Thermodynamics of the complex su 3 Toda theory

Ž . We present the first computation of the thermodynamic properties of the complex su 3 Toda theory. This is possible thanks to a new string hypothesis, which involves bound states that are non-self-conjugate solutions of the Bethe equations. Ž . Our method provides equivalently the solution of the su 3 generalization of the XXZ chain. In the repulsive regime, we confirm that the scattering theory proposed over the past few years – made only of solitons with non-diagonal S matrices – is complete. But we show that unitarity does not follow, contrary to early claims, eigenvalues of the monodromy matrix not being pure phases. In the attractive regime, we find that the proposed minimal solution of the bootstrap equations is actually far from being complete. We discuss some simple values of the couplings, where, instead of the few conjectured breathers, a Ž . very complex structure involving E , or two E of bound states is necessary to close the bootstrap. q 2000 Elsevier 6 8 Science B.V. All rights reserved. PACS: 72.10.-d; 73.40.Gk Ž . Integrable quantum field theories QFTs based Ž . on su 2 have been the subject of intensive studies over the past many years. New theoretical tools like w x the quantum Q-operators 1 , the Destri De Vega and w x Klumper and Pearce equations 2 , the connections ̈ w x with spectral determinant theory 3 , or the relations w x with elliptic curves and duality 4 have revealed mathematical structures of remarkable depth; they have also made possible the computation of quantities of experimental interest – and typical of strong interactions – in various low dimensional condensed w x matter systems 4 . Formal developments, as well as practical applications, would largely benefit from an extension of these results to the case of other Lie algebras, in Ž . particular su n . The situation here is somewhat embarrassing, however. Although the pillars of the Ž . su 2 case – the XXZ chain and the associated sine-Gordon model – have been under control for a Ž . long time, even the simplest su 3 case is only very partially understood. One of the difficulties here – and, from the field theory point of view, one of the most interesting issues at stake – has to do with unitarity. Indeed, the simplest integrable generalizations of the sine-Gordon 1 Ž . model are the complex affine su n Toda theories defined by the Lagrangian: n 1 a m a i ba .f j Ls E f E f yl e 1 Ž . Ž . Ž . Ý m 2 js0 1 Real Toda theories involve entirely different issues. For a w x recent review see 5 . 0370-2693r00r$ see front matter q 2000 Elsevier Science B.V. All rights reserved. Ž . PII: S0370-2693 00 00495-0 ( ) H. Saleur, B. Wehefritz-KaufmannrPhysics Letters B 481 2000 419–426 420 where l)0, a , . . . ,a form the root system of 1 ny1 the classical lie algebra a , a syÝa is the ny1 0 js1 j negative of the longest root. The conformal weights 2 Ž . of the perturbing operator in 1 are DsDsb r4p . ty 1 In the following, we shall parameterize Ds . t Ž . The theory described by 1 is obviously non-unitary at the classical level. The fascinating possibility was w x raised 6,7 that it could nevertheless describe a unitary field theory in a sufficiently strong quantum regime. This possibility was ruled out in the interestw x ing paper 8 , and we confirm and extend their observations here. From a practical point of view, unitarity is not such a key issue. In fact, the most interesting applications of complex Toda theories are potentially found in disordered systems of statistical mechanics, where Toda theories based on superalgebras natuw x rally seem to appear 9,10 , leading most likely to even stronger violations of unitarity. More crucial then are the questions of completeness of the bootstrap, the physical meaning of the bound states, and the calculation of physical quantities. The main progress in the study of complex Toda theories have been based on non-perturbative S maw x trix analysis, following the pioneering work of 6 . Ž . One of the difficulties in this approach for su n is the appearance of a large number of poles in the S matrix elements, whose signification is not entirely clear: it was argued, after careful analysis of several cases, that most of these poles were not physical, and occurred rather by mechanisms generalizing Colew x man Thun’s 11 . The issue of the completeness of the bootstrap in w x 6,7 could be settled by a study of the thermodynamics, and a computation of the central charge in the Ž . UV, using the thermodynamic Bethe ansatz TBA w x 2 12 . However, for the imaginary affine Toda theoŽ ries or, equivalently, the corresponding anisotropic . quantum spin chains , the TBA has never been written so far, because of the complexity of the set of solutions: no natural ‘‘string hypothesis’’ had been proposed, up to now. 2 w x The approach developed in 2 bypasses some difficulties in the study of the thermodynamics, but does not give much informaw x tion on the spectrum itself 13 . In this letter, we present the first solution of this Ž . vexing problem in the case of su 3 . This allows us, in particular, to show that the quantum theory deŽ . fined in 1 for ns3 is never unitary, even in the strong quantum regime, and that it presents considerably more bound states than expected. Our main technical progress is an understanding of the solutions of the Bethe equations for systems Ž . based on su 3 . Except in the exactly symmetric Ž case, these solutions the equivalent of the string . hypothesis for the XXZ chain had never been found. We hope that their understanding will spur new development in the area: generalizations of the works mentioned in the introduction, as well a calculations of physical properties in the super Toda case, seem particularly timely. The problem we want to tackle is largely equivaŽ . lent to the su 3 generalization of the XXZ spin chain. This integrable chain has been known for a w x long time 14 , and reads 3 3 r s sr r r r r Hsy e e qcosg e e Ý Ý Ý j jq1 j jq1 j r , ss1,r/s rs1 3 r r ss qising sign rys e e . 2 Ž . Ž . Ý j jq1 r , ss1 Relations of this system with the quantum field theory are two-fold. At the physical level, there is a simple integrable perturbation – obtained, in the quantum inverse scattering framework, by introducing heterogeneities of the spectral parameter, which Ž . amount to a staggered interaction – giving rise to 1 in the continuum limit, as can easily been shown w x using bosonization, or other arguments 15 . In that p correspondence, gs . At a more formal level, obt serve first that the scattering matrices proposed in w x 16 are non-diagonal. To study the thermodynamics of the gas of excitations in the S matrix approach, one needs to write wave functions, and impose their periodicity. This condition involves ‘‘passing’’ a particle through a set of other particles with which it scatters non-diagonally: the phases obtained in this way are, in the inverse quantum scattering framew x work, eigenvalues of a monodromy matrix 17 . This monodromy matrix is nothing but the transfer matrix Ž . Ž associated with the hamiltonian 2 actually, it involves a slight generalization of this hamiltonian ( ) H. Saleur, B. Wehefritz-KaufmannrPhysics Letters B 481 2000 419–426 421 with a mixture of the fundamental representation and . its conjugate , with however a renormalization of the p anisotropy parameter, which becomes gs . The ty 1 p Ž case gs , for instance the equivalent of the XX 2 . chain, which is not a free problem here however , is 1 a lattice regularization of the Toda theory for Ds ; 2 it also turns out in the monodromy problem for the 2 Toda theory at Ds . 3 This double correspondence is very useful. If one is able to solve say the field theory problem for a ty 1 particular value of Ds , one should also be able t p to diagonalize the lattice model for gs , since it is t just a discretization of this quantum field theory. But p Ž . knowing the spectrum of 2 for gs in turn means t knowing the spectrum of the monodromy matrix for this value of g . This is nothing but being able to t solve the Toda theory for Ds ! tq 1 Rather than dwell into the lengthy technical details, we would like to discuss first the solution of p the lattice model for gs . We consider the slightly 2 more general hamiltonian, where there is an arbitrary mixture of the fundamental representation and its w x conjugate. The lattice Bethe equations are 18 N3 1 sinh y q ipr2 Ž . j 2 1 0 sinh y y ipr2 Ž . j 2 1 sinh y yz y ipr2 Ž . j r 2 s , Ł 1 z sinh y yz q ipr2 Ž . j r 2 N3 1 sinh z q ipr2 Ž . r 2 1 0 sinh z y ipr2 Ž . r 2 1 sinh z yy y ipr2 Ž . r j 2 s , 3 Ž . Ł 1 y sinh z yy q ipr2 Ž . r j 2 where we have