numbers of U(l) groups and their neutral bound states. The two-body scattering matrix of fundamental particles is given by a tensor product of two sine-Gordon 5-matrices multiplied by a CDD-factor responsible for cancellation of double poles
o=l
where the S(X,6) is the soliton scattering matrix of the sine-Gordon model with A defined in Eq.(5). The mass of the bound state is given again by Eq.(5). 1.4
Physical interpretation
As we mentioned in the introduction, this models can be thought as describing very interesting physical situations. We present two possible interpretations. A common ingredient of both interpretations is the presence of a phononic background, represented in terms of Toda theories, coupled to some fermionic or bosonic degrees of freedom. In the first case, considering the model as a (l+l)-dimensional quantum field theory (QFT), we may interpret it as describing fermions moving on stripes and interacting via a two dimensional fermionic background. A schematic representation is presented in Fig.l. The problem of quasi one dimensional systems with electron-phonon interaction is a very important one and cannot be approached with perturbative methods, it is then very important to have exact results for some specific model. This
59
X
^
Figure 1. Schematic representation of one possible interpretation of Fateev models
interpretation of the model and the possible application to physical systems have been discussed in ref. 7 . On the other hand, the Euclidean version of the model can be thought as describing an effective Ginzburg-Landau action for two planar superconductors interacting with a phononic background. According to this interpretation the models of the first family describe a single superconducting layer deposited on an insulating substrate consisting of n layers. The superconducting order parameter interacts with the elastic modes of the substrate. The effect of the interaction is to shift the local transition temperature. Type II models describe the situation of a double layer where an insulator is sandwiched between two superconducting films. The latter ones may have different critical temperatures. This second interpretation was suggested in ref.10, and will be considered in details in the Sec. 3. 2 2.1
Thermodynamic Bethe Ansatz Thermodynamics of type I models
For type one models the S-matrix is diagonal and then the Thermodynamic Bethe Ansatz (TBA) equations can be easily obtained using the standard procedure 8 : Tln[l +
ti{8) T e
/]
- (adij + Gij) * Tln[l + e e ' w / r ] = m* cosh0 - hia (19) F/L = -TYJ7^
f M cosh6ln[l + e - £ ° ^ ' T ]
60
where the kernels are determined by G
'm = -^-^sij(e)
(20)
The indices i,j take values +, — for the elementary particles and a = 1,..., n — 1 for bound states. The constant a = 1 for the elementary particles and a = 0 for the bound states. The explicit form of the kernels dj can be found in Ref.5. It was shown there that in the limit n —>• oo the modes on the edges decouple from those of the Toda chains. They interact via an effective non local potential and satisfy the exclusion statistics 6 . In the limit T —> 0 only e + is negative for h > M and determined by the equation ,s e+(0) - / de'R(0-8')e+(O') J-B
= Mcosh9-ha
(21)
with e+(±B) = 0. The kernel will be reproduced in the next section. All the other energies are positive. 2.2
Thermodynamics of type II models
The TBA equations for type II models are much more complicated. In order to construct them we consider the case 1 — g = \jv, with integer v > 2, where the complex solutions of Bethe Ansatz equations (the strings) have the simplest classification 9 . For most of the results the fact that v is an integer is not important and one can generalize them for arbitrary g > 1/2 replacing v by (1 — g)~l. The TBA equations have been introduced in 1,y and discussed in 4 . The free energy of the system is expressed in terms of the excitation energies Ej{9) 1 ™_1
r M
F/L = -^-Yl n
decoshe l
H
+ e~Ei(e),T\
( 22 )
^ 3=0
where Mo = M and Mj are given by Eq.(5). To save space and make our notations more transparent we shall sometimes denote the kernels in the integral equations directly in terms of their Fourier transforms. For example, the convolution of two functions g and / with the function g having Fourier transform g(uj) will be written as + 00
/
dffgifi - 6')f(6') = g * f(9) = {g(u)} * f
(23)
61
The TBA equations for the bound states are Tln[l + eEiW/T]
- Gjk * Tln[l + e-Ek{8)/T] Eo
T
+Gj0 * Tln[l + e~ ^' l
= M3 cosh6>
(24)
(j.k = 1, ...n - 1)
We will not reproduce here the explicit form of the kernels, that can be found in 7 , 4 ; but we notice that GJQ > 0. Ej are directly coupled only to Eo, which is determined by the equation: Tln[l + e £ ° W / T ] - Ksym * Tln[l +
e~E^/T} E (9)/T
= M c o s h 9 - ( h + + /i_)/2 + Goj * Tln[l +
e-
'
]
ln[(l + e"'-<W T )(1 + e " ' - " <WT)]
-Tj^an* n=l
- Tav.,
* ln[(l + e ^ < 9 ) / T ) ( l + e — ^ > / T ) ]
(25)
The equations for e n and e_ n are e±„ = h±v/2 - s * Tln[l + e£±<-2>/T]
(26)
e± n = <J|„|,„-i/i±«//2 + s * Tln[l + e^-^T}[l +
<*„-2,M*
e±
+ e '±-+ 1 / T ]
T
£
(27)
T
* Tln[l + e- -/ ] + <S„,is * T ln[l + e - ° / ]
The diagrammatic representation of these equations is shown in Fig.2
o
<\o—o/ -v+2
O
E|
T o-Q-o fc
o
P K o—Q O
Figure 2. Diagrammatic representation of the TBA Eqs.(24-27)
We notice that the fields h± are linear combinations of the chemical potentials hit2 h± = hl±h2.
(28)
Let us now construct the ground state in presence of external fields. We assume that the fields h± are strong enough to make E0 negative in some interval [-B, B]. Then it is obvious from Eq.(24) that £q > 0. From Eqs.(27)we
62
Figure 3. Schematic diagram of the ground state energies in presence of chemical potential for h- < < /i-|_.
Figure 4. Schematic diagram of the ground state energies in presence of chemical potential for /i_|- « h-.
deduce that also all e± n with n ^ v are also positive. Then the ground state
63
is determined by the Eq.s ( sinhfi sinh[(»/ - 2)fl] \ , ( „) c ± i > = h+ + \ sinh vfl sinh vfl J 1
4+J + {: E,(+)
*E. (-)
(/1+ + /I-)
f sinh fi \ sinh vQ,
(29) (30)
* Fey + eL-jl
A schematic representation of the solution is presented in Fig.s 4,3 for two specific cases: a) h- « h+\ b) h+ « h-A. In the first case e„ > 0 and e_^ is negative inside an interval [—Q,Q], with Q » B and Q -> oo as /i_ ->• 0. In the second case e_„ > 0 and there is only one gapless mode. 3 3.1
Applications Exactly solvable Landau-Ginzburg Theories of superconducting order parameters coupled to elastic modes
In this section we show how Fateev's models can be interpreted as an effective Ginzburg-Landau free energy coupled to elastic modes. Let us consider the Euclidean version of the models (7). Rescaling the fields as: (•j/2)x —• A and 7 —> <j>, we obtain for type I models: i ) + 2 ( W
(31)
n-2
- m 2 e x p ( - 0 i ) + ^2 exp(0i - <j>i+1) + exp(^>„_1) and for n = 1 (i)
«5i
I
d2x
2 r1r — TV +A A + ™ AA
(32)
Analogously for the type II models we introduce two complex bosonic fields, A s = (•y/2)xs- The Euclidean action has then the form:
5(">
= -1 / d 2 J 2 Y.
d
: t t t +2m2A1A1exp(-(Al)
+ 2 m 2 A 2 A 2 e x p ( - 0 n ) + -(<9M<;
(33)
64
Insulator
Figure 5. Schematic representation of the system studied in Sec.3.
n-3
- m 2 < e x p ( - 0 i ) + ^2
ex
P(<^
_
&+i) + exp(^„_ 2 )
For n=2 the above expression has to be modified as
2
~W
duAsdMAs
dx
^ T T A :
2
+ m {AiAi + A 2 A 2 + 2(A 1 A 1 )(A 2 A 2 )}]
(34)
In terms of A s the conserved charges (here we work in Minkovsky spacetime), have the form: (AS<90AS Asd0As) da;(35) [1 + A S A S ] iJ where s = 1 for type I and s = 1,2 for type II models. Then in presence of external fields hs the Hamiltonian is modified: H = Ho — ]T)g hsQs. One can introduce new variables Qs =
2
A s -»• A,e" fc -
(36)
to remove the terms with one time derivative. Then we obtain the general form of the Ginzburg-Landau free energy for our layered superconductors 2 (37)
F^/T=^jd xf^
where /
(/)=2^A^A-ft2AA+2m2AAe_2,i 1 + AA
+
2 2 2 ^) -m
-0i -|- y ^ eii-i+i) _|_ e ^ » - i
(38)
65
and J
(in n
dgAsd^As - h2sAsAs + ASAS
y-
1
2OjiAsdi
2
(39)
+2 m 2 AiAiexp(-i) + 2m 2 A 2 A 2 exp(-„) n-3
m "~2
2exp(-„_2)
Clearly the fields A s can be interpreted as a superconducting order parameters coupled to the optical phonon modes „. We note that the elastic modes are not harmonic. This is necessary to make the model integrable, but in the limit 7 —> 0 we reproduce the conventional electron-phonon interaction (cfr Eq.s (7) and (13). The fact that the spectrum of the Toda chain coincides with the spectrum of harmonic phonons n gives us grounds to believe that the unharmonicity present in the model does not influence the qualitative validity of our results. At the same time it is interesting to have a model with phononic spectrum that is not restricted to the harmonic one. A schematic representation of the system in consideration is presented in Fig.5, In order to get a better understanding of the nature of the superconducting state, let us have a deeper look on the free energy for the case n = 0. For the type I models the free energy can be written as follows: F 0 ( / ) /T / *
<9UA<9UA T r . x A , , A A +y(A,A) 1 + AA
(40)
with F(A,A)=m2AA
,
AA 1 + AA
(41)
For m < h the effective potential for the order parameter has a minimum at |A| ^ 0 which signal the appearance of superconductivity. Indeed, expanding the action around the minimum we notice that the quantities r = (m 2 — h2)/y2 may be interpreted as the distance from the mean field transition (T/Tc — 1). Increasing h one can go from the disordered to the superconducting state. A more detailed justification of this interpretation on terms of a semiclassical analysis can be found in 10 . From the exact solution presented in the previous section one can see that a massless mode appears when h exceeds certain threshold value M, where M is a function of the coupling constant 7 and parameter m ( a similar condition is obtained for the type II models). In view of the above considerations naively one may identify h = M with the onset of superconductivity.
66
This interpretation is incorrect however, since it does not take into account vortices. Massless phases of the Fateev's models, like any other twodimensional critical theories with U(l), or [7(1) x £7(1), symmetry, may be unstable with respect to vortices. The reason being that the naive approach doesn't take into account non-analytic configurations of the fields, that on a lattice may give a finite contribution to the free energy. The real transition is of the Beresinskii-Kosterlitz-Thouless (BKT) type 12 and occurs at temperature below the mean field transition temperature established by the condition h = M. To take into account the effect of vortices one should add to the forms (38), (39) exponents of the dual field, 0 , (see for instance 13 ) f(na) -+f(na) + AcosQ
(42)
These terms are not contained in the models we consider, therefore the latter ones can provide an adequate description of the superconducting phase only below the BKT transition temperature, where the cosine term in (42) is irrelevant. In the next sections we will use the exact solution to study the relevance of vortices in our models. To do this we shall need to study more carefully the gapless state and extract from the Bethe Ansatz equations the scaling dimensions of the vortex operators. 3.2
The Berezinskii-Kosterlitz-Thouless models
transition in the monolayer
Let us consider the type-I models first. The Bethe Ansatz solution deals with the Minkovsky version of the models. In our analysis we shall benefit from the fact that the ground state energy of the (1 + l)-dimensional field theory with coupling constant 7 is equal to the free energy of the 2-dimensional classical theory with temperature T = j 2 F/T = £
(43)
In the Minkovsky version the appearance of the gapless state is related to the creation of a condensate of kinks. Then using the Bethe Ansatz solution one can express the ground state energy per unit area in terms of solution of the integral equation (21). We rewrite it here, omitting the subscript +, as e{6) - / d6'R(9 - 6»')e(6>') = M cosh 6-h J-B M fB F/T = £ = — d<9 cosh 0e(<9) 27T J_B
(44)
(45)
67
where e(0) < 0 for |0| < B and the integration limit B is defined by the condition e{±B) = 0. According to form:
J
the Fourier transform of the kernel,
(46) R(UJ),
has the following
1 - R(LO) - s ^ M 7 " ^ 1 ~ g)/ Q ] cosh[7r^(a + 2g)/2a] cosh(7rw/2) sinh(7rw/a) where 3 = —o r - a = —r 5— ( 48 ) 2 2 7 + 47r 4-rr + 7 with G, iJ depending on the model. For model (31) one has G = H = 2(n + l); G, iJ always scale with n when n —> oo. Another quantity that will be useful in the following is the dressed charge C(0)17-18 C(0) - / d0'ii(0 - fl')C(^') = 1(49) J-B The kernel R is defined in (47) and the limit B is determined by condition (46). As we have said, two-dimensional U(l)-symmetric critical points can be unstable with respect to the presence of vortex configurations of the order parameter field. The vortices constitute potentially relevant perturbations which appear in the effective action as exponents of the dual field (42). This operators become irrelevant if their scaling dimension, d@, is greater than two. When a critical theory includes just one U(l) field a, the scaling dimension of the order parameter A = exp(ia), O^A, is related to the scaling dimension of the vortex perturbation as follows: dAde = 1/4
(50)
Hence the superconducting regime exists at dA < 1/8.
(51)
In general the scaling dimension of a primary field $ is defined as d$ = A+ + A "
(52)
where the conformal dimensions A± determine the asymtotics of the correlation functions of primary fields. In (l+l)-dimensional critical theories the scaling dimension of primary fields are related to the finite size corrections to
68
the ground state energy 15 - 16 . For models which, as the model in question, have central charge c = l , conformal dimensions are given by 18 : 2A±(AN,D,N±)
= 2N±+(~±ZD\
.
(53)
The quantum number AN is characteristic of the local field under consideration; in the context of this model it represent the number of particles produced by the primary field in consideration. The quantum numbers D and N±, generate the tower of excited states, and represent respectively the number of particles that undergo back scattering processes from one Fermi boundary to the other and the number of particles added at B(N+) or —B(N~). While AN and D are fixed by the local field, iV^ must be chosen to give the leading asymptotics in the correlation functions, which is equivalent to minimize A^. The quantity Z is related to the dressed charge introduced above in the following way: 2 = COB)-
(54)
For the order parameter operator we have dA = A+(1,0,0) + A-(1,0,0) = l/[2Z]2.
(55)
In order to study the stability of our models with respect to vortices we then have to calculate the value of the dressed charge at the Fermi point. Then the phase diagram of the model can be constructed using (51). The results are presented in Fig. 6. It is clear that the insulating substrate strongly affects the superconducting transition, in particular the effective critical temperature increases with n. To our knowledge this is the first model that displays such characteristic behavior. 3.3
Effect of the interaction between superconducting order parameters: the two-layer model
The situation is considerably more complicated for the case of two-layer models. Since we want to focus our attention on the effects of the interaction between the order parameters we consider only the n = 2 version of the model, where no phonon modes are present. We have noticed elsewhere that in this case the two gapless modes have the same velocity 4 , and hence at this point the model is conformally invariant with the central charge c—2. This result is not general for the Fateev's models but valid only for the specific case n = 2. However, the results presented below can be generalized for a case when the two velocities are different. In the latter case the low energy sector is split into
69
Figure 6. Phase diagram of the type I models. The curves separate the superconducting region from the disordered one.
two quasi-independent sub-sectors. The system as a whole is not conformally invariant, but each sub-sector is (with its own velocity). Conformal symmetry of each sub-sector makes it possible to generalize the basic relations between the critical exponents and finite size scaling amplitude 20 ' 19 that we will use in the further discussion. The low-lying excitations for the type-II model with n = 2 are described by Eq.s (29,31). For \{h+ + /i_) > M the mode E0 becomes gapless and, depending on the relative values of h+ and /i_, it can also induce a second gapless mode. In the following we will consider h+ 3> M. In this case tv is always positive and decouples from the other equations. We rewrite Eq.s (29,31) in a more symmetric way, as
ei(0) = e?(0) + f J-B rQ
c-v{X)=t°_v+
(WRi(0 - 0 > i ( 0 ' ) + f
dXK^d - A)e_„(A) (56)
J-Q rB
dA'/e„(A-A')e_„(A')+ / dOK^X - 8)e1(9) J-Q J-B
(57)
70
Now the bare energies are e?(0) = M cosh(0) ~\{h++
h-),
e°±„ = h±
(58)
and we have defined the kernels _, rr.. sinhft „ . „ . sinh((f — 2)fl) smh(j/S2) sinh(j/0) r. ,™ sinh ft sinh On ,_ TTW.,
. „s
1-9
For h-
J
^
\ * e(+) + e(") = ^ /o | f
\2smhftcosh[(i/-l)ft]J
—'
_/
-"
*
]
:|:c (-)
x ' (60)
l2cosh[(i/-l)n]J
Using this form in Eq.(56) we get ^
^
n
r
-
M
c
o
s
h
^
^
-
j
^
^
j
.
-
(61)
where ^
=
^
j
smhftcosh[(3^-l)ft] 2sinhi/ftcosh[(i/-l)ft]cosh[(i/ + l)ft]
l
;
Since e_J is very small, to first approximation this reduces to ei(0)=e?(0)+ /
d0'R(6 - 6')ei(6')
(63)
where the new kernel i?(ft) is given by R(il) = 1 - K(il)
(64)
Clearly e_„ become gapless for /i_ < hc defined as 1 fB 2i/2 y _ B
sin — cosh ™ + cos ^
Already at this stage some interesting features emerge. Namely, in order to get the phase transition on the mean field level, one does not need both fields /ii,/i2 to exceed the critical value for a single superconductor. Instead of the condition \h\\ ~ \h2\ ~ M/2 we get a weaker condition hi > M/2 and h2 > hi - /i c . Now let us discuss how the mean field picture is modified by the vortices. For h- > hc when there is only one gapless mode we can calculate the scaling
71
dimension of the order parameter using the procedure of the previous section. However, for /i_ < hc when there are two gapless modes this procedure should be modified. In this case one needs to introduce the dressed charge matrix 19,20,21.
and the function ((#) is determined by the following equations: Cii(0) = l + / d8'R1(9-8')(n(0')+ J-B
AXK^O - A)6„(A)
C i „ ( A ) = / d\'Rv{\-\')Civ(\')+ J-Q rQ
U{0)=
(67)
J-Q [ dOK^X - e)Cn(9) J-B
(68)
[-B
dAAr1(0-A)G,„(A)+/
dOR.ie -e')u(o')
fB
rQ
(69)
U ( A ) = 1 + / d0#i(A - 0)C*i(0) + / dAJ2„(A' - A)U(A') (70) J-B J-Q In this general case the formula for the conformal dimensions are given by
2
^ A M n M ^ _ o , ± , - . ~ - • Z^AN! - ZluAN„ 7 1 2Af ( A N . D . N * ) = 27Vf + ( Z n D , + Z ^ A , ± " )( ) 2 d e t Z 2A±( A N , D, N ± ) = 27V± + {ZlvDx
+ Zvl/Dv
±
^i^_|^lML^
( 7 2 )
The quantum numbers A N = (ANi,ANv),T> = (Di,Dv) and N± = (N1 ,N^) were defined in the previous section. For two coupled Gaussian models with a total central charge c = 2, the scaling dimensions for primary fields are given by W D )
= (ZuDl
+ Zv,Dvf
+ (
Z
-
17 +7 DV 2 +i ^ {ZlvDn 1+ZvvDv)
A
^
t
^
A A r
- )
2
(73)
2 ^ (^AN ^ u-ZvlAN^
The vortices generate operators represented by exponents of the dual phases of the two gapless fields. We shall call the scaling dimensions of this operators d@z and d®v respectively. The conditions for irrelevance of vortices are de1 > 2 , efo„ > 2
(74)
72
To see the effect of the interaction between the order parameters on the stability of the superconducting state with respect to vortices we shall consider the dependence of the scaling dimension of the potentially relevant operators as a function of /i_ for fixed h+ 2> M. In particular we will compare the scaling dimensions in two extreme limits where Eqs (67-70) can be studied analytically. The limit of small h-, which corresponds to the physical situation in which the two superconductors have the same bare critical temperature, and the limit h- > hc. Having fixed h+ correspond to fix, for example, hi which is proportional to the mean field critical temperature of layer 1. We want to observe how the presence of the other superconductor affects the scaling dimension of the order parameter of this superconductor, i.e. his effective critical temperature. It turns out that for L « l the matrix problem given by Eqs. (67-70) greatly simplify and can be reduced by to scalar one 10 . The scaling dimensions of the dual fields turn out to be: 2 de1 = de2 = 2v = —— (75) Then in this particular limit the conditions (74) are always satisfied and the system is always stable with respect to vortices. For /i_ > hc one again has only one gapless mode and repeating the procedure of the previous section we get, for B » 1: de, (h > hc) = v,
(76)
in agreement with the results obtained for the type I models. Then the effect noticed at the mean field level survives a deeper analysis. The effective critical temperature of a superconductor is enhanced by the presence of another one. The effect is present only if the critical temperature of this superconductor is above some critical value and is maximal when the two superconductors have the same critical temperature. 4
Discussion
We have presented here some properties of two families of integrable models recently introduced by Fateev, and discussed some possible applications to physical systems. Fateev's models seem to be very rich and other applications may be found in the future. Here we mainly focused the attention on one possible interpretation of these models. Considering them as two-dimensional classical field theories, they can be seen as an effective Ginzburg-Landau action for two dimensional superconductors separated by and insulating stratum. We studied the dependence of the superconducting properties on the
73
number on insulating and superconduncing layers. The main result see to be that the effective critical temperature increases with the thickness of the insulating substrate. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
13.
14. 15. 16. 17. 18. 19. 20. 21.
V. A. Fateev, Nucl. Phys. B479 (1996) 594. V. A. Fateev, Nucl. Phys. B473 (1996) 509. V.A.Fateev, Phys. Lett. 357B (1995) 397. D.Controzzi and A.M.Tsvelik, Nucl. Phys. B572 (2000) 609. C. Pepin and A. M. Tsvelik, Phys. Rev. Lett. 82 (1999) 3859. F.D.M.Haldane, Phys. Rev. Lett. 66 (1991) 1529. C. Pepin and A. M. Tsvelik, cond-mat 9983180. Al.M.Zamolodchikov, Nucl. Phys. B342 (1990) 695; Nucl. Phys. B348 (1991) 497. M.Takahashi, Progr.Theor.Phys.46(1971)401; M.Takahashi and M.Suzuki, Progr.Theor.Phys. 48(1972)2187. D.Controzzi and A.M.Tsvelik, cond-mat /0004465; to appear in Phys.Rev.B. A.E.Arishtein, V.A.Fateev, and A.B Zamolodchikov, Phys. Lett. 87B (1979) 389. V.L.Beresinskii, Sov.Phys. JEPT 32(1971)493; J.M.Kosterlitz and D.J.Thouless, J. Phys. C6(1973) 1181; J.M.Kosterlitz,J. Phys. C 7(1974) 1046. A.O.Gogolin, A.A.Nersesyan and A.M.Tsvelik, Bosonization and Strongly Correlated Systems, Cambridge University Press (1999) A.A.Belavin, A.M.Polyakov and A.B.Zamolodchikov, Nucl. Phys. BB270 (() 1984 333. J.L.Cardy, Nucl. Phys. B270 (1986) 186. H.W.Blote, J.L.Cardy and M.P.Nightingale Phys. Rev. Lett. 56 (1986) 742; I.Affleck, Phys. Rev. Lett. 56 (1986) 746. N.M.Bogoliubov, A.G.Izergin and V.E.Korepin, Nucl. Phys. B275 (1986) 687. V.E.Korepin, N.M.Bogoliubov, A.G.Izergin, Quantum Inverse Scattering Method and Correlation Functions, Cambridge University Press (1993). A.G.Izergin, V.E.Korepin and N.Yu Reshetikhin, J.Phys A 22(1989)2616. F.Woyanarivich, J.Phys. A 22(989) 4243. H.Frahm and V.Korepin, Phys. Rev. B42 (1990) 10553.
74
THE ODE/IM
CORRESPONDENCE
PATRICK DOREY, CLARE DUNNING AND ROBERTO TATEO Department Department E-mails:
of Mathematical Sciences, University of Durham, Durham DH1 3LE, UK (FED and RT) of Mathematics, University of York, York YOl 5DD, UK (TCD) [email protected], roberto.tateoQdur.ac.uk, [email protected]
These lectures review a recently-discovered link between integrable quantum field theories and certain ordinary differential equations in the complex domain. Along the way, aspects of 'PT-symmetric quantum mechanics are discussed, and some elementary features of the six-vertex model and the Bethe ansatz are explained.
Introduction The study of integrable lattice models has been an intriguing part of mathematical physics since Onsager's solution of the two-dimensional Ising model, but it was probably only with Baxter's work at the beginning of the 1970s that the full richness of the subject began to be appreciated. Since that time interest in the subject has continued to grow, receiving a particular boost of late from the links which exist with integrable quantum field theories. Many different methods have been developed for the solution of these models, and a technique coming increasingly to the fore goes by the name of the 'functional relations' approach. The idea, initially put forward by Baxter, is to show that quantities of interest satisfy functional relations. When combined with suitable analyticity properties, these relations turn out to be highly restrictive and often allow exact formulae to be obtained. In a parallel chain of development which also dates back at least to the early 1970s, Sibuya, Voros and others have shown that functional relations have an important role to play in a rather more classical area of mathematics, namely the theory of Stokes multipliers and spectral determinants for ordinary differential equations in the complex domain. However, it is only recently that the existence of a precise link - an 'ODE/IM correspondence' - between these two areas has been realised. The aim of these notes is to provide an elementary introduction to this connection and its background. For most of the time the focus will be on the simplest example, which connects secondorder ordinary differential equations to integrable models associated with the Lie algebra SU{2). But towards the end, some recent work extending this to n t h order ODEs, and linking them to theories associated with SU(n), will also be mentioned.
75
Sections 1 and 2 set the scene by reviewing aspects of the theory of secondorder ordinary differential equations and integrable lattice models, while in section 3 a simple example of the connection between the two topics is established. Section 4 treats the generalisation to SU(n), and some other developments are discussed in section 5. This gives the notes the following structure: (1) 2 n d order ODEs (Schrodinger equations) (3) Connection (2) Functional equations in integrable models
'
^ (4) Generalisation to SU(n)
I (5) Further topics More on the ODE/IM correspondence can be found in [1-9], and these notes themselves are an extended version of the review article [10]. However it should be emphasised that all of this work rests on earlier studies by, among others, Sibuya [11], Voros [12], and Bender et al [13-15] (on the ODE side) and by Baxter [16], Kliimper, Pearce and collaborators [17,18], Fendley et al [19], and Bazhanov, Lukyanov and Zamolodchikov [20-22] on the integrable models side. 1
Schrodinger equations
The idea of a Stokes sector, and its relevance to eigenvalue problems defined in the complex plane, will be important in the following. To introduce this concept we begin by describing a class of problems much studied by Bender and collaborators in recent years. The story begins with a question posed by Bessis and Zinn-Justin: Question 1: What does the spectrum of the Hamiltonian H =p2 + ix3 look like? This is a cubic oscillator, with purely imaginary coupling i. (Strictly speaking, Bessis and Zinn-Justin, motivated by considerations of the Yang-Lee edge singularity, were interested in more general Hamiltonians of the form p2 + x2 +igx3, from which the above problem can be recovered as a strong-coupling
76
limit.) The corresponding Schrodinger equation is d2 ip{x) + ix^ipix) = Eip(x) dx2 and we will initially say that the (possibly complex) number E is in the spectrum if and only if, for that value of E, the equation has a solution I/J(X) on the real axis which decays both at x —> — oo and at x —> +oo: a
Notice that the differential equation forces the wavefunction ip(x) to be complex, even for real values of x. And since the Hamiltonian is not (at least in any obvious way) Hermitian, the usual arguments to show that all of the eigenvalues E must be real do not apply. Nevertheless, perturbative and numerical studies led Bessis and Zinn-Justin to the following conjecture: • the spectrum of H is real, and positive. In 1997 Bender and Boettcher [14] proposed a nice generalisation of this problem: Question 2: What is the spectrum of z H = p~
(ix)N
(N real, > 0) ?
Later, it will turn out that the passage from question 1 to question 2 corresponds to a change in a coupling constant in a sine-Gordon model, or equivalently to a change of a quantum group deformation parameter in a Bethe ansatz system. But for now, the generalisation is appealing because it unites into a single family of eigenvalue problems both the JV=3 case, for which we have the Bessis-Zinn-Justin conjecture, and the more easily-understood N=2 case, the harmonic oscillator. The Schrodinger equation is now - - T z^ ( s ) - (ix)N^(x) = E
77
details need extra care: for non-integer values of N, the 'potential' —(ix) " i s not single-valued; and when TV hits 4, the naive definition of the eigenvalue problem runs into difficulties. The first problem is easily cured by adding a branch cut along the positive imaginary z-axis, but the second is more subtle and will be discussed in greater detail below. This caveat aside, there is already a surprise while N remains below 4. Figure 1 is taken from [4], and it reproduces the results of [14]. Ignoring for a moment the region N > 4, it is clear that something strange occurs as N decreases through 2 - infinitely-many eigenvalues pair off and become complex, and only finitely-many remain real. By the time N reaches 1.5, all but three have become complex, and as iV tends to 1 the last real eigenvalue diverges to infinity. In fact, at iV=l the problem has no eigenvalues at all, as can be seen by solving the relevant Schrodinger equation in terms of an Airy function. For N > 2, the numerically-obtained spectrum is entirely real, and positive; this naturally generalises the conjecture of Bessis and Zinn-Justin. The 'phase transition' to infinitely-many complex eigenvalues at N=2 was interpreted in [14] as a spontaneous breaking of VT symmetry.
Figure 1: H = p2 — (ix)" : real eigenvalues as a function of N
78
Although figure 1 agrees with the plot in [14], it was obtained in [4] by an entirely different route - rather than making a direct numerical attack on the ordinary differential equation, a nonlinear integral equation for the relevant spectral determinant was solved. This method of solving such eigenvalue problems is a byproduct of the ODE/IM correspondence and appears to be new, though it owes much to earlier work of Voros [12]. Numerically, it is rather efficient - see for example the tables in [1] of eigenvalues of various anharmonic oscillators. Another idea motivated by the correspondence is the notion [4] to study the effect of an additional angular-momentum term l(l+l)x~2 on the BenderBoettcher problem. For — 1 < I < 0, this turns out to have a remarkable effect on the behaviour of the spectrum as the JV=2 phase transition is crossed.
Figure 2: H — p2 — (ix) + /(/+1) x : real eigenvalues as a function of N for / = -0.025, /(/-Hi) = -0.024735 Figure 2 zooms in on this part of the spectral plot for / = —0.025, and reveals that the picture has changed dramatically - the connectivity of the real eigenvalues has been reversed, so that while for /=0 (the original BenderBoettcher problem) the first and second excited states pair off, at / = —0.025 the first excited state is instead paired with the ground state, and so on up the spectrum. With this in mind, it may be a little hard to see how it is possible to pass between the sets of spectra depicted in figures 2 and 1 simply
79 by varying the continuous parameter / from —0.025 to zero. The two plots shown in figure 3, which are also taken from the paper [4], should help to resolve the puzzle. As I increases from its value in figure 2 towards zero, the level joined to the ground state moves up through the spectrum, leaving a restored connectivity pattern behind it.
I
„j
i
i N
•
•
!,
'
•
f
» ' -
•
1
i
•
S
—
f
N
Figure 3: H = p2 — (ix)N + 1(1+1) x~2 : real eigenvalues as a function of N for I = -0.0015 and / = -0.001 There remains one piece of unfinished business: what goes wrong at AT=4, and what can be done to resolve it? In the figures, the energy levels continue smoothly past iV=4, but in fact this can only be achieved by implementing a suitable distortion of the problem as originally posed. Consider the situation precisely at N=4 : the Hamiltonian is p2 — xA, an 'upside-down' quartic oscillator, and a simple WKB analysis (about which more shortly) shows, instead of the exponential growth or decay more generally found, wavefunctions behaving as a; -1 exp(±ix3/3) as x tends to plus or minus infinity. All solutions thus decay, albeit algebraically, and this complicates matters significantly. The problem moves from what is called the limit-point to the limit-circle case (again, see [23]), and additional boundary conditions should be imposed at infinity if the spectrum is to be discrete. While interesting in its own right, this is clearly not the right eigenproblem if we wish to find a smooth continuation from the region TV
80
The key is to examine the behaviour of solutions as \x\ —• oo along a general ray in the complex plane, even though the only rays that we initially need are the positive and negative real axes. The WKB approximation tells us that ip(x) ~ P(x) - 1 / 4 , as |a;| —»• oo, with P(x) = —(ix)N — E. (This is easily derived by substituting 4>(x) = f(x)e9^ into the ODE.) Since we set the problem up with a branch cut running up the positive-imaginary axis, it is natural to define a ray in the complex plane by setting x = pel6/i with p real:
Re
/e Figure 4: A ray in the complex x-plane For N > 2, the asymptotic is not changed if P(x) is replaced by —(ix)N, and substituting into the general formula we see two possible behaviours, as expected of a second-order ODE: ipd
-1/4
oi0(l
exp
-JV+21
+ N/2)
l + N/2
For most values of 6, one of these solutions will be exponentially growing, the other exponentially decaying. But whenever Re[eie(1+N/2^] = 0, the two solutions swap roles and there is a moment when both oscillate, and neither dominates the other. The relevant values of 9 are
e
7T
.
37T
,
±
57T
'N+2 ' N+2 ' 7V+2 ' (Confusingly, the rays that these values of 6 define are sometimes called 'antiStokes lines', and sometimes 'Stokes lines'.) Whenever one of these lines lies along the positive or negative real axis, the eigenvalue problem as originally
81
stated becomes much more delicate, for the reasons described above. Increasing N from 2, the first time that this happens is at N=4, the case of the upside-down quartic potential. But now we see that the problem arose because the line along which the wavefunction was being considered, namely the real axis, happened to coincide with an anti-Stokes line6. We also see how the problem can be averted. Since all functions involved are analytic, there is nothing to stop us from examining the wavefunction along some other contour in the complex plane. In particular, before N reaches 4, the two ends of the contour can be bent downwards from the real axis without changing the spectrum, so long as their asymptotic directions do not cross any anti-Stokes lines in the process. Having thus distorted the original problem, N can be increased through 4 without any difficulties. The situation for N just bigger than 4 is illustrated in figure 5 below, with the anti-Stokes lines shown dashed and the wiggly line a curve along which the wavefunction tp(x) can be defined.
Im
I'
ii
,;
Re "A
{
\ s
Figure 5: A possible wavefunction contour for N > 4. The wedges between the dashed lines are called Stokes sectors, and in directions out to infinity which lie inside these sectors, wavefunctions either grow or decay exponentially, leading to eigenvalue problems with straightforward, and discrete, spectra. Note that once TV has passed through 4, as in the figure, the real axis is once again a 'good' quantisation contour - but for a different eigenvalue problem, which is not the analytic continuation of the original N < 4 problem to that value of N. (For the analogue of figure 1 for this new problem, see figure 20 of [15].) There is a lesson to be drawn from all of this [11,13-15]. Associated with an ODE of the type under consideration there are many different eigenvalue ^as just mentioned, some would have called this a Stokes line.
82
problems, each defined by specifying a pair of Stokes sectors, and then asking for the values of E at which there exist solutions to the equation which decay exponentially in both simultaneously. For a given value of N, the two sectors which include the positive and negative real axes may appear to be the most natural choice, but if we want to discuss analytic continuation then all must be put on an equal footing. This picture will find a precise analogue on the integrable models side of the correspondence, but before describing this we need to review some more basic material. 2
Functional relations in integrable models
In this section the 'functional relations' approach to integrable lattice models will be outlined. The treatment will be sketchy, and for a more detailed discussion an excellent reference is the book [24] by Baxter. We shall discuss the six-vertex model, defined on an N x M lattice, with periodic boundary conditions in both directions and N/2 even. On each (horizontal or vertical) link of the lattice, we place a spin 1 or 2, conveniently depicted by an arrow pointing either right or left (for the horizontal links) or up or down (for the vertical links). Only those configurations of spins which preserve the 'flux' through each vertex are permitted.
A ^j3»i
v jjgg
">S
Figure 6: A typical configuration of spins for the six-vertex model Locally, flux conservation means that there are six options for the spins around each site, or vertex, of the lattice (hence the name of the model). To
83
each of these possibilities a number W, called a (local) Boltzmann weight, is associated. If we further restrict ourselves to the zero-field case, meaning that the Boltzmann weights should be invariant under the simultaneous reversal of all arrows, then just three independent quantities need to be specified:
i
t W
-
>
= W «-
•
w
->
=w
W
->
W
i
=b
<-
t
The relative probability of finding any given configuration is simply the product of the Boltzmann weights at the individual vertices. A first quantity to be calculated is the sum of these numbers over all possible configurations the partition function, Z. Very crudely speaking, a model is said to be integrable if it is possible to evaluate quantities such as Z (or even better, the free energy) exactly, at least in the limit where N and M both tend to infinity. The model under discussion turns out to be integrable in this sense for all values of a, b and c. Their overall normalisation factors out trivially from all calculations, and the remaining two degrees of freedom can be conveniently parameterised using a pair of variables v (the spectral parameter) and r\ (the anisotropy): a{v,rj) — sinh(i?7 — v) , b{v,rj) = sinh(iry + v) , c(v,ri) = sinh(2i?7) In calculations the anisotropy is usually held fixed, but it turns out to be useful to allow the spectral parameter to vary. The weights can then be drawn as in figure 7.
•
w
a'
a'
-
(3 $' _ a
(v) V
Vi
Figure 7: The local Boltzmann weights
84
This picture hints at the oft-found relationship between Boltzmann weights for integrable lattice models and S-matrix elements for integrable quantum field theories, with the spectral parameter of the Boltzmann weight proportional to the relative rapidity of two particles in the quantum field theory 0 . We won't explore this aspect much further here, but a nice discussion can be found in [25]. One approach to the computation of the partition function proceeds via the introduction of the so-called transfer matrix, T. Introduce multi-indices a =
(QI
, a-2 • • • OLN), a! = (a[ ,a'2- • • Q^) and set
T° M
= Ew
w
{/3.}
w\/33a'304
a
2
...w\0Na'N01
In pictures:
T(v)
Figure 8: The transfer matrix The job of T, a 2N x 2N matrix, is to perform the sum over a set of horizontal links. From the picture it is clear that the indices of T correspond to the spin variables sitting on the vertical links. These can now be summed by matrix multiplication, with a final trace implementing the periodic boundary conditions in the vertical direction. Thus: Z = Trace [T M ] . The next step is to compute via a diagonalisation of T. Suppose that the first few eigenvalues, to > t\ > ... are known, with eigenvectors \p(°), \I/(1', . . . : x
a
*Q'
—
l
J * a
•
Then, for example, the free energy per site in the limit M —> oo can be obtained as / = -j^jlogZ c
= - ^ l o g T r a c e [T M ] ~ - £ l o g t 0 •
In fact, in defining a, b and c we have also shifted the spectral parameter by ir\ from the value that would be appropriate for an S-matrix, so as to give later equations a more conventional form.
