Analytic theory of the eight-vertex model

We observe that the exactly solved eight-vertex solid-on-solid model contains an hitherto unnoticed arbitrary field parameter, similar to the horizontal field in the six-vertex model. The parameter is required to describe a continuous spectrum of the unrestricted solid-on-solid model, which has an infinite-dimensional space of states even for a finite lattice. The introduction of the continuous field parameter allows us to completely review the theory of functional relations in the eight-vertex/SOS-model from a uniform analytic point of view. We also present a number of analytic and numerical techniques for the analysis of the Bethe ansatz equations. It turns out that different solutions of these equations can be obtained from each other by analytic continuation. In particular, for small lattices we explicitly demonstrate that the largest and smallest eigenvalues of the transfer matrix of the eight-vertex model are just different branches of the same multivalued function of the field parameter.     

The asymmetric six-vertex model

The exact solution of the asymmetric six-vertex model, published nearly without derivation by Sutherlandet al. in 1967, is rederived in detail. The transfer matrix method and the Bethe Ansatz solution for the free energy (which can be calculated from an integral equation) are discussed. For some special cases (zero or maximal polarization) the integral equation can be solved exactly. In addition, an asymptotic analysis, valid for small but nonzero polarization, is carried out. The analytical properties of the results and their relevance for the BCSOS model are discussed.     

The matrix product ansatz for the six-vertex model

Recently it was shown that the eigenfunctions for the asymmetric exclusion problem and several of its generalizations as well as a huge family of quantum chains, like the anisotropic Heisenberg model, Fateev–Zamolodchikov model, Izergin–Korepin model, Sutherland model, t–J$t\text{–}J$ model, Hubbard model, etc, can be expressed by a matrix product ansatz. Differently from the coordinate Bethe ansatz, where the eigenvalues and eigenvectors are plane wave combinations, in this ansatz the components of the eigenfunctions are obtained through the algebraic properties of properly defined matrices. In this work, we introduce a formulation of a matrix product ansatz for the six-vertex model with periodic boundary condition, which is the paradigmatic example of integrability in two dimensions. Remarkably, our studies of the six-vertex model are in agreement with the conjecture that all models exactly solved by the Bethe ansatz can also be solved by an appropriated matrix product ansatz.     

The new formulas for the eigenvectors of the Gaudin model in the <Emphasis Type="Italic">so</Emphasis>(5) case

In our recent paper we proposed new formulas for eigenvectors of the Gaudin model in sl(3) case. Similarly in this paper we used the standard Bethe Ansatz method for finding the eigenvectors and the eigenvalues in the so(5) case in an explicit form.     

Gaudin model,Bethe Ansatz and critical level

We propose a new method of diagonalization oif hamiltonians of the Gaudin model associated to an arbitrary simple Lie algebra, which is based on the Wakimoto modules over affine algebras at the critical level. We construct eigenvectors of these hamiltonians by restricting certain invariant functionals on tensoproducts of Wakimoto modules. This gives explicit formulas for the eigenvectors via bosonic correlation functions. Analogues of the Bethe Ansatz equations naturally appear as equations on the existence of singular vectors in Wakimoto modules. We use this construction to explain the connection between Gaudin's model and correlation functios of WZNW models.     

Scalar Products in Generalized Models with SU(3)-Symmetry

We consider a generalized model with SU(3)-invariant R-matrix, and review the nested Bethe Ansatz for constructing eigenvectors of the transfer matrix. A sum formula for the scalar product between generic Bethe vectors, originally obtained by Reshetikhin, is discussed. This formula depends on a certain partition function Z({λ}, {μ}|{w}, {v}), which we evaluate explicitly. In the limit when the variables {μ} or ${\{v\}\rightarrow \infty}$ , this object reduces to the domain wall partition function of the six-vertex model Z({λ}|{w}). Using this fact, we obtain a new expression for the off-shell scalar product (between a generic Bethe vector and a Bethe eigenvector), in the case when one set of Bethe variables tends to infinity. The expression obtained is a product of determinants, one of which is the Slavnov determinant from SU(2) theory.     

