Connected correlations, fluctuations and current of magnetization in the steady state of boundary driven XXZ spin chains

TL;DR

This paper develops a method to derive PDEs for correlation functions in boundary-driven XXZ spin chains, revealing long-range correlations at the isotropic point without using algebraic representation theory.

Contribution

It introduces a PDE-based approach to analyze non-equilibrium steady states, bypassing the need for matrix algebra representations, and applies it to boundary-driven XXZ chains.

Findings

Connected correlations are long-range at the isotropic point Δ=1

Method avoids complex algebraic representations

Provides explicit PDEs for correlation functions

Abstract

We show how to exploit algebraic relations of operators (or matrices) which constitute the non-equilibrium matrix product steady state of a boundary driven quantum spin chain to find partial differential equations determining all the $m$-point correlation functions in the continuum limit. These partial differential equations, the order of which is determined by scaling of the non-equilibrium partition function, are readily solved if we also know the boundary conditions. In this way we can avoid resorting to representation theory of the matrix product algebra. We apply our methods to study the distributions, or moments, of the magnetization and the spin current observables in boundary driven open XXZ spin chains, as well as some connected correlation functions and find that the transverse connected correlation functions are thermodynamically non-decaying and long-range at the isotropic point $\Delta=1$.