Yang-Baxter Equation in Integrable Systems

The following sections are included:

• Preface

• CONTENTS

At an early stage the Yang-Baxter equation (YBE) appeared in several different guises in the literature, and sometimes its solutions have preceded the equation. One can trace basically three streams of ideas from which YBE has emerged: the Bethe Ansatz, commuting transfer matrices in statistical mechanics, and factorizable S matrices in field theory. For general reference about the first two topics the reader is referred to the books of Gaudin⁹⁶⁾ and of Baxter²⁰⁾ (see also ¹⁹⁾)…

The repulsive δ interaction problem in one dimension for N particles is reduced, through the use of Bethe's hypothesis, to an eigenvalue problem of matrices of the same sizes as the irreducible representations R of the permutation group S_N. For some R's this eigenvalue problem itself is solved by a second use of Bethe's hypothesis,in a generalized form. In particular, the ground-state problem of spin- fermions is reduced to a generalized Fredholm equation.

For N particles with equal mass, interacting with repulsive or attractive δ-function interaction of the same strength, the S matrix is explicitly given and shown to be symmetrical and unitary. The incoming and outgoing states may consist of bound compounds as well as single particles. The momenta of the particles and compounds are not changed in the scattering, but particles are exchanged, such as ABC+DE→ADC+BE. Only distinguishable particles are considered.

The partition function of the zero-field “Eight-Vertex” model on a square M by N lattice is calculated exactly in the limit of M, N large. This model includes the dimer, ice and zero-field Ising, F and KDP models as special cases. In general the free energy has a branch point singularity at a phase transition, with an irrational exponent.

The following sections are included:

• Introduction

• The general model

• Formulation as an eight-vertex model
  • Duality

• Star-triangle conditions
  • Corollaries
  • Quadrilateral theorem
  • Conditions for model to be solvable
  • Extended lattices

• Local thermodynamic properties
  • Free energy

• Elliptic function parametrization
  • Z-invariance conditions in terms of the elliptic angle parameters
  • Geometric model
  • Connection with Onsager's parametrization: the case K" = 0

• Expressions for f, M, P
  • Relation between the elliptic parametrizations in the ordered ferromagnetic phase
  • Phase boundaries

• Kagomé lattice eight-vertex model
  • Previous models as special cases
  • Classification of phases
  • Critical behaviour

• Two-spin correlations
  • Intra-row correlations in the Kagomé lattice
  • Commuting transfer matrices
  • Disordered ferromagnetic phase
  • Ising model case: K" = 0

• References

• Appendix A. Proof of Quadrilateral Theorem

The general properties of the factorized S-matrix in two-dimensional space-time are considered. The relation between the factorization property of the scattering theory and the infinite number of conservation laws of the underlying field theory is discussed. The factorization of the total S-matrix is shown to impose hard restrictions on two-particle matrix elements: they should satisfy special identities, the so-called factorization equations. The general solution of the unitarity, crossing and factorization equations is found for the S-matrices having isotopic O(N)-symmetry. The solution turns out to have different properties for the cases N = 2 and N ≥ 3. For N = 2 the general solution depends on one parameter (of coupling constant type), whereas the solution for N ≥ 3 has no parameters but depends analytically on N. The solution for N = 2 is shown to be an exact soliton S-matrix of the sine-Gordon model (equivalently the massive Thirring model). The total S-matrix of the model is constructed. In the case of N ≥ 3 there are two “minimum” solutions, i.e., those having a minimum set of singularities. One of them is shown to be an exact S matrix of the quantum O(N)-symmetric nonlinear σ-model, the other is argued to describe the scattering of elementary particles of the Gross-Neveu model.

This paper is a developed and consecutive account of a quantum version of the method of the inverse scattering problem on the example of the nonlinear Schrödinger equation. The method of R-matrices developed by the author is given basic consideration. The generating functions of quantum integrals of motion and action-angle variables for the quantum nonlinear Schrödinger equation are obtained. There is also described a classical version of the method of R-matrices.

