Factorization of the finite temperature correlation functions of the XXZ chain in a magnetic field

TL;DR

This paper conjectures a new algebraic structure for the finite temperature correlation functions of the XXZ chain in a magnetic field, simplifying their computation to polynomial expressions of two key functions.

Contribution

It introduces a conjecture that expresses the density matrix of the XXZ chain at finite temperature as a trace involving monodromy matrices and only two transcendental functions.

Findings

Correlation functions are polynomials in two functions and their derivatives.

The conjecture links the density matrix to algebraic and transcendental structures.

Provides a framework for calculating static correlations in the XXZ chain.

Abstract

We present a conjecture for the density matrix of a finite segment of the XXZ chain coupled to a heat bath and to a constant longitudinal magnetic field. It states that the inhomogeneous density matrix, conceived as a map which associates with every local operator its thermal expectation value, can be written as the trace of the exponential of an operator constructed from weighted traces of the elements of certain monodromy matrices related to $U_q (\hat{\mathfrak{sl}}_2)$ and only two transcendental functions pertaining to the one-point function and the neighbour correlators, respectively. Our conjecture implies that all static correlation functions of the XXZ chain are polynomials in these two functions and their derivatives with coefficients of purely algebraic origin.