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arXiv:nlin/0602003 · Journal of Statistical Mechanics Theory and Experiment

On the algebraic Bethe ansatz: periodic boundary conditions

A Lima-Santos

Published Jul 12, 2006• 3 citations
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Abstract

In this paper, the algebraic Bethe ansatz with periodic boundary conditions is used to investigate trigonometric vertex models associated with the fundamental representations of the non-exceptional Lie algebras. This formulation allows us to present explicit expressions for the eigenvectors and eigenvalues of the respective transfer matrices.

DOI
10.1088/1742-5468/2006/07/p07003

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In this paper, the algebraic Bethe ansatz with periodic boundary conditions is used to investigate trigonometric vertex models associated with the fundamental representations of the non-exceptional Lie algebras. This formulation allows us to present explicit expressions for the eigenvectors and eigenvalues of the respective transfer matrices.
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)
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).