We give the basic definitions connected with the Yang-Baxter equation (factorization condition for a multiparticle S-matrix) and formulate the problem of classifying its solutions. We list the known methods of solution of the Y–B equation, and also various applications of this equation to the theory of completely integrable quantum and classical systems. A generalization of the Y–B equation to the case of Z₂-graduation is obtained, a possible connection with the theory of representations is noted. The supplement contains about 20 explicit solutions.

Compared to (quantum) YBE, an important and simplifying feature of CYBE is that it can be formulated in the realm of Lie algebras, independently of the way it is represented by matrices. In the reprint [8] (see also ⁴²⁾), Belavin and Drinfel'd gave a detailed study of solutons of CYBE associated with complex simple Lie algebras. They succeeded in classifying all the non-degenerate elliptic and trigonometric solutions using the data from the Dynkin diagram. The paper ⁴³⁾ is a readable review of their theory. Analogous classification for Lie superalgebras is given in ¹⁵⁵⁾…

The following sections are included:

• Introduction

• Equivalence of Three Definitions of Nondegeneracy

• Weierstrass-Type Theorem

• Properties of Nondegenerate Solutions

• Elliptic Solutions

• Trigonometric Solutions

• Rational Solutions, not Having a Pole at Infinity

• LITERATURE CITED

Please refer to full text.

The following sections are included:

• Classical r-Matrices and Double Lie Algebras

• Relation with the Classical Formalism

• Yang-Baxter Equations

• Factorization Theorem

• Quadratic Poisson Brackets

• Two-Dimensionalization

• Classification of r-Matrices for Finite-Dimensional Semisimple Lie Algebras

• LITERATURE CITED

Through the development of QISM it became apparent that upon quantization of classical systems some of the structures undergo quantum deformation ^{108-109),145)}. In the course of constructing trigonometric solutions of YBE, Kulish and Reshetikhin ¹⁴¹⁾ introduced a deformation of the universal enveloping algebra of sl(2). In the reprint [11] Sklyanin defines an elliptic version in connection with the eight-vertex model, and found three series of representations of this algebra. These are the first occurrences of new algebraic objects, now called ‘quantum groups’…

The following sections are included:

• Classical Case

• Quantum Case

• Possible Applications and Generalizations

• LITERATURE CITED

The following sections are included:

• Representations of Series a)

• Representations of Series b) and c)

• Degenerate Cases

• Conclusions

• LITERATURE CITED

Let a be a given finite-dimensional simple Lie algebra over C with a fixed invariant inner product. According to [1], the function r(u) = u⁻¹I_μ ⊗ I_μ, where {I_μ} is an orthonormal basis in a (summation over repeated indices is always assumed to be carried out), satisfies the classical Yang-Baxter equation (CYBE). If, in addition, a representation ρ: a → End V is given, the question arises (see [4]) whether the quantum Yang-Baxter equation (QYBE) has a solution which can be written as a formal series

The following sections are included:

• What is a quantum group?

• Quantization

• Poisson groups and Lie bialgebras

• Classical Yang-Baxter equation

• Some historical remarks

• Quantization of Lie bialgebras (examples)

• Quantized universal enveloping algebras and h-formal groups

• The square of the antipode

• Obstruction theory

• Coboundary, triangular, and quasitriangular Hopf algebras

• QISM from the Hopf algebra point of view

• The pseudotriangular structure on Yangians and rational solutions of the QYBE

• The double. Trigonometric solutions of the QYBE

• Final remarks

• REFERENCES

A q-difference analogue of the universal enveloping algebra U(g) of a simple Lie algebra g is introduced. Its structure and representations are studied in the simplest case g = sl(2). It is then applied to determine the eigenvalues of the trigonometric solution of the Yang-Baxter equation related to sl(2) in an arbitrary finite-dimensional irreducible representation.

The following sections are included:

• Quantum Formal Groups

• Finite-Dimensional Example

• Infinite-Dimensional Example

• Deformation Theory and Quantum Groups

• References

The articles assembled in this section present explicit solutions of YBE, which are ‘quantization’ of the classical r matrices due to Belavin and Drinfel'd [8].