85
The eigenvalues to, t\ ... are all functions of v and r), and the remaining task is to find out what these functions are. One of the most popular techniques for doing this goes by the name of the (algebraic) Bethe ansatz (see, for example, [26-28]). The method works in two stages: (i) Guess a form for an eigenvector of T, depending on a finite number of parameters v\,... vn (the 'roots'). (ii) Discover that this guess only works if the {i^} together solve a certain set of coupled equations (the 'Bethe ansatz equations'). To make this more explicit we first introduce another auxiliary object, the monodromy matrix T:
Taa\v)ab =
Y.W
x
a
/?2
w
w Q2
Ql
Q3
...W
pNa'N
b
{ft} Again, this is a definition most easily digested with the aid of a picture:
r(v),ab {P,} Figure 9: The monodromy matrix In contrast to the transfer matrix T, the sum over one of the horizontal spins - /?i - has been omitted from the definition of 7". Of course, T can immediately be recovered once T is known, simply by performing this sum:
but it turns out that important data is also hidden in the off-diagonal elements of T. For convenience, we write its components as X-, 'ab
T^\
=
(A(V,V)
B(u,r,y
—
so that T(v) = A(v,ri) + D{v,V) We now come to a key property of the Boltzmann weights of the sixvertex model, a sufficient condition for integrability. There exists a collection of numbers R"^(v), making up the so-called R matrix, such that Ka^-^)W[c°Qb\(u)W
C
5
b
(i/) = W\a 5a c] (i/) W a„'
Q
o
„> c ly)K°.{v-v')
86
This relation 'intertwines' the local Boltzmann weights at spectral parameters v and v'. It is best handled diagrammatically, representing R^f as follows:
a'
^ d
Figure 10: The R matrix The placing of the indices reflects the interpretation of R as a matrix acting on the tensor product V\ <8> V2, where V\ and V2 are the two-dimensional vector spaces, spanned by —>• and +-, that are seen by the indices c and d respectively. The intertwining relation is then
Figure 11: Intertwining the local Boltzmann weights Apart from a shift in the spectral parameter, the entries of the R matrix are just the original Boltzmann weights. Explicitly, the nonzero entries are v
V
) == ( J Q ' RZX{v) = Rtti")
=
V
C$X.
)=v =
= a(v—ir},rf) = smh.{2ir]—v)
(
" X K^ "
R2W) = RZH") =
R5ZM=RZ5(v)= vV
vi V\
V = v: Li
ir], rj) = sinh(^)
r• =C(V-IT),
77) = sinh(2i?7)
87
Once the local intertwining relation is known, it is a simple matter to chain together N such equalities to find an analogous relation for the monodromy matrix. This is illustrated in figure 12. a,
b'
"JXs "1
a, ^2
oc, -!
~N
-
2
a~ .N
Figure 12: Commuting the monodromy matrices In equations: B£,{V
- ^')(T* ) c 6 H ( T « W ( z / ) = (Tsa)ac(v')(Tf')a,c,(v)R%(v
~ v') •
Two important consequences now follow. First, using the cyclic property of the trace it is easy to see that the transfer matrices at different values of the spectral parameter commute: [T(i/),T(i/)] = 0. This implies that they can all be diagonalised simultaneously, with vindependent eigenvectors. The second consequence is more technical. A key idea behind the algebraic Bethe ansatz method is to treat B{v) as a creation operator, and for this it is necessary to know how to pass the other components of the monodromy matrix through it. Taking the relations implied by figure 12 over all possible values of the vector (a,a',b,b') of free indices gives a total of 16 quadratic relations. These can be rearranged to give, amongst others, the following exchange relations: (->,->,<-,*-):
[B(i/),B(«/)]=0;
(-•, ->, <-, ->) :
A{v)B{y')
= g{v' - v)B{v')A{v)
- h{y' - v)B(v)A{v')
(->, <-, -», ->) :
D(y)B(v')
= g(v - v')B(v')D{v)
- h(v - v')B{v)D{v')
; .
The 'structure constants' g and h are found from the components of the R matrix: , > _ a(v - 177,77) _ sinh(2«77 - v) b{v — irj,r)) sinh(i^) h(u) =
0(77) b(v — irj, rj)
sinh(2i77) sinh(i>)
At last we can write down some eigenvectors. The first is easily found: it is the false ferromagnetic ground state |fi) = | "|"f . . . -\). In a vector notation,
This is an eigenvector of both A and D, A(v)\Sl) = a(v,r,)N\n)
D(V)\Sl) = b(v,T,)N\tl) ,
,
while C{v)\Q) = 0 and B(v)\Q) is, in general, a new state. For the simplest example, at TV = 1, |fi) = (J) and A, B, C and D are all 2x2 matrices:
^v=1 =
a 0 n 0 Ib
0 C U = i - ( 00
c
Q 0J
0 c
>B W = l >
^IJV=I
:
0 0
b 0 0 a
Thus A(u)\Q) = a(v)\Cl), £>(i/)|fi) = b(v)\Q), C(v)\fl) = 0, and B(v)\Sl) = c(i/)( 1 ). Notice, in addition to \il) being an eigenvector of T = A + D, the other eigenvector of T can be written as B(v)\Q). In fact, in this rather-trivial case it could equally have been written as B(v')\tl), for (almost) any value of v''. To generalise this observation, at larger values of TV one might search for eigenvectors of T of the form \9) = B{y1)B(y2)...B{yn)\to)
.
This is the 'guess' mentioned above as stage (i) of the algebraic Bethe ansatz method. For TV > 1 this guess (or ansatz) does not always work - extra conditions need to be imposed on the numbers v\ ... vn. To find these, we first compute the separate actions of A(v) and D(v) on our would-be eigenvector of T. Using the A—B exchange relation, A{u)\*)=A{y)B{y1)B(y2)...B{yn)\n) =
g{u1-v)B{y{)A{y)B{y2)...B{yn)\il) -h(v1-v)B{p)A{vl)B{v2)...B{vn)\n)
.
89
Continuing to pass A through the B operators, but only recording explicitly those terms resulting from the first term on the RHS of the A—B exchange relation, yields n
^Mi*)=n^-")o(".'?)jvi*> i=l n
- ftfo-i/) Y[g(vi - ^)a(iy,r,)NB^)B(u2)...
B(vn)\Sl)
i=2
+ further terms A neat way to reconstruct the further terms uses the observation that |*) depends on the parameters {v\ ...vn} in a symmetrical manner, since the B(VJ) commute. This enables us to deduce that n
A(z,)|*) = A+|*) + £ A + | * , ) k=\
where n
n
A + = a(v,T])N J J g{vj - v) ,
A^" = -a(uk,r])Nh(i/k
- v) J J g(vj - vk) >=
3= 1
1
and n 3= 1
(It is instructive to check this directly, at least for n = 2.) Likewise, n
Z?(i/)|*) = A"|*> + ^
Aj|* fc >
with n
A" = b{v,rj)N ng{v
n
- Vj) ,
A^ = -b(i/k,j])Nh{v
- vk) ~[[g(vk - v3) . i=
3=1
1
The ansatz |*) will be an eigenvector of T = A + D if and only if all terms proportional to I**), k = 1 . . . n, can be made to cancel between A(i/)\^) and D(v)\V). This requires A^~ + Ak = 0, or f
n
n TT sinh(2z7? - vk + ^ )
o ( ^ , r?)^
90
These are the Bethe ansatz equations (BAE) for the roots {vi ...vn}. It is important to realise that these equations do not have a unique solution, but a discrete set, matching the fact that the matrix T has many eigenvalues'*. For each solution, there exists an eigenvector I*) of T with eigenvalue t(y) = A+ + A" n
N
n
N
= <*(„,v) I I s("i - ") + &(".v) I I aiy - »J) • j=i
i=i
To select a particular eigenvalue, supplementary conditions should be imposed on the roots. In particular, for the ground state it turns out that n = N/2, and all of the v^ lie on the real axis. So much for the algebraic Bethe ansatz. There is a particularly neat reformulation of the final results, discovered by Baxter, that led him to an alternative way to solve the model. The first ingredient is the already-mentioned fact that the transfer matrices at different values of the spectral parameter v commute, implying that the eigenvectors are ^-independent. This allows us to focus on the individual eigenvalues t0(v), t\{v),... as functions of v. From the explicit form of the Boltzmann weights these functions are entire, and «7r-periodic. The other ingredient is simply the statement that, for each eigenvalue function t(u), there exists an auxiliary function q{v), also entire and (at least for the ground state) i7r-periodic, such that t(v)q{v) = a(i/, Jj)Nq{y + 2iV) + b{v, rt)Nq(v - 2iV).
(TQ)
We shall call this the T-Q relation, though this phrase should really be reserved for the corresponding matricial equation, involving T(i/) and another matrix Q(^), from which the above can be extracted when acting on eigenvectors. From one point of view, this is merely a rewriting of the expression just given for t(v). However, the earlier formula took as input a solution {v\ ... i/n) of the BAE. Baxter's key insight was that, when the formula is rewritten as in (TQ), there is no need to impose the BAE as a supplementary set of constraints. If we allow q(u) to be arbitrary, then these equations emerge as a consequence of the fact that both t and q are entire. Since Baxter was able to establish the T-Q relation by an independent argument, this provided him with an alternative treatment of the six-vertex model, which avoided the explicit construction of the eigenvectors. He was then able to generalise this to d
We won't go into the question of the completeness of the BAE solutions here; see [29] for a recent discussion.
91 the previously unsolved eight-vertex model, though we will not be elaborating this aspect any further here. The BAE are extracted from (TQ) as follows. Suppose, anticipating the final result in the notation, that the zeroes of q(v) are aXv\,...vn. Given that q(v) is 27r-periodic, it can be written as a product over these zero positions as n
q{v) = JJsinh^-i//) .
(Q)
From (TQ), t{v) is fixed by q{v), and from (Q), q(u) is fixed by the set {vi}To fix the {vt}, set v — V{ in (TQ). On the LHS we then have t(vi), which is nonsingular since t(v) is entire, multiplied by q{vi) which is zero by (Q). Thus the LHS vanishes, and rearranging we have a(i/j,T])N b(vi,r])N
q{vi - 2irj) _ q(vi+2iri) or, using (Q) one more time, ( 1
1i ij
nfTsinh(2^-t/i+^) fL\smh{2ir,-ul + ui)
=
a(vi,v)N
• '
KVUTI)"
,
%
This is exactly the Bethe ansatz equation for the problem, with the Vi the roots. The expression for t{u) implied by (TQ) then matches the formula t = A + + A~ resulting from a direct application of the algebraic Bethe ansatz. To make the match with the Schrodinger equations described in the last section, we will need to take the limit N —• oo in a specific way. For this, it will be more convenient to re-express the results just obtained using an alternative set of variables, setting Ei = e2vi
,
u = -e~2i7)
.
1 + uEj
\
The BAE become frf
E,-u2Ei\_
2n_N(
N
.
From now on we will concentrate on the ground state, for which, as already mentioned, all of the i>i lie on the real axis and n = N/2. This translates into all of the E{ being real and positive, and eliminates the factor ui2n~N from the RHS of the BAE. In addition, for the ground state the distribution of the Pi is symmetrical about v = 0, so each E, is paired with 1/E{ (this follows from the Perron-Frobenius theorem). We will also focus on the limit N -^ oo, in which the number of roots needed to describe the ground state diverges.
92 This complication is to some extent compensated by the fact t h a t the Bethe ansatz equations for the 'extremal' roots, those i/j lying to the furthest left or right along the real axis, simplify in this limit, a t least for 77 > 7r/4. T h e left edge of the root distribution tends to —00, as — ^ logiV. (This behaviour can be extracted, for example, from the results of [18].) Hence the lowestlying Ei scale to zero as JV -4 ''/ 71 ". To capture their behaviour, replace E{ with E[ = N-^^Ei in the BAE, and then hold the Et finite as N -> 00. T h e B A E simplify t o the following: ^r f Et- uj2Ei
\
(For TJ < 7r/4, the product must be regulated before the N —> 00 limit is taken, which complicates the story. We won't discuss this any further here, except to remark t h a t it corresponds t o the leaving of the 'semiclassical domain' mentioned in [20,21].) T h e simplified B A E result from the simplified T-Q relation t{u)q{v)
= q{v + 2irj) + q(v - 2irf) .
These equations control the distribution of the extremal roots in the largeN limit. Physically, they are important because they t u r n out to determine the leading finite-N corrections t o the ground state energy - see, for example, [18]. From our perspective their interest is t h a t they find a match in the theory of ordinary differential equations, as will be explained in the next section. So far we have been discussing the behaviour of lattice models. However, the 'large-TV' form of the T-Q system also arises directly in the context of continuum q u a n t u m field theory. In the papers [20,21], Bazhanov, Lukyanov and Zamolodchikov were able t o construct analogues of the T and Q operators using a free-field representation of the massless limit of the sine-Gordon model. The T-Q relation is then most usually written in terms of a variable A, on which the 'shifts' on the RHS act multiplicatively, as follows: T(\)A±{\) = e^'Mifa- 1 *) + e±2*ipA±{q\) 2
J7r/3
(TQ')
Here T and A± are entire functions of A , q = e with ft the sine-Gordon coupling, and p, an extra parameter compared t o the previous discussion, is related t o the possibility of adding a twist t o the periodic b o u n d a r y conditions (an option which also exists on the lattice). Note also t h a t q can b e interpreted as a q u a n t u m group deformation parameter. Since much of the current interest in the functional-relations approach t o integrable models is focussed on the continuum field theory applications, we will concentrate on this version of the T-Q relation in the remainder of these notes. However it
93 should be remembered that the link with the theory of ordinary differential equations applies equally to lattice models, so long as a large-./V limit is taken in the way described above. 3
The T Q / O D E connection
The goal now is to show that (TQ') also arises naturally in connection with the eigenvalue problems discussed in section 1. First, we need to develop our treatment of ordinary differential equations in the complex domain a little further, relying largely on the book by Sibuya [11]. Consider the ODE dx2
+ P(x) ijj{x) = 0
(*)
where P(x) — x2M — E, and M is real and positive. (This is the BenderBoettcher problem with N = 2M, x —> x/i and E —> —E-a, change which is made purely for convenience.) Then [30] the ODE (*) has a solution y(x, E) such that (i) y is an entire function of (x, E) [though x lives on a cover of C\{0} if 2M^Z ] (ii) as \x\ -> oo with | arg x\ < 3TT/(2M+2),
y~
z-^expj-^^]
2/ '~-a:
M 2
/ exp[-^^+ 1 ]
[ though there are small modifications for M < 1 ] These properties fix y uniquely; to understand where they come from we quickly recall the discussion of section 1. With the shift from x to x/i, the anti-Stokes lines for the current problem are arg(z) = ±77777 , ± 'N+2 ' N+2 '
"
and in between them lie the Stokes sectors, which we label by defining Sk — arg(z)
2-rrk
2M+2 < 2M+2
For M just larger than 2, three of these sectors are shown in the figure 13 below, a 90° rotation of figure 5 from section 1. The asymptotic given as property (ii) matches the WKB result in <5_i U <So U Si . Note however that the determination of the large \x\ behaviour of y
94
beyond these three sectors is a non-trivial matter (the continuation of a limit is not necessarily the same as the limit of a continuation!). This subtlety is related to the so-called Stokes phenomenon, and it can be handled using objects known as Stokes multipliers, to be introduced shortly.
s,
s Figure 13: Stokes sectors for the ODE (*), with M > 2. One more piece of notation: an exponentially-growing solution in a given sector is called dominant (in that sector); one which decays is called subdominant. It is easy to check that y as defined above is subdominant in So, and dominant in <S_i and Si. Note that subdominant solutions to a second-order ODE are unique up to a constant multiple; this is why the quoted asymptotics pin down y uniquely. Having identified one solution to the ODE, we can now generate a whole family using a trick due to Sibuya. Consider the function y(x,E) = y(ax,E) for some (fixed) a 6 C From (*), d2 + dx'1
a2M+2x2M
azE
y(x,E)=0.
(This is sometimes given the rather-grand name of 'Symanzik rescaling'.) If a 2 M + 2 = l , it follows that y(x,a~2E) solves (*). Setting ,,, _
e2ni/(2M+2)
and yk(x,E)=^2y(LO-kx,^E) we therefore have the key statements • yu solves (*) for all k 6 Z; • up to a constant, y% is the unique solution to (*) subdominant in Skfollows easily via the asymptotic of y. ]
[This
95 • each pair {yk,yk+i} forms a basis of solutions for (*). [This follows on comparing the asymptotics of yk and yu+i in either Sk or <Sjt+i.] We have almost arrived at the T-Q relation. Next, we expand y_i in the {2/o,2/i} basis: j / _ ! ( x , E ) = C(E)y0(x,
E) + C(E)Vl(x,E)
.
We will call this a Stokes relation, with the coefficients C(E) and C(E) Stokes multipliers. They can be expressed in terms of Wronskians. A quick reminder [31]: the Wronskian of two functions / and g is
W[f,g] =
fg'-f'g.
For two solutions of a second-order O D E with vanishing first-derivative term, W [f,g] is independent of x, and vanishes if and only if / and g are proportional. To save ink we set Wkuk2
=
W[ykl,yk2]
and record the following two useful properties: (E) = WklM(uj2E)
,
W0,i(E)=2i.
Now by 'taking Wronskians' of the Stokes relation first with yi and then with 2/o we find r
=
W-ltl W0,i
~ __
W-1<0
'
W0,i
and so the relation can be rewritten as C(E)y0(x,E)=y-1(x,E)+y1(x,E)
,
or, in terms of the original function y, as E) = cj-1/2y(ujx,
C{E)y(x,
UJ~2E)
+ uj1/2y{uj'lx,
u?E).
(CY)
This looks very like the T-Q relation. T h e only fly in the ointment is the x-dependence of the function y. But this is easily fixed: just set x to zero. We can also take a derivative with respect to x before setting it to zero, which swaps the phase factors w ± 1 / / 2 . So we define D-(E)
= y(0,E)
,
D+(E)=y'(0,E).
(The notation will be justified shortly.) Then the Stokes relation (CY) implies C{E)D*(E)
= ^1'2D^{LO~2E)
+ u}±x'2D*{w2E)
.
(CD)
96
Finally we are ready to make the comparison. If we set M+\
y
'
4M+4
then the match between (TQ') and (CD) is perfect, with the following correspondences between objects from the IM and ODE worlds: T A±
C «->
DT
The mapping could also have been made onto the limiting form of the Bethe ansatz equations for the six-vertex model that was obtained in the last section, with suitable twisted boundary conditions imposed on the lattice. In this case, the relationship between the anisotropy parameter rj of the lattice model and the parameter M appearing in the potential of the Schrodinger equation is _ 7T V
~
M
2M+1 '
How should we think about C and D? In fact they are spectral determinants. (The spectral determinant of an eigenvalue problem is a function which vanishes exactly at the eigenvalues of that problem. It generalises to infinite dimensions the characteristic polynomial det(M — AI) of a finite-dimensional matrix.) Recall that C{E) is proportional to W-i^(E). Thus C(E) vanishes if and only if W [ J / _ I , J / I ] = 0, in other words if and only if E is such that y-i and y\ are linearly dependent. In turn, this is true if and only if the ODE (*) has a solution decaying in the two sectors S-i and Si simultaneously, which is exactly the eigenvalue problem discussed in section 1, modulo the trivial redefinitions of x and E. This is enough to deduce that, up to a factor of an entire function with no zeroes, C{E) is the spectral determinant for the Bender-Boettcher problem 6 . Even this ambiguity can be eliminated, via Hadamard's factorisation theorem, once the growth properties of the functions involved have been checked; see [4] for details. To see that the functions D^ are spectral determinants is even more easy: first, we note that by their very definition the functions y(x,E) decay at (real) x —> oo for all values of E. If D~(E) = y(0,E) = 0, then this solution, decaying at +oo, also vanishes at x = 0, while if D+(E) = 0, then it has vanishing first derivative there. A moment's thought shows that this corresponds to there existing odd or even, e
Since we performed a variable change in this section compared with the discussion in section 1, it is in fact C( — E) which provides the spectral determinant for the BenderBoettcher problem as originally formulated.
97 respectively, wavefunctions for the equation on the full real axis with potential |a;| 2M . (It was for this reason that the functions D± were so labelled.) This insight allows us to fill in one gap in the correspondence. While the T-Q relation is very restrictive, as remarked in section 2 it does not have a unique solution. So to say that D~(E) is 'equal' to A+(\) begs the question: which A|_(A)? To answer, we first note that, in contrast to the BenderBoettcher problem, the full-line problem with \x\2M potential (or equivalently, the half-line problem with y(0) = 0 boundary conditions) is self-adjoint, and so all of its eigenvalues are guaranteed to be real. Back in the integrable model, the only solution to the BAE with all roots real is known to be the ground state, so the question is answered: the relevant A+(\) is that corresponding to the ground state of the model. As it stands, the correspondence is still not entirely satisfactory since, for each value of P2, it picks out just one value of p. A more complete mapping would find partners for the BAE at other values of the twist parameter as well. This was sorted out very shortly after the original observation of the correspondence in [1]: in [2], Bazhanov, Lukyanov and Zamolodchikov pointed out that the ODE (*) should be generalised to ax1
I/J{X) =
x1
0.
(This observation, combined with the discovery of the role of the T operator made in [4], provided the motivation to study the spectra shown in figures 2 and 3 of section 1.) The previous mapping between parameters becomes 0i
1 M+l
'
y
2Z+1 4M+4 '
and varying / away from zero allows us to explore the other values of p. This is a continuation through continuous values of angular momentum in a radial (three-dimensional) Schrodinger equation - in other words, non-relativistic Regge theory! A little more care is needed in the definition of D"^ once 1(1+1) is nonzero, since the equation acquires a regular singularity at the origin. The resolution is to match the solutions yk onto solutions ip± with simple scaling behaviours at the origin; the details can be found in [2,4]. Two more points deserve comment. First, studies of integrable models had already shown how to transform a T-Q relation into a nonlinear integral equation (NLIE), which in turn can be solved by numerical iteration rather easily [18,21,32]. The NLIE is particularly simple for the ground state, and it was this that allowed the spectral plots of section 1 to be obtained in [4], building on the checks for specific cases performed in [1].
98 Second, we should mention that there is another strand to the functional relations approach to integrable models, based on the so-called fusion hierarchy and its truncations (see for example [17,20]). This proceeds via the definition of fused transfer matrices Tj, j = 0, | , 1, §, . . . (with To = 1 and the original T identified with T x / 2 ). Ultimately, it leads to another set of nonlinear integral equations, often referred to as being of 'TBA type' [33]. Obviously it would be nice to find a role for these objects in the differential equations, and it turns out that this is possible. As explained in [4], the fused transfer matrices can be mapped onto the Wronskians Wkltk2 with \ki — k2\ > 2, just as the fundamental transfer matrix maps onto W_i,i. They therefore correspond to the other eigenvalue problems, found when other pairs of Stokes sectors are chosen for the quantisation contour, that were mentioned at the end of section 1 above. The full story is illustrated in figure 14.
i
/ T
__ ! \
V*Figure 14: General quantisation contours involved in the O D E / I M correspondence.
Once the fused transfer matrices have been understood in this way, truncation of the fusion hierarchy can be reinterpreted in terms of the (quasi-) periodicity (in A;) t h a t the functions y^ exhibit whenever M is rational. In the simplest cases (with M rational and / ( / + 1 ) = 0 ) this periodicity arises because the solutions to the O D E live on a finite cover of C \ { 0 } ; for other cases, the monodromy around x=0 needs a little more care, but the story remains essentially the same. All good correspondences need a dictionary, and to end this section we give a summary of the mapping between objects seen by the integrable model and by the Schrodinger equation.
99 Schrodinger equation
Integrable Model Spectral parameter
•H-
Energy
Anisotropy
•H-
Degree of potential
Twist parameter
•H-
Angular momentum
(Fused) transfer matrices
•H-
Spectral problems defined at |a;|=;oo
Q operators
•H-
Spectral problems linking |x|=oo and \x\-0
Truncation of the fusion hierarchy
<->
Solutions on finite covers of C\{0}
(The two classes of spectral problems mentioned in this table are related to the 'lateral connection' and 'radial connection' problems in general WKB theory - see, for example, [34].) Armed with the dictionary, the horizontal axis of figure 1 can be annotated to indicate which integrable models correspond to the various values of N in the Bender-Boettcher problem. Thus for N = 1,2,3,4 and 6, the relevant integrable models are the N=2 SUSY point of the sine-Gordon model, the free-fermion point, the Yang-Lee model, Z 4 parafermions and the 4-state Potts model respectively. It is amusing that the a;3 potential is related by the correspondence to the Yang-Lee model (or, strictly speaking, to the sineGordon model at the value of the coupling which allows for a reduction to Yang-Lee), thus returning by a very indirect route to a neighbourhood of the original thought of Bessis and Zinn-Justin. 4
Generalisations
The Bethe ansatz equations seen so far can all be written in terms of the variable E as
n(f^S|)=- ! ' +i . '-i.>where u = e 2 "/(2M+2) ] M is related to the quantum group deformation parameter, or anisotropy, of the lattice model, and I is related to the twist.
100
These are the n=2 cases of a general family of 5L r (n)-related Bethe ansatz 1 systems, relating n— 1 sets of unknowns {Ej," '}, with m = l,2...n—1 and k k= 1,2. ..oo:
nn
E(t)
L
*
1 = _ w «rn,+l
w =
e2*i/n(M+l)
t=i j=
As in the SU{2) case, M can be viewed as a deformation parameter, but this time there are not one but n—1 independent twists, TI, T2,.. . r n _ i . The indices m and £ should be thought of as living on an SU(n) Dynkin diagram, of which Cmt is the Cartan matrix:
n-\ To obtain these systems of equations using operators defined directly in a continuum quantum field theory, as achieved in [20,21] for the SU(2) case, appears to be a largely open problem, though the first steps have been undertaken in [35]. But even without this motivation, it is very natural to ask whether the correspondence described above can be extended to cover BA systems of these more general types. The answer is yes [8], and it turns out that one has to look to higherorder ordinary differential equations. Earlier but less complete results in this direction were obtained in [5,6]; aspects of the problem are also discussed in the recent article [9]. One of the main difficulties is to find a parametrisation of the higher-order differential operators which incorporates the twists in a manageable way. The solution found in [8] starts by defining an elementary first-order differential operator, D(g):
™=(E-! Basic properties are D(g)^ = —D(—g) and Z)((<7I) = Now, given a vector g = (g0, gi... gn-\), set
D(gi—l)D(g2).
D(S)=D(gn„1-(n-l))D(gn_2-(n-2))...D(g0) and impose ^2™=Q gi = n(n—1)/2 to ensure that the (n—l)th order derivative term vanishes (this allows various theorems about Wronskians to hold in their
101
simplest forms). With this notation in place, the ODE to consider is an immediate generalisation of those seen earlier: [{-l)n+1D(g)
+
(**\
P(x,E)]tl>{x)=0
with P(x, E) — xn - E. After some work, it turns out this ODE does indeed contain a hidden set of SU{n) Bethe ansatz equations. The parameter M in P(x,E) is equal to the M appearing in the SU(n) BAE quoted above, while the vector of parameters g is related to the twists in the BAE by
/ °1 \ + 2
(\—n 1 1
V
2—n 3—n 2—n 3—n 2 3-n
•
\
/ - i
7
1
\
Vn-J
To reveal the SU(n) \n-lj structure, one first introduces a special set {yk} of solutions to the n t h order ODE, with specific asymptotics in certain sectors in the complex plane. They generalise the yk employed by Sibuya in the secondorder case, described in the previous section. The next step is to consider a special set of Wronskians between these solutions, defining W,
(m)
W^[y
k,Vk+l,
•Vk+r,
where
h fi
/i
W
/(m)
[/i,/2,.../m]=Det
n
/r
&
Jm J m
/.
[m-l]
and f\ (x) = j-^fi{x). For m = 1 . . . n—1, these Wronskians are non-trivial functions of x, and can be associated with the nodes of the SU(n) Dynkin diagram. Furthermore, they can be shown to satisfy a set of functional equations in x and E that generalises the relation (CY) obtained for the SU(2) case above. A suitable expansion near to x = 0 then yields the SU(n) version of the T-Q relation (sometimes called the 'dressed vacuum form'), from which the SU(n) Bethe ansatz system follows in a simple way. The full story is a little too long to be included here, but can be found in the paper [8].
102
5
Duality and reduction
One important topic not covered so far is the new light that the correspondence sheds on some previously-conjectured duality properties of integrable models [2,4,5,8]. A sequence of transformations maps the potential xM to -xM, with M = —M/(M+1). The first step involves the following change of variables: z = \nx,
r/>(x) = e{n-l)z'2u{z)
.
This generalises the Langer transformation [36], to which it reduces for n = 2. When it is applied to the ODE (**) of the last section, the result is
(_ir.
(I - 7„_,)... ( i - 71) ( | - „) + ,«<.+»>< - E e » . ] „ w = o
where ~a = gt — (n—1)/2. Next, the role played by the two exponentials is swapped by sending z to M* + In jfijw, to obtain i[
'
\dz
M+l)---\dz
M+l)\dz
M+l)
\
[>
U
'
where E = - ( M + 1 ) " M £ , " < M + 1 ) . Finally, inverting the initial variable change, (**) has been transformed into [(-l) n + 1 £>(g) + P(x,E)]l>(x)
=0,
P(x,E)
= -
lM
-E
with T7 =
M
M "MTT'
_ _ , r_ _ S = {90,91,..-,9n-i},
_ gi + M ( n - l ) / 2 9i = jj^—j •
The final equation has the same form as the original ODE, the only differences being a reversed sign in front of the 'potential' term xnM, and the replacement of the parameters (M, g) by (M, g). It is also very simple to check that solutions of the radial connection problems in the original ODE are mapped, at each step of the transformation, to equivalent, yet apparently quite different eigenvalue problems for the transformed equations. For example, at n = 2 the exponential form of the equation corresponds to a Schrodinger problem for a Morse-type potential: (-•^
+ e22<M+1> - Ee2z\
1>{z, E, I) = K2t/,{z, E, I) .
103
In comparison to the original equation, E acts as a shape-changing parameter in the potential, while the role of the energy is taken by / via the relation re = -i(l+ i ) .
For positive E, the potential has a finite well of depth d = M(E/(M + i))(M+i)/M F o r i > _YJ2 and real, the original radial connection problem is precisely mapped onto a bound-state problem for this potential, while for re real we are dealing instead with a scattering problem. The reflection amplitude S(E, re) is expressible in terms of the spectral determinant
D~(E,l) as -D-^\+*K\.
S(E, = V K) '
D-{E,\-IK)
Completing the mapping back to the original form of the potential, the parameter q = e W(M+i) h a s b e e n replaced by q = e i7r /(M+i) = e "(M+i)_ T h i s is exactly the duality discussed in the quantum field theory context in, for example, [21]. The set of equivalent models related by simple variable and gauge transformations is certainly much larger than the three cases discussed above. However, the 'Bethe ansatz' approach to spectral problems reveals further spectral equivalences which seem to be rather more mysterious. Here, we will just mention one such equivalence, between pairs of ODEs of different orders. In the context of the ODE/IM correspondence this property was first noticed in [5]. We start by rewriting the SU(n) BAE in the form —w
splitting the Cartan matrix as Cmt = 28rnt—Arnt. {Amt is the incidence matrix of the Dynkin diagram of SU(n).) The key observation is that at M = 1,
104
uinM = — 1 and the infinite product on the RHS of the equation simplifies to one. We are still lacking a complete understanding of such 'collapses' at general n, but at n = 3 it turns out that the reduced BAE are the same as those for a Schrodinger equation with potential 6 2 , 1(1 + 1) x + ax H — . xz (The ODE/IM correspondence for this second-order problem was treated, for / = 0, by Suzuki in [7].) The original observation in [5] concerned the special cases with a = 0. The BAE then reduce to St/(2)-related systems of the sort discussed in these notes, at M — 3.
6
Conclusions
The main conclusion of these notes can be stated very simply: it is that the T and Q operators which arise in certain integrable quantum field theories encode spectral data, at least in their ground-state eigenvalues. This gives a novel perspective on the Bethe ansatz, and also a new way to treat spectral problems via the solution of nonlinear integral equations. There are many further problems to be explored, of which we list just a few. First, it will be interesting to find out how many other BA systems can be brought into the correspondence, beyond the An_ i -related cases described above, and whether more general polynomial potentials might also have a part to play. Some further generalisation is certainly possible - see [7] - but the problem of finding ODEs even for the D and E related BA systems remains open. Second, the correspondences established to date have all concerned massless integrable lattice models, in a 'field theory' limit where the number of sites, and of Bethe ansatz roots, tends to infinity. Correspondences for more general massive models, and for lattice models with a finite number of sites, are still lacking and would be very interesting. Even in the context of massless 5t/(2)-related systems in their continuum limits, so far we only have a 'spectral' interpretation for the ground state eigenvalues of the T and Q operators. The other eigenvalues, t\(v), t2(i') and so on, satisfy the same functional relations, but have more complicated distributions of zeroes, not matched by the eigenvalues of the Schrodinger problems we have been treating. Finally, we should admit that our observations remain at a rather formal and mathematical level. It is natural to ask whether there is a more physical explanation for the correspondence, but perhaps this question will have to wait until the answers to some of the other open problems have been found.
105 Acknowledgements We would like to t h a n k Davide Fioravanti, Junji Suzuki and Andre Voros for many useful discussions. P E D would like to t h a n k Changrim Ahn, Chaiho Rim and Ryu Sasaki, the organisers of the Cheju school, for all the work they put into organising such an enjoyable and interesting event. RT also t h a n k s Changrim Ahn for the invitation to present p a r t s of this material during a series of lectures at the A P C T P . P E D and RT t h a n k the UK E P S R C for an Advanced Fellowship and a Visiting Fellowship respectively; we were also supported, in part, by a T M R grant of the E u r o p e a n Commission, reference ERBFMRXCT960012. References 1. P. Dorey and R. Tateo, 'Anharmonic oscillators, the thermodynamic Bethe ansatz and nonlinear integral equations', J. Phys. A32 (1999) L419, h e p - t h / 9 8 1 2 2 1 1 2. V.V. Bazhanov, S.L. Lukyanov and A.B. Zamolodchikov, 'Spectral determinants for Schrodinger equation and Q-operators of Conformal Field Theory', h e p - t h / 9 8 1 2 2 4 7 3. J. Suzuki, 'Anharmonic Oscillators, Spectral Determinant and Short Exact Sequence of Uq(sl2)', J. Phys. A32 (1999) L183, h e p - t h / 9 9 0 2 0 5 3 4. P. Dorey and R. Tateo, 'On the relation between Stokes multipliers and the T-Q systems of conformal field theory', Nucl. Phys. B563 (1999) 573, h e p - t h / 9 9 0 6 2 1 9 5. P. Dorey and R. Tateo, 'Differential equations and integrable models: the S£/(3) case', Nucl. Phys. B571 (2000) 583, h e p - t h / 9 9 1 0 1 0 2 6. J. Suzuki, 'Functional relations in Stokes multipliers and Solvable Models related t o Uq(A{n])\ J. Phys. A33 (2000) 3507, h e p - t h / 9 9 1 0 2 1 5 7. J. Suzuki, 'Functional relations in Stokes multipliers - Fun with x6 + ax2 potential', q u a n t - p h y s / 0 0 0 3 0 6 6 8. P. Dorey, C. Dunning and R. Tateo, 'Differential equations for general SU{n) Bethe ansatz systems', J. Phys. A33 (2000) 8427, hep-th/0008039 9. J. Suzuki, 'Stokes multipliers, Spectral Determinants and T-Q relations', n l i n - s i / 0 0 0 9 0 0 6 10. P. Dorey, C. Dunning and R. Tateo, 'Ordinary differential equations and integrable models', J H E P Proceedings P R H E P - t m r 2000/034, Nonperturbative Quantum Effects 2000, h e p - t h / 0 0 1 0 1 4 8 11. Y. Sibuya, Global Theory of a second-order linear ordinary differential
equation with polynomial coefficient (Amsterdam: North-Holland 1975) 12. A. Voros, 'Semi-classical correspondence and exact results: the case of the spectra of homogeneous Schrodinger operators', J. Physique Lett. 43 (1982) LI; — 'The return of the quartic oscillator. The complex WKB method', Ann. Inst. Henri Poincare Vol XXXIX (1983) 211; — 'Exact resolution method for general ID polynomial Schrodinger equation', J. Phys. A32 (1999) 5993, math-ph/9903045 13. C M . Bender and A. Turbiner, 'Analytic continuation of eigenvalue problems', Phys. Lett. A173 (1993) 442 14. C M . Bender and S. Boettcher, 'Real spectra in non-hermitian Hamiltonians having VT symmetry', Phys. Rev. Lett. 80 (1998) 4243, physics/9712001 15. C M . Bender, S. Boettcher and P.N. Meissinger, lVT symmetric quantum mechanics', J. Math. Phys. 40 (1999) 2201, quant-phys/9809072 16. R.J. Baxter, 'Eight-vertex model in lattice statistics and one-dimensional anisotropic Heisenberg chain; l.Some fundamental eigenvectors', Ann. Phys. 76 (1973) 1; '2.Equivalence to a generalized ice-type model', Ann. Phys. 76 (1973) 25; '3.Eigenvectors of the transfer matrix and Hamiltonian', Ann. Phys. 76 (1973) 48 17. A. Kliimper and P.A. Pearce, 'Analytical calculations of Scaling Dimensions: Tricritical Hard Square and Critical Hard Hexagons', J. Stat. Phys. 64 (1991) 13 18. A. Kliimper, M.T. Batchelor and P.A. Pearce, 'Central charges of the 6and 19-vertex models with twisted boundary conditions', J. Phys. A24 (1991) 3111 19. P. Fendley, F. Lesage and H. Saleur, ' Solving 1-D plasmas and 2-D boundary problems using Jack polynomials and functional relations', J. Stat. Phys. 79 (1995) 799, hep-th/9409176; — 'A unified framework for the Kondo problem and for an impurity in a Luttinger liquid', J. Stat. Phys. 85 (1996) 211, cond-mat/9510055 20. V.V. Bazhanov, S.L. Lukyanov and A.B. Zamolodchikov, 'Integrable Structure of Conformal Field Theory, Quantum KdV Theory and Thermodynamic Bethe Ansatz', Commun. Math. Phys. 177 (1996) 381, hep-th/9412229 21. V.V. Bazhanov, S.L. Lukyanov and A.B. Zamolodchikov, 'Integrable structure of conformal field theory II. Q-operator and DDV equation', Commun. Math. Phys. 190 (1997) 247, hep-th/9604044 22. V.V. Bazhanov, S.L. Lukyanov and A.B. Zamolodchikov, 'Integrable
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23.
