Thermodynamics of Inozemtsev's Elliptic Spin Chain

TL;DR

This paper analyzes the thermodynamic properties of Inozemtsev's elliptic spin chain using Bethe ansatz, deriving TBA equations, and confirming the model's unique spectral features and equivalence with supersymmetric versions in the thermodynamic limit.

Contribution

It provides a comprehensive classification of solutions to the Bethe equations and derives the thermodynamic Bethe ansatz equations for Inozemtsev's elliptic spin chain.

Findings

Derived TBA equations and Y-system for the model.

Confirmed the non-selfconjugate solutions and their implications.

Numerically verified the equivalence of original and supersymmetric models in the thermodynamic limit.

Abstract

We study the thermodynamic behaviour of Inozemtsev's long-range elliptic spin chain using the Bethe ansatz equations describing the spectrum of the model in the infinite-length limit. We classify all solutions of these equations in that limit and argue which of these solutions determine the spectrum in the thermodynamic limit. Interestingly, some of the solutions are not selfconjugate, which puts the model in sharp contrast to one of the model's limiting cases, the Heisenberg xxx spin chain. Invoking the string hypothesis we derive the thermodynamic Bethe ansatz equations (TBA-equations) from which we determine the Helmholtz free energy in thermodynamic equilibrium and derive the associated Y-system. We corroborate our results by comparing numerical solutions of the TBA-equations to a direct computation of the free energy for the finite-length hamiltonian. In addition we confirm numerically the interesting conjecture put forward by Finkel and Gonz\'alez-L\'opez that the original and supersymmetric versions of Inozemtsev's elliptic spin chain are equivalent in the thermodynamic limit.