The group-invariant R matrices known at an earlier stage ^{182),207),265),274-276)} correspond to the simplest rational solutions in [8]…

We consider the equations of triangles (alias Yang-Baxter equations), which a factorized two-particle S-matrix obeys. These equations are shown to possess a symmetry which consists of discrete Lorentz transformations acting independently on states of particles with different momenta. It is this symmetry which ensures compatibility of the overconstrained equations of triangles. The use of it enables one to construct the factorized two-particle S-matrix requiring invariance (automorphity) with respect to discrete Lorentz transformations.

Using the method of commuting transfer matrices new exactly-solvable q-state vertex models and their associated spin-chain Hamiltonians are found. A partial review of related work is given.

We have obtained six new infinite series of trigonometric solutions to triangle equations (quantum R-matrices) associated with the nonexceptional simple Lie algebras: sl(N), sp(N), o(N). The R-matrices are given in two equivalent representations: in an additive one (as a sum of poles with matrix coefficients) and in a multiplicative one (as a ratio of entire matrix functions). These R-matrices provide an exact integrability of anisotropic generalizations of sl(N), sp(N), o(N) invariant one-dimensional lattice magnetics and two-dimensional periodic Toda lattices associated with the above algebras.

We report the explicit form of the quantum R matrix in the fundamental representation for the generalized Toda system associated with non-exceptional affine Lie algebras.

In statistical mechanics, there are usually three ways of formulating two dimensional lattice models. These are called vertex models, interaction-round-the-face (IRF) medels and (Ising-type) spin models, depending on the form of the interaction. The YBE is written indifferent forms accordingly. Mathematically the three formulations are mutually equivalent⁸⁾, but in each particular case one is often more natural and convenient than the others. The solutions of Section 4 all belong to the category of vertex models…

We obtain some simple eigenvectors of the transfer matrix of the zero-field eight-vertex model. These are also eigenvectors of the Hamiltonian of the one-dimensional anisotropic Heisenberg chain. We also obtain new equations for the matrix Q(v) introduced in earlier papers.

We establish an equivalence between the zero-field eight-vertex model and an Ising model (with four-spin interaction) in which each spin has L possible values, labeled 1, …, L,and two adjacent spins must differ by one (to modulus L). Such an Ising model can also be thought of as a generalized ice-type model and we will later show that the eigenvectors of the transfer matrix can be obtained by a Bethe-type ansatz.

We obtain the eigenvectors of the transfer matrix of the zero-field eight vertex model. These are also the eigenvectors of the Hamiltonian of the corresponding one-dimensional anistropic Heisenberg chain.

A series of solvable lattice models with face interaction are introduced on the basis of the affine Lie algebra X_n⁽¹⁾ = A_n⁽¹⁾, B_n⁽¹⁾, C_n⁽¹⁾, D_n⁽¹⁾. The local states taken on by the fluctuation variables are the dominant integral weights of X_n⁽¹⁾ of a fixed level. Adjacent local states are subject to a condition related to the vector representation of X_n. The Boltzmann weights are parametrized by elliptic theta functions and solve the star-triangle relation.

We show that a class of 2D statistical mechanics models known as IRF models can be viewed as a subalgebra of the operator algebra of vertex models. Extending the Wigner calculus to quantum groups, we obtain an explicit intertwiner between two representations of this subalgebra.

The self-dual solution of the star-triangle relations in Z_N models is presented. The corresponding partition functions are calculated.

In classical representation theory it is often useful to consider the decomposition of tensor products of fundamental representations. This affords in particular a way of realizing ‘higher’ representations explicitly. The fusion procedure is an analog of this technique which enables one to generate new solutions to YBE out of elementary ones. Systematic study of this sort has begun in the reprint [24] by Kulish, Reshetikhin and Sklyanin for the rational R matrix associated with gl(n). They showed that the products of R matrices, with the spectral parameters suitably shifted and followed by projecton operators, yield new R matrices corresponding to symmetric or anti-symmetric tensor representations. (Such an idea is also implicit in the paper²⁹⁰⁾.) As Cherednik shows in the reprint [26], this method can be generalized to include arbitrary irreducible representations. It is applicable further to the elliptic solutions of Baxter and Belavin…

The problem of constructing the GL(N, ℂ)-invariant solutions to the Yang–Baxter equation (factorized S-matrices) is considered. In case N = 2 all the solutions for arbitrarily finite-dimensional irreducible representations of GL(2, ℂ) are obtained and their eigenvalues are calculated. Some results for the case N > 2 are also presented.