24. 25. 26.
27.
28. 29. 30.
31. 32.
33.
34. 35.
36.
structure of conformal field theory III. The Yang-Baxter relation', Commun. Math. Phys. 200 (1999) 297, hep-th/9805008 M. Reed and B. Simon, Methods of Modern Mathematical Physics, I & //(Academic Press 1972, 1975); R.D. Richtmyer, Principles of Advanced Mathematical Physics I (Springer- Verlag 1978) R.J. Baxter, Exactly solved models in statistical mechanics (Academic Press 1982) A.B. Zamolodchikov, 'Factorised S matrices and lattice statistical systems', Sov. Sci. Rev., Physics, 2 (1980) 1 L.D. Faddeev, 'How Algebraic Bethe Ansatz works for integrable model', Proceedings of the 1995 Les Houches summer school, hep-th/9605187 V.E. Korepin, N.M. Bogoliubov and A.G. Izergin, Quantum inverse scattering method and correlation functions (Cambridge University Press 1993) H.J. de Vega, 'Yang-Baxter algebras, integrable theories and quantum groups', Int. J. Mod. Phys. A4 (1989) 2371 K. Fabricius and B.M. McCoy, 'Bethe's equation is incomplete for the XXZ model at roots of unity', cond-mat/0009279 P.-F. Hsieh and Y. Sibuya, 'On the asymptotic integration of second order linear ordinary differential equations with polynomial coefficients', J. Math. Analysis and Applications 16 (1966) 84 Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations, McGraw-Hill 1955 C. Destri and H.J. de Vega, 'New thermodynamic Bethe ansatz equations without strings', Phys. Rev. Lett. 69 (1992) 2313; — 'Unified approach to thermodynamic Bethe ansatz and finite size corrections for lattice models and field theories', Nucl. Phys. B438 (1995) 413 Al.B. Zamolodchikov, 'Thermodynamic Bethe ansatz in relativistic models: scaling 3-state Potts and Lee-Yang models', Nucl. Phys. B342 (1990) 695 F.W.J. Olver, Asymptotics and special functions (Academic Press 1974) D. Fioravanti, F. Ravanini and M. Stanishkov, 'Generalized KdV and quantum inverse scattering description of conformal minimal models', Phys. Lett. B367 (1996) 113, hep-th/9510047 R.E. Langer, 'On the connection formulas and the solutions of the wave equation', Phys. Rev. 51 (1937) 669
108
I N T E G R A B L E SIGMA MODELS PAUL FENDLEY Department of Physics University of Virginia Charlottesville, VA 22904-4714 fendleyOvirginia.edu
These lectures describe exact results for two-dimensional integrable sigma models. I describe the particles spectrum, the 5 matrix, and the computation of the free energy as a function of temperature and magnetic field. I also describe the effects of a theta term and a WZW term.
1
Introduction
Sigma models arise frequently in particle physics and condensed-matter physics as low-energy effective theories. Two-dimensional sigma models have been the subject of a huge amount of study because they are interesting toy models for gauge theories, because they often arise in experimentallyrealizable condensed-matter systems, because this is the highest dimension in which they are naively renormalizable, and because of the powerful theoretical methods applicable. One of the nice things about sigma models is that the same model can often describe completely different physics. The reason is that in many situations, the precise sigma model of interest follows mainly (or sometimes entirely) from the symmetries. For example, sigma models often arise in theories of interacting fermions invariant under some group G. If some fermion bilinear gets an expectation value manifestly invariant under some subgroup H, then the excitations at low energy can be described by a field taking values in G/H. Put another way, the expectation value gives the fermions mass at some scale M. One can then integrate out fermionic excitations, leaving only bosonic G/H excitations with masses below M. The sigma model describes the interactions of these low-energy excitations, and is independent of many of the details of the original theory. This is why vastly different theories may end up having the same low-energy physics. Two-dimensional G/H sigma models all have a global symmetry group G, even though the fields take values in the smaller space G/H. This is one big difference between two and higher dimensions. In higher dimensions, the symmetry G of these sigma models would be spontaneously broken to H, and in the effective low-energy-theory, the G symmetry is not manifest. In other
109
words, in higher dimensions the sigma model describes the physics of the massless Goldstone bosons. However, the Mermin-Wagner-Coleman theorem says that in two dimensions continuous symmetries cannot be spontaneously broken. The way these sigma models satisfy this theorem is to give the wouldbe Goldstone bosons a mass and keep the original global symmetry intact. In particular, many interesting sigma models in two dimensions are asymptotically free. This corresponds to the manifold G/H having positive curvature. At large energies the interactions are weak, but at low energies the interactions are strong. Naively, there seems to be no mass scale in the theory (the coupling constant g is dimensionless), but a mass scale \x appears in the theory as a result of short-distance effects which need to be renormalized. The effective coupling g(n) depends on this scale. At \i large, g((i) is small, so the theory is effectively free, while as /x decreases, g(fi) increases. In renormalization-group language, there is an unstable trivial fixed point at g — 0. For G/H sigma models, the manifold G/H has dimension dimG—dimH, so as g —> 0 the theory reduces to dimG—dimH free bosons. One extremely interesting question is if 5>6. To understand the regime where g(/j.) is large, one must utilize alternative techniques. Large-./V expansions are a common and useful tool. However, for most applications N is small. In fact, sigma models describing disordered
no systems are derived by using the replica trick. This trick requires sending N —> 0 at the end of the computation, so obviously large-iV expansions are not necessarily going to be reliable. Luckily, for two dimensions there are other non-perturbative methods applicable. A number of sigma models are integrable. This means that there are an infinite number of conserved currents. The resulting conserved charges constrain the system, making exact computations possible, even at strong coupling. The aim of these lectures is to attempt to discuss a number of aspects of integrable sigma models. I will develop the tools necessary to understand to derive the exact free energy at finite temperature and in the presence of a magnetic field. This makes it possible to compute the susceptibility and specific heat. It also makes it possible to understand exactly the effects of the theta term. The outline of these lectures is as follows. In section 2 I introduce the models to be studied, which are G/H sigma models on symmetric spaces. In section 3, I explain the effects of WZW terms and theta terms. In section 4, I give the exact particle spectrum for the known integrable sigma models, and in section 5, I give the exact S matrices. In section 6, I compute the energy at zero temperature in a background field, while in section 7 I compute the free energy at finite temperature. Big chunks (but not all) of these lectures have been lifted from my papers 5 , e . A survey of the applications of these and other results to disordered systems can be found in 7 . 2
The models
A sigma model is a field theory where the field takes values on a manifold with a metric gij. The action of a sigma model is independent of the coordinates X1 used to parameterize the manifold. In two spacetime dimensions with coordinates z and z, S = J d2z
j gij(X)d^X'(z,z)d"X (z,z).
(1)
The one and two-loop beta functions for all sigma models can be expressed in terms of the curvature of the field manifold *. If the metric gtj is Ricci flat (the manifold has vanishing first Chern class), then the model is conformally invariant and has no scale. A manifold with positive curvature like a sphere gives an asymptotically-free massive theory. These lectures will be devoted to sigma models where the fields take on values on a symmetric space. A symmetric space is a space G/H where G and H are Lie groups, and H is a maximal subgroup of G (no normal subgroup
111
other than G itself contains H). The important property of a sigma model on a symmetric space is that it contains only one coupling constant g in the action. In other words, the space G/H preserves its "shape" under renormalization, with only the overall volume changing. The effect of renormalization is to increase the curvature (increase g) at low energies. All symmetric spaces have been classified by mathematicians. The non-exceptional ones are displayed in Table 1. The action for sigma models on symmetric spaces is more conveniently written in terms of an matrix field $. The action is
S = -tr I dPxd^&d^
(2)
along with the constraints $t$ = j det{§) = ±1
(3)
where / is the identity matrix. A given model may have have additional constraints on the form of $ . Constraints like (3) can easily be obtained from theories without constraints by adding a potentials like A tr ( $ t $ — I)2. When A gets large, one recovers the restrictions (3). In theories with interacting fermions, this often results from introducing a bosonic field to replace fourfermion interaction terms with Yukawa terms (interactions between a boson and two fermions). Integrating out the fermions then results in such potentials and hence the sigma model. Sigma models on symmetric spaces have action (2), with additional restrictions of the matrix field $. The constraints are listed in Table 1. One model studied in detail here is the SU(N)/SO(N) sigma model, which is obtained by requiring that $ be a symmetric N x N unitary matrix with determinant 1. The other model studied in detail here, the 0(2P)/0(P) x 0{P) sigma model, is obtained by requiring that $ be a real, symmetric, orthogonal and traceless 2P x 2P matrix. This global symmetry G acts on the field $ as $ -> U$V,
(4)
where U and V are unitary matrices. This is the most general symmetry which keeps $ unitary. In the 0(2P)/0(P) x O(P) sigma models, U must be real as well, so G = 0(2P). The field $ in this case can be diagonalized with an orthogonal matrix U, so $ = UAUT
$ e 0(2P)/Q(P)
x O(P),
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where U is in 0(2P), and A is the matrix with P values + 1 and P values — 1 on the diagonal. Different U can result in the same $: the subgroup leaving $ invariant is H = 0(P) x O(P). This is why the space of symmetric orthogonal traceless matrices is indeed 0(2P)/0{P) x O(P). Similarly, field configurations in the SU(N)/SO(N) sigma model can be written in the form $ = UUT
$ e
SU(N)/SO(N)
where U is in SU(N). The subgroup H leaving $ invariant is SO(N). For example, $ = / for any real U in SU(N), i.e. if U is in the real subgroup SO(N) of SU(N). This is why H = SO(N) here. All sigma models on symmetric spaces can be written in the form (2,3). These results are collected for all symmetric spaces (other than exceptional ones) in table 1. The principal chiral models H x H/H for the Lie groups H = SU(N),SO(N) or Sp(2N) are obtained by letting $ be a member of the group H. The SU(2N)/Sp(N) sigma model corresponds to $ even-dimensional and antisymmetric. If it is even-dimensional, anti-symmetric and real as well, one obtains 0{2N)/U(N). One obtains Sp{2N)/U(N) if $ is even-dimensional, symmetric, and obeys the restriction tr J $ = 0, where J is the 2N x 2N matrix of the form
' - ( - ° , o)
<»>
The U(M + N)/U(M) x U(N) models follow if $ is hermitian and has m positive eigenvalues and n negative ones. The Sp(2(M + N))/Sp{2M) x Sp(2N) models have the same conditions plus t r J $ = 0, while the 0(M + N)/0(M) x O(N) follow from the same conditions plus $ real. These conditions are summarized in table I. sigma model SU{N)/SO{N) 0{2N)/0{N) x O(N) Sp{2N)/U(N) U(2N)/Sp(2N) 0(2N)/U(N) U(M + N)/U(M) xU(N) 0(M + N)/0(M) x 0(N) Sp(2M + 2N)/Sp(2M) x Sp(2N) H xH/H
additional conditions on $ symmetric symmetric, real, traceless symmetric, tr J $ = 0 antisymmetric antisymmetric, real $ = UKM'NW,U G U{M + N) $ = UAM'NUT, U e 0(M + N) $ = Uh.2M>2NUt,U e Sp(2M + 2N) €H
Table 1: Summary of the field description of sigma models on symmetric spaces. J is the antisymmetric matrix in (5), and AM,7V is a diagonal matrix with M entries +1 and N entries — 1.
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The action for models of the form G(M + N)/G{M) x G(N) can often be written in simpler forms. For example, U(N + l)/U(N) x U(l) are known as the CPN models and can be written in terms of an N + 1-dimensional complex vector z obeying z*z\ — 1 (and an extra gauge-type interaction), while U{2N)/U(N) x U(N) are known as the Grassmanian sigma models and are equivalently described by making $ Hermitian and traceless. The most famous sigma model on a symmetric space is probably the 0(N)/0(N — 1) model, which can be written in terms of an ^-dimensional real vector v obeying ViVi = 1. In other words, the manifold 0(N)/0(N — 1) is the N — 1-dimensional sphere. 3
Theta terms and W Z W terms
In this section I will discuss two terms one can add to the action for some sigma models. These terms, the theta term and the WZW term, can cause the appearance of a non-trivial stable critical point. 3.1
WZW term
Two-dimensional sigma models with a manifold of the form H x H/H are called principal chiral models, and have been widely studied. They correspond to taking $ in the action (2) to be an element of the group H. The chiral H x H symmetry corresponds to taking U and V in (4) to each be elements of if. The principal chiral models are massive asymptotically-free field theories. However, there is an additional term which can be added to the action (2) which changes the excitations from massive to massless. This is called the Wess-Zumino-Witten term. To write it out explicitly, first one needs to consider field configurations 3>(x,y) which fall off at spatial infinity, so that one can take the spatial coordinates x and y to be on a sphere. Then one needs to extend the fields $(x, y) on the sphere to fields $(x,y,z) on a ball which has the original sphere as a boundary. The fields inside the ball are defined so that $ at the origin is the identity matrix, while $ on the boundary is the original $(x,y). It is possible to find a continuous deformation of $(x,y) to the identity because ^{H) = 0 for any simple Lie group. Then the WZW
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term is kT, where r = | g
f dxdydztv [ ( $ - 1 5 a $ ) ( $ - 1 5 6 $ ) ( $ - 1 a c $ ) ] .
(6)
The coefficient k is known as the level, and for compact groups must be an integer because the different possible extensions of $(x,y) to the ball yield a possible ambiguity of 2mr in T. The WZW term changes the equations of motion and beta function, but only by terms involving $(x, y): the variation of the integrand is a total derivative in z. The two-dimensional sigma model with WZW term has a stable fixed point at 1/g = 16ir/k 8 , so the model is critical and the quasiparticles are gapless. The corresponding conformal field theory is known as the Hk WZW model 9 . The WZW terra is invariant under discrete parity transformations (e.g. x -¥ —x, y —• y) if $ —> <j>-1 under this transformation. For there to be a WZW term in a parity-invariant theory, some of the low-energy fields must be pseudoscalars. The WZW term can appear in sigma models derived as the low-energy effective theories for interacting fermions. This effect was understood in particle physics some time ago. The application of these results to disordered systems was the topic of 10 ' 11 . The sigma model arises when integrating out the fermions in this action, leaving a low-energy effective action in terms of $, which is some fermion bilinear. Upon doing so, one easily obtains the ordinary sigma model action of the form (2). The WZW term arises for a subtler reason. To perform a consistent low-energy expansion, one must change field variables, which results in a Jacobian in the path integral 12 . In these two-dimensional cases, the Jacobian is precisely the WZW term (6). The coefficient k is not an arbitrary parameter, but rather is determined by the original theory, just as the G/H symmetry structure is. In fact, one can determine without explicit computations whether or not the WZW term will appear in the low-energy effective action. The reason is some very deep physics known as the chiral anomaly. In the models which admit a WZW term, the fermions have a chiral Hi x HR symmetry. As is well known, chiral symmetries involving fermions are frequently anomalous. Noether's theorem says that a symmetry of the action gives a conserved current with d^j^ = 0, but this is only true to lowest order in perturbation theory. An anomaly is when the current is not conserved in the full theory (although the associated charge is still conserved). In the case of massless fermions in 1 4- 1 dimensions, this was shown in detail in 8 . The WZW term is the effect of the anomaly on the low-energy theory. Even though the fermions effectively become massive when the fermion bilinear gets an expec-
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tation value, their presence still has an effect on the low-energy theory, even if this mass is arbitrarily large. This violation of decoupling happens because the chiral anomaly must be present in the low-energy theory. In other words, the anomaly coefficient does not renormalize. This follows from an argument known as 't Hooft anomaly matching 13 . One imagines weakly gauging the anomalous symmetry. It is not possible to gauge an anomalous symmetry in a renormalizable theory, but one can add otherwise non-interacting massless chiral fermions to cancel the anomaly. Adding these spectator fermions ensures that the appropriate Ward identities are obeyed, and the symmetry can be gauged. In the low-energy effective theory, the Ward identities must still be obeyed. Because the massless spectator fermions are still present in the low-energy theory, there must be a term in the low-energy action which cancels the anomaly from the spectators. This is the WZW term. To determine whether a chiral anomaly and hence a WZW term is present, one usually needs to do only a simple one-loop computation. For chiral symmetries, it is customary to define the vector and axial currents Jv = J% + J^, JA = J*L ~ JR'I f° r these theories J^ = e^Jvv One then computes the correlation functions (Jv(x,y)Jv(0,0)) = C"u'. If d^W ^ 0 or ea0daC0,/ jt 0 for some v, then there is an anomaly. An important characteristic of the anomaly is that it is independent of any continuous change in the theory, as long as the chiral symmetry is not broken explicitly. Thus to find the anomaly, one can compute C " using free fermions (where the only contribution is a simple one-loop graph, see e.g. 8 ) . 3.2
The theta term
A famous example of a theta term arose in the study of integer and halfinteger spin chains 2 . The spin-1/2 Heisenberg quantum spin chain is exactly solvable, and from Bethe's exact solution it is known that the spectrum is gapless. The obvious guess for the field theory describing the spin chain in the continuum limit is the sphere sigma model. This sigma model is an 5[/(2)-symmetric field theory where the field takes values on a two-sphere. The two-sphere can be parametrized by three fields (^1,^2,^3) obeying the constraint (vi)2 + (v2)2 + (v3)2 = 1. The Euclidean action is 3
-J2
f dxdy (d^Vi)2
(7)
This field theory, however, is not the correct continuum description of the spin chain, because it is also exactly solvable, and its spectrum is gapped 14 . It was proposed that that when this sigma model is modified by adding an extra
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term called the theta term, the spectrum can become gapless 2 . The theta term is inherently non-perturbative: it does not change the beta function derived near the trivial fixed point at all. Nevertheless, the physics can be dramatically different from when 8 = 0 15 . In two dimensions, this can result in a non-trivial fixed point. Not only does the spectrum become gapless at the non-trivial critical point, but in addition, charge fractionalization occurs: the charges of the quasiparticles of the theory are fractions of the charge of the fields in the action. To define the theta term, field configurations are required to go to a constant at spatial infinity, so the spatial coordinates (x, y) are effectively that of a sphere. Since the field takes values on a sphere as well, the field is therefore a map from the sphere to a sphere. An important characteristic of such maps is that they can have non-trivial topology: they cannot necessarily be continuously deformed to the identity map. This is analogous to what happens when a circle is mapped to a circle (i.e. a rubber band wrapped around a pole): you can do this an integer number of times called the winding number (a negative winding number corresponds to flipping the rubber band upside down). It is the same thing for a sphere: a sphere can be wrapped around a sphere an integer number of times. An example of winding number 1 is the isomorphism from a point on the spatial sphere to the same point on the field sphere. The identity map has winding number 0: it is the map from every point on the spatial sphere to a single point on the field sphere, e.g. (vi(x,y),V2(x,y),vz(x,y)) = (1,0,0). Field configurations with non-zero winding number are usually called instantons. The name comes from viewing one of the directions as time (in our case, one would think of say x as space and y as Euclidean time). Since instanton configurations fall off to a constant at y = ±oo, the instanton describes a process local in time and hence "instant". Therefore, the field configurations in the sphere sigma model can be classified by an integer n. This allows a term S$ = in6 to be added to the action, where 6 is an arbitrary parameter. Since n is an integer, the physics is periodic under shifts of 27r in 6. Haldane argued that when 9 = IT, the sphere sigma model flows to a non-trivial critical point 2 . Around the same time, a similar proposal arose in some more general sigma models 3 . These models are most easily formulated by having the field take values in the coset space G/H, where G and H are Lie groups, with H a subgroup of G. In this language, the two-sphere is equivalent to 0(3)/0(2): while the vector (^1,^2,^3) c a n be rotated by the 0(3) symmetry group, it is invariant under the 0(2) subgroup consisting of rotations around its axis.
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Thus the space of distinct three-dimensional fixed-length vectors (the sphere) is the coset 0 ( 3 ) / 0 ( 2 ) . In order to describe two-dimensional non-interacting electrons with disorder and a strong transverse magnetic field, one takes G = U(2N) and H = U(N) x U(N) 3 . For N — 1, one recovers the sphere sigma model. Pruisken conjectured that in the replica limit N —>• 0, this sigma model has a critical point at 9 = n. This critical point possibly describes the transitions between integer quantum Hall plateaus. Subsequently, a number of conjectures for non-trivial fixed points in sigma models have been made. For a survey of the applications of these conjectures to disordered system, see 7
One argument for the flow in the sphere sigma model at 0 = n goes as follows 16 . First one uses Zamolodchikov's c-theorem, which makes precise the notion that as one follows renormalization group flows, the number of degrees of freedom goes down. Zamolodchikov shows that there is a quantity c associated with any two-dimensional unitary field theory such that c must not increase along a flow. At a critical point, c is the central charge of the corresponding conformal field theory 18 . At the trivial fixed point of a sigma model where the manifold is flat, the central charge is the number of coordinates of the manifold. For the sphere, this means that c = 2 at the trivial fixed point. This if the sphere sigma model flows to a non-trivial fixed point for 6 = 7r, this fixed point must have 0(3) ss SU(2) symmetry and must have central charge less than 2. The only such unitary conformal field theories are SU(2)k WZW theories for k < 4. (The central charge of SU(2)k is 3k/(k + 2); in general, the central charge of Hk is kdim(H)/(k + h), where h is the dual Coxeter number of H.) One can use the techniques of 9 to show that there are relevant operators at these fixed points, and at k = 2 or 3, no symmetry of the sphere sigma model prevents these relevant operators from being added to the action 16 . So while it is conceivable that the sphere sigma model with 6 = n could flow near to these fixed points, these relevant operators would presumably appear in the action and cause a flow away. However, there is only one relevant operator (or more precisely, a multiplet corresponding to the WZW field w itself) for the SU{2)\ theory. The sigma model has a discrete symmetry (vi,V2,V3) —> (—vi, —V2, —vz) when 9 = 0 or IT; the winding number n goes to — n under this symmetry, but 9 = IT and 6 = —ir are equivalent because of the periodicity in 9. This discrete symmetry of the sigma model turns into the symmetry w —> —w of the WZW model. While the operator trw is SU(2) invariant, it is not invariant under this discrete symmetry. Therefore, this operator is forbidden from appearing in the effective action. The operator (trw) 2 is irrelevant, so it is consistent for the sphere sigma model at 9 = IT to have the 5C/(2)X WZW model as its low-energy fixed point. A variety of
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arguments involving the spin chain strongly support this conjecture 2 , l e . An important question is therefore whether the existence of these nonperturbative fixed points in sigma models at 9 = n can be established definitively. The fact that the sphere sigma model has a non-trivial fixed point at 9 = n was proven in 4 ' 17 . This proof does not involve the spin-1/2 chain which motivated the result. Rather, it is a statement about the non-trivial fixed point in the sphere sigma model at 9 = n. This proof utilizes the integrability of the sphere sigma model at 9 = 0 and n. Integrability means that there are an infinite number of conserved currents which allow one to find exactly the spectrum of quasiparticles and their scattering matrix in the corresponding 1 + 1 dimensional field theory. The quasiparticles for 9 = 0 are gapped and form a triplet under the SU(2) symmetry, while for 6 = n they are gapless, and form SU(2) doublets (left- and right-moving) 14,4 . This is a beautiful example of charge fractionalization: the fields {vi,V2,v3) form a triplet under the SU{2) syrnmetry, but when 9 = IT the excitations of the system are doublets. To prove that this is the correct particle spectrum, first one computes a scattering matrix for these particles which is consistent with all the symmetries of the theory. From the exact S matrix, the c function can be computed. It was found tr^at at high energy c indeed is 2 as it should be at the trivial fixed point, while c = 1 as it should be at the SU(2)i lowenergy fixed point 4 . As an even rriore detailed check, the free energy at zero temperature in the presence of a magnetic field was computed for both 9 = 0 19 and 7r 17 . The results can be expanded in a series around the trivial fixed point. One can identify the ordinary perturbative contributions to this series, and finds that they are the same for 9 = 0 and ir, even though the particles and S matrices are completely different 17 . This is as it must be: instantons and the 9 term are a boundary effect and hence cannot be seen in ordinary perturbation theory. One can also identify the non-perturbative contributions to these series, and see that they differ. Far away from the trivial fixed point, non-perturbative contributions can dominate which allow a non-trivial fixed point to appear when 9 = IT even though there is none at 9 = 0. This sort of fascinating behavior is not unique to the sphere sigma model. The question of whether a theta term is possible in a given sigma model was answered long ago mathematicians (for a discussion accessible to physicists, see 2 0 ) . In mathematical language, the question is whether the second homotopy group TT2{G/H) is non-trivial. The second homotopy group is the group of winding numbers of maps from the sphere to G/H, so for the sphere it is the integers. The general answer is that ^{G/H) is the kernel of the embedding of 7Ti (H) into 7Ti (G), where 7Ti (H) is the group of winding numbers for maps of the circle into H. The rubber band on a pole example means that 7Ti (H) is
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the integers when H is the circle = U{\) = 50(2). The only simple Lie group H for which m is nonzero is SO(N), where m(SO(N)) = Z 2 for N > 3 and Z for JV=2. Thus there are models with integer winding number, some with just winding number 0 or 1, and some with no instantons at all. Integer winding number means that 9 is continuous and periodic, while a winding number of 0 or 1 means that 9 can be either 0 or IT (just think of 9 as being the Fourier partner of n). The SU(N)/SO(N) and 0(2P)/0{P) x 0(P) sigma models have instantons with Z2 winding number, and so 9 can be zero or w. Later in these lectures I will find the non-trivial fixed points for the SU(N)/SO(N) and 0(2P)/0{P) x 0(P) sigma models. A solution for all N is important in order to have any hope of being able to take the replica limit N —»• 0. The SU(N)/SO(N) sigma model reduces to the sphere sigma model when N = 2, while 0(2P)/0(P) x 0(P) reduces to two copies of the sphere when P = 2. Thus these sigma models are the generalizations of the sphere sigma model with SU(N) and 0(2P) symmetry respectively. One difference, however, is that in general they do not allow a continuous 6 parameter. Other models admit a theta term, but it has not yet been conclusively established if at 9 = 7r there is a stable fixed point. All the 0(M + N)/0(M) x O(N) sigma models admit a Z2 theta term for M > 1 and N > 1. The U(M + N)/U(M) x U{N), Sp(2N)/U(N), and 0(2N)/U(N) models all have a continuous theta term because of the f/(l) factor in the denominator. 4
T h e particle content
Even though the sigma models are originally defined in two-dimensional Euclidean space (x,y), it is very convenient to continue to real time t — iy. In other words, I will treat these models in the formalism of 1 + 1 dimensional field theory. The reason I do so is in order to treat the field theory as a particle theory. A fundamental property of many (if not all) field theories is that all states of the theory can be written in terms of particles. In other words, the space of states is combination of one-particle states, usually called a Fock space. If the theory has a stable non-trivial fixed point at low energy, then the particles should be massless: the energy is linearly related to the momentum: E = \P\. If there is no fixed point, then the particles are massive: E = VP2 + M2 in a Lorentz-invariant theory. In condensed-matter physics, the particles are often called "quasiparticles", to emphasize the fact that the particles may not be the same as the underlying degrees of freedom: just because a system is made up of electrons does not mean that the collective excitations are electrons, or even resemble them. I will give numerous
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examples of this phenomena below. The particles in a G/H sigma model in two dimensions must form representations of the global symmetry group G. In this section I discuss the particle content of all the known integrable symmetric-space sigma models. In particular, I explain which representation of G they lie in. An extremely interesting open question is if one can classify which sets of representations of G are consistent with being the particles of some integrable theory. Before giving the results for the sigma models, it is useful to first discuss the Gross-Neveu models. These are not sigma models, but are closely related: they are asymptotically-free field theories with a global symmetry G. For groups O(N) and U(N) they were originally formulated in terms of fermions with a four-fermion interaction. For O(N), the action was first written in terms of N Majorana (real) fermions with a four-fermion interaction: 21
s
= J E {ri^i+titidRtfa)
+ x I f; ^R j .
(8)
For SU(N) the action can also be written in terms of fermions (this is sometimes known as the chiral Gross-Neveu model 2 2 ) , but for general G the GrossNeveu models are most easily defined in terms of the G Wess-Zumino-Witten model at level 1. The WZW model has a local symmetry GL X GR generated by chiral currents J£ and J^. The action of the G Gross-Neveu model is then a perturbation of the WZW action at level 1: S = Swzw
+ A 2_^
JLJR
(9)
The perturbation breaks the chiral symmetry, but the diagonal subgroup G is still preserved. A well-known special case is the SU{2) case, which is the sine-Gordon model as the coupling /32 —>• 8TT. The coupling A is naively dimensionless, but it has lowest-order beta function /?(A) oc A2, where the constant of proportionality is negative. This means that for positive A the coupling is marginally relevant 2 1 . In other words, the WZW fixed point in (9) is unstable and (8) defines a massive field theory called the Gross-Neveu model. For negative A, the coupling is marginally irrelevant and the WZW fixed point is stable, a fact which we will see becomes important for sigma models with 6 = ir. As one would expect, there are particles in the O(N) Gross-Neveu model with the same quantum numbers as the fermions in (8). These are in the defining (^-dimensional vector) representation oiO(N). For general G, this is also true: there are particles in the defining representation of G. However, there is more. For 0(N), there are kinks as well, as follows from a semi-classical
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analysis of the action (8). These particles are in the spinor representations of 0{N) 23 . For even n, there are two spinor representations, of dimension 2JV/2-1. £ or Q ^ J n there is one spinor representation, of dimension 2I27. For 0(2N), for example, all particles in the spinor have mass M. For the other representations, with highest weight /i a , a = 1 . . . N—2, there is a multiplet with mass 2Msink/(2N - 2) for k = a, a + 2,... (N - 2). There are also singlets with masses 2Msink/(2N — 2) for k odd. The representations of the global symmetries of sigma models are somewhat more elaborate than that in the Gross-Neveu model. The results are different for each model, and also can change dramatically if a WZW term or theta term is are added to the action (2). The work on the principal chiral models H^ x HR/H is summarized in 2T .
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Since the global symmetry is HL X HR, the particles must transform under both groups. The result is that the particles are in the tensor products a ® a of fundamental representations of HL and HR. The mass spectrum in all cases is the same as that of the H Gross-Neveu model, but the multiplicities are very different. As discussed in the last section, the principal chiral models can be modified by including the WZW term. If one includes a Wess-Zumino term in the principal chiral model, the spectrum dramatically changes. Instead of being a massive field theory, the model flows to a stable fixed point at long distances, the Hk WZW theory 8 . The particles become massless, and so are either left-moving or right-moving. The left movers are in the fundamental representations of HL and trivial under HR, while the right movers are in the fundamental representations of HR and are trivial under HL 4 . Another well-known example of an integrable sigma model is the 0(N)/0(N — 1) x 0(1) model. This model is the generalization of the sphere sigma model (7) to a vector Vi, where i = 1 . . . n. It was the first sigma model to have its exact S matrix found 14 . Here the particle content is very simple: there is just one set of particles, in the vector of 0(N). This is not completely surprising, given that the action is written in terms of a vector i>j. However, it is not completely obvious either, because of the non-linear constraint of the action. For example, 0(3)/0(2) « SU(2)/U(1), so the N = 3 model is equivalent to the CPl model. The model can be written in terms of a twodimensional complex vector field z defined by v1 — z*a%abzi,, where the a1 are the Pauli matrices. The constraint on (v1)2 + (v2)2 + (v3)2 = 1 turns into Nil 2 + \z2\2 = 1. The action (7) turns into 2
r
^2 a=l
J
Kd.-A^l2
A^z*bd»zb
The action can be written like a gauge theory because the system is invariant under the gauge symmetry za ->• eiaza, which changes z but leaves v and hence the action invariant. At first glance at the action involving z, one might be tempted to included that the particles form doublets under the global SU(2) symmetry instead of the three-dimensional vector. One must appeal to the exact solution of 14 to know that the latter is true. In gauge theory language, this means that the z excitations are confined into triplet mesons 28 . The sphere sigma model allows a 9 angle, but otherwise the 0(N)/0(N 1) models allow neither a 9 term nor a Wess-Zumino term. For arbitrary 9, the sphere sigma model does not appear to be integrable. However, for 9 = ix (the only value other than 9 = 0 where the model is time-reversal and parity invariant), it is integrable 4 . As discussed above, when 9 = IT, the 0(3)/0(2)
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sigma model flows to the SU(2)i WZW theory 16 . The particles thus are massless and, as in the principal chiral model with the k = 1 WZW term, form left- and right-moving doublets 4 . Roughly speaking, the particles at 9 = -K are like a massless version of the SU(2) Gross-Neveu model. Thus at one can think of the effect of 9 = n as being deconfmement, or more precisely, oblique confinement 15 . In condensed-matter physics, this effect is often called charge fractionalization. This is because under the 1/(1) subgroup of SU(2), the fields vl in the action have charge — 1,0,1. However, the excitations when 9 = 7T are doublets under the SU(2), so their charges are ± 1 / 2 . There are three other known hierarchies of integrable sigma models. Two (SU(2N)/Sp(2N) and SU(N)/SO(N)) were discussed in 29 , and one (0{2P)/0(P) x 0(P)) was recently uncovered 5 ' 6 . The latter two are particularly interesting because they allow 6 = n 5 . The particle content for the first two can be guessed from the action, and is consistent with the non-local conservation laws of the model derived in 29 . For SU(2N)/Sp(2N), the action is written in terms of an antisymmetric unitary matrix field $. Not surprisingly, there are no particles in the 2Ndimensional vector representation of the global symmetry SU(2N), but rather the smallest representation is the N(2N — l) dimensional antisymmetric tensor (and its conjugate for N > 2). As with the principal chiral and Gross-Neveu models, there are bound states, this time corresponding to antisymmetric products of the antisymmetric tensor. Thus the states consist of all Young tableau with one column and an even number of boxes (antisymmetric tensors with an even number of indices). No 9 terms is allowed here. For SU(N)/SO(N), the action is written in terms of a symmetric unitary matrix field $. Thus as opposed to the Gross-Neveu model, one expects there to be particles forming the symmetric representation of SU(N), which has dimension N(N + l ) / 2 . Indeed, it was long ago established that for AT = 2 (the sphere sigma model), the particles are in the symmetric (triplet) representation of SU{2) 14 . For any N, there are non-trivial non-local conserved currents in this sigma model 29 . These conserved currents are consistent with particles in the symmetric representation, and also with bound states in all representations of SU(N) with Young tableau with two columns and rectangular (i.e. the same number of boxes in each of the columns). In group-theory language, these are representations with highest weight 2p? (the fundamental representations arising in the Gross-Neveu model have highest weight /J J ). The mass spectrum is the same as that of the SU(N) Gross-Neveu model, Each representation appears just once with mass M simrj/N. The SU{N)/SO(N) model allows 9 = w for any N. (When N = 2 it reduces to the CPl = SU(2)/U(1) = 0(3)/0(2) = sphere sigma model, and
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allows a 6 angle). To find the particles and scattering matrices when 6 — IT, it is first necessary to understand the non-trivial fixed point 5 . Generalizing from N=2, it is natural to assume that it is SU(N)\. In fact, the argument from 16 ' 30 can be adapted to show that this is completely consistent with the symmetries of the problem. To apply the consistency argument here, it is first useful to study the symmetries of the SU(N) Gross-Neveu model. To avoid confusion with the sigma model, I denote the field in the Gross-Neveu model as w. With the definition (9) as a perturbed WZW model, w must be an SU(N) matrix. Therefore, it transforms under the SU(N) symmetry a s w - > UwU\ where U is an element of SU(N). The currents also transform as J —> UJU^. (In other words, w and J are in the adjoint representation of SU(N).) The action (9) is manifestly symmetric under the symmetry SU(N)/Zpf. The ZJV is the center of SU(N), consisting of of matrices fll, where fl is a JVth root of unity, and / is the identity matrix. The matrices fll commute with all elements of SU(N). The reason this discrete subgroup is divided out is that if U = fll, w and j are left invariant. Thus the symmetry acting manifestly on the action is SU{N)/ZN, not SU(N). However, the full symmetry of the SU(N) Gross-Neveu model is larger. This can be seen in several ways. First of all, note that there are particles in the model in the vector representation. The full SU(N) group acts on the vector representation non-trivially. Indeed, one can assign an extra ZJV charge to the particles of the model, defined so that the particles in the j-index antisymmetric representation have charge Qi. Another way of seeing this extra charge is by examining terms which do not appear in the action (9). The SJ7(iV)/Z/v-symmetric operators in the unperturbed SU(N)i WZW theory are tr wi, where j = 1 . . . N — 1. Some of these are relevant operators, and if added to the action would change the physics considerably. However, they are forbidden from the action if the model is required to be invariant under the symmetry w -> flw. This discrete symmetry is not part of the SU(N) acting on the w field, but rather makes the full symmetry of the model ZN x SU(N)/ZN. This symmetry of the action and the full SU(N) of the particle description are completely consistent with each other if the particles are kinks in the w field. (They are kinks in the 1 + 1dimensional picture, vortices in the 2 + O-dimensional picture.) The presence of kinks is easy to see. The field values w = I and w = fl are not related by a continuous symmetry, so field configurations with w(x = —oo,t) = I and w(x = oo, t) = fll are topologically stable. These are the kinks. The extra ZJV symmetry is then indeed the discrete kink charge. This sort of symmetry should be familiar from the sine-Gordon model, where the soliton charge is not an explicit symmetry of the action. The symmetries of the SU(N)/SO(N)
sigma model at 6 = n are similar
125
to those in the Gross-Neveu model. The Z/v center acts non-trivially on the SU(N)/SO(N) matrix 3>, as seen in (4). Thus the symmetry of the sigma model action is the full SU(N) (or to be precise, for even N it is Z 2 x SU(N)/Z2)Therefore it is consistent for the low-energy fixed point of the SU(N)/SO(N) sigma model to be SU(N)i. The reason is the same as in the SU(N) Gross-Neveu model: the full SU(N) symmetry of the action in terms of $ implies a ZJV x SU(N)/ZN symmetry of the low-energy effective action involving w. This forbids the relevant operators from being added to the SU(N)i action. This argument shows that the effective action near the low-energy fixed point contains only ^[/(AO-invariant irrelevant operators. The operator JLJR is SU(N) invariant and of dimension 2. Thus the effective action here is like the Gross-Neveu action (9), except here the coupling A is negative so that the perturbing operator is irrelevant. The coupling A is related to the scale M; when M —>• oo, g —>• 0 from below, and the model reaches the low-energy fixed point. Sine the low-energy fixed point is the SU{N)\ conformal field theory, the particles must be massless and in representations of SU(N). As will be confirmed below, the left- and right-moving particles are in the fundamental representations oiSU(N). Thus the SU(N)/SO(N) model at 8 = ir resembles a massless version of the SU(N) Gross-Neveu model. The model 0(2P)/0(P) x O(P) is like the SU(N)/SO(N) case. The field 4> is symmetric and unitary (and also real and traceless), so again there are particles in the symmetric representation of 0(2P) (which is P(2P + 1) — 1 dimensional), and its various bound states. However, like the 0(2P) GrossNeveu model and principal chiral models, there are also particles which do not follow obviously from the action. These are in what I call "double-spinor" representations, formed by the symmetric product of two spinor representations. In group theory language they have highest weight 2fis, where /J,S is the highest-weight of a spinor representation; they are (2N — \)\/P\{P — 1)! dimensional. These particles presumably are kinks like in the Gross-Neveu model, but it is not clear how to extract this information directly from the action. Thus there are particles in representations with highest weight 2^ 1 , like the SU(N)/SO(N). The mass for the particles in representations 2yuJ in this sigma model as the same as for the fundamental representation /i-7 in the 0(2P) Gross-Neveu model. Moreover, as will be seen from the S matrix given in the next section, there are even more degeneracies: there are particles in fundamental representations as well. The behavior when 9 = n in the 0(2P)/0(P) x O(P) model is also reminiscent of the SU(N)/SO(N) model. The model has the 0(2P)i WZW model (2P free Majorana fermions) as a stable low-energy fixed point. Charges fractionalize: the left- and right-moving
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particles are in the fundamental representations of 0(2P). This section is summarized in Table 2.