not written a crucial but complicated sign on the right hand sides, and the energy reads 1 1 Esy2Ý y2Ý . A combination of numericosh z cosh y cal studies and analytical arguments led to the identification of the following sets of roots in the thermodynamic limit. The y and z can both be real, or both Ž . have an imaginary part equal to p antistring . In addition, it is possible to have complexes, with a two string z centered on an antistring y, that is: ysrq p Ž ip , zsr" i here r is a real number, and we 2 p p . identified rq3i and ry i , and the same thing 2 2 with y and z reversed. The existence of these complexes is easy to understand. Suppose for instance that z has a positive imaginary part, so the lhs Ž . of the second equation in 3 blows up as N TM`. It 3 is then necessary to have the rhs also blow up, which can be accomplished if there exists a y such that p yszq i . The same goes if z has a negative 2 imaginary part, so the complexes proposed are the minimal possible structures leading to real Bethe equations. Together with the real and antistring solutions, they reproduce the 3q3 degeneracy expected for the fundamental solitons. In addition however, Ž another type of complex which we call yz in the 3p . following is possible, of the form zsrq i , ysr 4 5p q i , together with the conjugate. Usually, one 4 would expect that these two complexes actually come glued together to ensure reality of the Bethe equations and the eigenvalues, resulting in a sort of w x ‘‘quartet’’ 19 . This does not seem to be the case here. Rather, to reproduce the correct entropy or central charge, one needs to treat their densities as independent. These complexes are the ones which dominate the thermodynamics. As usual, there are many other solutions to the Bethe equations. The existence of solutions which are not invariant under complex conjugation being rather unusual, we illustrate it briefly. Fig. 1 shows the yz complex obtained by a Ž . numerical solution of 3 for N s0 and for different 3 values of N . In this example – which seems the 3 simplest possible – the half sum of the imaginary parts of y and z goes to p in the thermodynamic p limit, but their difference does not go to . We 2 conjecture that this would however be the case for the overwhelming majority of such complexes: we checked that only in this case, are the correct scattering theory and thermodynamics recovered. The lattice model without heterogeneities has a continuum limit which is a conformal field theory, made of two bosons compactified on a triangular lattice. The latter is unitary, and we have checked numerically that the imaginary parts of the eigenen( ) H. Saleur, B. Wehefritz-KaufmannrPhysics Letters B 481 2000 419–426 422 Ž . Fig. 1. A numerical solution of 3 containing a yz complex for N s0 for different values of N s6,9, . . . ,39. For 39 sites we show all 3 3 roots belonging to this particular state, whereas for all other lattice lengths only the yz complex is shown. Ž . ergies of 2 scale to zero faster than their real parts, ensuring reality of the conformal weights, and, actually, unitarity of the conformal field theory, despite the non-hermicity of the hamiltonian. Meanwhile, the lattice model with heterogeneities does have a continuum limit described by a complex Toda the1 ory, with Ds . In this case, the complexes corre2 spond to physical particles, the bound states disw x cussed in 6,7 . The fact that they are described by solutions of the Bethe ansatz which are not self conjugate corresponds to the shocking fact that, in the quantum field theory, their S matrix is not a pure Ž phase and ‘‘naive’’ unitarity, in the sense of the . scattering theory, is broken . Accordingly, a crucial feature related with the yz complexes is that they give rise to non-real kernels in the continuum Bethe ansatz. For instance, the kernel for the scattering of a real z through a yz 3p 5p 1 d complex zsrq i , ysrq i is lnFsy1r 4 4 i dz ip cosh zyrq . Although at equilibrium the densiŽ . 