Nearest-neighbour correlation functions for the supersymmetric XYZ spin chain and Painlev\'e VI

TL;DR

This paper derives exact expressions for nearest-neighbour correlation functions in the supersymmetric XYZ spin chain using Painlevé VI tau functions, revealing deep connections between quantum spin models and integrable systems.

Contribution

It establishes a novel link between supersymmetric XYZ spin chain correlations and Painlevé VI tau functions, providing explicit formulas and Hamiltonian interpretations.

Findings

Correlation functions expressed via Painlevé VI tau functions

Connection between spin chain correlations and integrable systems

Hamiltonian interpretation of correlation functions

Abstract

We study nearest-neighbour correlation functions for the ground state of the supersymmetric XYZ spin chain with odd length and periodic boundary conditions. Under a technical assumption related to the $Q$-operator of the corresponding eight-vertex model, we show that they can be expressed exactly in terms of the Painlev\'e VI tau functions $s_n$ and $\bar s_n$ introduced by Bazhanov and Mangazeev. Furthermore, we give an interpretation of the correlation functions in terms of the Painlev\'e VI Hamiltonian.