sigma model
0(2P)/0{P) x 0(P) SU(N)/SO(N) H xH/H 0(N)/0{N - 1) U(2N)/Sp(2N) Gross-Neveu
particle content double-spinor, symmetric, . . . symmetric, . . . fundamental (g> fundamental vector antisymmetric, . . . fundamental
with theta or WZW fundamental fundamental fundamental not possible not possible not possible
Table 2: Summary of the particle content of known integrable sigma models with no topological or Wess-Zumino term. All representations in the second and third columns are of the global symmetry G (the numerator in the first column)
5 5.1
The S m a t r i c e s Generalities on exact S matrices
One finds the scattering matrix of an integrable model by utilizing a variety of constraints: unitarity, crossing symmetry, global symmetries, the consistency of bound states, and the factorization (Yang-Baxter) equations. A final check is given by computing the free energy from the S matrix and ensuring that it agrees with the correct results in various limits. The latter computations are described in sections 6 and 7. It is convenient to write the momentum and energy of a massive particle in terms of its rapidity /?, defined by E = m cosh /?, P = msinh/3. Lorentz invariance requires that the two-particle S matrix depends only on the rapidity difference (3\ — /?2 of the two particles. The invariance of the G/H sigma model under the Lie-group symmetry G requires that the S matrices commute with all group elements. The 5 matrix can then be conveniently written in terms of projection operators. A projection operator Pk maps the tensor product of two representations onto an irreducible representation labelled by k. By definition, these operators satisfy VhPi — SkiVk- Requiring invariance under G means that the S matrix
127
for a particle in the representation a with one in a representation b means that the S matrix is of the form
SabU3) = Y,fk"Wp'>
(10)
k
where ft = fta - ftt is the difference of the rapidities, and the f£b are as of yet unknown functions. The sum on the right-hand side is over all representations k which appear in the tensor product of a and b; of course Ylk^k = 1. In an integrable theory, the functions f%b(ft) are determined up to an overall function by requiring that they satisfy the Yang-Baxter equation. This stems from the requirement that the .S-matrix be factorizable: the multiparticle scattering amplitudes factorize into a product of two-particle ones. There are two possible ways of factorizing the three-particle amplitude into two-particle ones; the requirement that they give the same answer is the Yang-Baxter equation. There have been hundreds of papers discussing how to solve this equation, so I will not review this here. For a detailed discussion relevant to the sigma models here, see e.g. 25.27>32. Solutions arising in the sigma models will be given below. I define the prefactor Fab(ft) to be the coefficient / ° 6 in (10) where the highest weight of the representation c is the sum of the highest weights of the representations a and b. The Yang-Baxter equation does not give this prefactor. One needs to impose additional requirements. Two basic ones are that the S matrix be unitary and that it obey crossing symmetry. With the standard assumption that the amplitude is real for ft imaginary, the unitarity relation S*(ft)S{ft) = 7" implies S(ft)S(-ft) = I. The latter is more useful because it is a functional relation which can be continued throughout the complex ft plane. Crossing symmetry is familiar from field theory, where rotating Feynman diagrams by 90° relates scattering of particles at and bj to the scattering of the antiparticle
= CaSab(ft)Ca,
(11)
where C° is the charge-conjugation operator acting on the states in representation a. Multiplying any 5 matrix by function F(ft) which satisfies F(ft)F(-ft) = 1 and F(iir — ft) = F(ft) will give an S matrix still obeying the Yang-Baxter equation, crossing and unitarity. This is called the CDD ambiguity. To determine F(ft) uniquely, one ultimately needs to verify that the S matrix gives the correct results for the free energy. How to do this will be described in the following sections. Before doing this calculation, one must make sure an additional criterion holds: the poles of the S matrix are consistent with
128
the bound-state spectrum of the theory. The S matrices of bound states are related to those of the constituents by the bootstrap relation, which can be formulated as follows 25>27. Poles of Sab matrix elements at some value P = Pab with Pab imaginary and in the "physical strip" 0 < Im(Pab) < T are usually associated with bound states. Each of the functions f%b(P) has a residue at this pole Rk- Then there are bound states in representation A; if RkVk < 0, where T)k = ± 1 is the parity of states in representation k 33 (poles which do not correspond to bound states give bound states in the process obtained by crossing). These bound states (ab) have mass (m(ab))2 = (m a ) 2 + (mb)2 +
2mambcosh(3ab
The bound states are not in a irreducible representation of G if more than one of the residues Rk is non-zero. Then the S matrix for scattering the bound state (ab) from another particle c is given by S(ab)c =
/ £ rfRjpA 5-(/? + i^f)
where the projection operators Vu act on the states in representations a, b. Note that the matrices in (12) are not all acting on the same spaces, so this relation is to be understood as multiplying the appropriate elements (not matrix multiplication). All of the above considerations apply to massless particles as well, with a few modifications and generalizations 4 ' 3 1 . The theories being studied are along a flow into their stable low-energy fixed point. Even though the quasiparticles are gapless, there is still a mass scale m describing the crossover: m —> 0 is the unstable high-energy fixed point, while m —> oo is the stable low-energy one. Rapidity variables are still useful for parameterizing the energy and momentum of massless particles: E = p = mae® for a right mover, and E = —p = mae~® for a left mover. The parameter m here is not the mass of the particle, but rather is the scale (analogous to AQCD) which parameterizes the interactions. In condensed-matter language, it is the crossover scale. With these definitions, the rapidity difference is still an invariant in a collision. In a collision between a right mover and a left mover, the invariant is (Ei +E2)2 — (pi +P2)2 = m 2 e / 3 l _ ^ 2 . In "collisions" between two right movers, the invariant is E\ /E2 = e01 ~02. I put collisions in quotes because the S matrix is properly interpreted here as a matching condition on the wavefunction, as in (41). For more details on the 5 matrix approach to massless theories,
129
see 4 ' 3 1 . Thus in the massless case, the S matrix still depends only on rapidity differences. Note in particular that the S matrices for two left movers SLL or two right movers SRR depend on the ratio do not depend on the mass scale m at all. These are determined solely by properties of the low-energy fixed point. On the other hand, SLR depends on the physics of how one flows into this fixed point. Massless particles still can have bound states like massive ones. These bound states show up as poles in SLL and SRR, but it does not seem possible to have poles in SLR in the physical strip 4 . 5.2
The Gross-Neveu models
The Gross-Neveu models are integrable, so their exact S matrices can be derived. For SU(N), the tensor product of two vector representations contains the symmetric tensor and the antisymmetric tensor. Thus the S matrix for two particles in the vector representation is of the form Svv
= f)(vVs
+ fXvVA,
(13)
where the projection operators on the two-index symmetric (S) and antisymmetric (A) representations are explicitly Vs{ai(pa)bj{pb))
= \
(aiiPaMPri+ajiPJbitft))
PAiOiiPaMPb)) = \ {ai{Pa)b0b) - ajiPaMfo)) .
(14)
The subscripts in a^ and bj represent the i and j particles in the vector multiplets, so the S matrix in (10) is indeed a matrix. Requiring that the S matrix obey the Yang-Baxter equation means that SY_ _ P + 2niA f%v (3-2iziA'
{0)
where A will be determined shortly. The simplest solution for the overall function consistent with crossing symmetry and unitarity for particles in the vector representation of SU(N) is fgV = F^Jn, where
r f i - J U r (^- + A) Fvv
_
V 27 W V27" / r ( i + ^ ) r ( - / . + A)
nfi N
This is called the minimal solution, because the resulting S matrix has no poles in the physical strip: a zero in (16) cancels the pole in (15). It is not the unique solution, because of the CDD ambiguity.
130
The Gross-Neveu model has bound states and so the 5 matrix must have poles in the physical strip. This means that / J v = F ^ X ( / 3 ) , where X(fi) contains the poles. If there are to be bound-state particles with in the antisymmetric representation of SU(N), but not in the symmetric representation, then there must be a pole in f\v but not in f^v. The relation (15) means the pole must beat P = 2wiA and so particles in the antisymmetric representation have mass mA/mv = 2cos7rA. TO determine A, one must ensure that the bootstrap closes. This means that one does not generate an infinite number of particles: the poles in all the S matrices correspond to a finite number of particles. For SU(N), the bootstrap closes if A = 1/TV. For general groups, the Gross-Neveu bootstrap closes if A = 1/h, where h is the dual Coxeter number. In the rest of these lectures, A = 1/h, with h appropriate for the case at hand. Putting (13,14,15,16) and the bootstrap together means that the S matrix for the scattering of vector particles in the Gross-Neveu model is22 -
S^tf)
vta\-cVVto\
fr>
, /? + 2i7rA
= X(P)F^n(f3) \VS + p_2inA
TAJ
(17)
where X{P)
~ sinh((/3-27riA)/2)'
(18)
The extra factor X(/3) indeed reinstates the pole at /? = 27n'A canceled by Fmin- Using this bootstrap to compute the S matrices of the bound states, one finds for example that the scattering matrix SAV for a particle in the vector representation with one in the antisymmetric representation has a pole at /? = 3A/2, leading to particles with mass m^/my = sinh(3A7r)/sinh(A7r). These particles are in the representation with highest weight fi^ (the Young tableau with 3 boxes and one column, or equivalently the three-index antisymmetric tensor). One keeps repeating this bootstrap procedure and finds particles in all the antisymmetric representations, ending up with the particles in the conjugate representation TV representation arising from the bound state of TV — 1 vector particles. This provides a substantial check on the S matrix, because the crossing relation (11) relates Fvv to Fvv. One can indeed check that the S matrices built up from (17) satisfy this relation. For the SU(N) GrossNeveu model, there is one copy of each fundamental representation 34 . The dimension of the antisymmetric tensor with a indices (a boxes in the onecolumn Young tableau) is N\/a\(N - a)\, so there are 2 ^ - 2 particles in all. Each multiplet of particles has mass M sin -na/N, where M is an overall scale
131
related to the coupling constant g. The particles in the N — a representation are the antiparticles of those in the a representation. One can check that there are no additional particles by computing the energy in a background field as done in subsequent sections 39 , or by computing the free energy at non-zero temperature 57 . Moreover, the SU(N) Gross-Neveu model can be solved directly using the Bethe ansatz 22 ; the results agree with those above.
The S matrices for the O(N) Gross Neveu model are derived in the same manner as those with SU(N) symmetry, but there are a number of additional complications. The scattering of particles both in the vector representation of O(N) is of the form svv
= TvvVs
+ TVVVA
+ Tvvv^
(19)
where the projection operators on the symmetric (S), antisymmetric (A) and singlet (0) representations are explicitly N 1 1 Vsiciibj) - - (ciibj + a,jbi) - —Sij ^2
a
kb,
k=l
VA(aibj)
= - (ciibj -
djbi)
N 1 V0(aibj) = -r;kj'Ylakhk
( 20 )
k-l
where I have suppressed the /3 dependence. For example, the scattering process ai6i -> mbi has S matrix element ((N - 2)T^V + NT\V + 2T%V)/2N, while the process aibi -> o 2 6 2 has element (TQV -TgV)/N. The extra term in (19) as compared to (13) stems from the fact that the trace Sijdibj is an SO(N) invariant. Requiring that the S matrix obey the Yang-Baxter equation means that 14
*r_/>+2*iA F%v
(2l)
0 - 2-KIA '
T%v _ /3 + 2?r»A P + iw TYV ~~ /3-2niA P-in'
(22)
132
The minimal solution (no poles in the physical strip) for the overall function for O(N) with particles in the vector is TgV = J ^ , where
^y(!).
r
( 1 - A ) r ( i + & ) r ( A + A ) r ( | - & + A) r(i + A ) r ( | - A ) r ( - £ + A ) r ( | + & + A)
(23)
The 5 matrix with / J y = T^in defined by (19,20,22,23) is the S matrix for the 0(N)/0(N - 1) sigma model 1 4 . The S matrix does not contain any poles because the 0(N)/0(N — 1) model has no particles in any representations other than the vector, and hence no bound states. The O(N) Gross-Neveu model has bound states. Just like in SU{N), there are particles in all the antisymmetric representations, whose S matrices follow from the bootstrap. These representations are self-conjugate: there is no N representation. That means that both the bound-state (s-channel) pole at j3 = 2iriA and the crossed pole (^-channel) channel at /? — in — 27riA appear in J7^, yielding ^
P + 2niA^
Vs + s
/3 + 2-rriA B + iir^\ VA +
P- 2niA
A
,
——VQ
j3- 2iriA 0 - in
,
(24)
J
{
'
where X(/3) is as in (18) with A = 1/h, with the dual Coxeter number here h = N — 2. This value of A ensures the bootstrap closes. The particles obtained by fusing the vector particle j times therefore have mass mj/mv — 2sin(j7rA)/sin(7rA). One difference between the O(N) and SU(N) GrossNeveu models is apparent in (24): if there is a pole in the antisymmetric channel at f3 = 2niA, there is also one in the singlet channel. This means that there are extra bound states in the O(N) Gross-Neveu model. For example, there are N(N - l ) / 2 + 1 particles with mass m 2 : ./V(./V - l ) / 2 of them transform in the two-index antisymmetric representation of O(N), while one is a singlet under O(N). In general, at mass rrij there are particles in all the /c-index completely antisymmetric representations k = 0, 2 , . . . j for even j and k = 1,3,... j for odd j 25 . Despite this additional complication, one can in principle obtain the S matrices for all of these states by using the fusion technique discussed below. This was done for k — 2 in 32 . The S matrix for the kinks in the spinor representation in the O(N) GrossNeveu model is given in 25>27>26. It can be written out explicitly, but since the answer is somewhat complicated I will not give it here. It is derived by using the fact that kink-kink bound states contain the particles in the antisymmetric representations as bound states; in fact is a sort of reverse bootstrap. The mass of the spinor states is given by mspinor/mv = l/sin(7rA).
133
In general, the particles of the G Gross-Neveu theory are in fundamental representations of G. Their masses and multiplicities are known from considering the bound-state properties of the exact S matrix. 5.3
The SU{N)/SO(N)
sigma model
The representations of the global symmetries of sigma models are somewhat more elaborate than that in the Gross-Neveu model. The results are different for each model, and also change dramatically when 6 — ir. The sigma model S matrices can be derived from the Gross-Neveu ones by a method called fusion. This is a way of finding new solutions of the YangBaxter equation from known ones 35 . One starts with a solution where the states are in some representation of a symmetry algebra G. Then one can find new solutions in other representations, just as one takes tensor products of representations. The usual place fusion appears in the study of exact S matrices is in the bootstrap (12), relating the S matrices of a particle and its bound states. However, fusion is a more general procedure than just the bootstrap. It can be used to relate S matrices of different models. This fact will prove very useful here, because when the S matrices are related, the TBA equations for the free energy are related as well. This observation enables the computation of the TBA equations for the SU(N)/SO(N) later in these lectures. In this section, I will give the solution of the Yang-Baxter equation 5 S S for particles in the symmetric representation. With the appropriate prefactor, this is the S matrix for particles in the SU(N)/SO(N) sigma model. Formally speaking, fusion relies on the observation that at certain values of /?, the coefficients of some of the projectors in the S matrix vanishes. This means that some particles can be treated as being composites: they are composed of "constituent" particles at specific rapidities. I avoid calling the composite particles bound states, because this implies that the composites and the constituents are both particle states in the same theory. This is the not case in general. For example, the only particles in SU(2) Gross-Neveu model are in the spin-1/2 representation of SU(2), while in the sphere sigma model, the only particles are in the spin-1 representation of SU(2). Fusion means that the S matrices are related, even though the theories are different: the spin-1 particles are composites of the spin-1/2 ones. This equivalence turns out to be a manifestation of confinement in two-dimensional gauge theories 28,36
The two-particle S matrix for two particles in the iV-dimensional vector representations of SU(N) (17) contains two terms: one involving the projector Vs onto the symmetric representation, the other VA onto the antisymmetric
134 representation. This is because the tensor product of two symmetric representations in SU(N) decomposes into the irreducible symmetric (N(N + l ) / 2 dimensional) and antisymmetric representations (N(N — l)/2 dimensional): (TV) ® (TV) = (N(N - l)/2) © (N(N + l)/2). For SU(2), the antisymmetric representation is the singlet, so this statement means that two spin 1/2 representations tensored together is the sum of the spin-0 and the spin-l representations. At /? = — 2ni/N, SVV in (17) involves only the projector onto the symmetric representation. The variable f3 in the S matrix is the difference of the rapidities of the two particles. The fusion procedure means that particles of rapidity j3s in the symmetric representation can be treated as being composed of two constituents in the vector representation, of rapidities /?s — ITT/N and /3s + in/N. The reason this works is described in 35 . Because the two particles have rapidity difference 2ni/N, the antisymmetric combination is effectively projected out. More precisely, if the incoming states have particles only in the symmetric representation, the Yang-Baxter equation ensures that even after scattering, the particles in the final states will still be in the symmetric representations. This is true for any number of particles and scatterings. Because particles in the symmetric representation are composed of vector constituents, the S matrices are related as well. The S matrix for scattering two particles in the symmetric representation has three terms. In the language of weights, the symmetric representation has highest weight 2/J.I, and the tensor product is (2m) x (2pi) = (4/ii) + (2//J + /i 2 ) + (2/i2) In Young tableau, these are 4
/*i = I I I I I
2
A*i + A«2 =
2^2
This fusion procedure gives the S matrix particles in the symmetric representation of SU(N) to be S
-F
\V4tll + p _
i n i A
? V 1 + „ 2 + p_2mA
p_47TiA
? V 2 ) .(25)
The projection operators onto these representations can be given explicitly. Labelling particles in the symmetric representation with two indices and the constraint Oy = ctj,, the projectors are ? V i {aijbki) = ^ {a-ijbki + atkbji + aubjk + akibij + ajkbu + ajibik)
135
^ 1 + ^ (fly &*;/) = - (dijbkt
^^(aijhi)
+ CLklbij)
= - {2aijbki + 2akibij - aikbji - aubik - ajkbu - ajibik) (26)
This means, for example, that the 5 matrix element for scattering an initial state of a,\2 and 6 i3 to a final state of a\2 and 613 is (2/iE + 3 / * | + M + / £ ) / 6 =
FSS(9)j-L^,
while the element for scattering the same two particles and getting a n and 623 in the final state is ( / i ^ " &/S =
F5S
W(/?_27riA)(/?-47riA)-
In cases where the composites are bound states of the constituents (all are particles in the same theory), then the bootstrap procedure relates the prefactors of composite scattering to those of constituent scattering. However, the fusion does not make such a requirement in general: the prefactor Fss((3) does not necessarily follow from Fvv(/3). All the fusion procedure does is determine the overall form of the S matrix for the composite particles and ensure that it obeys the Yang-Baxter equation. Although one might expect that F s s (/3) = Fss(j3 + 2iTr/N){Fss(l3))2Fvv{p - 2in/N), it is not true here. In another words, the CDD ambiguity may be resolved in different ways in the constituent and composite theories. The particles in the SU(N)/SO(N) sigma model at 9 = 0 have bound states at the same masses as in the Gross-Neveu model. Therefore the S matrices must have a poles in the same values of 0. This gives the prefactor to be Fss(p)
= F%fn(P)X{P)
(27)
where X(/3) is the same as in the Gross-Neveu model (18). The minimal solution Fgfn is FSS
-
( W
_ o-2, ri Ar(i- s f ; )r( 5 j ;
- * +wAr(i
+
+
2A)
&)r(-£ + » A ) '
< '
The pole in the S matrix at (3 = 2wiA results in a bound state transforming in the representation with highest weight 2fi2, with mass m2fi2/ms = 2cos7rA. In fact, the factor X here ensures that the entire spectrum of bound states is that of the SU(N) Gross-Neveu model. Building the S matrices for the bound
136
states in the SU(N)/SO(N) sigma model proceeds just like the SU(N) GrossNeveu model. For example, the prefactor in (25) ensures that Fss determined by the bootstrap is the same as that following from crossing. The general prefactor Fab(P) is given in the appendix. To close the bootstrap, A = 1/h as before, with h = N for SU(N). There are particles in all representations with highest weight 2// a . Each representation appears just once with mass is Msimra/N. For N = 2, this model reduces to the 0(3)/0(2) model (7), and the S matrix (25) becomes that of a massive 0(3) triplet, familiar from 14 .
The SU(N)/SO(N) sigma model with no 9 term is therefore an integrable generalization of the sphere sigma model to all SU(N). I will now show that the behavior at 6 = n also generalizes. The form of the effective action near the low-energy fixed point gives the S matrices SLL and SRR immediately, because as explained above they are independent of M and therefore follow solely from SU(N)i. They must be in fact those of the massless limit of the Gross-Neveu model. Written as functions of rapidity they are Sl"L(P)
= SaRbR(P) = SaGbN(p).
(29)
Thus the spectrum of these gapless particles is the same as that of the massive particles of the Gross-Neveu model: there are particles in any antisymmetric representation, a left-moving and right-moving set for each representation. Even though the particles here are gapless, the mass ratios of the Gross-Neveu model still appear in the definition of rapidity. For example, a right mover in the representation fia has energy E = mae®. The charge has fractionalized, as compared to 9 = 0. For N=2, the S matrix SLR(P) as a function of rapidity is the same as 4 SLL(P) - This S matrix satisfies the Yang-Baxter equation, and has the appropriate SU(2) (not SU(2)L X SU(2)R) symmetry. However, the situation is not quite as simple for general N, because there are poles in SLL and SRR in the physical strip, resulting in the bound states. These poles are forbidden in SLR- However, it is easy to remove these and still have a sensible S matrix. The bound-state S matrices SGbN are labeled with a, b = 1...N—1 corresponding to the antisymmetric representations with a and b indices. For example Sn = Svv and S 2 1 = SAV. The unwanted poles in Sab all arise from the factor X n (/3) = X(/3) and its fusions. The matrix Sg,^ contains the overall function a
X^
b
(P) = ]H[X(f3 »=ij=i
+ l2(i + j - l ) - a -
b]7riA).
(30)
137
For example, X[^} = X(p + mA)X{P - TTJA). The prefactor X{a"] contains all the poles in the physical strip in SQ 6 W . It is easy to check that X^\p)X^\-P) = 1 ^ d that xgH/3) = X$\in - /?). Therefore SlbRm = SgN(0)/X%\(l)
(31)
satisfies crossing, unitarity, the Yang-Baxter equation, and has no poles in the physical strip. Thus this is the 5 matrix for left-right scattering in the SU(N)/SO(N) sigma model at 0 = TT. For TV = 2, X = 1, and the result reduces to that in 4 . To prove that this picture is correct, in the next section I will show that these S matrices give a model which is SU(N)i in the low-energy limit, but is the SU(N)/SO{N) sigma model at high energy. Moreover, in 6 and in section 7 the corresponding c function is calculated, and indeed flows from the cuv = (N — l)(N + 2)/2 (the value at the trivial high-energy fixed point in SU{N)/SO(N)) tocIR = N-l (the central charge for SU{N)i). 5.4
0{2P)/0(P)
x O(P) sigma model
As with the Gross-Neveu models, the 0(2P)/0(P) x O(P) sigma model is a slightly more complicated version of its SU(N) analog, the SU(N)/SO(N) sigma model. The field $ is symmetric and unitary (and also real and traceless), so again one expects particles in the symmetric representation of 0(2P) (which is P(2P + 1) — 1 dimensional), and its various bound states. This is what happens for P = 2, where the model reduces to two copies of the sphere sigma model: there are six particles: three in each of the symmetric representations. Here, for P > 2, the non-local conserved currents have not been found, although some interesting results for the local currents in the classical model (g small) have been found 3 7 . The S matrices for the 0(2P)/0(P) x O(P) model can be constructed by fusing the 0(2P) Gross-Neveu results. The S matrix for particles in the symmetric representation of 0(2P) (highest weight 2//i) is of the form 32 ess S
_ B f « L P + 4TT»A / - R{P) [V4tll + j^r^K (P2-+/? + 4?riA /? + 2?nA P - 4?riA j3 - 2-rviA /3 + iiT + 2TTIA 2M2 +
p-iw-
2wiA
+
/? + »7r + 27riA /3-i7r-27riA
. V
^ }
(32)
p + i-K 4- 2iriA P + in . M2 +
P-iiT-
2iriA p - i-K
where A = 1/h as always, with h = 2P — 2 here. The minimal solution (no
138
poles in the physical strip) for the prefactor is ss
min\P)
X
r(i
+
/? - 2iriA p + 2lxi^
^)r(|- 1 f I )r(-^
+
2A)r(i + ^
+
2A)-
Like the 0 ( 2 P ) Gross-Neveu model, to get bound states there must be poles at /? = 27riA and /? = iri(l - 2A). This means that the prefactor of Sss is R(0)=X(l3)X{iir-l3)S%!n(/3).
(34)
Because the factor X(f3)X(iir — /?) is the same, the bootstrap for the 0(2P)/0(P) x 0(P) model gives bound states with the same spectrum as the 0 ( 2 F ) Gross-Neveu model. The S matrices for these kinks in the double-spinor representations with highest weight 2p s and 2fi3 is quite complicated, since many representations appear in the tensor product of two double-spinor representations. It can presumably be obtained by fusion of the spinor S matrices, or by the reverse bootstrap like the Gross-Neveu kinks. The presence of these particles is confirmed by studying the free energy at non-zero temperature 6 , as described in section 7 below. Therefore, in both series of integrable sigma models there are particles in all representations with highest weight 2/j,a. However, in the 0(2P)/0(P) x O(P) there are even more degeneracies and more representations appearing. Since there is a pole in (33) at j3 = 2iriA, then there must be bound states not only in the representation with highest weight 2/i2, but also in the singlet and in the antisymmetric representation (highest weight ^2)- Thus charge is fractionalized even at 6 = 0. For example, in 0(8), the vector and spinor representations are 8-dimensional, the antisymmetric representation has dimension 28, the symmetric and double-spinors dimension 35, and the representation with highest weight 2/^2 has dimension 300. Thus in the 0(8) Gross-Neveu model, there are 8 particles in the vector (mass 2Msin7r/6 = M) 8 particles in each of the spinor representations (mass M), and 2 8 + 1 particles of mass 2Msin7r/3 = \/?>M. Thus there are 53 stable particles in the 0(8) Gross-Neveu model. In the 0(8)/0(4) x 0(4) sigma model, there are 35 particles in each of the vector and two double-spinor representations, all of mass M. There are 300 + 28 + 1 particles of mass V3M, giving 434 particles in all.
139
The behavior when 9 = % in the 0{2P)/0(P) x O(P) model is also reminiscent of the SU(N)/SO(N) model at 9 = •K. The arguments follow in the same fashion. The model has the 0(2P)\ WZW model as a stable low-energy fixed point. The 0{2P)\ WZW model is equivalent to IP free Majorana fermions, or equivalently 2P decoupled Ising models. The word "free" is slightly deceptive, because just as in a single 2d Ising model, one can study correlators of the magnetization or "twist" operator, which are highly nontrivial. The consistency argument is simpler here: the only relevant 0[2P) symmetric operators are the fermion mass and the magnetization operator; neither is invariant under the symmetry <£—>•—$ of the sigma model. The c-function is computed in 6 , and indeed flows from c = P2 to c = P as it must. The S matrix for the quasiparticles follows from the Gross-Neveu model. The same arguments applied above to the SU(N)/SO{N) sigma model at 9 = TX show that the massless left- and right-moving particles have the same spectrum as the 0(2P) Gross-Neveu model. Charges fractionalize: the leftand right-moving particles are in the fundamental representations of 0(2P). The S matrices for the particles in antisymmetric representations are S&(0)
= SRR(0)
= SGN(P)
b
Sl R{P) = S&M/ixfiHfiXlPi™ ~ P))
(35)
(36)
For particles in the spinor representations, the poles can easily be removed as well; see 27 for the definition of Xab for the spinor representations. Even though the field theory at the low-energy fixed point is free fermions, the S matrix is factorizable away from the fixed point only in the basis related to the Gross-Neveu model. 6
Matching perturbative expansions
In this section I compute the energy of these sigma models at zero temperature in a background magnetic field. This is very useful for several reasons. First of all, it allows a direct comparison of the S matrix to perturbative results. This in particular ensures that the 5 matrices are correct as written in the last section. For example, it eliminates the possibility of extra CDD factors and/or extra bound states. Second, because the effect of a 9 term is nonperturbative, the perturbative expansions for the sigma models at 9 = 0 and 7r must be the same. Thus even though the S matrices for 9 = 0 and 7r are very different, the energy must have the same perturbative expansion. I verify that this is true for the above S matrices.
140
The abelian subgroup of the group G is U(l)r, where r is the rank of the group. Thus a model with a global symmetry G has r conserved charges. These charges can be coupled to a background field, which is constant in spacetime. In the sigma models 0(2P)/0{P) x O(P) and SU(N)/0(N) where $ is a symmetric matrix, this means that the Euclidean action (2) is modified to f d2x (<9"$+ - AT&
S=-tr
- &AT\
(d^
+i$ + $i)
(37)
where A is a matrix in the Cartan subalgebra of G (the Cartan subalgebra is comprised of the generators of the abelian subgroup of G). For SU(N)/SO(N), A is diagonal with entries (Ai,A2, • • • AN) and the constraint Y^i^i = 0. For 0{2P)/0(P) x O(P), the matrix can be written in the form Ajk
=
12l = l Al(5j^2l-lSk,2l — <5j,2i<Sfc,2i-l)Because A has dimensions of mass, the energy depends on the dimensionless parameters Ai/M or Ai/M. The strength of the background field controls the position of the theory on its renormalization group trajectory and, in particular, in the limit of large field the theory is driven to the ultraviolet fixed point. To be near the UV fixed point, all non-zero Ai/M must be large in magnitude, so let A be one of the non-zero components. The perturbative expansion around this fixed point therefore is an expansion for large A/M. The computation of this expansion is a fairly standard exercise in Feynman diagrams; for details closely related to the cases at hand, see 39 . The only effect of the sigma model interactions resulting from the non-linear constraints (3) to the one-loop energy comes through the running of the coupling constant. Specifically, the two-loop beta function is of the form
P(g) = / ^ s t e ) = - / W M - /W(AO - • • • where f}\ and /?2 are model-dependent, but universal for these sigma models. Solving this equation means that - i - = ft In A/A + ^ ln(ln A/A)) + 0{\/ \n{A)) 9(A) Pi where the scale A depends on the perturbative scheme used. temperature energy for SU(N)/SO(N) through one loop is
E(A)-E(0)
=
(38) The zero-
-^^(Aj)2-±^(Ai-AJf(\n\Ai-AA-^+O(g),
141
with a similar formula for 0(2P)/0(P) x O(P). For both cases, this means that at large A/M, the energy is of the form E(A) - E{0) oc A2{\n{A/M)
+ jp-
ln(ln(A/M)) + ...)
(39)
In particular, note that the ratio of the first two terms is universal. These logarithms are characteristic of an asymptotically free theory, where the perturbation of the UV fixed point is marginally relevant. I now explain how to calculate the ground-state energy E(A) — E(0) directly from the 5-matrix, giving the promised check. In this picture, the model is treated as a 1 + 1 dimensional particle theory at zero temperature. Turning on the background field has the effect of changing the one-dimensional quantum ground state. If one were working in field theory or in a lattice model, this would change the Dirac or Fermi sea. In the exact 5-matrix description, a similar thing happens: the ground state no longer is the empty state — it has a sea of real particles. For example, a particle a, in the vector representation of SU(N) has its energy shifted by Ai in a magnetic field. If the total energy is negative, then it is possible for such a particle to appear in the ground state. Because the technicalities are slightly different, I treat the cases of massive and massless particles separately. 6.1
Massive particles
For simplicity, I first treat only the case where the magnetic field is chosen so that only one kind of massive particle of mass m and charge q appears in the ground state. In integrable models obeying the Yang-Baxter equation, the particles fill levels like fermions: only one can occupy a given level. Then the ground state is made up of particles with rapidities
-B
for some maximum rapidity B. When the length L of the system is large, these levels are very close together, so the density of particles per unit length p(/3) is defined so that p((3)Ld(3 is the number of particles with rapidities between /? and /3 + dp. The energy density of the ground state is then
rB E{A)~E(0)
= y2 c
d/3 p((3)(-A+
m coshP).
(40)
J
-B
I have normalized A so that the charge q — 1. The exact 5-matrix allows us to derive equations for the p(/3) and B. This is because a diagonal two-particle 5 matrix is the boundary condition
142
the phase shift in the wave function: ip(xux2) i>(xux2)
=eipiXl+ip2X2 = eip^+ip^S(Pl,p2)
iorx1<^x2 for:n»x2
(41)
The quantization condition for a multiparticle state is derived by requiring that one-dimensional space of length L be periodic, and that the wavefunction be invariant under sending any of the coordinates x —> x + L. It is convenient to write this equation in terms of rapidity instead of momentum. Imposing periodic boundary conditions on the box of length L requires that the rapidities Pi of the particles in the ground state all satisfy the quantization conditions gimsinhfti Y[S(Pi-Pj)
=l
(42)
for all i. One can think of this intuitively as bringing the particle around the world through the other particles; one obtains a product of two-particle Smatrix elements because the scattering is factorizable. This is an interactingmodel generalization of the one-particle relation msmhPk = pk = 2nrik/L to the case where the particles in the ground state scatter elastically from each other with S matrix element S(P). In the large L limit, taking the derivative of the log of (42) gives an integral equation for the density p(P): p(P) = ^coshp + I
dp'P{p')cj>{p-p'),
(43)
where m
_
*" dlnS(P)
=
-to—dp—
This equation is valid for \P\ < B; for |/?| > B, p(P) = 0. The maximum rapidity B is determined by minimizing the energy equation (40) with respect to B subject to the constraint (43). If the particles were non-interacting, the density would be m cosh ft and m cosh B = A, but the effect of the interactions is quite substantial. These equations can be put in a convenient form by defining the "dressed" particle energies e(/3) as fB
e(p)=A-mcoshp+
dp'4>(p - P')e(P'). J-B Substituting this into (40) and using (43) yields
(44)
B
/
dp cosh/J e(P). -B
(45)
143
In this formulation, B is a function of A/m determined by the boundary condition e(±B) = 0. Consider the 0(2P)/0(P) x 0{P) sigma model, and choose the magnetic field to be Ai = A, Ai — 0 for i 7^ 1. The massive particles used above (dij = ctji) are not eigenstates of the matrix magnetic field A, but one can easily change basis to particles which are. This merely requires taking linear combinations of particles of the same mass. The state which has the largest eigenvalue of this magnetic field is d = an — a of the integral equation (44) is then proportional to the derivative of log R(/3). It is convenient to write this in Fourier space as oo
eiLJ0(l-IC(Lo))
dij
/
(46)
-OO
2 cosh((l - 2A W 2 ) sinh(27rAM) (47) — T— cosh(7rw/2) where A is as always 1/h, with the dual Coxeter number h = 2P — 2 for Mw) = e
_nMui
0(2P). For the SU(N)/SO(N) sigma model, the states a,ij are eigenstates of the magnetic field operator, with eigenvalue Ai + Aj. If we chose the magnetic field to be Ax = A, Aj = —A/(TV - 1) for j > 1, then the particle a n has maximum charge. Again making the assumption that this is the only particle in the ground state, the Fourier transform of the resulting kernel is 2c -7rAla>|
sinh
(( 1 ~ A)7r|o;|) sinh(27rAcu) sinh(7rw)
However, it is convenient here to instead choose the fields to be A\ = —A2 = A, Aj = 0 for j > 2 (this choice of field was useful also in the supersymmetric CPn model 38 ) . Then there are two particles with largest eigenvalue: a n and the antiparticle (in the N representation) 022- Thus they both appear in the ground state. These two particles scatter diagonally among themselves and each other. The S matrix element Si for a n o n —>• a n a n and 022^22 —>• 022^22
144
The element S2 for a n 022 —• 0,110.22 is found easily from (25) by using crossing. It is 52(/3) - X^
- P)Smin(rn
- /J) ^
_ ^ _ ^ ^
_
p
_ ^
Because of the symmetry between the two kinds of particles, their groundstate densities must be the same. The ground-state energy then follows from a simple generalization of the above analysis: the equations (44) and (45) still apply, with ^(/3) = - ^ ^ ( l n 5 1 ( / J ) + l n 5 2 ( ^ ) ) here. Plugging in the explicit expressions of the functions yields the useful fact that Si(/?)52(/?) = R(P), where R(/3) is the function appearing in (34). Thus with these choices of magnetic field, the 0(2P)/0(P) x O(P) and the SU(N)/SO(N) sigma models can be treated by using the kernel (47): the only difference is the A = 1/(2P — 2) in the former and A = l/N in the latter. The linear integral equation (44) cannot be solved in closed form. However, there is a generalized Weiner-Hopf technique which allows the perturbative (and non-perturbative) expansion for large A/M to be obtained 40 . I discuss this technique in the next subsection. In order to compare the energy in the 9 = 0 sigma model with the perturbative computation (39), I can rely on the results of 39 . There the first few terms in the large A expansion for models with kernels like (47) are given. The technique requires that the kernel K,{u>) be factorized into £(w) =
* (48) /C+(w)/C_(w) where /C+ (w) has no poles or zeroes (and is bounded) in the upper half plane Im(u>) > 0, while /C_(w) = /C+(—w) has no poles or zeroes and is bounded in the upper half plane. Then ,2iAu) K+ (w) = \j
• A , r ... r ( - 2 i A w ) r ( i -i{\ - A)w) e™A l n (*")+v _J ' V2 12 U..
