An exact solution to asymptotic Bethe equation

TL;DR

This paper provides an exact analytical solution to the asymptotic Bethe equations for a weakly anisotropic Heisenberg spin chain, linking it to classical mathematical problems and field theory.

Contribution

It introduces a novel exact solution to the asymptotic Bethe equations, connecting quantum spin chain excitations to classical mathematical frameworks.

Findings

Solution describes low-energy excitations above ferromagnetic ground state.

Links to generalized Jacobi polynomial and Stieltjes problem.

Continuous limit relates to Riemann-Hilbert problem and classical Landau-Lifshitz solutions.

Abstract

We present an exact solution to the asymptotic Bethe equation of weakly anisotropic Heisenberg spin chain, which is a set of non-linear algebraic equations. The solution describes the low-energy excitations above ferromagnetic ground state with fixed magnetisation, and it has a close relation to generalised Jacobi polynomial. It is equivalent to a generalised Stieltjes problem and in the continuous limit, it becomes a Riemann-Hilbert problem closely related to the finite-gap solutions of classical Landau-Lifshitz field theory.