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arXiv:1707.02584 · Journal of Integrable Systems

The algebraic Bethe Ansatz and combinatorial trees

R. S. Vieira, A. Lima-Santos

Published Jan 1, 2019• 3 citations
arXivHTMLPDF

Abstract

We study in this article the algebraic Bethe Ansatz from a combinatorial perspective.The commutation relations are represented by combinatorial diagrams so that the action of the transfer matrix over the nth excited state can be found through the analysis of some combinatorial trees.This allows us to compute the eigenvalues and eigenstates of the transfer matrix, as well as the respective Bethe Ansatz equations, in a straightforward way.As examples, we consider the six-vertex model with both periodic and non-periodic boundary conditions.In parallel, we also present a comprehensive introduction to the usual algebraic Bethe Ansatz (for both the periodic and the non-periodic cases).

DOI
10.1093/integr/xyz002

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We study in this article the algebraic Bethe Ansatz from a combinatorial perspective.The commutation relations are represented by combinatorial diagrams so that the action of the transfer matrix over the nth excited state can be found through the analysis of some combinatorial trees.This allows us to compute the eigenvalues and eigenstates of the transfer matrix, as well as the respective Bethe Ansatz equations, in a straightforward way.As examples, we consider the six-vertex model with both periodic and non-periodic boundary conditions.In parallel, we also present a comprehensive introduction to the usual algebraic Bethe Ansatz (for both the periodic and the non-periodic cases).
The algebraic Bethe ansatz is one of the main approaches to solving integrable models in statistical mechanics. This thesis is a detailed introduction to aspects of this technique. In chapter 1, following Takhtajan1, we fully explain all preliminaries that are required by this technique in the context of the 8-vertex model, including the solution of the Yang-Baxter equations in terms of Boltzmann weights that are parameterized by elliptic function. In chapter 2, also following Takhtajan, we restrict our attention to the special case of the 6-vertex model, with Boltzmann weights parameterized by hyperbolic trigonometric functions, formulate the model in detail on a finite lattice with periodic boundary conditions, and show how the algebraic Bethe ansatz can be used to characterize the eigenvectors and eigenvalues, in the homogeneous case (all vertical rapidities are equal, and all horizontal rapidities are also equal). In chapter 3, following Bogoliubov et al2, we consider the 6-vertex model in inhomogeneous case (all rapidities are now independenti variables) but with domain wall boundary conditions. Using the algebraic Bethe ansatz, and Izergin’s result for the partition function, we calculate boundary 1-point functions of the six-vertex model on a finite lattice and represent them in determinant form. Takhatajan, L. J. Introduction to Agebraic Bethe Ansatz, Lecture Notes in Physics, Exactly solvable problems in condensed matter and relativistic field theory 2 Boundary correlation functions of the six-vertex model J. Phys. A: Math. Gen. 35, 5525-5541 (2002)
Introduction The algebraic Bethe Ansatz is presented in this chapter. This is an important generalization of the coordinate Bethe Ansatz presented in Part I, and is one of the essential achievements of QISM. The algebraic Bethe Ansatz is based on the idea of constructing eigenfunctions of the Hamiltonian via creation and annihilation operators acting on a pseudovacuum. The matrix elements of the monodromy matrix play the role of these operators. The transfer matrix (the sum of the diagonal elements of the monodromy matrix) commutes with the Hamiltonian; thus constructing eigenfunctions of τ(µ) determines the eigenfunctions of the Hamiltonian. The basis of the algebraic Bethe Ansatz is stated in section 1. The commutation relations between matrix elements of the monodromy matrix are specified by the R-matrix. The explicit form of the commutation relations allows the construction of eigenfunctions of the transfer matrix (the trace of the monodromy matrix). (Recall that the Hamiltonian may also be obtained from the transfer matrix via the trace identities.) Further developments of the algebraic Bethe Ansatz necessary for the computation of correlation functions are given in section 2. The general scheme is illustrated with some examples in section 3. The NS model, the sine-Gordon model and spin models are considered in detail. The Pauli principle for interacting one-dimensional bosons plays an important role in constructing the ground state of the system and is discussed in section 4. The eigenvalues of the shift operator acting on the monodromy matrix are calculated in section 5.
In this paper, we review a few known facts on the coordinate Bethe ansatz. We present a detailed construction of the Bethe ansatz vector $\psi $ and energy $\Lambda $, which satisfy $V\psi =\Lambda \psi $, where VVV is the transfer matrix of the six-vertex model on a finite square lattice with periodic boundary conditions for weights a=b=1a=b=1a=b=1 and c>0c>0c>0. We also show that the same vector $\psi $ satisfies $H\psi =E\psi $, where HHH is the Hamiltonian of the XXZ model (which is the model for which the Bethe ansatz was first developed), with a value EEE computed explicitly. Variants of this approach have become central techniques for the study of exactly solvable statistical mechanics models in both the physics and mathematics communities. Our aim in this paper is to provide a pedagogically-minded exposition of this construction, aimed at a mathematical audience. It also provides the opportunity to introduce the notation and framework which will be used in a subsequent paper by the authors [5] that amounts to proving that the random-cluster model on Z2\mathbb{Z}^{2}Z2 with cluster weight q>4q>4q>4 exhibits a first-order phase transition.