^
^ ( 2 * 2 )
where /1 = 2Aln(2A) + {\~ Z>
A ) l n ( i - A) + \ In2 - A. Zi
£
, v (49)
145
The factor elufi ensures that /C+ is bounded appropriately as \u>\ -> oo in the upper half plane. When for small £, K+(i£) goes as
K + (»0 = ^ e ' « l n « ( l + .. the energy at large A is
39
E(ff)-.E(0) = - ^ L 2
M£)+(' + H t o ( £ ) ) +
(50)
This expansion is of the same form as (39). The two are equal if
<&-4
(51)
The explicit kernel (47) yields s = A, while perturbative computations (see e.g. SU(N)/SO(N)
fa Wi)2 while for 0(2P)/0(P)
41
and references within) give for
N + N2/2 N2
x O(P) the ratio is /?2
_ 2P 2 - 2 P
(ftF " (2P - 2)2 Using the relations A = 1/N and A = 1/(2P —2) respectively, one indeed sees that that the condition (51) holds. This is a substantial check, and coupled with the fact that 5 matrix also gives the correct central charge, leads me to be completely convinced that the S matrices discussed above are indeed the correct sigma model S matrices. 6.2
Comparing the perturbative expansions at 0 = 0 and TT
In this subsection I analyze the equations for the ground-state energy in a magnetic field in more detail. The final result will be that the entire large-A perturbative expansions for 9 = 0 and for 9 = ix are identical, but that nonperturbative contributions differ. This effectively confirms the S matrices and spectrum above, and the identity of the low-energy fixed points.
146
The generalized Weiner-Hopf technique is discussed in detail in the appendix of 4 0 . The equation to be solved is of the form £(/?)- f 4>{/3 - 0')e(J3') = g(J3). J-B
where e(/3) and g(/3) are both vanishing for \/3\ > B, and (/?) = (-/?). Denning the Fourier transforms e(u), g{to) and /C(w) (the latter related to 0 as in (46)) gives
f
e-^{e(tj)K(o;)-g(Lj)}=0.
This equation is valid for \/3\ < B. Fourier-transforming this gives ?(w)/C(w) - g(u) = X+(u})eiuB
+ X_e~iujB.
(52)
The functions X± arise, roughly speaking, from the analytic continuation of e(0) to |/3| > B. The extra factors e , w B ensure that X+(OJ) is analytic in the upper half plane, while X-(ui) = X+(—ui) is analytic in the lower half plane. The relation (52) can be split into two equations, one involving poles in the upper half plane and the other the lower. To split a given function, one uses
/ H = [/M] + + [/H]_ where u
v
n
2TT* J_00
UJ'
-uj^iS
where 8 is a positive real number tending to zero. The functions [/]± are analytic in the upper (+) and lower (—) half planes. Because e(/3) and B, the functions e±(u)=e(u;y"B
g±(u)=g(u)ei"B
are similarly analytic in the upper half and lower half planes. Equations for the functions X± can be derived by exploiting these analyticity properties and the factorization (48). Namely, (52) implies the equations p^r
= [ j i M ^ H J i + [X±(W)/CTMe±2-B]±,
(53)
while the X± are given by X±(w)/C±M + [XT{u)K,±{uj)e^B}±
= -[ff T (w)/C±M] ±
(54)
147
Unfortunately, it is not possible to solve the equations for X± explicitly. However, this form does allow the large-A expansion to be systematically developed. To study the large A expansion, it turns out to be much easier to consider the more general kernel ^'
cosh(77rAw) 2cosh((l - 2A)7rw/2) sinh(2A7rw) ~ sinh((7 + 1)TTAW) cosh(?rw/2) '
(
'
In the limit 7 —>• 00, K goes back to the sigma model kernel /C. In the N=2 case, this deformation corresponds to deforming the sphere sigma model into the sausage sigma model 17 . This kernel factorizes as, K = 1/(K+K_), giving K+{uj)
-\/^-ie
•(56)
r H ( 7 + i)A u )r(i-i|)
where v = fi + 7A ^ ( 7 / ( 7 + 1)) - A In A + A. The equations (54) can be written as a single one by exploiting the relation X+(ui) = X-(—u). Denning K+(u))
Lj + 10
2
u) — 1
they become , .
B IS. (4\ imeoB K+(i)
iAK+(0) UJ
+ 10
2
u) •
r
/ Jc+
02uiB
;
r;a(b}')v(u}')-— (57) v ' K '2m v ;
U + GJ' + IS
where a(oj) is defined as
and the integration contour circles all singularities on the positive imaginary axis. The boundary condition e(±B) = 0 becomes iAK-{Q)-^-K.(-i)= 2
f e" B a ( U )«( U )^ Jc+ 2ni
(58)
while the energy (45) becomes. E(A)-E(0) =
^K_(-i) mpB
AK+(0) - ^-K+(i) 4
r
/
(,2uiB
Jc+ UJ — 1
e
J. ,
—a(U)v(uJ)^ 2TTI
(59)
148
The contour here includes the double pole at w = i, and the poles in a(w). The function v(ui) has no poles in the upper half plane except the explicit one at u) = i. These equations are convenient for deriving the expansion for large A. The reason is that when A is large, the range of rapidities allowed in the ground state is large, so B is large (recall that if the particles were free, A =TOcosh B). In this limit the integrals in (57,58,59) are small corrections, and their effect can be treated iteratively. This iterative expansion is worked out in detail in 42 . The first correction to v(u) comes from approximating the integral using the leading pieces of V(OJ) in the integrand. The boundary condition (58) relates B to A. Thus the contributions to this integral come from the pole in v(to) at u = i and the poles in a(u). The poles of a are at UJ - i{2n - 1)/(2 7 A), u = (2n - l)/i/(2A + 1) and w = m/(2A), for n a positive integer. Using the iterative procedure it is straightforward to see the form of the expansion for large A: it is a power series of the form
The first series is a result of the first series of poles in a; the coefficients a.j depend on the residues at these poles. The order A2 corrections are a result of the pole at w = i and the other poles in a. Recall that 7 is large, so these contributions are much smaller. The double pole at UJ = i results in an A-independent piece, which is identified as E(0) 17 . The sigma model result of interest is recovered in the limit 7 —• 00. This complicates matters considerably for the power series in (60), since the exponent is going to zero. This is precisely what happens for example in the anisotropic Kondo problem as the anisotropy is tuned away 43 . What happens in this limit is that the power series turns into an expansion in ln(A),\n(\n(A)),.... Note from (38) that \jg oc ln(A) + ..., so the series depends on powers of g, while the order A2 corrections depend on e-consila_ Thus the first series consists of the perturbative contributions to the free energy, while the second part is non-perturbative: the latter will never be seen in standard sigma model perturbation theory in g. The exact form of the perturbative expansion cannot be displayed in closed form, but one can build it piece by piece; the first pieces are displayed in (50).
The analogous equations for E(A) for the sigma models with 6 = IT follow from their S matrices. I will show that the large AjM expansion is of the
149
form (60), with the identical an. This effectively proves that these S matrices are those of the sigma model at 9 = IT. The flow to the WZW model then follows immediately, because the S matrix manifestly becomes that of the appropriate WZW model in the large-M limit. The integral equations for massless particles in a background field are similar to the ones for massive particles. The massless particles are left and right moving, and because they are gapless, they begin filling the sea for arbitrarily field A. Therefore, right moving particles with rapidities —oo < P < B and left moving particles with rapidities — B < (3 < oo fill the sea. For one species of right mover, and one species of left mover, the analysis at the beginning of this section can be repeated to derive the equations
<*(/*) = A - -e0 + / Z
/ M
*UP) =A- —e-* + / *
MP~ P'^nW'W
J -oo -oo /•OO oo
MP-P')*L{P'W
(61)
-B
fB
MP-
P'MP'W
J -oo
oo
/
MP-pyup'w
(62)
-B
valid for — oo < /? < B and — B < /3 < oo respectively, with the boundary conditions
eR{B) =
tL(-B)=0.
The kernels are defined as
The energy is EM(A)
- £W(0) = - — /
e"efi(W
(63)
where I have used the symmetry £L(/3) = €R (—/?). The two equations (61,62) can be made into one by exploiting the leftright symmetry. In terms of Fourier transforms, ?R(W) = ?£,(—w), giving £R(w)*;i(ij)-£fi(-«)t2(u)-jfl(a;) =
Y+{u)eiuB
where the symmetry k-2(-u>) = ^ ( w ) is used, and <7.R(/?) — A - me /3 /2. Note only one term is need on the right-hand side, because the integral in /? space
150
runs all the way to — oo. Getting rid of the efl(w)
€R(—UJ)
gives
^(u) - Y — -Sfi(u) - 5 i j ( - u ) fci(w) / ^i(w)
= Y+(uj)e^B + r _ ( u ; ) e - ^ B ^ 4
(64)
where as always F_(w) = F+(—w). This is now an equation of the form (52), and the same generalized Weiner-Hopf analysis can be applied. Defining ki(u))
and factorizing it as k(u) — l/(fc+(w)fc_(w)) in the usual way, one finds an equation just like (57), namely . ,
+ /
iAk+(0)k!(0) to + iS ^ ( 0 ) 2 iB "
JC+
LO
imeB 2
K+(i)kAi) — i k2(i) <M_ ; =n(u/)v(LS 2ni + LO' +l8 UJ
(65)
where
n(w)
k-(uj) ki(uj)
k+(u)
k2{uj)'
The boundary conditions and energy follows with the same substitutions. Now I can show that the sigma models have the same perturbative expansions at 0 = 0 and IT. Using the magnetic field described in the last subsection, the kernels <j>\ and 0 2 follow simply from the Gross-Neveu 5 matrix and the relations (29,31,35,36). As before, it is assumed that only particles with largest eigenvalue of this magnetic field occupy this state. For the 0(2P)/0(P) x 0(P) models at 9 = 7r, there is only one kind of particle (left and right moving) in the ground state. The Fourier transforms of the kernels are ~ sinh(( 7 + lfrAhQ cosh((l - 2 A ) ^ / 2 ) ki{u) = 1 - i(w) = —-7—^ r~7 77T\ v v ; ' smh(77rA) cosh(7rw/2) ~ sinh(( 7 - l)nAu) cosh((l - 2 A ) T T ^ / 2 ) fcolW) = ©2(w) = y
—.
T-T
—,
77^
(00) ("')
v ; ' wv ' sinh(77rA) cosh(?rw/2) where I have again included an extra parameter 7 to simplify the analysis. The sigma model kernels are recovered in the limit 7 —> 00, giving the exponential factors e7rAlul and e - 7 r A l"l respectively. Just like the massive case, the
151
SU{N)/0(N) model has two particles in the ground state, and ends up with the same kernels. From these explicit forms, one finds remarkably enough that k(u) in the massless case is identical to K(u) in (55) in the massive case. The only difference between the equations and (57) and (65) is the extra function fciM &2(w)
=
sinh((7 + l)Airu)) sinh((7 — l)A7ixd)
,Qg.
in all three terms. This indeed means that the energy at 6 = 7r is not the same as 6 = 0. However, this extra piece has no effect on the perturbative contributions, so the energy is still given by (60). This is because the extra piece introduces no new poles in the integrand in (65) (the poles in (68) are canceled by zeros in k-/k+). Moreover, the residues at the poles at UJ = i(2n + 1)/(2A7) are the same in the massive and massless cases (up to an overall sign), because M i ( 2 n + 1)/(2A 7 )) fc2(i(2n + l)/(2A 7 ))
=
These residues are what determine the coefficients an in (60), so indeed the perturbative expansions at 6 — 0 and 8 = TT are completely identical. This was established for the sphere sigma model in 17 ; here I have extended this proof to two infinite hierarchies of models. This completes the identification of the massless 5 matrices with the sigma models at 8 = TT. They give the identical perturbative expansion as the sigma model at 0 = 0, but the non-perturbative pieces differ. As an additional check, I have also computed the c-function by computing the free energy with no magnetic field but at non-zero temperature 6 . This gives the correct behavior, thus completely confirming this identification. This computation is described in the next section. 7
The free energy at finite temperature
Once the exact S matrix is known, the exact free energy at any temperature can be computed by using the thermodynamic Bethe ansatz (TBA) 4 4 4 5 . This enables one, for example, to compute thermodynamic quantities like the susceptibility. It also allows a very substantial check on the assumption of integrability. The reason is that at a critical point, the free energy is known exactly - it is related to the central charge of the corresponding conformal field theory 4 6 . Thus the free energy computed from the TBA must give this
152
result in the limit where the mass of the particles goes to zero, and the system is at the unstable UV fixed point. The results of this section appear in my paper 6 . 7.1
The free energy of an integrable theory
To derive the TBA equations, one first needs a relation between the density of states of the particles to the actual particle density. This relation is called the Bethe equation. If the particles are free, this is trivial: the density of states is independent of the particle density. The simplest case is where there is only one species of particle with mass m and S matrix 5(/3). The Bethe equation is written in terms of the density of states per unit length P(/3) and the density of rapidities per unit length p(P). The former is defined so that the number of allowed states with rapidities between /? and /3+df3 is P(fi)Ldp, while the number of states actually occupied in this interval is p(/3)Ld/3. The quantization condition (42) relates the two. Taking the derivative of the log of (42) yields
P(/3) = — cosh p + /
4>(P - P'W)
(69)
where
This is the generalization of (43) to allow for the fact that at finite temperature, any state may either be filled or unfilled. This is easily generalized to the situation where there is more than one particle in the spectrum, as long as the scattering is diagonal. Defining densities Pa and pa, where a indexes the type of particle, the Bethe equations are oo
/
*ab(P - P')pb(P').
(70)
where
with Sab the S matrix element for scattering a particle of type a from one of type b. It is convenient to put the factors of 2ir in a different place in this section, hence the slightly different definitions of $ a ;, vs. 4>Once the Bethe equations are known, the TBA equations and hence the free energy can be derived. This is done by minimizing the free energy, using
153
(70) as a constraint. In all known integrable particle theories of this type, it is either proven or assumed that the particles fill levels like fermions: at most one particle in a level. The result is most conveniently written in terms of the dressed particle energies ea(/3) defined as Pa(0) _ 1 P„(/3) l+e-*«/r'
(71)
Notice that as T —• 0, p/P is either 0 or 1: a level is either filled or empty. The resulting TBA equations are 44 ' 45 ea
(/?)=mQcosh/?-£T|°° |^$a6(/?-/?')ln(l+e-^)/T)
(72)
If the particles were free, $ a 6 = 0, and the ea just are the particle energies. For simplicity, I have set all chemical potentials and background fields to be zero, but the analysis can easily be extended to include them. The free energy per unit length F is given in terms of these dressed energies ea It is F(m,T) = -TYma
f°
^ cosh/? In (l + e~t'^T)
(73)
In the IR limit m —¥ oo, the gas of particles becomes dilute, and interactions can be neglected. The free energy becomes ma / ° ° — cosh P e'm" ™MP)/T
lim F(m, T) = -TY m->oo
*—<
J_OQ
(74)
2lT
This integral can be done, yielding a Bessel function. Calculating the free energy using the TBA allows an extremely non-trivial check on the exact S matrix. In the limit of all masses going to zero, the theorem of 46 says that the free energy per unit length must behave as 7TT 2
HmF= m—>0
— cuv
(75)
D
where cuv is the central charge of the conformal field theory describing this UV limit. The number Cuv can usually be calculated analytically from the TBA, because in this limit the free energy can be expressed as a sum of dilogarithms 4 7 . The cuv computed from the TBA must of course match the cuv from the field theory. This provides an extremely non-trivial check not only of the S matrix, but of whether the entire spectrum is known. All particles contribute to the free energy, so if some piece of the spectrum is missing, the TBA will not give the correct cuv-
154
The TBA computation is much trickier if the scattering between particles is non-diagonal, as is the situation for the models of interest here. The Bethe equation is much harder to derive, roughly speaking because imposing periodic boundary conditions means that we must know what happens as one particle goes around the periodic world. If there is non-diagonal scattering, the particle changes its state (and the state of the rest of the particles) as it scatters though the other particles. What is needed is the eigenstates for bringing a particle around the world. This requires introducing the "transfer matrix" T for bringing the a given particle through the others. When the scattering is not diagonal, T is not diagonal. To define T explicitly, I first introduce the scattering matrix Tab{/3) for bringing a particle of type a and rapidity /3 through J\f particles and ending up with a particle of type b. Thus Tab is a set of s2 s x s matrices, where s is the number of different types of particles. The scattering is completely elastic, so the rapidities do not change even though the scattering is not diagonal. This means Tab(/3) depends on the rapidities /3i . . . /3^/ as well as /?. The components of Tab can be written in terms of the S matrix elements as
(Tab((3))dc: = Yl
Saci->dih{p-h)SflC^d2h{P-fo)...StKCM^d„b{P-M
all/,, where Sab^cd is the two-particle elastic 5 matrix element for scattering particles of types a and b and ending with types c and d. The matrix T follows by exploiting the fact that all the scattering at zero relative rapidity permute the colliding particles. In other words, the S matrices here all satisfy Sab^cd(0) — -5aC8bd, Thus setting (3 = /3a effectively turns the a t h particle so that it scatters through all the others. This is precisely what is needed for the TBA. To put periodic boundary conditions on the system, one sums Taa over all a. The result is that r ( ^ a | ^ l , . . . ^ ) = 53ra„(/3 = ^a|)8a+l,---/3jV,|8l,---,/5a-l)a
(76)
This is a s ^ - 1 x s^~l matrix. If the scattering is diagonal, T reduces to the product of 5 matrix elements times the identity matrix. The important consequence of integrability is that the Yang-Baxter relation ensures that the T(/3) commute for different /3. This means it can be simultaneously diagonalized for all ft by a /3-independent set of eigenvectors; only the eigenvalues depend on /3. The TBA requires finding the eigenvalues A(/3 a |/3i,.. .Ptf) of T. The quantization condition (42) is eim"sinh0°LA(l3a\l3u...M
= 'L-
(77)
155
In the limit of large number of particles Af, A depends instead on the particle densities instead of the individual rapidities. Henceforth I will just write A(/3). For the cases of interest here, finding the eigenvalues A(/3) is quite difficult, but has been done in 48>49.50. The Bethe equations are still of the form (70), and the TBA equations are still of the form (72). However, extra particles, known as "pseudoparticles" or "magnons", enter the equations. These particles enter the equations just as if they were a particle species, but with ma = 0. I will give many examples of the explicit form of these equations below.
7.2
The sphere sigma model
The TBA equations for the sphere sigma model were derived originally by taking the limit of certain integrable fermion models 51>52) and conjectured on different grounds in 53 . I will rederive the TBA equations here directly from the 5 matrix, because this is the method which generalizes most simply to the more general sigma models of interest. Since this 5 matrix is non-diagonal, one needs to diagonalize the transfer matrix as described in the last section. The way to do this is to first solve the problem for particles in the spin-1/2 representation of SU(2) (the SU(2) Gross-Neveu model), and then use fusion to find the answer for spin 1. For particles in the spin-1/2 representation of SU(2), the Bethe equations were derived 70 years ago, in the original paper by Bethe himself 54 . The reason is that the transfer matrix for the spin-1/2 representation of SU(2) as defined in (76) precisely corresponds to the transfer matrix of the Heisenberg spin chain. In the limit of large number of particles TV, the eigenvalues of the transfer matrix follow by adopting the "string hypothesis". This means that the eigenvalues A(/3) of the transfer matrix defined in (76) are expressed in terms of densities f>k(0), with k = 1 . . . oo. These are the pseudoparticles discussed above: they enter the TBA equations as if they were real particles with no mass term. (I have somewhat abused the conventional notation: most authors would not use the "here, but it makes subsequent relations less confusing.) The other density entering the equations is the density of particles Po(/3). This is the total particle density, with contributions of both spin up and spin down particles. Bethe's result for the eigenvalues is •
oo
2
-lnA(/?) = r< > *Po(/?) + 52^°°)*pj{0)
(78)
156
where convolution integrals are defined as oo
dp'f(P - P')g(/3). /
-oo
The kernels are given explicitly in the Appendix. The kernel F W comes from the prefactor of the S matrix. This only affects the coupling to the total particle density, and not the pseudoparticles, because it contributes an overall factor ria=i Fvv(f3 — /3a) to the transfer matrix. Now I can write down the first of the Bethe equations, by taking the derivative of the log of (77). This gives oo
2TTP0 (/?)
= m cosh /? + Y<2> * Po (/?) - ^
a^
* pj (/3)
(79)
where m is the mass of the particles. Po is the total density of states for the particles. The other Bethe equations relate the densities of states for the pseudoparticles to particle and pseudoparticle densities. They are oo
2*Pj(P) = a™ * po(/?) - Y, ATi] * Pi^)
(8°)
where the density of string states Pj is Pj = Pj + Pj •
Note that all the Bethe equations are of the form (70), with no mass term for the pseudoparticles. Using the identities in the appendix, all the Bethe equations (including that for P 0 ) can be written in the compact form oo
2nPj(P) = Sj0m cosh f3 + £
.
/j,
oo)
J <&'cosh{p
^
_ pf*?)-
^
Here the indices j and I in the incidence matrix A = <$J,J+I + <^,/-i run from 0 , 1 , . . . , oo. Note that the right-hand-side involves the densities p, not p. This Bethe equation is conveniently represented by the diagram in figure 1. With these equations, it follows from the standard TBA calculation that the TBA equations (72,73) hold, with j(oo)
157
a—o—o—o—o— Figure 1. The incidence diagram for the SU(2) Gross-Neveu model (the sine-Gordon at 02 —> 8JT). The circles represent the functions e a ; the filled node represents the fact that the equation for to has a mass term.
and rrij =
Sjomcoshfi.
These equations were first derived in the context of the sine-Gordon model at /32 —> 8TT in 55 . One can easily check that the free energy has the correct properties. In the UV limit m/T —• 0, one obtains the correct central charge cuv — 1 from (75). This follows from the standard dilogarithm analysis (see e.g. 47-45>31). In the IR limit, the generalization of (74) to the case with pseudoparticles is F = m W l + e- £ l < 0 0 >y / 2 r V
' -6
^cosh/?e-™coshW/T
7_oo 27T
00
For particles with rrij ^ 0, e -^ ) vanishes. However, the pseudoparticles have no mass term, and here one finds that e - "^ 0 0 ) = (j + l ) 2 — 1 for j > 1. This means that the free energy in the IR limit is that of 2 particles of mass m, as it must be. Now we need to apply fusion to get the S matrices and TBA for the sphere sigma model. This is fairly simple to do, now that everything has been set up. The fusion procedure says that the spin-1 particles in the sphere sigma model can be viewed as having the spin-1/2 particles as constituents. As explained above, a spin-1 particle (in a representation with highest weight 2/ii) is composed of a pair spin-1/2 particles (each in a representation with highest weight fii) with rapidities Pi+iir/2 and Pi—iir/2. The transfer matrix for TV/2 spin-1 particles is related to that for the N spin-1/2 particles (highest weight T 2 ^ ( / ? Q | f t , . . ./3AA/2) ex T"-G8Q + r?|/3i + V,Pi ~ V, • ..Pjsr/2 + V,PM/2 - V) *T'"(0a-T,\l31+Ti,l31-T,,...Pjsr/2+Ti,0Ar/2-ri) (82) where j] = iir/2 here. The reason for the proportionality instead of the equality is that the prefactors of the S matrices need not satisfy the exact fusion
158
relation, as discussed above. Given this relation between transfer matrices, the eigenvalues obey the relation Asphere{f3)
=
C(/
J)A(/3
+
^(fi
_ ^
(83)
The constant of proportionality is
cm
FSSW-M
= TT
Because the two transfer matrices are related in this way, the Bethe equations for the sphere sigma model follow from those above after a few modifications. The eigenvalue of the sphere sigma model transfer matrix follows from the spin-1/2 eigenvalue (78), and the fusion equation (82). It is A
°°
— In A""""(/3) = Z<2> * Po(P) + ^ t P
]
* PM
(84)
j=i
where T^tf)
= a^iP
+ in A) + a ^ \ p
- ITTA)
with A = 1/N. The first term in (84) arises from the prefactor of the sphere S matrix. The explicit expressions for r^ s ' and Z W are given in (111) and (108) in the appendix. Using this expression for the eigenvalue in (77) gives oo
2TTPO(/3)
= m cosh0 + Z^
* Po(f3) - £
r j o o ) * Pj((3).
(85)
The Bethe equations for the densities of states of the pseudoparticles (80) are modified because the real particles come in pairs with rapidities ft ± i7r/2. Thus for the sphere sigma model oo
2-npj(P)=r{jco)*p0(P)-J2
4 ^ * *^)
( 86 )
for j > 1. By using the identities in the appendix, the Bethe equations (86, 85) can be put in the unified form oo
2nPj(0) = Swoosh? + ^ i j ^
.
-
J ^ ' C Q s h ( / ? _ pfW)
(8?)
The indices j and I here run from 0 . . . oo. Above, the incidence matrix / ' s ' was associated with SU(s). Here, the incidence matrix 1^ is associated with
159
Figure 2. The incidence diagram for the sphere sigma model,
0(2s): iff Explicitly, ZjT
= 28ji - C ° ( 2 s ) , where C°(2s> is the Cartan matrix for 0{2s). =
^J'.'+I + dj,i-i + ^',2^,o + SjtoSi,2 - Sj,iSifi - Sj,oSi,i
(88)
This Bethe equation is conveniently represented by the diagram in figure 2. With these equations, it follows from the standard TBA calculation that the TBA equations (72,73) hold, with j(oo)
and rrij = 8jomcoshf3. One can easily check that this TBA system has the correct properties to describe the sphere sigma model. 5 3 . In the UV limit m/T —> 0, one obtains the correct central charge cuv = 2. by the standard dilogarithm analysis. In the IR limit, one finds that
mT\(l
+e - ^ Y
2
[
^cosh
?e
-mcosh(/3)/T
As with the spin-1/2 system, the functions obey e - ^ ' 0 0 ' = (j + l ) 2 — 1 for j > 1. This means that the free energy in the IR limit is that of 3 particles of mass m, the spin-1 triplet. 7.3 SU(N)
Gross-Neveu models
As in the N = 2 case, it is best to first perform the analysis for the vector particles in the Gross-Neveu model and then use fusion for the more complicated sigma model.
160
Computing the Bethe equations for the SU(N) Gross-Neveu computation looks extremely difficult or impossible, because all the S matrices are not even known in closed form. Remarkably, the computation has already been done in 47,48,49,50 j j e r e t k e g g ^ g equations are found for any simply-laced Lie algebra G, when the particles are in any representations with highest weight nfii where p\ is a fundamental weight of G, and n is an integer. This work was generalized to non-simply-laced groups in 56 , but this is not necessary for these lectures The fusion procedure gives functional relations like (82) for all the Ta(Pa\Pi, • • • , / V ) 49- The label a here indicates that the a t h particle is in the representation with highest weight p,a. These functional relations relate various Ta. The prefactors Fab(8) need to be computed, but the explicit S matrix is not needed: all the relevant physics is contained in the representation theory and in the fusion. From the functional relations and a few mild analyticity assumptions, the eigenvalues of Ta and the Bethe equations can be derived in the limit of a large number of particles. The Bethe equations for the general case require the introduction of pseudoparticle densities and densities of states into the Bethe equation (70). Here the pseudoparticle densities paj and densities of states Paj(/3) are labelled by two indices. The index a runs from 1 to N — 1 for SU(N). For the N = 2 case treated above, this index takes only one value can be suppressed. The index j is the same index as before, running from 1 , . . . oo for the pseudoparticles. The real particle densities are labelled by pa,o, and densities of states Pa,oThe density pafi is defined as the density of all the particles in states in the representation pa. It is consistent to define separate densities for each representation, because the particles cannot change representation when scattering. For all values of a and j , Paj = paj + paj. The computation of the TBA equations for the SU(N) Gross-Neveu model was done in 57 . The eigenvalues of the transfer matrix Ta are 49 ' 50 ,
N-\
oo
^ l n A ^ G S ) = J2 Ya?] *P»,o(P) + E C T i ° 0 ) *P°AP) "
(89)
j=l
6=1
where the kernels are given explicitly in the Appendix. The kernel Y^b comes from the prefactor Fab of the S matrix. It couples the density of states of real particles in representation a to the density of particles in representation b. The first of the Bethe equations follows from (77), and is JV-l
27rPa,0(/3) = ma cosh/? + £ 6=1
oo
Y^
* p M (/3) - £ j=l
a^
* paij(0).
(90)
161
The other Bethe equations relate the densities of states for the pseudoparticles to particle and pseudoparticle densities. They follow from 49 ' 50 as well, and are N-l
oo
2 ^ 0 9 ) = <7<°°> * pa,0(P) - J2 E At] * ^T * P*M
(91)
6=1 1=1
where Paj = pa,j+Pa,j- Explicit expressions for these kernels are given in the Appendix. Note how all these equations reduce to those in the last subsection by setting N = 2. By using the fact that A and K are inverses, and the identities in the appendix, all the Bethe equations (90,91) can be written in the combined form 57 J V - 1 oo
27rpaJ (/?) = 6j0ma cosh /? - Y, E
K
i™] * AaT * PM (0) •
( 92 )
6=1 / = 0
Here the indices j and / run from 0 , 1 , . . . , oo. With these densities, the dressed energies eaj(/3) are defined as in (71). It follows from the standard TBA calculation that the TBA equations (72,73) hold, with * a M G 8 ) = 2n6il6ab6(P) - K™
*
A^tf)
and maj = 6joma cosh/3. The TBA equations can be rewritten in a much more elegant form by using the fact that A and K are inverses. The result is "
t[
l-oo ^
_ T y^ j(oo) r ° dfj[ U
1
2cosh(iV(/3 - /J')/2)
N
y-oc27r2cosh(iV(/3-^)/2)
/ V1 +
e
V
+ C
{S')/T\
)
, ) •
(93)
This is a substantial simplification because of the incidence matrices are simply /-, = Sjj+i + 8j,i-\- Thus the equation for ea,j only involves "adjacent" functions e 0 ,j±i a n d ea±i,j- These equations are displayed schematically in figure 3. The dashed and unbroken lines account for the different minus signs in (93). Note that the masses do not appear in rewritten TBA equations (93), although they appear in the original ones. When using the form (93), the asymptotic conditions ea,oC# ->• oo) —> ma cosh /?
162
f—
I
I
•••
I
|—<^—r}—r^I
I
I
I
•
i
i
i
i
i
i
i
...
*—6—6—6Figure 3. The incidence diagram for the SU(N) and an infinite number of columns.
Gross-Neveu model. There are N — 1 rows
must be imposed. One can check that the free energy of the SU(N) Gross-Neveu model has the correct properties. In the UV limit m/T —> 0, one obtains the correct central charge c = TV — 1. from the dilogarithm analysis. In the IR limit, one finds that each representation contributes one term to the free energy, with the correct multiplicity (e.g. N for the vector representation a = 1, N(N — l)/2 for the antisymmetric representation a = 2). 7.4 SU{N)/SO(N)
sigma models
Here I find the TBA equations for the SU(N)/SO(N) sigma model, generalizing the analysis for the sphere sigma model. The Bethe equations for the SU(N)/SO(N) sigma model are obtained from those of the Gross-Neveu model by fusion. There are two resulting modifications in the kernels. The kernel Y^b coming from the S matrix
163
prefactor is replaced with Z{ab ', while the kernel er^ defined by
is replaced with r]
r j s ) (/?) = af (/? + in/N) + a j s ) (/? - iw/N).
,
(94)
The sigma model version of (90) is 7V-1
oo
2irPa,o(P) = m Q c o s h / 3 + E Z™ * pb,0(P) - E 6=i
7
^
* p a j (/3)
(95)
j=i
while the Bethe equations for the pseudoparticles are J V - l oo
2npaj(0) = r^ * pafi(P) - E E At] * ^ 6=1
* ?M(/^ 06)
/=1
Explicit expressions for these kernels are given in the Appendix. Note how all these equations reduce to those of the sphere sigma model by setting N = 2. The different kernels in the Bethe equations of course mean that the TBA system is modified, as with the sigma model. All the modifications involve the couplings of the functions of pa,o(/3) to the other pbj. After using the identities in the appendix, one finds that the net effect is to remove couplings between ea,o to eaji in the Gross-Neveu TBA (93), and replace them with a coupling between eOi0 to ea,2- The SU(N)/SO(N) TBA equations are
e
.(B)-TY"I{N)
~ - r f j N
h f°° ^L
r
J-°°
^ -
(N(P
H
\n(l + ei"-'^lT\
-
2n 2cosh
- P')/2)
V+
]n(l + e-f'-'lf>')/T)
) (97)
The asymptotic conditions are the same as for the Gross-Neveu model. In fact, the only difference is that the second incidence matrix J^00' is replaced with l(°°). These equations are displayed schematically in figure 4. Both cases can be conveniently summarized in the language of Dynkin diagrams: the Gross-Neveu model in figure 3 is described by {AN, A^), while the SU(N)/SO(N) sigma model is described by ( A ^ A x . ) The latter TBA system was previously discussed in 58 , but without the association with the sigma model. As with all previous cases, one can check that the UV and IR limits of the TBA equations agree with known results, namely the central charge c = (N + 2)(N — l ) / 2 and the particles being in the representations 2/ia-
164
W
'
I
I
I
Figure 4. The incidence diagram for the SU(N)/SO(N) rows and an infinite number of columns.
7.5
The 0(2P)
sigma model. There are N — 1
models
As with the models with SU(N) symmetry, I will start with the 0{2P) GrossNeveu models. Luckily, the Bethe equations for 50(2F)-type systems were also found in 49 - 50 . These were more or less conjectured based on analogy with the SU(N) case, but were proven up to some technical assumptions in 56 . Basically, they amount to doing the computation by replacing the SU(N) incidence matrix 1^ with the SO(2P) incidence matrix l^p\ The details for proving this are given in the appendix. The TBA equations for the 0(2P) sigma models are
e (8)-T^Tl{P)
f°° ^-
^—^
Xnil +
e^^A
165
*—
I
I
•'•
I
\
•o
0
6 -
Figure 5. The incidence diagram for the 0(2P) Gross-Neveu model. There are P rows and an infinite number of columns.
*Pi n
J —C
P
~
l
Ya(\ + e-<"'W'VT\
27TCOSh[(P-l) (/?-/?')]
V
(98) )'
[
'
These equations are displayed schematically in figure 5; the indices a and b now run over the nodes of a Dp Dynkin diagram. The correct central charge c = P is obtained in the UV limit. This system was discussed in 58 , but the appendix contains the first proof (as far as I know) of its connection to the 0(2P) Gross-Neveu models. The TBA equations for 0{2P)/0{P) x O(P) sigma models follow from the 0(2P) Gross-Neveu model calculation, just as the SU(N)/SO(N) calculation follows from that of the SU(N) Gross-Neveu model. They are P c
„vm=rVl(f) r
—
P
~
1
lnfl + e ^ C 3 ' ) / ^
166
h
1
y-oo27rcosh[(P-l)(/?-/?')]IH1
+ e
J"
{
™>
The kernels and identities for this derivation are discussed in the Appendix. The TBA equations for the 0(2P)/0(2P — 1) sigma models are also known 59 . Because the spectrum consists of a single multiplet of N particles in the vector representation, a in pa0 can only be 1. The other eaj still run from 1 ...P. Because there are no bound states, the prefactor J711 is not the same is in the Gross-Neveu models: the poles need to be removed from the prefactor 14 . The kernel appearing in the TBA equations is therefore y?\P){P) - 2TT6(/3) + A[\P)(I3). Using this with the above Bethe equations gives the TBA equations given in 59 7-6 Sigma models with 9 = TT Here I find the TBA equations for these sigma models when 9 = n. Since the S matrices are almost the same as the Gross-Neveu models the derivation of the TBA equations is easy. The pseudoparticles are identical, so we have densities paj labeled by two indices as before. However, in scattering, left movers stay left moving, and right movers stay right moving. Thus instead of densities pa>0, now there are both p 0v j, and pa,R- For the SU(N)/SO(N) case, the first of the Bethe equations (90) is replaced with the two equations N-l
^PaAP)
= mae0 + E
Y
a?] *
P»M
6=1
JV-1
oo
+ E ( ^ - S*»W) + AiT) * P»M ~ E ^°0) * ? « # )
(100)
3=1
6=1
N-l
2vPaM
13
= rriae- + £
Y™ * pb,L(P)
6=1
JV-1
oo y
+ E
( i f > - SaMP) + A™) * Pb,R(P) - E
6=1
^
* P*A0)-
(101)
J= l
The Bethe equations for the pseudoparticles (91) become N-l
27rpaJ(P) =
*{PaM+P*M)
~ E
oo
T,AP
6 = 1 1=1
* * » r *MP).
(102)
167
Using the identities in the appendix gives the combined TBA equations e
(8)-Y^I{N)
" hi i
£
Z
»
f°° ^
'-<*>2w
]n(l + ee*-tWr)
-
2 C0Sh(7V(/? /?,)/2)
-
y_oo27r2cosh(iV(/3-/J0/2)lnl1
" + e
' J(1°3)
where j takes the values L, R, 1 . . . 00. These equations for the SU(N)/SO(N) sigma model at 6 = IT are identical to those for the SU(N)/SO(N) sigma model at 9 = 0 (97), once the labels are redefined (there j takes the values 0 , 1 . . . 00). However, that does not mean the solutions are the same. Because the 6 = 0 theory is massive and the 6 = TT theory is massless, the asymptotic conditions are different. Namely, as (5 —> ±00, for the massive theory: ea0(/3 -> 00) —> ma cosh(/3) while for the massless theory as /? —> +00 e a i(/? -» 00) —> mae^ ean(/3 —>• 00) —> constant and as /? —»• —00 eai(/? —> — 00) —• constant £
-> -00) —• m a e _ / 3
The free energy per unit length (73) is modified in the massless case to F™{m,T) = -TY,ma
f°° ^
[e" In (l + e " ^ > / T ) + e-s\n(l
+ e-^(0)/T^Y
(104)
The equations for the massless theory are pictorially depicted in figure 6. The different asymptotic conditions do not affect the free energy in the ultraviolet limit m/T —> 0. Thus the free energy is the same in massive and massless cases, corresponding to that of a conformal field theory of central charge cuv — (N+2)(N—l)/2. This of course is the dimension of the manifold SU(N)/SO(N). In fact, because the TBA systems are identical except for the asymptotic conditions, the entire UV perturbation theory is identical in both cases. This is as it must be: instantons are a non-perturbative effect, and so the effect of the instanton coupling j3 cannot be seen in perturbation theory. Unfortunately, it is not known how to compute the perturbative expansion at non-zero temperature, except for the leading logarithmic correction 4 .
