Hidden Grassmann Structure in the XXZ Model III: Introducing Matsubara direction

TL;DR

This paper develops a method to compute temperature-dependent correlation functions in the XXZ spin chain using a fermionic basis, expressing results through transfer matrix eigenvalues and a differential related to deformed Abelian integrals.

Contribution

It introduces a novel approach to evaluate temperature correlation functions in the XXZ model using a fermionic basis and connects the results to deformed Abelian integrals.

Findings

Derived expressions for correlation functions in terms of transfer matrix eigenvalues and a differential.

Established a connection between the correlation functions and deformed Abelian integrals.

Provided a framework for calculating finite-temperature properties in the XXZ model.

Abstract

We address the problem of computing temperature correlation functions of the XXZ chain, within the approach developed in our previous works. In this paper we calculate the expected values of a fermionic basis of quasi-local operators, in the infinite volume limit while keeping the Matsubara (or Trotter) direction finite. The result is expressed in terms of two basic quantities: a ratio $\rho(\z)$ of transfer matrix eigenvalues, and a nearest neighbour correlator $\omega(\z,\xi)$. We explain that the latter is interpreted as the canonical second kind differential in the theory of deformed Abelian integrals.