Orthogonality and completeness of the Bethe Ansatz eigenstates of the nonlinear Schroedinger model

A rigorous proof is given of the orthogonality and the completeness of the Bethe Ansatz eigenstates of theN-body Hamiltonian of the nonlinear Schroedinger model on a finite interval. The completeness proof is based on ideas of C.N. Yang and C.P. Yang, but their continuity argument at infinite coupling is replaced by operator monotonicity at zero coupling. The orthogonality proof uses the algebraic Bethe Ansatz method or inverse scattering method applied to a lattice approximation introduced by Izergin and Korepin. The latter model is defined in terms of monodromy matrices without writing down an explicit Hamiltonian. It is shown that the eigenfunctions of the transfer matrices for this model converge to the Bethe Ansatz eigenstates of the nonlinear Schroedinger model.     

Symmetry relations in the sixteen-vertex model

We explore the consequences of the existence of a group of linear symmetry transformations for the sixteen-vertex model. An arbitrary sixteen-vertex model is shown tobe equivalent to a standard model, described in terms of ten parameters having an invariant significance. The general eight-vertex model is obtained when four of the invariant parameters vanish. Two further subciasses of the sixteen-vertex model are found to be equivalent to the eight-vertex model. The invariant conditions leading to the solved cases (i.e the general six-vertex, the free-fermion and the symmetric eight-vertex models) are given.     

Completeness of the Bethe Ansatz for the Six and Eight-Vertex Models

We discuss some of the difficulties that have been mentioned in the literature in connection with the Bethe ansatz for the six-vertex model and XXZ chain, and for the eight-vertex model. In particular we discuss the beyond the equator, infinite momenta and exact complete string problems. We show how they can be overcome and conclude that the coordinate Bethe ansatz does indeed give a complete set of states, as expected.     

具有非对角开边界的SU(2)不变Thirring模型的精确解

利用量子反散射方法研究了1+1维时空中具有非对角开边界条件下的SU(2)不变Thirring模型. 于辅助空间引入独立于谱参量的规范变换,找到了适当的Fock真空态. 通过Bethe Ansatz方法得到了系统相应转移矩阵的本征值和本征态,及其谱参数所满足的Bethe Ansatz方程,并讨论了体系的边界自由度. 关键词： SU(2)不变Thirring模型')" href="#">SU(2)不变Thirring模型 非对角开边界 量子反散射方法     

Diagonalization of theXXZ Hamiltonian by vertex operators

We diagonalize the anti-ferroelectricXXZ-Hamiltonian directly in the thermodynamic limit, where the model becomes invariant under the action of . Our method is based on the representation theory of quantum affine algebras, the related vertex operators and KZ equation, and thereby bypasses the usual process of starting from a finite lattice, taking the thermodynamic limit and filling the Dirac sea. From recent results on the algebraic structure of the corner transfer matrix of the model, we obtain the vacuum vector of the Hamiltonian. The rest of the eigenvectors are obtained by applying the vertex operators, which act as particle creation operators in the space of eigenvectors. We check the agreement of our results with those obtained using the Bethe Ansatz in a number of cases, and with others obtained in the scaling limit—thesu(2)-invariant Thirring model.Dedicated to Professors Huzihiro Araki and Noboru Nakanishi on the occasion of their sixtieth birthdays     

Novel magnetic properties of the Hubbard chain with an attractive interaction

The magnetic properties of an attractive Hubbard chain are considered. Based on the Bethe Ansatz equations of the problem, exact analytic expressions are derived for the magnetization and susceptibility. These formulae can be evaluated after solving certain derivatives of the Bethe Ansatz equations. These derivative equations are also given. We give the magnetization and susceptibility curves for several values of the interaction-strength and bandfilling. We find that the susceptibility at the onset of magnetization (at the critical field) isfinite for all bandfillings, except for the cases of half filled and empty bands, and in the limit of vanishing interaction. We argue that the finiteness of the initial susceptibility is due to the fermion-like behavior of the bound pairs. We also give the gap (what is equal to the critical field) and the initial susceptibility as functions of the interaction-strength and bandfilling for the cases of nearly half filled and almost empty bands as a function of the interaction, and in the weak coupling limit as a function of the bandfilling. To our knowledge, this is the first Bethe Ansatz calculation for the gap in this latter limit.     

Algebraic Bethe ansatz for the eight-vertex model with general open boundary conditions

By using the intertwiner and face-vertex correspondence relation, we obtain the Bethe ansatz equation of the eight-vertex model with open boundary conditions in the framework of algebraic Bethe ansatz method. The open boundary condition under consideration is the general solution of the reflection equation for the eight-vertex model with only one restriction on the free parameters of the right side reflecting boundary matrix. The reflecting boundary matrices used in this paper thus may have off-diagonal elements. Our construction can also be used for the Bethe ansatz of SOS model with reflection boundaries.     