168
?-> I
I
Figure 6. The incidence diagram for the SU(N)/SO(N) sigma model with 9 = n. There are N — 1 rows and an infinite number of columns The cross-hatched circles represent eaL and eaR.
On the other hand, the physics for 9 = 7r is radically different from that at 6 = 0 in the low-energy limit m/T —> oo. In the massive case the free energy in this limit is merely that of a dilute gas of massive particles, as in (74). However, the particles are massless when 6 = ir because the system flows to a non-trivial field theory in the low-energy limit. This flow is immediately apparent from the 5 matrix point of view, because the two-particle Lorentz invariant for a left and a right mover is oc m 2 , so the S matrix goes to a constant 5(0). The right-right and left-left matrices remain non-trivial, however, since the Lorentz invariant here is independent of m. Thus in the low-energy limit, the left and right sectors decouple from each other, but remain non-trivial. This is the behavior of a conformal field theory. The free
169 energy must obey a relation like t h a t of the UV limit, namely lim
F =
7rT 2 T-cm.
(105)
Here this gives CJR = N — 1. This is the central charge of SU{N)\, confirming the flow, as discussed above and in 5 . In fact, since the left and right movers decouple in the IR limit, the T B A system for the right movers in this limit is obtained merely by removing the terms involving eaL from the equations. T h e resulting system is identical to t h a t of the SU(N) Gross-Neveu model (93); only the asymptotic condition changes from e a o(/? -> oo) —> m a c o s h / J t o eaR{(3 -> oo) —• m a e ^ . T h e T B A system for the left movers is the same, with the replacement /?—•—/?. This close relation is a consequence of the fact discussed in 5 , t h a t the effective field theory for the SU(N)/SO(N) sigma model at 6 = TV in the low-energy limit is t h a t of the SU(N) Gross-Neveu model at negative coupling. T h e sign change changes the sign of the b e t a function, meaning t h a t while t h e GrossNeveu model is an asymptotically-free massive theory, the critical point in the sigma model is stable. In another language, the different signs correspond to marginally-relevant and marginally-irrelevant perturbations respectively. Not surprisingly, the 0{2P)/0(P) x O(P) sigma model behaves in the same fashion. T h e T B A system in (99) applies t o both massive and massless cases. Only the asymptotic conditions differ, as with the SU(N)/SO(N) model. As a consequence, the same CJJV — P2 is obtained for both 6 = 0 and 9 = it. In the massless case, the flow is to a conformal field theory with cm = P, and the equations in t h e IR limit are those of the 0(2P) GrossNeveu model. T h u s indeed the flow is to the 0{2P)\ conformal field theory, confirming the results of 5 . 8
Conclusion
In these lectures I have described some of the techniques of integrability applicable t o two-dimensional sigma models. One technique I have not discussed is the form-factor approach. This allows one to compute large-coupling correlators fairly accurately by expanding them in terms of the massive particle basis. Some recent applications of this approach to the sphere sigma model and Haldane-gap spin chains are given in 6 0 . T h e big question is if other sigma models are integrable. T h e grail in particle physics is probably the CPN models. They have been widely studied, because they are tractable in large N, and they allow instantons. (The models
170
studied above have a parameter N and have instantons, but they are difficult to treat in large N. The reason is that they are matrix fields: the number of fields at large TV grows as N2, not as N.) In particular, the CPN models allowed Witten to conclude that instantons were not important in real-world QCD 28 . It would be very interesting to prove Witten's results directly, instead of relying on large N. Virtually all the symmetric-space sigma models have arisen in various condensed-matter applications, but the grail here is the U(2N)/U(N) x U(N) model. The reason is that in the replica limit N —> 0, this is believed to describe the transition between quantum Hall plateaus 3 . This transition is experimentally realized, and good measurements have been made of critical exponents. These critical exponents should arise in some conformal field theory, but it is still not known which one. Solving the sigma model as a function of TV would presumably solve this problem. So why are sigma models integrable? In some sigma models (see e.g. 61 ' ) , one can find non-local conserved currents. In this context, non-local means that the conserved currents do not have integer spin under the 1 + 1 dimensional Lorentz group or the two-dimensional rotation group. The best known example of such a current is supersymmetry, where the currents are of dimension 3/2. More generally, these non-local currents are often associated with quantum-group or Yangian symmetry algebras. Although the existence of non-local currents does not prove integrability, it is a good indicator. In the 0(N)/0(N — 1) models, one can prove the the non-local currents of 61 are the generators of an infinite-dimensional symmetry algebra called the Yangian 62 . This proves the integrability of these sigma models. Unfortunately, this result is difficult to extend to other sigma models. So are other sigma models integrable? An old result (see e.g. 29 ) suggested that the only integrable symmetric-space G/H sigma models are those where H is a simple Lie group. The reason is that they found that the non-local conserved currents coming from the classical sigma model (the limit of g small) are not conserved once loop corrections are included. This certainly does not prove the model is not integrable, because it is possible that some or all of the classical conserved currents can be modified so that they are conserved in the full theory. A simple Lie group has only one factor. Thus the symmetric spaces with H simple are 0{N)/0(N-l), SU(2N)/Sp(2N) and SU(N)/SO{N), and the principal chiral models H x H/H. All of these models are indeed integrable. However, the 0(2P)/0(P) x O(P) models are integrable, but H is not simple! Thus the suggestion of 29 is in not true here. It is not clear whether this is a fluke of this model, or other symmetric-space sigma models are integrable
171
as well. It would be most interesting to construct the non-local conserved currents here explicitly, to understand how and they remain conserved even in the full theory. The results described above certainly imply that there is a Yangian symmetry to all known integrable sigma models. In fact, this is the reason for the extra particles in the models with 0(2P) symmetry. The representations of the Yangian of 0(2P) are larger than that of its subalgebra 0{2P). The particles at a given mass are in a reducible representation of 0(2P), but in an irreducible representation of the Yangian. This poses an interesting question: is there any way of telling which representations of the Yangian yield the particles and S matrices for an integrable field theory? And if so, what are these theories? Unfortunately, the technology of Yangians does not seem developed enough yet to answer these questions. However, the results of 59 suggest an alternate approach to the issue of non-local conserved currents in sigma models. In 59,6 , it is conjectured that G/H sigma models can be obtained as the limit of certain perturbed conformal field theories called coset models. Here, one can use standard conformal techniques to determine if non-local currents remain conserved even in the massive theory. For example, it was noted in 59 that there are (at least to lowest order in perturbation theory) conserved non-local currents in the perturbed coset models which give the CPN model in a limit. This is certainly an issue which should be explored in further detail. I think it is a good time to study integrable sigma models. The techniques for exploiting the integrability are definitely progressing. But most importantly, the strong-coupling physics the sigma models describe is extremely interesting. My work is supported by a DOE OJI Award, a Sloan Foundation Fellowship, and by NSF grant DMR-9802813. Appendix A A.l
Kernels and identities SU(N)
One set of kernels I use comes from the prefactors of the S matrices. These kernels are defined as
172
r(N),^
Y£>{P) =
. d
-i-\nFgN<Ji)
^ ( 0 ) = -*—lnF°6(/3). df3 The reason for the extra factor in the definition of A b will become apparent below. The kernel appearing in vector-vector scattering is defined as y W = FX1 . It is most useful to give the kernels in Fourier space. To make the equations look a little nicer, I define the Fourier transform with normalization f(u) = -
/
dw e^V'fiP)
(106)
I use this definition of Fourier transformation for any kernel in a model with SU(N) symmetry. A fact useful for obtaining TBA equations back in rapidity space is that if /(w) = l/cosh(w) then /(/5) = N/(2cosh{NP/2)). For the Gross-Neveu models, by using the S matrices in 22 one finds after some after some manipulation 57
? r M = sab - eH^((*-«)")y(M ao
sinh(iVw) smh(w) for a > b, with Y^b ' = Yba ' To find the kernels Fab appearing in the SU(N)/SO(N) sigma models requires even more work. Using the results of 5 for the S matrices, I find _|u,4cosh(o;)sinh((iV-aV)sinh(M
9{N) •>ab ao
W
) = °*<> ~
e
• u IM
\
sinh(iVa;)
Notice how the Fourier transforms are related: ZaUH") ~ *««. = e " 2 " smh(2Lo)(Y^\uj)
- 8ab).
This relation is useful in proving various identities. The kernel A^ab' arises in several places. The function X ( N ' is the S matrix for the SU(N) parafermion theories, and appears as part of the prefactor in the Gross-Neveu and SU(N)/SO(N) sigma models. A(ab' also arises in the Bethe ansatz diagonalization. It is £(*),, jl
=
2sinh((s - j)u) cosh(cj) sinh(Zu;) sinh(w) sinh(sw)
for j > I, with A\- = AW. Other kernels arising in the Bethe ansatz diagonalization are ^ sinh((s-,>) 3 sinh(sw)
173
in the Gross-Neveu models, and 2sinh((s-jHcosh(w) Hs) T i M = r-T7—x °H (m) J sinh(sw) in the sigma models. Notice that r and a are related via (94). Naively, this seems to imply f- (w) = 2cosh(w)cr)- (w), but this is not quite true. The 5ji appears in (111) after a careful analysis of the Fourier transforms; note that the correct forms vanish as ui -* oo. The inverses of the matrices AV are very useful. By using the Fourier transforms, it is simple to derive the identity
k=i
where 11
2 cosh(w)
where i\-( is the incidence matrix for the algebra SU(s), defined as /];» = Sj,,-! + 8j,,+1
j,l = l...N-l
(113)
More generally, the incidence matrix for a simply-laced Lie algebra is twice the identity minus the Cartan matrix, and is conveniently pictured by the Dynkin diagram. I denote the incidence matrix for SO(2s) as l ' s ' . Other useful identities are
N £4 s ) *^) = ^2;cosh(./V/V2) and s-l
g^-l-'M'S^m Useful identities involving the S matrix prefactors are \
;
M - d„6 = ^41,, ' M |
—— - 1 2 cosh(w)
and 7 W f {u)-5 J; _ A(N),( ,o (JJT%O)_ Ml Z , ) ^ ^ — ^ - l, \j ab ab-Aab
The extra Sj\ in (111) is crucial to obtaining the right identities.
174
A.2
0(2P)
The prefactors for the 0(2P) Gross-Neveu models can be found by using the S matrices given in 25 , while those for the 0(2P)/0(P) x 0{P) sigma ss 5 model can be derived from T {6) in . The kernels are defined as y b and (P) Z\h respectively. The Fourier transform used below is that of (106) with N replaced with 2P — 2. For a,b = 1 . . . P — 2, the Gross-Neveu kernels are closely related to the SU(2P - 2) kernels, namely (, ,\ yv(P) ab K^) -
Y< > (
ab
= 5ab
2p 2
- ) I, ,\ _i_ < > ( 2 p - 2 )
,. ,\ yu) + r2P_2-a b{u) . I cosh((P — 1 — a)to) smh(boj) - e cosh((P — l)u>) sinh(w)
for a > b, with yba = yab . For those involving the spinor representations s and s (the nodes labelled P — 1 and P), the kernels are }(P) _ v( p )
yPP
— yp-\
p-i
- ^< p )
— -ypp-i
- i _ >l
—±
sinh(Pa,) 2sinh(2w)cosh((P- \)LU)
V(-P)=v(p) = r1"1 sinh(Qa;) sap J'oP-i 2sinh(w)cosh((P-l)w) where in the latter a = 1 . . . P — 2. The result of 49 ' 50 for the Bethe equations for 0{2P - 2) says that the equation for the eigenvalue (89) and the first Bethe equation (90) are CO
P
27rPa,0(/3) = ma cosh/? + £
Y(J] * pbfi(P) - £
cr;(oo) * pa,,(/3).
(114)
1=1
6=1
These are virtually identical to those for SU(2P - 2), with Ya(6 replaced by Y P 2) al ~ - I n Particular, the kernel c r ^ " 2 ) is still given by (110). The other Bethe equations are now P
27rpOJ-(/3) = ^
oo
* Pa,0(/3) - J2 E 4-r' * < r ) * PtAP)
(H5)
6=1 /=1
where
^^=(6><-v^r\) Jt
<116>
\ 2cosh(w) I where I\ ' is the incidence matrix for the algebra SO(2P), defined above in (88). The reason for the P—j and P — l indices is that above it was convenient
175
above to define the spinor nodes as 0 and 1, whereas here I have defined them as P and P — 1. The proof of the TBA equations is now basically identical to that done for the SU(N) Gross-Neveu model. The reason is that the kernels here satisfy basically the same identities as the SU(N) case. Namely, one can define the matrix inverse A of /C, just like A is the inverse of K. One finds that
~W•n [xWVDxWfa d
A[Pb\p)
- /?)] = 2n6abS((3) -
Then
(
(oo)
\
Using this and the identities in the first appendix gives the 0(2P) Gross-Neveu TBA equations in (98). For the 0(2P)/0(P) x O(P) models, the proof is the same as for the SU(N)/SO(N) models. The only new identity needed is
IT<«>-«- = ^ < » > ( ^ - i ) From the prefactor given in 5 , it follows that this identity holds for a = b = 1. However, I have not been able to prove it in general. The reason is that the S matrices for particles in the representations 2/is and 2fis are not known explicitly, so it has not been possible to work out the prefactors involving these particles. However, I have checked that if they obey the above identity, then they also are consistent with the massless 6 — TT S matrices and resulting TBA derived in section 4. By consistent, I mean that the perturbative expansion of the free energy is the same for 9 = 0 and 0 = ir. I have also checked this consistency for the energy at zero temperature in a magnetic field, extending the analysis of 5 to the particles in representations fis for the massless case and 2/is in the massive case. References 1. D. Friedan, Ann. Phys. 163 (1985) 318 2. F.D.M. Haldane, Phys. Lett. 93A (1983) 464; Phys. Rev. Lett. 50 (1983) 1153; J. Appl. Phys. 57 (1985) 3359; I. Affleck, Nucl. Phys. B257 (1985) 397. 3. A. Pruisken, Nucl. Phys. B235 (1984) 277
176
4. A.B. Zamolodchikov and Al.B. Zaraolodchikov, Nucl. Phys. B379 (1992) 602 5. P. Fendley, "Integrable sigma models with 9 = TT" , to appear in Phys. Rev. B. [cond-mat/0008372] 6. P. Fendley, "Integrable sigma models and perturbed coset models", [hepth] 7. P. Fendley, "Critical points in two-dimensional replica sigma models," [cond-mat/0006360] 8. E. Witten, Comm. Math. Phys. 92 (1994) 455 9. V. Knizhnik, A. Zamolodchikov, Nucl. Phys. B247 (1984) 83 10. P. Fendley and R. Konik, Phys. Rev. B62 (2000) 9359 [condmat/0003436] 11. A. Altland, B. Simons and M.R. Zirnbauer, [cond-mat/0006362] 12. K. Fujikawa, Phys. Rev. D29 (1984) 285 13. G. 't Hooft, in Recent Developments in Gauge Theory (Plenum 1980). 14. A.B. Zamolodchikov and Al.B. Zamolodchikov, Ann. Phys. 120 (1979) 253. 15. G. 't Hooft, Nucl. Phys. B190 (1981) 455 16. For a review see I. Affleck in Fields, Strings and Critical Phenomena (North-Holland 1988). 17. V. Fateev, E. Onofri, Al. Zamolodchikov, Nucl. Phys. B406 (1993) 521 18. A.B. Zamolodchikov, JETP Lett. 43 (1986) 730 19. P. Hasenfratz, M. Maggiore and F. Niedermayer, Phys. Lett. B245 (1990) 522 20. S. Coleman, Aspects of Symmetry (Cambridge 1985) 21. D. Gross and A. Neveu, Phys. Rev. D10 (1974) 3235. 22. N. Andrei and J. Lowenstein, Phys. Rev. Lett. 43 (1979) 1698; Phys. Lett. B90 (1980) 106. 23. E. Witten, Nucl. Phys. B142 (1978) 285 24. R. Cahn, Semi-Simple Lie Algebras and their Representations, (Benjamin-Cummings, 1984), available for free at http://wwwphysics.lbl.gov/~rncahn/book.html 25. M. Karowski and H. Thun, Nucl. Phys. B190 (1981) 61 26. P. Fendley, "Kinks in the Gross-Neveu model", in preparation. 27. E. Ogievetsky, N. Reshetikhin and P. Wiegmann, Nucl. Phys. B280 (1987) 45 28. E. Witten, Nucl. Phys. B149 (1979) 285. 29. E. Abdalla, M. Abdalla, M. Forger, Nucl.Phys. B297 (1988) 374 30. I. Affleck, Nucl. Phys. B305, 582 (1988). 31. P. Fendley and H. Saleur, "Massless integrable quantum field theories and
177
32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59.
massless scattering in 1+1 dimensions", in the Proceedings of the Trieste Summer School, 1993 and in the Proceedings of Strings 93, Berkeley (World Scientific) [hep-th/9310058] N. MacKay, Nucl. Phys. B356 (1991) 729 M. Karowski, Nucl. Phys. B153 (1979) 244 B. Berg and P. Weisz, Nucl. Phys. B146 (1979) 205; R. Koberle, V. Kurak and J. A. Swieca, Phys. Rev. D20 (1979) 897 P. Kulish, N. Reshetikhin and E. Sklyanin, Lett. Math. Phys. 5 (1981) 393 P. Fendley, "Confinement and oblique confinement in two-dimensional gauge theories", to appear eventually. J. M. Evans and A. J. Mountain, Phys. Lett. B483 (2000) 290 [hepth/0003264]. J. M. Evans and T. J. Hollowood, Nucl. Phys. Proc. Suppl. 45A, 130 (1996) [hep-th/9508141]. J. Balog, S. Naik, F. Niedermayer, P. Weisz, Phys. Rev. Lett. 69 (1992) 873; J. Balog, S. Naik, F. Niedermayer, Phys. Lett. B283 (1992) 282 G. Japaridze, A. Nersesian, P. Wiegmann, Nucl. Phys. B230 (1984) 511 S. Hikami, Phys. Lett. B98 (1981) 208 Al. Zamolodchikov, Int. J. Mod. Phys. A10 (1995) 1125 A.M. Tsvelick and P.B. Wiegmann, Adv. Phys. 32 (1983) 453 C.N. Yang and C.P. Yang, J. Math. Phys. 10 (1969) 1115 Al.B. Zamolodchikov, Nucl. Phys. B342 (1990) 695 H. Blote, J. Cardy and M. Nightingale, Phys. Rev. Lett. 56 (1986) 742 A. Kirillov and N.Yu. Reshetikhin, J. Phys. A20 (1987) 1587 V.V. Bazhanov and N.Yu. Reshetikhin, Int. J. Mod. Phys. A4 (1989) 115 V. Bazhanov and N. Reshetikhin, J. Phys. A A23 (1990) 1477 V.V. Bazhanov and N.Yu. Reshetikhin, Prog. Theor. Phys. Suppl. 102 (1990) 301 P. B. Wiegmann, Phys. Lett. B152 (1985) 209. A.M. Tsvelik, Sov. Phys. JETP 66 (1987) 221 V. Fateev, Al. Zamolodchikov, Phys.Lett. B271 (1991) 91 H. Bethe, Z. Phys. 71 (1931) 205. M. Fowler, X. Zotos, Phys. Rev. B26 (1982) 2519 A. Kuniba, T. Nakanishi, J. Suzuki, Int. J. Mod. Phys. A9 (1994) 5215 [hep-th/9309137]; 5267 [hep-th/9310060]; T. J. Hollowood, Phys. Lett. B320 (1994) 43 [hep-th/9308147]. E. Quattrini, F. Ravanini and R. Tateo, hep-th/9311116. P. Fendley, Phys. Rev. Lett. 83 (1999) 4468 [hep-th/9906036]
178
60. R.M. Konik and P. Fendley, to appear soon. 61. M. Luscher, Nucl. Phys. B135 (1978) 1. 62. D. Bernard, Commun. Math. Phys. 137 (1991) 191.
179
LORENTZ LATTICE GASES A N D SPIN CHAINS
M.J. M A R T I N S Departamento de Fisica Universidade Federal de Sao Carlos Caixa Postal 676, 13565-905, Sao Carlos, Brazil We survey a theoretical framework to study the diffusive behaviour of Lorentz lattice gases. These systems can be mapped on integrable spin chains which allow us to determine the geometrical critical behaviour of the paths. We discuss two classes of such spin chains and study the thermodynamic limit and the finite size properties. The massless regime of one of the models is an example of conformally invariant theories with central charges c = 3 and c = 2. This system also describes coupled Potts models whose low-energy properties hide the N = 1 supersymmetry. The other spin chain is an interesting case of a dimerized spin-orbital model in which the energy gap can be exactly determinated.
1
Introduction
The process of self-diffusion in crystals may be described by repeated scattering of mobile particles by impurities. One of the simplest models in which this type of process occurs is the so-called Lorentz lattice gases, see e.g. refs. l'2. In these models, the lattice sites are occupied by randomly placed scatterers and the particles move along the lattice bonds. One question we are interested to answer is: what is the influence of the collision rules to the macroscopic behaviour of the lattice gas such as the statistical behaviour of long orbits?. In this paper we investigate this question and its ramifications in the case of surface diffusion on a square lattice. We consider the simplest non-trivial scattering rules one can think of: the scatterers are either right and left mirrors or rotators and their collision rules are illustrated in figure 1. Note that the behaviour of a particle hitting a rotator does not behave exactly the same as when it hits a mirror, providing us a simple example in which the dependence of the scattering rules can be studied in details. We shall study this system with toroidal boundary conditions and therefore every particle follows a closed trajectory. Since we are mainly interested in the statistics of such paths it is convenient to assign a fugacity q to every closed orbit. In this way we are able to define a more general system, i.e the corresponding loop model 3 ' 4 whose partition function is Z=
Y, scatter
configurations
w^w^w^w^q*paths
(1)
180
where Wi(w3) and W2{wi) are the weights probabilities of right and left mirrors(rotators) and rii(i = 1,2,3,4) are their respective number in a given configuration. If we set q = 1 we recover the original Lorentz lattice gas model. This formulation is appealing because it allows us to make a connection between (1) and the partition function of solvable vertex models. This approach bears the well known equivalence between the Potts model, the sixvertex model and the XXZ spin chain in the sense that all of them share the Temperley-Lieb (TL) algebra 5 ' 6 . In our case, however, the underlying algebraic structure behind the rotators/mirrors scattering rules is similar to a multi-colored variant of the TL algebra denominated Fuss-Catalan algebras 7 . In the next section we introduce this algebra and show how it can be used to generate integrable manifolds for the weights Wj. We mention that such solvable systems can be seen as finely tuned coupled or decoupled XXZ spin1/2 chains. In section 3 we discuss their exact solution, the thermodynamic limit behaviour and the corresponding finite-size corrections in the massless regimes. In particular, this allows us to determine the scaling behaviour of self-avoiding correlators of the q = 1 Lorentz lattice gas itself. Our conclusions are presented in section 4. We recall that a brief summary of part of our results has appeared in refs. 8 ' 9 . 2
Integrable loop models and spin chains
One way to search for possible integrable manifolds Wi is first to establish the underlying algebraic structure of the mirrors and rotators collision rules. In the absence of rotators, the partition function (1) can be seen as a graphical representation of the <72-state Potts model. This means that the algebra associated with mirrors is the well known TL algebra and this already provides us a clue how to proceed. The rotators do not behave exactly as mirrors and this little difference can be overcome by doubling the number of the TL generators. More precisely, the algebra governing the rotators consists of two commuting TL algebras with alternating generators 8 . If we associate to right and left rotators the elements Ri and Li, acting on sites i and i + 1 of a chain of size L, they satisfy the following relations R* = qRi, L\ = qLu LiRi±\Li
= Li, RiLi±iRi
— Ri,
[Ri,Li} = 0 [Ri,Ri±i] = [L,,Li±i] = 0
[Li, Lj] = [Ri,Rj] = {Rt,Lj] = 0 \i - j \ > 2
(2) (3) (4)
181
The formulation is completed by identifying the left and the right mirrors with the TL operator Et = RiLi and the identity /,, respectively. To make further progress we rewrite the partition function (1) as a trace of a row-torow transfer matrix whose local Boltzmann weights are given in terms of the matrix elements of the operators Ri, Li, Ei and the identity / j . That is, the transfer matrix elements are given by
K\::baLLto)= sca (v>i)s£z to)... s £ - i a i
to)
(5)
while the Boltzmann weights S ^ t o ) are Sliim)
+ u,2El:db + W3Rli
= W,raf
+ mLcadb
(6)
In order to find integrable manifolds one has to solve the Yang-Baxter equation for 5^' 6 with the help of the algebraic relations (2-4). After performing such Baxterization we found three distinct integrable manifolds x given by8 sinh(6> - A) w2{\) - l / w i ( A ) , 103(A) = w 4 (A ) = 1 (7) sinh(A) cosh(6>-A) sinh(6> - A) w 2 (A) = , w 3 (A) == 104(A) -= 1(8) (H) W l (A) = cosh(26/ — A) sinh(A) (I)
Wl (A)
=
(III) «7!(A) =
9 e2X - 1 '
_ 1
^ qi-l-e2*'
iu3(A) = l,w 4 (A) = 0 or io3(A) = 0,w4(A) = 1
(9)
where A is the spectral parameter of the Yang-Baxter equation. For later convenience we also have parametrized the fugacity by q — 2 cosh(#). Let us now pause to comment on the origin of these Yang-Baxter solutions. We first note that solution (I) factorizes into two independent models since we can rewrite it as Si = (w\Ii + Ri)(Ii + Li/wi). Considering that 101 is the parametrization entering in the TL solution it is easy to conclude that solution (I) indeed corresponds to two decoupled six-vertex models. It is possible to show that solution (II), after a canonical transformation, can be mapped onto the A\ vertex model found by Jimbo 10 . One way to see this connection is to observe that the algebra (2-4) allows us to define the braid operator hi = Ri + Li — e6'Ei — e~eIi which together with its inverse and the monoid Ei satisfy the so-called the Birman-Wenzel-Murakami(BWM) algebra n . Taking this into account it is not difficult to verify that the eingenvalues of the braid 1 Here we excluded the case of null rotators weights since this will lead us to the standard TL solution.
182
operator lies precisely on the curve of the BWM algebra that produces the A\ solution 12 - 13 . The third solution is apparently genuine of the mirror/rotator algebra and it has a number of unusual properties: the asymptotic braid limits Si{\ = ±00) are not invertible and for special values q = ± 1 we find a connection to a singular point of the Hecke algebra 2 . Finally, we remark that this solution appeared originally in integrable coupled Potts models 15 and recently has been also rediscovered in the context of Fuss-Catalan algebras 16
Next we turn to the quantum spin chains associated with the above loop models. For even lattice sizes these Hamiltonians are better represented in terms of two commuting TL operators 171(a) E
i
+ , Q —,Q =(J
i
a
i+l
,
—,Q +,a , COSXl^t7j ,
+ai
CJ
i+l+
2
^
i
z a
z,a\ ai
+^
, SHin[o) +
2
z a
(ai+l~ai
z,a\
>
(10) where \ai 'a,a*'a}, a = 1,2 are two commuting sets of Pauli matrices. In terms of these operators we find that the Hamiltonians are
HU(q) = -J2[E^+E^} 8
2
q2 - 2
(11) (12)
l
L/2
H <™> = - £ [Eg + E%U] /q-Y: E^E^/tf i=l
- 2)
(13)
i=l
In order to assure that closed loops on the cylinder pick up the correct Boltzmann weights we still need to set the appropriate boundary conditions for these Hamiltonians. This is accomplished by imposing twisted boundary conditions defined by ^L+l =
e
a
\
> aL+l ~ a\
(14)
where ip^ are appropriate angles. For instance, if we impose periodic boundary conditions on the loop models one needs to set ip^ = —2i6 in the corresponding spin chains. 2
This is seen by defining the operator Of1 — Ri =F Ei for q = ±1, respectively. Then it is easy to verify that these operators satisfy the Hecke relations \Oi ] 2
ofof±1of = 0.
183
Clearly, we see that model (I) corresponds to two decoupled XXZ spin chains and the results concerning this system can be read off from previous work in the literature, see e.g ref. 14 . The remaining systems can be seen as coupled spin-1/2 chains and they will be investigated in the next section. 3
Thermodynamic limit and finite-size behaviour
The purpose of this section is to study the infinite volume properties and the large L behaviour of massless regimes of models (II) and (III). We begin by describing the Bethe ansatz solution of model (II) and its possible critical universality classes. Exact results for model (III) are obtained by establishing the corresponding inversion relation. 3.1
Model (II)
The Bethe ansatz solution of this system can be related to that of the A\ vertex model 17 . Adapting this solution to include toroidal boundary conditions we find that the eigenvalues of H^IV> (q) are given by 8 E^(L)
= -t
25inh2W
(15)
(1) 1)
^cosh(e)-cosh(A^ ) and the corresponding Bethe Ansatz equations are sinh(A}-72 - 0/2)L l(l)/o OI^ r. I i n w i ( l ) „ _ x(l) iW7 2 + 0/2) _sinh(A). i i 1 s m h ( A ; i ) / 2 - A i 1 ) / 2 + 0) ^ Pl
sinh(A<.1) - A<2) + 6)
"gs^^M^-
] smh{Xf - A<;2) + 20) v i 0i(V>i-2) TT (2) i(2) jfj sinh(Af - \\<> - 26)
i-1
ri
'-'
pr sinh(A(.2) - A<x) + 6)
The numbers ra labels the many possible disjoint sectors of the Hilbert space of HW(q). They are related to the eigenvalues n ' a ' of the number a of spins JV< > = £ f = 1 o\>a by n = n™ + n^ and r 2 = n*1* or r 2 = n<2>. The Bethe ansatz solution allows us to investigate the thermodynamic limit
184
of .ff^n)(q) and the massless region lies in the region q € [2, —2]. If we restrict ourselves to positive values of the fugacity we find two possible massless regimes. By setting 6 = i-y they are given by 3 (A) 0 < 7 <
TT/4
(B)
TT/4
< 7 <
TT/2
(18)
In the context of the loop models these two regimes are associated with distinct behaviour of the weights probabilities u>j. In fact, in regime (B) all the weights can be made positive while in regime (A) either w^ or w4 is necessarily negative. In terms of the Bethe ansatz equations, this means that we have two different behaviours for the roots X"1' that describe the ground state and the low-lying excitations of H^(q). More precisely, we find that regime (A) has three independent branches of roots
Af) = £f
(19)
while regime (B) has the following complex root structure A ^ = $» ±
.(TT/2
- 7) + 0(e~L),
xf
= tf} + in/2
(20)
where £;. are real numbers. In order to study the thermodynamic limit we substitute the roots ansatz (19,20) in the Bethe ansatz equations and after taking the L —> oo we end up with coupled integral equations for the density of roots of the rapidities (a)
&
. These equations are solved by using standard Fourier transforms and
from equation (15) the ground state energy per site ea» is calculated to be sinh ^ - 7)*] riffrff ~ sinh2M e(n,A) = _ 4 s i n ( } f dx (21) TO J0 sinh[7ra;] sinh[47a;] egf-B) 00
=
_4sin( 7 ) [°° d / i n h [ ( , - 7)x]smh[( 7 r/2 - 7 ) , ] K "J0 sinh[7ra;] sinh[(27 - ?r/2)a;]
v
;
We now turn to the study of the finite-size corrections of the ground state energy Egs (L). In the regime (A) all the Bethe ansatz roots Qa' are real for finite L and we can use a method introduced by De Vega and Woynarovich 18 to estimate the large L behaviour. Here we shall omit technical details and 3 Positive (negative) values of q correspond to the antiferromagnetic (ferromagnetic) regime due to the property H ( " ) ( 7 ) = —H^11\-K — 7). We further remark that for q > 2 the excitations of model (II) are massive.
185
present only the final results. We find that the ground state in the presence of seams tp^ behaves as &9S
V-kJ _ „(II,A)
L
6
°
7TfS
~6L2
(1 - 27/TT) l v 2TT
7
v
2?r
v
2TT
2TT ' 2TT
where v s = 7r ""^ 7 ' is the sound velocity of the three possible branches of massless excitations. Comparing this result with the behaviour of the eigenvalues of a conformally invariant system on a strip of size L 20 ' 21 one can read off the central charge of the loop model in regime (A). By substituting ip^ = 2-y = 2n/p we obtain c(n,A) = 3 ( 1
_ _ 8 ^ 4 (23) P(P - 2) Interesting enough this result is double of the conformal anomaly of the N = 1 supersymmetric minimal models 19 . Therefore, in regime (A) there exists a hidden supersymmetry and the system cannot be understood in terms of weakly coupled Heisenberg chains. We can also use the same approach to investigate the finite-size scaling of the excitations which will lead us to determine the conformal dimensions of the primary operators. In this regime the excited states are characterized by the spin-wave vector n = (711,712,713) and by the vorticity vector m = (7711,7712,7723) corresponding to vacancies in the ground state sea and multi-string configurations of the three inde(a)
pendent rapidities £• , respectively. Denoting the corresponding weight by ^ r i T n T ^ ' ' ' ' ^ 2 ' ) anc ^ a g a m omitting technicalities we find that such conformal dimension is given by
< : n ~ 3 ) ^ ( 1 ) ^ ( 2 ) ) = £ m < W 4 + *<#*,•
(24)
where z = (mi + ^ - , m 2 + ^—^—,7773 + \—) and the matrix elements Cij are C=
/2(1-7/TT) -(1-27/TT) V -2 7 /7T
-(1-27/7T) 2(1 - 27/TT) _(l-27/TT)
-27/TT \ - ( 1 - 2-y/Tr) 2(1 - 7 / T T ) /
(25)
We remark that the integer numbers 71$ and mi are not arbitrary but satisfy certain constraints typical of nested Bethe ansatz equations, e.g. see
186
ref. 2 2 . Here such restrictions can be determined by noticing that the isotropic limit 7 -> 0 of model (II) corresponds to the integrable 5(7(4) exchange spin chain 2 3 . This information can be used to find the appropriate numbers m and rrii in Eq.(24) that should reproduce, in the isotropic limit, the known operator content of the SI/(4) spin chain 22 . Taking this into account we find that the lowest spin-wave and vorticity excitations should be associated with the sectors n = (1,2,1) and m = (1,0,0) or m = (0,0,1), respectively. For example, in the SU(4) limit these excitations correspond to primary operators with dimension 1 and 3/4, respectively. One possible application of these results is the computation of the anomalous dimension Xt of the "energy" operator of the loop model. This dimension is associated with the gap between the two lowest lying levels in the ground state sector n = (0,0,0). Considering the true ground state of the loop model we find Xt = x g $ > ( 2 7 > 2 7 ) - X a g ( 2 7 > 2 7 ) = f £ ± §
(26)
which is precisely double of the conformal dimension of the energy operator /i2,i °f the supersymmetric minimal models 19 . We now turn our attention to the regime where the Lorentz lattice gas sits. In this case the roots A^ are complex for finite L and we have studied the finite-size corrections numerically. In table 1 we present our estimates for the effective central charge for some values of the angles xjj^ and 7 = 7r/3. We observe that the finite-size corrections of the ground state is identical to that found for two decoupled XX Z spin chains 14 with twisted boundary conditions V' 1 ' and V^ 2 \ namely Els
'(L) _ L
(IIB) =
°°
_TTV. 6L
6
( '^ + (1 - 7/71-);LV [2TT
v
2TT
(^]
(27)
where in this regime vs = Z sl"l/4) • This behaviour has also been checked for other values of 7. By substituting ijj^ = 2n/p, the conformal anomaly of the loop model in regime (B) is c(ii,B) =
2(1 _
6
) 2< p<4 (28) P(P ~ 1) which corresponds to the one of two decoupled minimal models. Similar decoupling is observed in the finite-size corrections of the low-lying excitations and consequently the coupling becomes asymptotically irrelevant in regime (B). We recall that this is the same scaling behaviour found for manifold (I). The fact that our findings are the same for quite distinct manifolds (I) and (II) strongly suggest that w3 = w4 should be a critical surface in
187
the phase space of the Lorentz lattice gas model. In particular, one can use the above results to determine the scaling of the correlators measuring the probability that two or more points on the lattice belongs to the same loop 24 25 ' . Let rii +7i2 be the number of such loop segments, then we find that the anomalous dimension X n i „ 2 for q = 1 is 8 (l-^n,,0) ^7li,7l 2 — 2^,
12
(29)
Now we reached a point in which we can answer the question posed in the introduction. A single path should be governed by the lowest conformal dimension X i o = Xo,i = 1/4, leading us to the fractal dimension df = 7/4. This is the same fractal dimension of percolation 25 and yet the same value found in numerical simulations of a fully occupied lattice of either mirrors or rotators 26 . However, the situation changes when one considers the scaling properties of multi-loops. For instance, the second lowest dimension in the case of mixed rotator-mirror model is X^i = 1/2 which leads us to a fractal dimension dj = 3/2. This value is double of that expected when only mirrors are present since this system follows the percolation universality class 25 . This means that a single particle cannot distinguish the rotators and the mirrors in a full lattice, but two particles in different orbits are capable of making such distinction. Therefore only the scaling behaviour of one loop orbits is expected to be universal. Probably this is enough when one is concerned with numerical simulations and the corresponding continuum limit of these systems. 3.2
Model (III)
We first observe that this system consists of a coupled spin-1/2 chains in the presence of a dimerization background. This is best seen in the limit q —>• 2 where it is possible to rewrite the model as
i=l
a=l,2 L
- 2£(S? ) .S£ ) 1 ) x (SP.SH\) - 3 i / 8
(30)
i=l
where S i , a — 1,2 are half of commuting sets of Pauli matrices (spin-1/2 operators).
188
To our knowledge this is a rare example of interacting and dimerized spin chain where exact results can in principle be derived due to its integrability. The presence of dimerization induces extra difficulties and, for example, the coordinate Bethe ansatz solution of H^ul^ (q) has eluded us so far. However, we can still make further progress on this problem thanks to the inversion relation trick 27>28>29. We shall see that this approach will provide us the means to compute the ground state and the excitations properties at least for q > \[2. Let us describe how to implement such important tool for model (III) 9 . The first step is to notice that, for even L, the corresponding row-to-row transfer matrix can be written in the usual way T(m>(A) = TrA[C<£A(\)C™_liAW £jA(\)
• • • Cil\(X)C^(X)]
denotes the local Boltzmann weights.
