Exact solutions of exactly integrable quantum chains by a matrix product ansatz

TL;DR

This paper introduces a matrix product ansatz as an alternative to Bethe ansatz for solving exactly integrable quantum chains, providing a unified framework for various models with explicit solutions.

Contribution

It proposes a novel matrix product ansatz that simplifies and unifies the exact solution process for a range of integrable quantum chains, extending applicability to open boundary conditions.

Findings

Exact solutions for several quantum chains including XXZ, Hubbard, and t-J models.

Unified matrix product framework for integrable models.

Extension to chains with open boundaries and surface magnetic fields.

Abstract

Most of the exact solutions of quantum one-dimensional Hamiltonians are obtained thanks to the success of the Bethe ansatz on its several formulations. According to this ansatz the amplitudes of the eigenfunctions of the Hamiltonian are given by a sum of permutations of appropriate plane waves. In this paper, alternatively, we present a matrix product ansatz that asserts that those amplitudes are given in terms of a matrix product. The eigenvalue equation for the Hamiltonian define the algebraic properties of the matrices defining the amplitudes. The existence of a consistent algebra imply the exact integrability of the model. The matrix product ansatz we propose allow an unified and simple formulation of several exact integrable Hamiltonians. In order to introduce and illustrate this ansatz we present the exact solutions of several quantum chains with one and two global conservation laws and periodic boundaries such as the XXZ chain, spin-1 Fateev-Zamolodchikov model, Izergin-Korepin model, Sutherland model, t-J model, Hubbard model, etc. Formulation of the matrix product ansatz for quantum chains with open ends is also possible. As an illustration we present the exact solution of an extended XXZ chain with $z$-magnetic fields at the surface and arbitrary hard-core exclusion among the spins.