Invariant measures for integrable spin chains and integrable discrete NLS

TL;DR

This paper investigates invariant measures for discrete integrable models related to classical dispersive PDEs, proving measure invariance and solution uniqueness for the discrete Heisenberg spin chain and Ablowitz--Ladik system on the infinite lattice.

Contribution

It establishes the existence, uniqueness, and invariance of Gibbs and white noise measures for discrete integrable spin chains and NLS models, extending continuum results to discrete settings.

Findings

Gibbs measure invariance for the discrete Heisenberg model

White noise measure invariance for the focusing Ablowitz--Ladik system

Connection between models via discrete Hasimoto transform

Abstract

We consider discrete analogues of two well-known open problems regarding invariant measures for dispersive PDE, namely, the invariance of the Gibbs measure for the continuum (classical) Heisenberg model and the invariance of white noise under focusing cubic NLS. These continuum models are completely integrable and connected by the Hasimoto transform; correspondingly, we focus our attention on discretizations that are also completely integrable and also connected by a discrete Hasimoto transform. We consider these models on the infinite lattice $\mathbb Z$. Concretely, for a completely integrable variant of the classical Heisenberg spin chain model (introduced independently by Haldane, Ishimori, and Sklyanin) we prove the existence and uniqueness of solutions for initial data following a Gibbs law (which we show is unique) and show that the Gibbs measure is preserved under these dynamics. In the setting of the focusing Ablowitz--Ladik system, we prove invariance of a measure that we will show is the appropriate discrete analogue of white noise. We also include a thorough discussion of the Poisson geometry associated to the discrete Hasimoto transform introduced by Ishimori that connects the two models studied in this article.