(31)
Defining / ^ " ^ ( A )
=
a
Pi,i+iRl i+1(X) where Pi,i+i is the four dimension space exchange operator, the i?-matrix is given by p(«)
(\\-T
l
|
p(") , ^ z ( A )
i? M+1 (A) - / M + 1 + ^ £ ,
+ ^ E
(1) {
(2)
Et
,
,
(32)
The inversion identity follows from a combination between the unitary property, C\a'2,(\)C2 i (—A) = 1^2, and a "mixed" crossing relation for the weights £ j 2'(A). These operators are not independently crossing symmetric, instead they satisfy the following relations 4?2>(A) = ^
^
M [c[3r]}t2(S
- A) M, a = 1,2
(33)
where S = ln[q2 — 1] and tk denotes the transpose on the space k. The matrix elements of M are M=
0 0 0
0 0 ". ee
29
0
0 ee
"0 0
1 0 0 0
04)
The crossing relation (34) together with unitarity leads us to the following identity for the transfer matrix T(iii)(A)T(iii)(A +
(5)
UJI
{5 + A)
W
A
L
ld + 0(e~L)
(35)
L i(- ) J where the last term denotes exponentially small matrix elements. In the infinite volume limit this term vanishes, providing us a functional relation for
189
the transfer matrix eigenvalues. In particular, the largest eigenvalue Ags (A) per site satisfies
K1s1)WA^(S + X) = ^(±^-
(36)
and due to unitarity we also have A(I.")(A)A(»I)(-A) = 1
(37)
It is possible to solve such functional equations under plausible analyticity assumptions in the region q > 1. The solution is given by
If we recall that the ground state energy per site eio .
A
(III) ,
to A^s ' by eio e,
~
d i n A ™ (A).
(III)
=
^
\x=o, we obtain
2-j ( e (2j+l)(5 _ 1)(e2;<5 _ 1) 2615
is directly related
, , .
2 ^ ( e ( 2 j + l)<5 - l)( e (2>+2)<5 _ 1)
e2S
(39) e - 1 + e~2* -^1 Before proceeding it is wise to check this result and its physical content. In order to do that we have numerically diagonalized Hamiltonian H^lI>(q) up to L = 16 sites by using a Lanczos type algorithm. In table 2 we exhibit our numerical results for the ground state energy as well as their extrapolations towards the thermodynamic limit. We find that these results are in good agreement with the exact values for q > \[2. In the region 1 < q < y/2, however, the numerical results tell us that the exact value (39) corresponds to the ground state of -i/^ 111 ' (q) instead. For this reason our next result for the excitations should be restricted to the regime q > \f2. It turns out that the inversion relation imposes stringent constraint to the form of the low-lying excitations 2 9 . From (36) and standard trigonometric periodicity we conclude that the next-largest eigenvalue should be described in terms of meromorphic functions with periods 25 and 2-KI. Following ref.29, this observation permits us to calculate the dependence of the energy excitations e(p) on the momenta p. The result we found for the dispersion relation is d
e(p) = ^-^L^j{l-k)2+Aksm2{p)
(40)
190
where K(k) and K (k) are periods of elliptic function of modulus k satisfying KLy = |-. From this result we conclude that the elementary excitations are massive whose gap value is A(III) =
2K(k)(l-k)
=
~
-(l_e-^/2)^2
(1 + e-J'/ 2 )
(41)
In conclusion, for q > %/2 the system has a finite correlation length and the geometrical correlators should decay exponentially at large distances. Preliminary results in the region 1 < q < y/2 suggest that model (III) is critical but we have not yet identified its universality class. 4
Concluding Remarks
Motivated by the study of motion in random environments, we have discussed integrable models whose building blocks are TL operators and therefore they can also be viewed as coupled g 2 -state Potts systems. Such generalized Potts models were introduced long ago by Domany and Riedel 30 to model the adsorption of molecules on a crystal surface. Since recently there has been renewed interest on this subject, see e.g. refs. 31 ' 32 , it is worthwhile to rephrase some of our results in this context. ^From eq.(12), we see that model (II) may be interpreted as coupled self-dual g 2 -state Potts models by a finely tuned energy-energy interaction. For instance, when two 3-state(4-state) Potts models are coupled in such a way, the critical properties of the resulting theory is governed by two coupled c = l(c = 3/2) N = 1 supersymmetric field theories. These are clear examples in which one cannot go continuously from weak to strong coupling due to hidden symmetries in between. Acknowledgements I would like to thank B. Nienhuis for collaborating on refs. 8 ' 9 and C. Ahn, C. Rim and R. Sasaki for organizing such a nice conference and the hospitality of the APCTP. I would also like to thank the hospitality of the Institute of Advanced Study, Princeton, where part of this work was carried out. This work was partially supported by Brazilian Agencies Fapesp and CNPq. References 1. T.W. Ruijgrok and E.G.D. Cohen, Phys.Lett.A 133 (1989) 415 2. J.M.F. Gunn and M. Ortuno, J.Phys.A. 18 (1985) L1035
191
3. R.J. Baxter, S.B. Kelland and F.Y. Wu, J.Phys.A:Math.Gen. 18 (1985) L1035 4. B. Nienhuis, Int.J.Mod.Phys.B 4 (1990) 929; B. Nienhuis, S.O. Warnaar and H.J. Blote, J.Phys.A:Math.Gen.26 (1993) 477 5. H.N.V. Temperley and E. Lieb, Proc.R.Soc.A 322 (1971) 251 6. A.L. Owczarek and R.J. Baxter, J.Stat.Phys. 49 (1987) 1093; V. Pasquier, Nucl.Phys.B 285 (1987) 162 7. D. Bisch and V. Jones, Inv.Math.128 (1997) 89 8. M.J. Martins and B. Nienhuis, J.Phys.A:Math.Gen.31 (1998) L723 9. M.J. Martins and B. Nienhuis, cond-mat/0004238 10. M. Jimbo, Commun.Math.Phys. 102 (1986) 537 11. J. Birman and H. Wenzel, Trans. A.M.S. 313 (1989) 249; J. Murakami, Osaka J.Math. 24 (1987) 745 12. V. Jones, Comm.Math.Phys. 125 (1989) 459 13. M.-L. Ge, Y.S. Wu and K. Xue, Int.J.Mod.Phys.6A (1991) 3735 14. F.C. Alcaraz, M.N. Barber, and M.T. Batchelor, Phys.Rev.Lett. 58 (1987) 771; Ann.Phys. 182 (1988) 280 15. H. Au-Yang and J.H.H. Perk, Inter.Mod.Phys.A 7 (1992) 1025 16. P. Di Francesco, Nucl.Phys.B 532 (1998) 609. 17. N.Yu. Reshetikhin, Lett.Math.Phys. 14 (1987) 235 18. H.J. de Vega and E. Lopes, Nucl.Phys.B 362 (1991) 261 19. Z. Qiu, Nucl.Phys.B 270 (1986) 205 20. J.L. Cardy, Phase Transitions and Critical Phenomena vol 11 edited by C. Domb and J.L. Lebowitz (Academic, New York, 1987 ) 21. H.Blote, J.L. Cardy and M.P. Nightingale, Phys.Rev.Lett. 56 (1996) 742; I. Affleck, ibid. 56 (1996) 746 22. F.C. Alcaraz and M.J. Martins, J.Phys.A.Math.Gen. 23 (1990) L1079 23. B. Sutherland, Phys.Rev.B 12 (1975) 3795 24. B. Nienhuis, Phase Transitions and Critical Phenomena vol 11 edited by C. Domb and J.L. Lebowitz (Academic, New York, 1987 ) 25. B. Duplantier, J.Phys.A 19 (1986) L1009; B. Duplantier and H. Saleur, Phys.Rev.Lett. 59 (1987) 539; Nucl.Phys.B 290 (1987) 291 26. R.M. Ziff, Phys.Rev.Lett. 56 (1986) 545; Meng-She Cao and E.G.D. Cohen, J.Stat.Phys. 87 (1997) 147 27. Yu.G. Stroganov, Phys.Lett.A 74 A (1979) 116; R.J. Baxter, J.Stat.Phys. 28 (1982) 1 28. P.A. Pearce, Phys.Rev.Lett. 58 (1987) 1502 29. A. Kliimper, J.Phys.A:Math.Gen. 23 (1990) 809; Europhysics Lett. 9 (1989) 815 30. E. Domany and E.K Riedel, Phys.Rev.Lett. 40 (1978) 561; Phys.Rev.B
192 19 (1979) 5817 31. A. LeClair, A. Ludwig and G. Mussardo, Nucl.Phys.B 512 (1998) 523 32. V. Dotsenko, J.L. Jacobsen, M.A. Lewis, M. Picco, Nucl.Phys.B 546 (1998) 505
L 8 16 24 32 40 Extr.
Ipl =4>2 = 7r/3
Ipl = tp2 = 7r/4
tpl = 7T/3, 1p2 = 7r/4
1.55667 1.51857 1.51021 1.50683 1.50506 1.5001(2)
1.78890 1.63017 1.73159 1.72737 1.72516 1.7186(2)
1.67253 1.63017 1.62083 1.61705 1.61507 1.6092(2)
Table 1. Conformal anomaly estimators {—Eg3 ' '/L + e^c'
L 4 6 8 10 12 14 16 Extrap. Exact
q = 2cos(7r/5) -3.895 078 -3.601 417 -3.509 362 -3.468 787 -3.447 357 -3.434 672 -3.426 548 -3.400 939 -3.400 9298
q = 2cos(7r/7) -2.521 191 -2.357 366 -2.306 182 -2.283 681 -2.271 822 -2.264 814 -2.260 332 -2.246 275 -2.246 2710
)§L2/(-KVS)
for 7 = 7r/3.
q=2 -2. -1.885 455 -1.849 743 -1.834 071 -1.825 822 -1.820 954 -1.817 844 -1.808 126 -1.808 1265
Table 2. Finite size, extrapolated and exact results for the ground state energy of model
194
Figure 1. Scattering rules for (a) right mirrors; (b) left mirrors; (c) right rotators; (d) left rotators on the square lattice .
195 Q U A N T U M CALOGERO-MOSER MODELS FOR A N Y ROOT S Y S T E M RYU SASAKI Yukawa Institute
for Theoretical Physics, Kyoto Kyoto 606-8502, Japan E-mail: ryu6yukawa.kyoto-u.ac.jp
University,
Calogero-Moser models and Toda models are best known examples of solvable many-particle dynamics on a line which are based on root systems. At the classical level, the former (C-M models) is integrable for elliptic potentials (WeierstraB p function) and their various degenerate limits. The latter (Toda) has exponential potentials, which is obtained from the former as a special limit of the elliptic potential. Here we discuss quantum Calogero-Moser models based on any root system. For the models with degenerate potentials, i.e. the rational with/without the harmonic confining force, the hyperbolic and the trigonometric, we demonstrate the following for all the root systems: (i) Construction of a complete set of quantum conserved quantities in terms of a Lax pair, (ii) Liouville integrability. (iii) Triangularity of the quantum Hamiltonian and the entire discrete spectrum, (v) Algebraic construction of all excited states in terms of creation operators. These are mainly generalisations of the results known for the models based on the A series, i.e. su(N) type, root systems. Solution methods of quantum Calogero-Moser models are expected to give some clues for understanding dynamical symmetries, non-perturbative methods for field theories, etc.
1
Introduction
Calogero-Moser models 1 ' 2,3 are one-dimensional multi-particle dynamical systems with long range interactions. They are integrable at both classical and quantum levels. In this lecture we consider quantum models. The simplest example is described by the following Hamiltonian with r degrees of freedom:
j=l
jjtk
KH}
HK
>
in which g is a real coupling constant (assumed to be g > 1 for simplicity) and q = (qi,-..,qT) are the coordinates and p = {pi, • • • ,pr) are the conjugate canonical momenta obeying the canonical commutation relations: [Qj,Pk] = iSjk,
[qj,Qk] = \pj,Pk] = 0,
The Heisenberg equations of motion read q3 =i[H,qj]
=Pj,
j,k = l,...,r.
(1.2)
196
Qj = P] = i[H,Pj] = 2g(g - 1) ^
——
* ^ j
(1.3)
(9j ~ 9 f c ) 3 '
T h e repulsive 1/(distance) 2 potential cannot be surmounted classically and t h e relative position of t h e particles on t h e line is not changed during the time evolution. It is remarkable t h a t these equations of motion can be expressed in a matrix form (Lax pair) i[H, L] = ^ = LM-ML at
gj=Pj,
= [L, M]
w = 2g(g-i)s; t o _ 1 g t ) „
(1.4)
k±i in which L and M are defined by /
Pi
J£_
'.9
P2
L = ig
\
91-92 92-9r
_Ja_
Pr
92
(1.5) )
and /
j.9
V
«9
m2
(92-91)2
M =
»9 \ (9l"9r)2 \
1.9 (91 ~ 9 2 ) 2
mi
(92-9.)2
mr
«.9
(9.-91)*
(9r-92)2
(1.6)
J
The diagonal elemeiit nij is given by rtij = ig
(1.7)
2 M i (Qj ~ Qk) '
T h e matrix M has a special property r
r
J=I
A=I
(1.8) It is straightforward to derive quantum sum of powers of Lax matrix L:
[H,Qn} = 0,
conserved
Qn = Ts(Ln) = J2(Ln)jk, j,k
quantities
from the total
n = l,...,
(1.9)
197 since
= J2 ({Ln)jiMtk - Mje(Ln)ek)
= 0,
(1.10)
in which (1.8) is used at the last equality. It is elementary to verify that the Hamiltonian is equivalent to Q2, % ex Q2 + const. In other words, the Lax matrix L is a kind of "square root" of the Hamiltonian. These are fundamental results for the integrability of the model (1.1). Similar integrable quantum many-particle dynamics are obtained by replacing the potential in (1.1) by sinh'2(qj-qk)
1 (Qi - QkY
' • ,
(1.11)
inn.
1
2
sin' (qj-qk)
The quantum equations of motion are again expressed by Lax pairs if the following replacements are made: 1 ili ~ Qk)
l
coth(qj - qk) in L,
-J-
cot(qj - qk)
sinh2(qj-qk)
1 (Qj
-Qk)2
in M.
-> < „
sin2(qj-qk)
(1-12) Since the property (1.8) holds in these cases, we obtain quantum conserved quantities in the same manner as above (1.9) for the models with the trigonometric and hyperbolic interactions. The integrability or more precisely the triangularity of the quantum Hamiltonian was first discovered by Calogero1 for the model with inverse square potential plus a confining harmonic force and by Sutherland 2 for the particles on a circle with the trigonometric potential. Later classical integrability of the models in terms of Lax pairs was proved by Moser.3 Olshanetsky and Perelomov4 showed that these models were based on AT-\ root systems, i.e. qj — qk = a • q and a is one of the root vectors of Ar_\ root system (A.6). They also introduced generalisations of the models 5 based on other root systems including the non-crystallographic ones.
198
In this lecture we discuss quantum Calogero-Moser models with degenerate potentials, that is the rational with/without harmonic force, the hyperbolic and the trigonometric potentials based on any root system. We demonstrate that various results known for the quantum Ar models can be generalised universally to the models based on any root systems as well. 6 ' 7 ' 8 ' 9,10 They are (i) Construction of a complete set of quantum conserved quantities in terms of quantum Lax pairs and other methods, (ii) Universal proof of quantum Liouville integrability for the rational, hyperbolic and trigonometric potential models. Namely, these quantum conserved quantities commute among themselves, (iii) Triangularity of the quantum Hamiltonian for all the models. In other words, the Hamiltonian is demonstrated, in certain bases, to be decomposed into a sum of finite dimensional triangular matrices. Thus any eigenvalue equation can be solved by finite steps of linear algebraic processes only. This also gives the entire discrete spectrum of the models, (iv) Equivalence of the quantum Lax pair method and that of so-called differentialreflection (Dunkl) operators. 11 (v) Algebraic construction of all excited states in terms of creation (annihilation) operators for rational models with harmonic confining force. For the classical Liouville integrability see Ref.12 For the Ar models, the Lax pairs, conserved quantities and their involution were discussed by many authors with varied degrees of completeness and rigour, see for example Refs. 5 ' 1 3 - 2 2 The point (iv) was shown by Wadati and collaborators 20 and point (v) was initiated by Perelomov14 and developed by Brink and collaborators 19 and Wadati and collaborators. 20 A rather different approach by Heckman and Opdam 23 ' 24 to Calogero-Moser models with degenerate potentials based on any root systems should also be mentioned in this connection. For physical applications of the Calogero-Moser models with various potentials to lower dimensional physics, ranging from solid state to particle physics and supersymmetric Yang-Mills theory, we refer to recent papers 7 ' 8 ' 9 and references therein. This lecture is organised as follows. In section two, quantum CalogeroMoser Hamiltonian with degenerate potentials is introduced as a factorised form (2.5). Connection with root systems and the Coxeter invariance is emphasised. Some rudimentary facts of the root systems and reflections are summarised in Appendix A. A universal Coxeter invariant ground state wavefunction and the ground state energy are derived as simple consequences of the factorised Hamiltonian. In section three we show that all the excited states are also Coxeter invariant and that the Hamiltonian is triangular in certain bases. Complete sets of quantum conserved quantities are derived from quantum Lax operator L in section four. Instead of the trace, the total sum of Ln is conserved. That is Ts(L") = H Mil/e 7j (£");«>, l n which TZ is a set
199 of R r vectors invariant under the action of t h e Coxeter group. They form a single Coxeter orbit. T h e details of the complete set for each root system are given in Appendix B. In section five t h e creation and annihilation operators for t h e rational models with harmonic force are derived. In section six, t h e equivalence of the Lax pair operator formalism and t h e so-called differentialreflection (Dunkl) operators is demonstrated. Another form of the q u a n t u m conserved quantities are given in terms of t h e differential-reflection (Dunkl) operators. In section seven an algebraic construction of excited states in terms of t h e differential-reflection (Dunkl) operators for rational models with harmonic force is presented. T h e complete sets of explicit eigenfunctions for t h e rank two models are derived. Section eight gives a universal proof of t h e Liouville integrability for models with rational (without the confining force), hyperbolic a n d trigonometric potentials. T h e final section is for summary, comments and outlook. 2
Quantum Calogero-Moser Models
A Calogero-Moser model is a Hamiltonian system associated with a root system A of rank r, which is a set of vectors in R r with its s t a n d a r d inner product. A brief review of t h e properties of t h e root systems and t h e associated reflections together with explicit realisations of all t h e root systems will be found in t h e Appendix A. 2.1
Factorised
Hamiltonian
T h e dynamical variables of the q u a n t u m Calogero-Moser model are t h e coordinates {qj} and their canonically conjugate m o m e n t a {pj}, with the canonical commutation relations: [qj,Pk]=iSjk,
[qj,qk} = \pj,Pk] = 0,
These will be denoted by vectors in R q = (qi,...,qr),
j,k = l,...,r.
(2.1)
r
P = (Pi, • • • ,Pr)-
(2.2)
T h e m o m e n t u m operator pj acts as differential operators Pi = ~ i ^
i = l,-,r.
As for t h e interactions we consider only t h e degenerate potentials, t h a t is t h e rational (with/without harmonic force), hyperbolic and trigonometric poten-
200
tials: i/(p-g)2, V(p q) = t a 2 /sinh z2 a(p a(p • q), [ a 22/sin 22 a(p • q), q),
type I, type type II, II, type III III, type
p £ A,
(2.3)
in which a is an arbitrary real positive constant, determining the period of the trigonometric potentials. They imply integrability for all of the CalogeroMoser models based on the crystallographic root systems. Those models based on the non-crystallographic root systems, the dihedral group l2(m), H$, and H4, are integrable only for the rational potential. The rational potential models are also integrable if a confining harmonic potential ^wV,
w>0,
typeV,
(2.4)
is added to the Hamiltonian. Since we will discuss the universal properties and solutions applicable to all the interaction types as well as those for specific interaction potentials, let us adopt the conventional nomenclature for them. We call the models with rational, hyperbolic, trigonometric and rational with harmonic force the type I, II, III and V models, respectively. (Type IV models have elliptic potentials which will not be discussed in this lecture.) The Hamiltonian for the quantum Calogero-Moser model can be written in a 'factorised form': «,
1v- (
V
®w\
-^{"-'oi)\ ' iA/2
'y'
(
P
2
fdw\ \
+
.dW\
'^)-
,„„,
(2 5)
'
2
i^d w
+ l T.9W(llM-VW2V{p-q)
+ (Y«2)-<*
(2-7)
PSA +
It should be noted that the above factorised Hamiltonian (2.7) consists of an operator part %, which is the Hamiltonian in the usual definition, and a constant £Q which is the ground state energy to be discussed later:
n = H-£0,
(2-8)
^ = \P2 + \ E PSA+
9\P\{9\P\ - 1)H 2 V(p • q) + ( y q2).
(2.9)
201
The real positive coupling constants <7|p| are defined on orbits of the corresponding Coxeter group, i.e. they are identical for roots in the same orbit. That is, for the simple Lie algebra cases one coupling constant g\p\ = g for all roots in simply-laced models and two independent coupling constants, 9\P\ — 9L for long roots and g\p\ = gs for short roots in non-simply laced models. For the him) models, there is one coupling if m is odd, and two independent ones if m is even. Let us call them ge for even roots and g0 for odd roots. The H3 and #4 models have one coupling constant g\p\ = g, since these root systems are simply-laced. It should be noted that the operator part of the Hamiltonian % is strictly positive for g\p\ > 1, which we will assume throughout this lecture for simplicity of the arguments. Throughout this lecture we consider the coupling constants at generic values. The simplest way to introduce the factorised form is through supersymmetry, 10 ' 25 in which function W is called a superpotential:
Wiq)=
^ P€A
gM\n\wip-q)\
+ i--q2),
gM > 1,
w > 0.
(2.10)
+
The potential V{a) (2.3) and the function w(u) are related by V U
^ '
=
d luX^U''
dwiu) du 'W^U'
=
X U
^ '' 0 rational, Viu) = —yiu) = x2iu) + a2 x ^ - 1 hyperbolic, 1 trigonometric. The following Table I gives these potential type rational I& V hyperbolic II III trigonometric
(2.11) (2.12)
functions for each potential: ar(u) w(w) yiu) u 1/u -1/u2 sinh au acoth au -a2 / sinh2 au a cot au -a2/ sin2 au sin cm
Table I: Functions appearing in the Lax pair and superpotential. For proofs that the factorised Hamiltonian (2.5) actually gives the quantum Hamiltonian (2.7) for any root system and potential see Refs. 5 ' 13 ' 10 It is easy to verify that for any potential Viu), the Hamiltonian is invariant under reflection of the phase space variables in the hyperplane perpendicular to any root nisa(p),saiq))=Hip,q), with sa defined by (A.2).
Va€A
(2.13)
202
The main problem is to find all the eigenvalues {A} and eigenfunctions {tp} of the above Hamiltonian: •Hip = \t/>.
(2.14)
Some remarks are in order. For any root system and for any choice of potential (2.3), the Calogero-Moser model has a hard repulsive potential ~ l / ( a • q)2 near the reflection hyperplane Ha = {q e R r , a • q = 0}. The strength of the singularity is given by the coupling constant g\a\(g\a\ - 1) which is independent of the choice of the normalisation of the roots. (Thus for rational models with/without harmonic force there is equivalence, A2 = ^ ( 3 ) , B2 = J 2 (4), G2 = / 2 (6), Br S Cr ^ BCr.) In other words, (2.14) is a second order Fuchsian differential equation with regular singularity at each reflection hyperplane Ha. That is any solution of (2.14) is regular at all points except those on the union of reflection hyperplanes UaeA+Ha (and possibly at infinity for type V models). Near the reflection hyperplane Ha, the solution behaves ip ~ (a • g) S | Q | (l + regular terms),
or
ip ~ (a • q)1*9^* (1 + regular terms).
The former solution is chosen for the square integrability. This determines the form of the ground state wavefunction, as we will see in subsection 2.2. This repulsive potential is classically insurmountable. Thus the motion is always confined within one Weyl chamber. This feature allows us to constrain the configuration space to the principal Weyl chamber (II: set of simple roots, see Appendix A) PW = {9 £ R r | a-q>0,
aeU},
(2.15)
without loss of generality. In the case of the trigonometric potential, the configuration space is further limited due to the periodicity of the potential to PWT = {9 £ R r | aq>0,
a en,
ah-q
(2.16)
where a^ is the highest root. The fact that the classical motions are confined in the Weyl chamber (alcove) PW(PWT) does not necessarily mean that the corresponding quantum wavefunctions vanish identically outside of the region a. On the contrary, as we will see soon, the ground state wavefunction (subsection 2.2) and all the other excited states wavefunctions (section 3) are Coxeter invariant, reflecting the Coxeter invariance of the Hamiltonian (2.13). In the early years "Remember the quantum mechanical system defined in a finite interval on a line with the periodic boundary condition. It is described by a Fourier series which does not vanish outside of the interval.
203
of Calogero-Moser models in which those based on the AT root system were mainly discussed, these Coxeter invariant solutions were considered as totally symmetric states of bosonic systems. We will not, however, adopt this interpretation, for in the models based on the other root systems the reflection is not the same as particle interchange. The quantum theory we are discussing is the so-called first quantised theory. That is, the notions of identical particles and the associated statistics are non-existent. 2.2
Ground state wavefunction and energy
One straightforward outcome of the factorised Hamiltonian (2.5) is the universal ground state wavefunction which is given by
$ 0 ( 9 ) = e ff (') = Yi \w{p • q)\9M e~^q\ LJ
(2.17)
2
The exponential factor e~^q exists only for the rational potential case with the harmonic confining force. It is easy to see that it is an eigenstate of the Hamiltonian (2.5) with zero eigenvalue: **»(*) = \ ±
(p3 - i9^)
(Pi + i ^ )
*o(*) = 0,
(2.18)
since it satisfies
By using the decomposition of the factorised Hamiltonian into the operator Hamiltonian (2.9) and a constant, we obtain
Hew=(±p> \
+± £ P6A+
9w{g\p\-\)\p?V{p-q)
ew = S0ew.
+ (£M J
(2.20) In other words, the above solution (2.17) provides an eigenstate of the Hamiltonian operator % with energy £Q. The fact that it is a ground state (for type I, III and V) can be easily shown within the framework of the supersymmetric model 10 thanks to the positivity of the supersymmetric Hamiltonian. It should be stressed that £o is determined purely algebraically,10 without really applying the operator on the left hand side of (2.20) to the wavefunction. This type of ground states has been known for some time. It is derived by various methods, see for example Refs. 3 ' 5 , and also by using supersymmetric quantum mechanics for the models based on classical root systems. 25 ' 18
204
The ground state energy for the rational potential cases are f
0
type I,
°"i-(i+Ep6A+%|)
< 2 - 21 )
typeV.
The same for the hyperbolic and trigonometric potential cases are £0 = 2a 2 , 2 x ( - 1 hyperbolic ^ 1 trigonometric,
x
'
in which
? = ^ E 9\P\P
(2.23)
can be considered as a 'deformed Weyl vector'. 5 ' 24 Again these formulas are universal. That is they apply to all of the Calogero-Moser models based on any root system. A negative £Q for the obviously positive Hamiltonian of the hyperbolic potential model indicates that the interpretation of ew as an eigenfunction is not correct. This function diverges as \q\ —¥ oo for the hyperbolic and the rational potential cases, destroying the hermiticity of the Hamiltonian. Obviously we have f
e2W(g)
JPW (PWT)
, _ f oo : type I and II,
I nnite
: tv
Pe
n i and V
>
in which PW and PWT denote that the integration is over the regions defined in (2.15) and (2.16). Naturally, most existing results in quantum CalogeroMoser models are for the models with trigonometric potential and the rational potential with harmonic force which have normalisable states and discrete spectra. It should be remarked that the universal ground state wavefunction $o and W are characterised as Coxeter invariant: s p $o = $o,
§PW = W,
VpeA,
(2.25)
in which sp is the representation of the reflection in the function space. For an arbitrary function / of q, its action is defined by (*„/)(«) = /M
v5eGA.
(2.27)
In the rest of this lecture we mainly discuss the type III and V models which have normalisable states and discrete spectra.
205
3
Coxeter invariant excited states, triangularity and spectrum
In this section we show that all the excited states wavefunctions are Coxeter invariant, too. In other words, the Fock space consists of Coxeter invariant functions only. With the knowledge of the ground state wavefunction ew, the other states of the Calogero-Moser models can be easily obtained as eigenfunctions of a differential operator ti obtained from H by a similarity transformation: n = e-wUew e
\P2 + \ E 9M(9M ~ 1)\P\2V(P • q) + (y
n^x = X^x
/
^
n^xew
= \^\ew.
(3.2)
Since all the singularities of the Fuchsian differential equation (2.14) are contained in the groundstate wavefunction ew the function $ A above must be regular at finite q including all the reflection boundaries. Thanks to the factorised form of the Hamiltonian 7i (2.5), (2.6), the transformed Hamiltonian ti takes a simple form:
The Coxeter invariance of W implies those of Ti and ti: SpUsp = H,
sptisp
=ti,
Vp £ A.
(3.4)
For type V and III models we introduce proper bases of Fock space consisting of Coxeter invariant functions and show that the above Hamiltonian ti (3.3) is triangular in these bases. This establishes the integrability of the type V and III models universally b and also gives the entire spectrum of the Hamiltonian, see (3.12), (3.13) and (3.44).
^Triangularity of the AT type V and III Hamiltonians was noted in the original papers of Calogero 1 and Sutherland. 2 That of rank two models in the Coxeter invariant bases was shown in Refs. 2 4 , 2 6
206
3.1
Rational potential with harmonic force
First, let us determine the structure of the set of eigenfunctions of the transformed Hamiltonian "H, for the type V models: d n = ujq--
1 —^ -
^ >
g\p\ d - ^ - p - —.
(3.5)
As remarked above, the eigenfunctions of % have no singularities at finite q. Thus we look for polynomial (in q) eigenfunctions. The situation is very similar to that of the harmonic oscillators to which the type V CalogeroMoser models reduce in the free limit g\p\ —> 0. Obviously a constant and a quadratic polynomial uiq2 — £O/LJ are its eigenfunctions with eigenvalue 0 and 2u, respectively. Let us suppose that a polynomial P{q) is an eigenfunction oiU: •HP(q) = XP(q).
(3.6)
Due to the Coxeter invariance of H (3.4), we know that spP together with the difference
Q=
(l-s„)P
are also eigenfunctions with the same eigenvalue: -HQ(q) = XQ(q),
(3.7)
if the latter is not identically zero. Since Q is a polynomial which is odd under reflection sp §pQ(q) =
-Q(q),
Q(q) = (p-qfn+1Q(q),
Q
it can be factorised as # 0,
(3.8)
pq=o
with a non-negative integer n and a polynomial Q. By substituting (3.8) into (3.7) and using the explicit form of H near the reflection hyperplane p-q = 0, we obtain -(p • q)2n-1(2n
+ l)(n + gM)p2Q + 0[(p • q)2n] = \(p • q)2n+1Q,
which would imply the vanishing of Q on the reflection hyperplane Q
= o,
(3.9)
207
an obvious contradiction. Thus we are led to the conclusion that the eigenfunctions are Coxeter invariant polynomials and that the Hamiltonian % (3.5) maps a Coxeter invariant polynomial to another. An obvious basis in the space of Coxeter invariant polynomials is the homogeneous polynomials of various degrees. This basis has a natural order given by the degree. For a given degree the space of homogeneous Coxeter invariant polynomials is finite-dimensional. The explicit form of Iri (3.5) shows that it is lower triangular in this basis and the diagonal elements are u x degree as given by the first term. Independent Coxeter invariant polynomials exist at the degrees fj listed in Table II: fj
=
l + eh
j = l,...,r,
(3.10)
in which {e^}, j = 1 , . . . , r, are the exponents of A. Let us denote them by Zj(nq) = Kf> Zj(q).
zi{q),---,zr{q);
A Ar Br L^r
Dr E6 E7
fj = 1 + e i 2,3,4,...,r + l 2,4,6,....2r 2,4,6,...,2r 2,4,...,2r-2;r 2,5,6,8,9,12 2,6,8,10,12,14,18
A E8
(3.11)
fj = 1 + ej 2,8,12,14,18,20,24,30 2,6,8,12 2,6 2,m 2,6,10 2,12,20,30
Fi
G2 h(m) H3 Hi
Table II: The degrees fj in which independent Coxeter invariant polynomials exist. Thus we arrive at: the quantum Calogero-Moser models with the rational potential and the harmonic confining force is algebraically solvable for any (crystallographic and non-crystallographic) root system A. (See Ref.27 for algebraic linearisability theorem of the classical Calogero-Moser system.) The spectrum of the operator Hamiltonian T-L is uN + £0,
(3.12)
with a non-negative integer N which can be expressed as r
N
= J2njfj>
riJeZ+,
(3.13)
and the degeneracy of the above eigenvalue (3.12) is the number of different solutions of (3.13) for given N. This is generalisation of Calogero's original
208
argument for the Ar model 1 to the models based on any root system. Now let us denote by N the set of non-negative integers in (3.13): N = (m,n2,...,nr),
(3.14)
and by ${q) the homogeneous Coxeter invariant polynomial determined by N and the above basis {ZJ} (3.11): r
(3-15)
^(<7)=rR^)-
As shown above, there exists a unique eigenstate ip${q) for each (j)fi(q): ^AK?) = <M?) + ] C diV'<^V'(9)>
d
N> • COnsti
(316)
N'
Hi^ffiq) = uN^${q).
(3.17)
It satisfies the orthogonality relation (^,4>t},)=0, N'
(3.18)
with respect to the inner product in PW: (tl>, 2, can be expressed universally in terms of Laguerre polynomials in q2. This will be discussed at the end of section 5 and in subsection 7.2. 2. The other Coxeter invariants corresponding to the degrees f2,---,fr could be interpreted as special 'angular' variables of a unit sphere 5 r _ 1 (q2 = 1), with the first Coxeter invariant y/q2 being the radial coordinate. These would provide proper variables for describing solutions. Solutions in terms of separation of variables are in general possible only for the
209 simplest cases, t h a t is the rank two models, which will be demonstrated in subsection 7.3. 3. For A = A\, the simplest root system of rank one, the Hamiltonian ti can be rewritten in terms of a Coxeter invariant variable u = uiq2 as: d
Id2
g d
„
f
d2
,
1
s
d \
.„ „„.
T h e Laguerre polynomial satisfying the differential equation
provides an eigenfunction with eigenvalue 2um, which corresponds to the eigenvalue 2un + £o of H. This is a well-known result. 3.2
Trigonometric
potential
Here we consider those root systems associated with Lie algebras. I n order to determine the excited states of the type III models, we have t o consider the periodicity. T h e superpotential W and the Hamiltonian W are invariant under the following translation: W(q')
= W(q),
q' = q + lvir/a,
n(p,q')=n(p,q),
(3.22)
in which / v is an element of t h e dual weight lattice, t h a t is /V
=£
JJ-TXj'
lj e Z
'
aj
€ n
'
^ '
Xk = Sjk
•
(3 23)
'
As is well known in q u a n t u m mechanics (Bloch wavefunctions) with periodic potentials, the wavefunctions diagonalising the translation operators are expressed as e2ian-g
J j ^ bae2iaaq \aei(A)
I ew, /
ba : const,
L(A) : root lattice,
(3.24)
in which a vector n G R r is as yet unspecified. In other words, u p t o the overall phase factor e 2 m M q, this is a Fourier expansion in t e r m s of the simple roots. Let Pr(q) be a polynomial in e±2iaai-q, aj G II and suppose t h a t a function {q)=e2ia^PT{q),
fieKr,
(3.25)
210
is an eigenfunction of %: H0(q) = \(q),
(3.26)
in which the explicit form of T-L is given by ^
= _
a
2 ^ a ? "
^
ffHcot(a/>-(?)p-
—.
(3.27)
As above, due to the Coxeter invariance of T-L (3.4), we know that spcp together with the difference =(! -sp)(j> are also eigenfunctions with the same eigenvalue: H
(3.28)
Since ip is odd under reflection sp Spf{q) = -
(3.29)
in a neighbourhood of the reflection hyperplane p • q = 0. In this neighbourhood, the singularity structure of Ti for the trigonometric potential is the same as that of T-L for the rational potential discussed in the previous subsection. Thus we obtain, as before, a contradiction
(3.30)
Since sp(j>{q) = e2ias^"spPT(q)
=
e2ia^q-pV^q)spPT{q),
the following condition is necessary, but not sufficient, /•fiez,
VpeA
(3.31)
for the Coxeter invariance of 4>. Thus we arrive at (3.30). Let us introduce a basis for the Coxeter invariant functions of the form (3.25). Let A be a dominant weight r
A = y j m j A j , fflj £ Z + ,
(3.32)
211
and W\ be the orbit of A by the action of the Weyl group: W, = {peA(A)\p
= g(\), V 9 e G A } .
(3.33)
We define
«M
e2iaM?
(3-34)
'
M6VVX
which is Coxeter invariant. The set of functions {<^A} has an order >: |A|2>|Af
=*
4>x>y-
(3-35)
Next we show that ti is tower triangular in this basis. By using (3.27) we obtain Ufa = 2a2\24>x - 2ia2 Y
Y
gM cot (ap • q){p • p)e2ia^.
(3.36)
First let us fix one positive root p and a weight p. in W>, such that p • p ^ 0. Then /*' = Sp{n) = n - (pv -n)peW\,
p • p = -p • p..
(3.37)
Without loss of generality we may assume pv-p
= k>0,
keZ.
(3.38)
The contribution of the pair (p,p') in the summation of (3.36) reads \p • p\e2a^q
{I - e-2aikpi)
cot(ap • q)
= i\p - p\ [ e2ai^q + e2aif'q + 2 Y
c 2«(/*-J»-9
J
(3.39)
;
which is the generalisation of Sutherland's fundamental identity eq(15) in Ref.2 to any root system. The summation in the expression correspond to <j)\i with A' being lower than A. Thus (3.36) reads H(f>x=2a2\2<j>x+2a2
Y E 9\„\\p • ^e2^" PeA+ newx
+ Y |A'|<|A|
c
vAv,
(3.40)
in which {cv}'s are constants. It is easy to see that (p = g(X), 3g € G A )
Y p6A+
9\P\\p• p\ = Y peA+
9\p\\g(p) • M = ( Y pSA+
9
\p\ti
x = 2
e-\
(3.41)
212
which is independent of /J,. Thus we have demonstrated the triangularity of H4>x = 2a2(A2 4- 2g • A)0A +
£
c v \>,
(3.42)
|A'|<|A|
or that of ri Hxew = 2a2(\ + g)24>xew +
£
cx.<j>x.ew,
(3.43)
|A'|<|A|
with the eigenvalue 2a2(A +
(3.44)
In other words, for each dominant weight A there exists an eigenstate of % with eigenvalue proportional to A(A + 2g). Let us denote this eigenfunction by 4>\{q)ip\ (q) = \ (
d
v«/»A'(9),
dy : const,
(3.45)
|A|'<|A|
Hi;x(q) = 2a2X(X + 2g)xpx(q), and call it a generalised Jack polynomial.28~31 relation. (tl>x,M=0,
(3.46) It satisfies the orthogonality
|A|'<|A|,
(3.47)
with respect to the inner product in PWT(t/>,
r{qMq)e2W{q)dq.
