The matrix product ansatz for the six-vertex model

TL;DR

This paper develops a matrix product ansatz for the six-vertex model, extending the approach used for one-dimensional quantum chains to a two-dimensional integrable system, supporting the conjecture of universality of the method.

Contribution

It introduces a matrix product ansatz formulation for the six-vertex model with periodic boundary conditions, bridging integrability in 2D with algebraic eigenfunction representations.

Findings

The ansatz aligns with Bethe ansatz solutions for the six-vertex model.

Supports the conjecture that all Bethe-ansatz solvable models can be solved via matrix product ansatz.

Provides a new algebraic framework for 2D integrable models.

Abstract

Recently it was shown that the eigenfunctions for the 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 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.