Finite lattice Bethe ansatz systems and the Heun equation

TL;DR

This paper explores the connection between complex domain P"oschl-Teller equations, Bethe ansatz systems, and the Heun function, revealing new classifications and interpretations in quantum spin chains and field theory limits.

Contribution

It introduces a classification of Bethe ansatz equations via four integers and links them to finite lattice spin chains and the Heun function, extending previous results.

Findings

Explicit solutions for Q in terms of Heun functions.

Classification of models by four integers.

Connection to finite lattice XXZ quantum chains.

Abstract

We study the P"oschl-Teller equation in complex domain and deduce infinite families of TQ and Bethe ansatz equations, classified by four integers. In all these models the form of T is very simple, while Q can be explicitly written in terms of the Heun function. At particular values there is a interesting interpretation in terms of finite lattice spin (L-2)/2 XXZ quantum chain with Delta= cos(pi/L) (for free-free boundary conditions), or Delta=-cos(pi/L) (for periodic boundary conditions). This result generalises the findings of Fridkin, Stroganov and Zagier. We also discuss the continuous (field theory) limit of these systems in view of the so-called ODE/IM correspondence.