The following sections are included:

• Introduction
  • The fusion models
  • The corner transfer matrix (CTM) method
  • Linear difference equations
  • The 1D configuration sums as modular functions
  • Notations

• The Fusion Models
  • Fusion of SOS models
  • Restricted SOS models
  • Vertex-SOS correspondence

• One Dimensional Configuration Sums
  • Boltzmann weights in the conjugate modulus
  • Regime II, III
  • Regime I, IV

• Combinatorial Identities
  • Fundamental solution for the linear difference equation
  • Various representations for
  • Fundamental solution for the modified linear difference equation
  • 1D configuration sums as superpositions of the fundamental solutions

• One Dimensional Configuration Sums as Modular Functions
  • Regimes III and IV
  • Regime II

• Appendix A. Minimum/Maximum Configurations
  • Regime III
  • Regime IV

• Appendix B. Ground State Configuration in Regime II

• Appendix C. Branching Coefficients and String Functions
  • Products of theta functions
  • Branching coefficients and string functions
  • Branching coefficients for m₂ = 3, 4

• Appendix D. Free Energy
  • Unitarity and crossing symmetry
  • The free energy of the fusion vertex models and the restricted SOS models
  • Critical behavior

• References

The following sections are included:

• R-matrices

• Young projections

• Examples

• Deformation of U(gl_N)

• Maximal commutative subalgebra of A

• BIBLIOGRAPHY

The eigenvalues of the unitarizable and factorized S-matrices and their transfer matrices are presented under the form of the Bethe-ansatz equations expressed in terms of the simple root system and of the automorphisms of the corresponding simple Lie algebra.

This last section concerns two important directions of generalizing YBE.

In the reprint [28] Zamolodchikov introduced the three dimensional analog of YBE, called tetrahedron equations, and postulated an explicit solution to it in trigonometric parametrization. That it indeed solves the tetrahedron equations was later proved in the reprint by Baxter [29], who also computed its free energy²³⁾. Bazhanov and Stroganov³²⁾ formulated higher dimensional analogs of YBE (called d-simplex equations) and showed that they imply the commutativity of the transfer matrices. They also studied the free-fermion model in higher dimensions³³⁾. See also ^166-167). Related discussions on the higher dimensional geometry can be found in¹⁷³⁾. Because of the complexity of the equations, it remains a challenge to find non-trivial solutions other than the examples mentioned above…

The quantum S-matrix theory of straight-strings (infinite one-dimensional objects like straight domain walls) in 2+1-dimensions is considered. The S-matrix is supposed to be “purely elastic” and factorized. The tetrahedron equations (which are the factorization conditions) are investigated for the special “two-colour” model. The relativistic three-string S-matrix, which apparently satisfies this tetrahedron equation, is proposed.

The tetrahedron equations arise in field theory as the condition for the S-matrix in 2+1-dimensions to be factorizable, and in statistical mechanics as the condition that the transfer matrices of three-dimensional models commute. Zamolodchikov has proposed what appear (from numerical evidence and special cases) to be non-trivial particular solutions of these quations, but has not fully verified them. Here it is proved that they are indeed solutions.

We present new explicit N-state solutions of the star–triangle relations for a nearest-neighbour two-spin interaction model. The solutions include families with real and positive Boltzmann weights. They are given in terms of two rapidities associated with two lines, which cross through each edge. The rapidities are 4-vectors, restricted to lie on the intersection of two Fermat surfaces. The usual difference property is not present.

We observe that N-state integrable chiral Potts model can be considered as a part of some new algebraic structure related to six-vertex model. As a result we obtain a functional equation which determine all the eigenvalues of the chiral Potts model transfer matrix.

The following sections are included:

• LIST OF REFERENCES