(3.48)
J PWT
In the Ar model, specifying a dominant weight A is the same as giving a Young diagram which designates a Jack polynomial. It should be emphasised, however, that {tpx} are not identical to the Jack polynomials even for the Ar root systems, because of different treatments of the center of mass coordinates. Thus we arrive at: the quantum Calogero-Moser models with the trigonometric potential are algebraically solvable for any crystallographic root system A. The spectrum of the Hamiltonian ti is given by (3.44) in which A is an arbitrary dominant weight. This is generalisation of Sutherland's original argument 2 to the models based on any root system. Some remarks are in order.
213
1. The weights /x appearing in the lower order terms {4>\> }'s are those weights contained in the Lie algebra representation belonging to the highest weight A. 2. As a simple corollary we find that for a minimal weight A, vewx is an eigenfunction of H. A minimal representation 7 consists of a single Weyl orbit and all of its weights \i satisfy py • fi — 0, ± 1 , Vp € A. 3. If A = a^, the highest root of a simply-laced root system, W\ is the set of roots itself. Then the lower order terms are constants only. We find that ( c o s (2aP ' tf) + 9P2/ah • (ah + 20))
T/W (?) = 2 E is an eigenstate of Iri.
4. If A = ash, the highest short root of a non-simply laced root system, W\ is the set of short roots itself. The lower order terms are constants, too. Similarly as above, we find that c o s 2a
i>ash (q) = 2 E peAL
( P' q)
+
+2 Y^
{cos(2ap-q)+gsPs/ash-(ash
+ 2Q))
peAs+
is an eigenstate of ti. Here AL(S) is the set of long (short) roots. 5. If —A ^ W\ then there is another set of functions containing the weight — A which belongs to the same eigenvalue. 6. The Coxeter invariant trigonometric polynomials specified by the fundamental weights {Aj} \j{q) = E newx.
e 2l °' 1 ' 9 ,
Aj:: fundamental weight,
j =
l,...,r (3.49)
are expected to play the role of the fundamental variables. 24 ' 26
214
7. Let us consider the well-known case A = A\, the simplest root system of rank one. By rewriting the Hamiltonian ti in terms of the Coxeter invariant variable z = cos(apq), we obtain 1 d2
n
/ a C0t
=-w - ^
xd
a2
M 2 f,,
(1
2,d2 z
„
^d-q = "IT { - fe "
, d ) (1 + 2
^Tz) •
(3.50) The Gegenbauer polynomials, 5 a special case of Jacobi polynomials P« provide eigenfunctions: pL9~h'9~i}(cos(apq)), The Jacobi polynomial P„
£ = a2\p\2(n + g)2/2,
n € Z+.
(3.51)
(z) satisfies a differential equation
{ d - , 2 ) ^ + (/3- Q -(2 + a + ^ ) | + n(n + a+p+l)\pia'^(z)
= 0.
(3.52)
Here we follow the notation of Ref.32 They form orthogonal polynomials with weight e2W = | sin(apq)\2g in the interval q G [0,Tr/ap], (2.16). 8. Triangularity of type II models follows from the same algebraic reasoning. 4
Quantum Lax pair and quantum conserved quantities
Historically, Lax pairs for Calogero-Moser models were presented in terms of Lie algebra representations 3,5 , in particular, the vector representation of the AT models. However, the invariance of Calogero-Moser models is that of Coxeter group but not that of the associated Lie algebra, which does not exist for the non-crystallographic root systems. Thus the universal and Coxeter covariant Lax pairs are given in terms of the representations of the Coxeter group. 4-1
General case
Here we recapitulate the essence of the quantum Lax pair operators for the Calogero-Moser models with degenerate potentials and without spectral parameter. The quantum Lax pair in this subsection applies to all the degenerate potential cases except for the case of the rational potential with the harmonic
215
force, which will be treated separately in subsection 4.2. For details and a full exposition, see Ref.10 The Lax operators without spectral parameter are L(p, q)=p-H
+ X{q),
M(q) = \ Y .
X(q) = i ^
a\p\ \P\2 y(p •(i)'sp-\
gM (p • H)x(p • q)sp,
Jl 9\P\ \P\2 y(p • g) x /,
(4.1)
(4-2)
in which / is the identity operator and {sa, a G A} are the reflection operators of the root system. In contrast with {sa} operators (2.26) which act in function space, {sa} act on a set of R r vectors 7c = {/it'*' G R r , k = 1 , . . . ,d}, permuting them under the action of the reflection group. The vectors in 7c form a basis for the representation space V of dimension d. The operator M satisfies the relation
Y, MM„ = ] T M„„ = 0,
(4.3)
which is essential for deriving quantum conserved quantities. The matrix elements of the operators {sa, a G A} and {Hj, j = 1 , . . . , r } are defined as follows: (sP)„v =
<5 M , S „H
= K,Sp(n),
{Hj)>»> = Nsn">
P € A,
n,v G 7^.
(4.4)
The form of the function x depends on the chosen potential, and the function y are defined by (2.11), (2.12). Note that these relations are only valid for the degenerate potentials (2.3). The underlying idea of the Lax operator L, (4.1), is quite simple. As seen from (4.10), L is a "square root" of the Hamiltonian. Thus one part of L contains p which is not associated with roots and another part contains x(p-q), a "square root" of the potential V(p-q), which being associated with a root p is therefore accompanied by the reflection operator sp. Another explanation is the factorised Hamiltonian % (2.5). We obtain, roughly speaking, L ~ \ff[ ~ p + i^-s and the property of reflection s 2 = 1 explains the sign change in the first term in (2.5). It is straightforward to show that the quantum Lax equation jtL
= i[H,L] = [L,M],
(4.5)
is equivalent to the quantum equations of motion derived from the Hamiltonian (2.7). From this it follows: ±(Ln)llv=i[H,(Ln)llv]
=
[Ln,M]ltv
216
= J2 ({Ln)»\MXv - Mp.x(Ln)Xv\ ,
n = 1,....
(4.6)
Thanks to the property of the M operator (4.3):
we obtain quantum conserved quantities as the total sum. (Ts) of all the matrix elements of L™ c : Q n = Ts(L n ) = £
(i/V,
[W,Qn]=0,
n = l,....
(4.7)
The Lax pair is Coxeter covariant: L {sP(p), *„(g))M„ = L (p,g)M,„, , M (s p (g)) Mi/ =
M(q)Wv.,
1
\J! = Sp(n), v = sp{i>),
(4.8)
which ensures the Coxeter invariance of the conserved quantities. Independent conserved quantities appear at such power n that n = 1 + exponent
(4.9)
of each root system. These are the degrees at which independent Coxeter invariant polynomials exist. There are r exponents for each root system A of rank r. Thus we have r independent conserved quantities in Calogero-Moser models. We list in Table II these powers for each root system. In particular, the power 2 is universal to all the root systems and the quantum Hamiltonian (2.7) is given by H = - ^ - T s ( L 2 ) + const,
(4.10)
2C-R
where the constant C-R is the quadratic Casimir invariant, which depends on the representation. It is defined by TT(HjHk) = £
(HjHk)^ = Y, W*
Some remarks are in order. c
This type of conserved quantities is known for Ar models 1 8 , 2 0 .
= Cn V
(4-11)
217
1. Lax pairs can be written down in various representations and the quantum conserved quantities Qn do depend on the representations, in general. If necessary, we denote by Q1^ the explicit representation dependence. 2. The availability of plural representations of the Lax pair and the conserved quantities is essential for the completeness of the set of conserved quantities as polynomials of the momentum operators. For example, let us consider the case of Dr with even r, which has two independent conserved quantities at power r, see Table II. At least two different representations of the Lax pair are necessary in order to represent them in the form of (4.7). Those based on the vector, and the (anti)-spinor representations give two independent conserved quantities. For D4 case, we obtain 4
2
4
A
QI = Y.P V j=\
Qt-Qi
= ul[pj,
(412)
j=l
in which v, s and a stand for the vector, spinor and anti-spinor representations and we have set g = 0 for simplicity. If two conserved quantities are independent for zero coupling constants, surely they are so at nonvanishing couplings. 3. If a representation 1Z contains a vector yu and its negative — ft at the same time, then we have Ts(Lodd) = 0. In such a case the corresponding Lie algebra representations are called real. In order to construct the odd power conserved quantities appearing in Ar for all r, Dr for odd r, E§ and I{m) for odd m, we need a Lax pair in non-real representations. For Ar all the fundamental representations corresponding to the fundamental weights Xj, j = 1 , . . . , r except for the middle one \(r+1y2 for odd r are non-real. For Dr with odd r, the spinor and anti-spinor representations are non-real. For EQ the 27 and 27 are non-real, him) is the symmetry group of a regular m-sided polygon. The set of m vectors with 'half angles of the roots (see (A.15)) to be denoted by Vm (B.2), provides a non-real representation when m is odd. 4. In Appendix B we list for each root system how the complete set of independent conserved quantities are obtained by choosing proper representations of the Lax pair.
218
4-2
Rational potential with harmonic force
The quantum Lax pair for the type V models needs a separate formulation. The explicit form of the Hamiltonian is n = \P2 + \ ^
2
+\ £
9M(9M-l)j^-£o.
(4.13)
The canonical equations of motion are equivalent to the following Lax equations for L±: jL*
=i[U,L±]
= [L±,M]±iuL±,
(4.14)
in which (see section 4 of Ref.8) M is the same as before (4.2), and L*1 and Q are defined by L±=L±iujQ,
Q = q-H,
(4.15)
with L, H as earlier (4.1), (4.4). If we define hermitian operators £ i and L% by d=L+L~,
C2=L-L+,
(4.16)
they satisfy Lax-like equations Ck=i[H,Ck]
= [Ck,M],
k = l,2.
(4.17)
From these we can construct conserved quantities Ts(£?),
j = l,2,
n = l,2,...,
(4.18)
as before. Such quantum conserved quantities have been previously reported for models based on Ar root systems. 18 ' 20 It should be remarked that Ts(£2) is no longer the same as Ts(£") due to quantum corrections. It is elementary to check that the first conserved quantities give the Hamiltonian (4.13) 7i oc Ts(£x) = Ts(£ 2 ) + const.
(4.19)
This then completes the presentation of the quantum Lax pairs and quantum conserved quantities for all of the quantum Calogero-Moser models with nonelliptic potentials.
219 5
Algebraic construction of excited states I
In this section we show that all the excited states of the type V CalogeroMoser models can be constructed algebraically. Later in section 7 we show the same results in terms of the I operators to be introduced in section 6. The main result is surprisingly simple and can be stated universally: Corresponding to each partition of an integer N which specify the energy level (3.12) into the sum of the degrees of Coxeter invariant polynomials (3.13), we have an eigenstate of the Hamiltonian "H with eigenvalue cuN + £$: r
r
I[(B+)n>ew,
N = J2nJfJ,
n3 e Z + ,
(5.1)
in which the integers {fj}, j = 1 , . . . ,r are listed in Table II. They exhaust all the excited states. In other words the above states give the complete basis of the Fock space. The creation operators Bt and the corresponding annihilation operators'* BJ are defined in terms of the Lax operators L^ (4.15) as follows: Bf^TsiL*)*,
j = l,...,r.
(5.2)
They are hermitian conjugate to each other
{Bfy=Bl
(5.3)
with respect to the standard hermitian inner product of the states defined in PW: (if>,
P(QMq)dq.
(5.4)
Jpw
We will show later in section 6, (6.15) that the creation (annihilation) operators commute among themselves: [B+,B+] = [B-,Bf}
= 0,
k,l€{fj\j
= l,...,r},
(5.5)
so that the state (5.1) does not depend on the order of the creation. The proof is very simple. By using (4.14) we obtain | ( L ± ) " = i[H, ( i ± ) " ] = K ^ ) " , M) ± ino;(L ± ) n ,
(5.6)
from which [%,B$] = ±m,B$, d
(5.7) 5 14
We adopt the notation by Olshanetsky and Perelomov. '
220
follows after taking the total sum. This simply says that B^ creates (annihilates) a state having energy noj. In other words we have n n
(Bpn> ew = U
j=i
+ a ; £ > * / * ) f[ (B+)n> ew. k=i
\
J j=\
Moreover, it is trivial to show that
^
Pq
V
P^+
v
')
dqJ
(5.8) which implies that the ground state is annihilated by all the annihilation operators B~ew = 0, j = l,...,r. (5.9) Some remarks are in order. 1. In most cases the energy levels are highly degenerate. The above basis is neither orthogonal nor normalised. 2. The independence of the creation-annihilation operators can also be shown in a similar way to that of the conserved quantities. As with the conserved quantities, plural representations are necessary to define the full set of creation-annihilation operators in some models. This aspect will be discussed in later sections in connection with the I operators. 3. Reflecting the universality of the first exponent, / i = 2, the creation and annihilation operators of the least quanta, 2w, exist in all the models. They form an s/(2,R) algebra together with the Hamiltonian V.: [ft, 6±] = ±2u>bf,
[b+, 62"] = -uj-'n,
(5.10)
in which bf are normalised forms of B^ '• bf = £
(L^Ui^Cn).
(5.11)
The s/(2,R) algebra was discussed by many authors (see, for example, 14,33 ' 19 ' 23 and others) in connection with the models based on classical root systems. We will show later in subsection 7.2 that the states created by B% (&J) only can be expressed by the Laguerre polynomial: (b+)n ew = nlL^Hojq2)
ew,
£0 =
£0/UJ.
(5.12)
221
It is trivial to verify that L„ °
{toq2) is an eigenfunction of % (3.5)
HLgo-'Huq2)
= 2nuL^-1\uq2).
(5.13)
The normalisation of the state \\(b+)new\\2=nW0/r(n
+ £0),
M> = \ew\2r{£0),
(5.14)
is also dictated by the si(2, R) relations. The Laguerre polynomial wavefunctions appear as 'radial' wavefunctions in all the cases 34 . This will be shown explicitly for for the rank two models given in subsection 7.3. 4. As is emphasised by Perelomov14 and Gambardella 33 the sl(2, R) algebra and the corresponding Laguerre wavefunctions are more universal than Calogero-Moser models. They arise when the potentials are homogeneous functions in q of degree —2 with the confining harmonic force. 5. The operators {Qn} and {B^} do not form a Lie algebra. They satisfy interesting non-linear relations, for example, [[B+,b-],b+] = nB+,
[[£>-, &+],&2~] = nB~.
(5.15)
This tells, for example, that although f?+ and b\ create different units of quanta n and 2, they are not independent
[ 5 + , 6 2 - ] ^ 0 ^ [£",&+]. 6
£ operators
In this section we will show the equivalence of the quantum conserved quantities obtained in the Lax operator formalism of section 4 and those derived in the 'commuting differential operators' formalism initiated by Dunkl 11 and followed by many authors. Again the equivalence is universal, applicable to the models based on any root system. We propose to call the operators in the latter approach simply '£ operators', since they are essentially the same as the L operator in the Lax pair formalism and that they are not mutually commuting, as we will show presently, when the interaction potentials are trigonometric (hyperbolic), (6.14). Although these two formalisms are formally equivalent, the £ operator formalism has many advantages over the Lax pair one. Roughly speaking, the 'vector-like' objects £M's are easier to handle than the matrix LM1/. Let us fix a representation 7Z of the Coxeter group GA and define for each element fx E TZ the following differential-reflection operator l„=e-IJ.=p-H
+ i Y^ 9\P\ (p • M) x(P • Q)sP,
p&TZ.
(6.1)
222
It is linear in fi and Coxeter covariant = £M + 4 ,
W
SplpSp = ltp(lt),
Vp e A.
(6.2)
They are hermitian operators, ft = £M, with respect to the standard inner product for the states (5.4). It is straightforward to show that the quantum conserved quantities Qn derived in the previous section (4.7) can be expressed as polynomials in the £ operators as follows:
in which ip is an arbitrary Coxeter invariant state, spip = V- This also illustrates the Coxeter invariance of Qn clearly, since sp(^2 en£")sp = E/iew *?,(,.) = £ W T C * H - F o r n = 1 it is trivial, since
53 (L W^ = \P • M + i 53 9M (p ' ^ X(P ' q^ 53 ( S ")^ I ^ -
[P • A* + i \
in which
X^GTC^PW
=
53 #H ^' v) X(P • , (6-4)
P€A +
•*• an<^
*P^
=
/
^
a r e use
d- Let us assume that
53(L n w = *>.
(6-5)
is correct, then we obtain +i
£(L"
= 53
p
w = 5 3 LMA(in)A^ = 53 w w > ,
' ^" A
+
A6TC \
* 53 5W ^ ' ^ ^
' 9)(«P)MA 1 ^AV-
P6A+
/
In the second summation only such A as A = s p (/i) contributes and we find
Thus we arrive at
53(L"+W> = ^ +1 V, veil
(6-6)
223
and the equivalence of the two expressions of the conserved quantity (6.3) is proved. Commutation relations among I operators can be evaluated in a similar manner as those appearing in the Lax pair, 8 ' 10 that is, by decomposing the roots into two-dimensional sub-root systems. We obtain
{
0 rational, - 1 hyperbolic,
(6.7)
1 trigonometric. One important use of the £ operators is the proof of involution of quantum conserved quantities. For type I models Heckman 23 gave a universal proof based on the commutation relation (6.7): [Qn,Qm]il)= £ [ C > C W > = 0, rational model. (6.8) This was the motivation for the introduction of the commuting differentialreflection operators by Dunkl. 11 In fact, Dunkl's and Heckman's operators were the similarity transformation of £M by the ground state wavefunction ew:
4 =e " V
= P • /i + i £
9\P\ 7 ^ 4 ( ^ - 1)-
(6-9)
As for type V models, we define £ ± corresponding to L^ (4.15): t±=Ck-li=p-H±
iu(q -/i)+i
£ gM j^ap, /*£K. (6.10) [P q PGA+ ' > They are linear in /j,, Coxeter covariant and hermitian conjugate of each other with respect to the standard inner product (5.4):
The conserved quantities are expressed as polynomials in t^ operators: Ts(£?)V=
£ (L+L-)^ =£(#*;;)>, ti.i'eK veil
(6.12)
Ts(£?)v= £ ( L " i+ )2> =£(*;;#)>• Likewise the creation and annihilation operators B^ (5.2) are expressed as £ ± V = T s ( L ± ) " ^ = £ (L* W p,v€U
= £ (£)>. H&TI
(6.13)
224
The commutation relations among ^ ± operators are easy to evaluate, since I operators commute in the rational potential models (6.7):
P£A+
V
(6.14) From these it follows that the creation (annihilation) operators B^ do commute among themselves: [B+,B+]rp=[B-,B-)ij
= 0.
(6.15)
It is also clear that £^/^/2ui are the 'deformation' of the creation (annihilation) operators of the ordinary multicomponent harmonic oscillators. In fact we have £+ ew = 2iu(ji • q) ew
and
l~ ew = 0.
(6.16)
In the next section we present an alternative scheme of algebraic construction of excited states of type V models by pursuing the analogy that t^ are the creation and annihilation operators of the unit quantum. This method was applied to the Ar models by Brink et. al and others. 19 ' 17 ' 20 7 7.1
Algebraic construction of excited states II Operator solution of the triangular Hamiltonian
In subsection 3.1, we have shown that an eigenfunction of 7i with eigenvalue NCJ is given by (pN{q) + PN-2(qj)
ew,
(7.1)
in which PN (q) is a Coxeter invariant polynomial in q of homogeneous degree N and -P/v-2(
= [^(O.S] H-iVwPjv^),
(7.6)
in which the first term in r.h.s. vanishes due to the Coxeter invariance of P/y. Thus we arrive at the desired commutation relation
[•u,pN(e+)} = NcoPN(e+) + £ p6A+
9M
f 1 lp{2 2{P.qy
,PN(£+)
(sP - 1),
(7.7)
and the eigenvalue equation nPN(£+)ew
= NuPN(£+)ew.
(7.8)
Since the action of the creation operators on the ground state is C ' •' C
eW
= [&u)N(q
• /ii) • • • (9 • /ijv) + lower powers of q] ew,
(7.9)
our assertion (7.3) is proved. It should be stressed that in this formalism the Coxeter invariance of the polynomial P is important but not how it is obtained.
226
Like the above Hamiltonian (7.4), the t operator formulas of higher conserved quantities (6.12) contain extra terms:
Ts(£?) = Y. (L+L~)^ = E ( W + VT-
(7.10)
Here VT stands for vanishing terms when they act on a Coxeter invariant state. The same is true for most formulas derived in section 6. 7.2
States Created by £ +
Here we derive the explicit forms of the subseries of eigenstates obtained by multiple applications of the least quanta creation operator £?+ (5.2), or its normalised form &+ (5.11). It is convenient to work with the similarity transformed operator (7.11) lien
in which i^=p-H
+ 2iuj(q-fi)+i
J2 peA.,
(7.12)
—(s„-l). p-q
Let f(u) be an arbitrary function of u = uiq2, then it is Coxeter invariant. We find !+f(u) = 2ico(q • /x)(l - — )/(u), du
u = ojq2,
(7.13)
and b+f{u) = - u [ l - - ) (£o-i),
Since 6+1 = £0 - u = L f 0 -
-
f 0
(l
d_ du
/
(
«
)
•
(7.14)
> ) we assume
{bt)nl=n\L^-l\u).
(7.15)
By using the Laguerre differential equation (3.21) and the recurrence formulas of the Laguerre polynomial L„ (u), U
- 4
a
nL^iu)
' H = nL^{u) + (u-2n-a
- (n + a J L ^ u ) , + l J L ^ u ) + (n + a-
(7.16) l ) 4 - 2 ( « ) = 0,(7.17)
227 we can show u(l
du
du'
L(f»-1)(U) = (n+l)L^r 1 ) («)-
(7-18)
Thus the induction is proved and we arrive at (5.12). The orthogonality of the states ((B+)new,(B+)mew)
= 0,
n^m
(7.19)
can be easily understood as the du part of the measure e2Wdrq = e~uu£°~1dudQ,
dfl : angular part,
is the proper weight function for the Laguerre polynomial Lyn ° 7.3
(u).
Explicit solutions of the rank two models
For rank two models, the Liouville integrability, or the involution of conserved quantities is automatically satisfied since the second conserved quantity is already obtained. For rank two type V models, the complete set of orthogonal wavefunctions can be written down explicitly in terms of separation of variables by using the Coxeter invariant polynomials. These are based on the dihedral root systems I2(m), with A2 = 7 2 (3) 34 , B2 = J 2 (4) and G2 = i 2 (6) 3 5 . The Coxeter invariant polynomials exist at degree 2, i.e. q2 and m which is m
J } («;•«). where {VJ} is a set of vectors given in (B.2). dimensional polar coordinates system e for q
(7-20) If we introduce the two-
q = r(sin6,cos8),
(7.21)
then the principal Weyl chamber is PW : 0 < r 2 < o o ,
0 < 6 > < n/m.
(7.22)
The two Coxeter invariant variables read: m q2=r\
'[[(vj-q)=2(r-)m
cosm6,
(7.23)
e We believe no confusion arises here, between the radial coordinate variable r and the rank of the root system r, which in this case is 2 of Ii^m).
228
and the latter variable varies the full range, — 1 < cosmO < 1 in the PW. Thus solving the eigenvalue equation for TL (3.6) by separation of variables in the polar coordinate system is compatible with Coxeter invariance. We adopt as two independent variables u = uir2,
z = cosm9.
(7.24)
The solutions consist of a Gegenbauer (Jacobi) polynomial in cos m6 times a Laguerre polynomial in ujr2. The former we have encountered in the A\ Sutherland problem, subsection 3.2 and the latter in the A\ Calogero problem subsections 3.1 and 7.2. Let us demonstrate this for odd m with a single coupling constant and for even m with two independent coupling constants, in parallel. In terms of the Coxeter invariant variables (7.24) the ^(m) Hamiltonians take surprisingly simple forms: -
d dr
=-2w
mj g 1 9 1 d ( d r 2r dr \ dr \h(9o + 9e)j dr 1 d2 ( 2g cot m6 1 d_ ~272 dO2 + m I -9o tan rf + ge c o t M J 86 u— 2 + (£o-u) du
— du d2
^ f e + \\9o-9e)
2u
(7.25)
+ 2g 11 + ge + 9o
dz
The z part admits polynomial solutions {, J 9 dz' "
\\9o-9ef
\l+9e+9o)
,+ 2 p i
—( L :J) >
J dz\
l
r (9-i-
-4>'
(s)
p 26)
"»• -
in which / is the degree of the polynomial. After substituting them, the radial part of the Hamiltonian %T reads Ur = -2w
u
dV2
+ {£
°
u) a du
m l[l 4M
+ ( 29 )) {9e+9o) J
(7.27)
229
By similarity transformation in terms of umll2 oc rml, which is the radial part of the highest term of the polynomial p(a>0>(rm cos mO), it reads nl 2
l Urumll2
= -2OJ u a
d^
+
/
,
nl +
~
V
\ dd
£
ml ml
u
(7.28)
°- )d^-T
This is the main part of the differential equation for the Laguerre polynomial (3.21):
, £ + („, + &-,)!-»] 4-**-»<.)-a Thus the eigenstates of the Hamiltonian are obtained: nrum,^L^+i°-1){u)
= w(2n + ml)uml/2L^nl+i°-1\u),
(7.29)
7£ u mJ/2 L (mJ+£o-l)( u ) p^"-^"-^\z)
(7.30) r U-i.9-i) |
2
+£
1
= (v(2n + ml)+£0)U^ LW °- \u)Pt
<''-*•'•-*%).
It is instructive to note that the Hamiltonians % look also simple: 1 2 , Ti = -u! r2 2
9(9-1)
sin 2 m.9
Id2 m2 2 2 1 2r d6 2r2
1 d ( d \ r— 2rdr\dr
> . (7.31) 4 cos
2
^
2
4 sin ^
J
5
Olshanetsky and Perelomov obtained the above solutions starting from these formulas. 8
Universal proof of involution of quantum conserved quantities for type I, II and III models
Here we present a proof of involution of quantum conserved quantities {Q„} derived from the universal Lax pair in subsection 4.1 for type I, II and III models. The proof is applicable to all models based on any root system. Though a universal proof of involution for type I models is given by Heckman 23 as recapitulated in section 6, we believe the universal proof applicable to type II and III models as well is new. It depends on a theorem by Olshanetsky and Perelomov. 13 Our own contribution is that we have provided a universal Lax pair and conserved quantities satisfying all the requirements of the theorem. Liouville's theorem states the complete integrability as the existence of an involutive set of conserved quantities as many as the degrees of freedom.
230
We have already given conserved quantities {Qn} (4.7) independent and as many as the degrees of freedom (see Appendix B). They have the following properties: 1. Coxeter invariance Qn(sp(p),sp(q))
= Qn{p,q),
VpeA.
(8.1)
2- Qn(p,q) is a homogeneous polynomial of degree n in variables (p1,...,pr,x(p-q)). 3. Scaling property for those of type I models: 'Qn{K-lp,
nq) = K~n 'Qn{p, q)
(8.2)
as a consequence of the above point. 4. For type II and III models, the asymptotic behaviour near the origin: Qn(p,q) = IQn(p,q)(l
+ 0(\q\)),
for
| 9 | -> 0.
(8.3)
We need to show the vanishing of Jlm = [Ql,Qm],
(8-4)
which is a polynomial in {p} of degree s s
(8.5)
Let us decompose J( m into the leading part and the rest: Jlm = Jfm + J[mst,
J,°m = £
c* •-•'•' (q)Pjl ...
Pja
(8.6)
and J[^st is a polynomial in {p} of degree less than s. From Jacobi identity and conservation [H, Qi(m)] = 0, we obtain [H, Jim] = 0.
(8.7)
Considering the explicit form of the Hamiltonian (2.7) (ui = 0), the leading (i.e. of degree s + 1 in {p}) part of [H, Jim] comes only from the free part
b2,4°J and it vanishes if the following conditions are satisfied:
J2-S-^--k-(q)=0,
(8.8)
where the sum is taken over all permutations of indices a(t,k\,...,ks) = (ji,... ,js+i)- In Ref.36 it is proved (Lemma 2.5, p. 407) that the system
231
(8.8) has only polynomial solutions. Then Olshanetsky and Perelomov argue that for type I models the scaling property tells that ckl'-'k'(nq) = Ks~l~mckl'-,k,(q). Since s < I + m (8.5), it follows that the only polynomial solution satisfying the condition is the null polynomial. Thus we obtain c7'1''"'•'* (q>) = 0 => J°m = 0 and J; m = 0. The same results follow for type II and III models by considering the asymptotic behaviour for \q\ —> 0. Thus the involution of all the conserved quantities {Qn} is proved. This result also implies the involution of classical conserved quantities by taking the classical limit (h -*• 0). See Ref.12 for the classical Liouville integrability of the most general Calogero-Moser models with elliptic potentials.
9
Summary, comments and outlook
Various issues related to quantum integrability of Calogero-Moser models based on any root system are presented. These are construction of quantum conserved quantities and a unified proof of their involution, the relationship between the Lax pair and the differential-reflection (Dunkl) operators formalisms, construction of excited states by creation operators, etc. They are mainly generalisations of the results known for the models based on Ar root systems. Integrability of the models based on other classical root systems and the exceptional ones including the non-crystallographic models are also discussed in Refs. 35,38 ~ 41 There are still many interesting problems to be addressed to: The structure and properties of the eigenfunctions of the trigonometric potential models, which are generalisations of the Jack polynomials. 2 8 - 3 1 Comprehensive treatment of Liouville integrability of rational models with harmonic force. Understanding the roles of supersymmetry and shape invariance in CalogeroMoser models. 42 ' 41 Formulation of various aspects of quantum CalogeroMoser models with elliptic potentials; Lax pair, the differential-reflection operators, 43 ' 44 conserved quantities, supersymmetry and excited states wavefunctions.
Acknowledgements I thank S. P.Khastgir and A. J. Pocklington for enjoyable collaboration. This work is partially supported by the Grant-in-aid from the Ministry of Education, Science and Culture, priority area (#707) "Supersymmetry and unified theory of elementary particles".
232
Appendix A: Root Systems In this Appendix we recapitulate the rudimentary facts of the root systems and reflections to be used in the main text. The set of roots A is invariant under reflections in the hyperplane perpendicular to each vector in A. In other words, SQ
(/?)£A,
VQ,/3 6 A ,
(A.l)
where sa((3) =P-{a
v
-/?)a,
av=2a/|a|2.
(A.2)
The set of reflections {sa, a £ A} generates a group G A , known as a Coxeter group, or finite reflection group. The orbit of /? £ A is the set of root vectors resulting from the action of the Coxeter group on it. The set of positive roots A + may be defined in terms of a vector U £ R r , with a • U / 0, Va £ A, as those roots a £ A such that a • U > 0. Given A + , there is a unique set of r simple roots II = {otj, j = 1 , . . . , r} defined such that they span the root space and the coefficients {a,j} in /3 = X^=i ajaj f° r P £ A+ a r e a n non-negative. The highest root a/,, for which X]j = 1 a.j is maximal, is then also determined uniquely. The subset of reflections {sa, a £ II} in fact generates the Coxeter group C?A- The products of sa, with a £ II, are subject solely to the relations (sas0)m^
= 1,
a,/?€ll.
(A.3)
The interpretation is that saS0 is a rotation in some plane by 2n/m(a,P). The set of positive integers m(a,0) (with m(a,a) = 1, Va £ II) uniquely specify the Coxeter group. The weight lattice A(A) is defined as the Z-span of the fundamental weights {Xj}, j — 1 , . . . , r, defined by a]-Xk=djk,
ajGU.
(A.4)
The root systems for finite reflection groups may be divided into two types: crystallographic and non-crystallographic. Crystallographic root systems satisfy the additional condition av-/3eZ,
VQ,/3€A,
(A.5) r
which implies that the Z-span of II is a lattice in R and contains all roots in A. We call this the root lattice, which is denoted by £(A). These root systems are associated with simple Lie algebras: {Ar, r > 1}, {Br, r > 2}, {Cr, r > 2}, {Dr, r > 4}, E6, ET, Eg, F4 and G2. The Coxeter groups for these root systems are called Weyl groups. The remaining non-crystallographic root systems are Hz, H4, whose Coxeter groups are the symmetry groups of
233
the icosahedron and four-dimensional 600-cell, respectively, and the dihedral group of order 2m, {^{m), m > 4}. Here we give the explicit examples of root systems. In all cases but the Ar, {ej} denotes an orthonormal basis in R r , ej € R r , e,- • ek = 6jk. 1. Ar: This root system is related with the Lie algebra su(r + 1). It is convenient to have the r + 1 dimensional realisation: l , . . . , r + l | e i eKr+\ej-ek
A = { ±(ej-ek),
j^k=
n = {ej -ej+1,
j = l,...,r}.
=Sjk}, (A.6)
2. Br: This root system is associated with Lie algebra so(2r + 1). The long roots have (length) 2 = 2 and short roots have (length) 2 = 1: A = {±ej ±ek,
±ej, j ^ k = 1 , . . . ,r},
n = {ej-ej+1,
j = l,...,r-l}U{er}.
(A.7)
3. Cr: This root system is associated with Lie algebra sp(2r). roots have (length) 2 = 4 and short roots have (length) 2 = 2: A = {±ej ±ek,
±2ej, j,k = l , . . . , r } ,
U = {ej-ej+1,
j = l,...,r-l}u{2er}.
The long
(A.8)
4. Dr: This root system is associated with Lie algebra so(2r): A = {±ej ±ek, II = {ej -ej+i,
j ^ k = l,...,r}, j = l,...,r1} U {e r _i +eT}.
(A.9)
5. EQ- All the roots have the same (length) 2 , which is chosen to be 2: A = {±ej±ek,
j ^ k ^ l , . . . , 5 } U { - ( ± e i - - - ± e 5 ± % / 3 e 6 ) , (even+)},
II = { - ( e x - e 2 - e 3 - e 4 + e 5 - V3e 6 ), e 4 - e 5 , e 3 - e 4 , e 4 + e 5 , - ( d - e 2 - e 3 - e 4 - e 5 + V3e6),
e2 - e 3 } .
(A.10)
6. E7: All the roots have the same (length) 2 , which is chosen to be 2: A = { ± e , ± e f c , j^k=
I,...,6}U{±V2e7}
U { - ( ± e i • • • ± e 6 ± \/2e 7 ), (even + ) } ,
(A.ll)
IT = { e 2 - e 3 , e 3 - e 4 , e 4 - e 5 , e 5 - e 6 , - ( e i - e 2 - e 3 - e 4 - e 5 + e 6 - V2e7),
V2e7, e 5 + e 6 } .
234
7. E$: All the roots have the same (length) 2 , which is chosen to be 2: A = {±ej n
± ek, j ? k = 1 , . . . , 8} U { i ( ± e i • • • ± e 8 ), (even + ) } ,
= { o ( e i - e 2 - e 3 - e 4 - e 5 - e 6 - e 7 + e 8 ), e 7 + e 8 } U{c,--ei+i,
i = 2,...,7}.
(A.12)
8. F 4 : The long roots ((length) 2 = 2) are those of D4 and the short roots ((length) 2 = 1) are the union of the vector, spinor and anti-spinor weights of D4: A = {±ej ±ek,
± e j , - ( ± e 1 - - - ± e 4 ) , j ^ k = 1 , . . . ,4},
II = {e 2 - e 3 , e 3 - e 4 , e 4 , - ( e x - e 2 - e 3 - e 4 )}.
(A.13)
9. G2: The G2 root system consists of six long roots and six short roots, and the sets of long and short roots have the same structure as the A2 roots, scaled |as| 2 /|o:z,| 2 = 1/3 and rotated by 7r/6. They are
A = {(±^0),(±/f,±-L),(0,±/|),,±i,±-iI)), n={(v^,0), ( - i , i ) }
(A.14)
10. him): This is a symmetry group of a regular m-gon. For odd m A consists of a single orbit, whereas for even m it has two orbits. In both cases we have a representation in which all the roots have length unity A = {(cos((j - l)7r/m), sin((j - l)ir/m)), j = 1 , . . . , m}, II = {(1,0), (cos((m - l)n/m), sin((m - l)n/m)))} (A.15) 11. H^. Define a = COSTT/5 = (1 + VE)/4 , b = COS2TT/5 = ( - 1 + >/5)/4. Then the H\ roots are generated by the following simple roots 45 : Ql
= U,-i,6,oJ,
«3 = ( 2 ' f e ' _ a ' ° ) >
a2= (-a,l,b,Oj , Q
(A.16)
4 = ( -2,_a'0'''
The full set of roots of H4 in this basis may be obtained from (1,0,0,0), ( i , i , i , | ) , and (a, i,6,0) by even permutations and arbitrary sign changes of coordinates. These 120 roots form a single orbit.
235
12. H3: A subset of (A.16), {01,02,03} is a choice of simple roots for the H3 root system. In this basis, the full set of roots for H3 results from even permutations and arbitrary sign changes of (1,0,0) and (a, | , 6 ) . These 30 roots also form a single orbit. Appendix B: Conserved quantities Here we list for each root system how the complete set of independent conserved quantities is obtained by choosing proper representations of the Lax pair. We choose those of the lowest dimensionality for the convenience of practical calculation. Of course there are many other choices of representations giving equally good sets of conserved quantities. The independence of the conserved quantities can be easily verified by considering the free limit: 9\P\ -> 0. 1. Ar: For all powers, the vector representation ( r + 1 dimensions) is enough. 2. Br: For all powers, the representation consisting of short roots {±ej : j = 1 , . . . , r}, (2r dimensions) is enough. 3. Cr: For all powers, the representation consisting of long roots {±2ej : j = 1 , . . . , r } , (2r dimensions) is enough. 4. Dr: For all even powers, the vector representation (2r dimensions) is enough. For the additional one occurring at power r, the (anti)-spinor representation (2r~1 dimensions) would be necessary. They are minimal representations. 5. E6: For all powers, the weights of the 27 (or 27) dimensional representation of the Lie algebra is enough. They are minimal representations. 6. £7: For all powers, the weights of the 56 dimensional representation of the Lie algebra is enough. This is a minimal representation. 7. Eg: For all powers, the 240 dimensional representation consisting of all the roots is enough. This is not the same as the adjoint representation of the Lie algebra. 8. -F4: For all powers, either of the 24 dimensional representation consisting of all the long roots or the short roots is enough. These are not Lie algebra representations.
236
9. G2'- For all powers, either of the 6 dimensional representations consisting of all the long roots or the short 