Differential equation for a correlation function of the spin-${1\over2}$ Heisenberg chain

TL;DR

This paper derives an integrable system of equations describing the probability of finding a string of aligned spins in the ground state of the antiferromagnetic Heisenberg chain, linking quantum correlations to integrable models.

Contribution

It introduces a novel integrable integro-difference equation system for spin correlation probabilities in the Heisenberg chain, with a Lax pair and Riemann-Hilbert formulation.

Findings

System is completely integrable

Quantum correlation function is a τ-function of the system

Provides a new mathematical framework for spin correlations

Abstract

We consider the probability to find a string of $x$ adjacent parallel spins in the antiferromagnetic ground state of the model (in a magnetic field). We derive a system of integro-difference equations which define this probability. This system is completely integrable, it has Lax representation and a corresponding Riemann-Hilbert problem. The quantum correlation funtion is a $\tau$-function of this system.