Heisenberg XXZ Model and Quantum Galilei Group

TL;DR

This paper explores the symmetry properties of the 1D Heisenberg XXZ spin chain using the quantum Galilei group, revealing how this symmetry determines magnon excitations and bound states, and deriving their energies explicitly.

Contribution

It introduces the quantum Galilei group symmetry into the analysis of the XXZ model, providing a new algebraic framework for understanding its excitations and bound states.

Findings

Magnon excitations are characterized by the algebra.

Bound state energies are expressed via Tchebischeff polynomials.

The symmetry induces the Bethe Ansatz formulation.

Abstract

The 1D Heisenberg spin chain with anisotropy of the XXZ type is analyzed in terms of the symmetry given by the quantum Galilei group Gamma_q(1). We show that the magnon excitations and the s=1/2, n-magnon bound states are determined by the algebra. Thus the Gamma_q(1) symmetry provides a description that naturally induces the Bethe Ansatz. The recurrence relations determined by Gamma_q(1) permit to express the energy of the n-magnon bound states in a closed form in terms of Tchebischeff polynomials.