4 ties of the two types of conjugate complexes are equal, it is necessary to let them vary independently to get the correct entropy. The physical results will be the same as the ones that would be obtained with a different theory, where the scattering kernels would be real, and given by the real part of the true kernels. This is what we will consider in the following, to make the notations simpler. Ž . The lattice model whose equations are 3 occurs also in the problem of diagonalizing the monodromy 2 matrix for Ds . The complexes just discussed have 3 no physical meaning then: they just correspond to ( ) H. Saleur, B. Wehefritz-KaufmannrPhysics Letters B 481 2000 419–426 423 pseudo particles of zero mass. However, the fact that they are not self conjugate means that eigenvalues of the monodromy matrix will not, in general, have w x modulus one. Contrary to early claims 6 , unitarity is strongly violated in the theory, even in the absence of bound states. The physical implications of having ‘‘monodromies’’ which are not pure phases are not clear to us: presumably S matrix elements have to be considered as formal objects used to build wave functions, and cannot be given a reasonably meaning in terms of scattering processes . This basic structure generalizes easily to the case p t integer. For gs , the Bethe equations contain in t addition a term of interaction between y roots, and between z roots. The solutions are the usual y or z real, antistrings, or 2, . . . ,ty1 strings, plus y-t strings centered on z antistrings, and the same thing with y and z reversed. In addition, the yz comp plexes now are of the form zsrq ipy i , ysr 2 t p q ipq i , and the same with y and z exchanged. 2 t To proceed, one has to use these results to solve ty 1 the monodromy problem. At Ds , passing a t soliton through a set of N , N solitons at various 3 3 rapidities gives rise to a ‘‘phase’’ which is an eigenvalue of the monodromy matrix. The latter is essenŽ . tially the transfer matrix associated with 2 , but for p gs . Its eigenvalues are obtained by generalizing ty 1 slightly the Bethe ansatz equations, so the arguments Ž . in the lhs of 3 are shifted by the corresponding rapidities, and using the generalized string hypothesis described previously. A few technical details are involved, which are Ž . completely equivalent to what happens in the su 2 w x case 20 . We will only describe the end result, which is quite simple. For the case tTM`, the TBA w x has been known for a long time 21,22 , and has a Ž . structure that mimics the known one for su 2 , with an infinity of massless nodes corresponding to the usual strings of the Bethe equations. Like for the Ž . su 2 case, the introduction of anisotropy truncates this to a finite number of strings: the truncation has to be completed by the proper ‘‘end structure’’ of the Ž . diagram. Like in the su 2 case, this structure, for p Ž . su 3 , is given by the Bethe roots for gs . After 2 3 A somewhat related problem occurs for instance in the massless description of conformal field theories. some manipulations, the results are the following. We call s ,s h the densities of the solitons at rapidity u , and m the mass of the fundamental soliton. Then 3 3h 3h 3h s qs smcoshuqfw r qs , . . . , Ž . 1 3 3h 3h 3h 3 r qr sfw r qr qr . . . 4 Ž . Ž . n n ny1 nq1 n and similarly for 3, with r 's . We recognize here 0 w x the standard equations for the minimal model 16 . In addition, in this untruncated case, we need the ‘‘closure’’ relations, which are the key to the whole problem. They read here 3 3h 3h 3h 3 3 3 r qr sfw r qr qr qr qr ž / ty3 ty3 ty4 ty2 ty3 a a 1 2 3 3 qcw r qr , Ž . b b 3 3h 3h 3 3 3 r qr sfw r qr yr yr ž / ty2 ty2 ty3 ty2 a a 1 2 3 3 ycw r qr , Ž . b b r 3 qr 3h sr 3 qr 3h 5 Ž . a a ty2 ty2 i i and similarly for 3; finally 3 3h 3h 3 3 3 r qr scw r qr yr yr q3TM3 ž / b b ty3 ty2 a a 1 2 3 3 yfw r qr . 6 Ž . Ž . b b In these equations, the kernels are defined by their 1 cosh xr2 4 ˆ ˆ fourier transforms fs and cs . The 2cosh x 2cosh x subscripts a , is1,2 stand for antistrings or t strings i centered on antistrings; b stands for yz complexes. We set e 0 r T ssrs , e n r T sr rr , ns i i 1, . . . ,t y 2, e a r T s r rr h , e b r T s r rr h a a b b i i Ž . color labels are kept implicit here . Introducing the 4 1 yi n xu rp Ž̂ . Ž . We define f x s Hf u e du . 2p ( ) H. Saleur, B. Wehefritz-KaufmannrPhysics Letters B 481 2000 419–426 424 usual variables x sea r , the TBA in the UV a Ž . limit reduces to the system x s1rx : ty2 a y1r2 1 1r2 x s 1qx 1q , . . . , Ž . 0 1 ž / x0 1r2 1r2 x s 1qx 1qx Ž . Ž . n ny1 nq1 = y1r2 1 1q , . . . , ž / xn y1r2 1 1r2 3r2 x s 1qx 1qx 1q Ž . Ž . ty3 ty4 a ž / xty3