Integrable time-discretisation of the Ruijsenaars-Schneider model

An exactly integrable symplectic correspondence is derived which in a continuum limit leads to the equations of motion of the relativistic generalization of the Calogero-Moser system, that was introduced for the first time by Ruijsenaars and Schneider. For the discrete-time model the equations of motion take the form of Bethe Ansatz equations for the inhomogeneous spin-1/2 XYZ Heisenberg magnet. We present a Lax pair, the sympletic structure and prove the involutivity of the invariants. Exact solutions are investigated in the rational and hyperbolic (trigonometric) limits of the system that is given in terms of elliptic functions. These solutions are connected with discrete soliton equations. The results obtained allow us to consider the Bethe Ansatz equations as ones giving an integrable symplectic correspondence mixing the parameters of the quantum integrable system and the parameters of the corresponding Bethe wavefunction.Supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO)     

Existence of Solutions to the Bethe Ansatz Equations for the 1D Hubbard Model: Finite Lattice and Thermodynamic Limit

In this work, we present a proof of the existence of real and ordered solutions to the generalized Bethe Ansatz equations for the one dimensional Hubbard model on a finite lattice, with periodic boundary conditions. The existence of a continuous set of solutions extending from any U>0 to U=∞ is also shown. We use this continuity property, combined with the proof that the norm of the wavefunction obtained with the generalized Bethe Ansatz is not zero, to prove that the solution gives us the ground state of the finite system, as assumed by Lieb and Wu. Lastly, for the absolute ground state at half-filling, we show that the solution converges to a distribution in the thermodynamic limit. This limit distribution satisfies the integral equations that led to the Lieb-Wu solution of the 1D Hubbard model.     

The eight-vertex model: Spectrum of the transfer matrix and classification of the excited states

We study the eigenvalue problem for the transfer matrix of the eight-vertex model by analyzing the Bethe Ansatz equations. Using functional equations and complex variable theory the excited states of the model are completely classified in terms of strings of complex moments. The different possible strings are determined. The main result of this paper is the derivation of a basic set of equations from which string positions can be calculated in terms of excitation parameters. The case of 2- and 4-particle excitations is analyzed in detail.Work performed within the research program of the Sonderforschungsbereich 125, Aachen-Jülich-Köln     

Order and disorder lines in systems with competing interactions. III. Exact results from stochastic crystal growth

The methods presented in the first two articles of this series are simplified and generalized by growing stationary stochastic crystals from a given Ansatz layer. On the disorder trajectory the free energy, correlation functions, and multicritical points are calculated explicitly for a large class of models with competing interactions, including the staggered eight-vertex model, the general sixteen-vertex model, theq-state Potts model on a triangular lattice, a generalZ(q) model, and restricted spin glass models in two dimensions.     

Row transfer matrix spectra of cyclic solid-on-solid lattice models

The eigenvalue spectra of cyclic solid-on-solid (CSOS) row transfer matrices are studied. An equivalence is established between the inversion identity and the Bethe ansatz functional equations and these equations are solved in the thermodynamic limit by a Wiener-Hopf perturbation technique for the bands of leading excitations. TheL-state CSOS model, with crossing parameter=s/L, possesses a 2(L – s)-fold degenerate largest eigenvalue corresponding to the 2(L – s) coexisting phases. The expressions for the largest eigenvalue and free energy coincide with those of the eight-vertex model. The string excitations for 2s < L and 2s > L admit different classifications and are treated separately. The correlation length is calculated in both regimes, yielding the critical exponentv=L/2s, in agreement with the scaling relations.     

The inversion relation method for some two-dimensional exactly solved models in lattice statistics

The partition-functions-per-site of several two-dimensional models (notably the eight-vertex, self-dual Potts and hard-hexagon models) can be easily obtained by using an inversion relation for local transfer matrices, together with symmetry and analyticity properties. This technique is discussed, the analyticity properties compared, and some equivalences (and nonequivalences) pointed out. In particular, the critical hard-hexagon model is found to have the same as the self-dualq-state Potts model, withq=(3 + 5)/2 = 2.618 .... The Temperley-Lieb equivalence between the Potts and six-vertex models is found to fail in certain nonphysical antiferromagnetic cases.