XX Heisenberg Spin Chain and an Example of Path Integral with "Automorphic" Boundary Conditions

TL;DR

This paper introduces a novel fermionic path integral representation for the XX Heisenberg spin chain's correlators, incorporating automorphic boundary conditions, and computes correlation functions explicitly at nonzero temperature.

Contribution

It presents a new fermionic Gaussian path integral approach with automorphic boundary conditions for the XX Heisenberg spin chain, extending traditional methods.

Findings

Expressed generating functions as determinants of matrix operators.

Calculated partition function and correlators using zeta-regularization.

Obtained explicit correlation functions at nonzero temperature.

Abstract

New representation for the generating function of correlators of third components of spins in the XX Heisenberg spin chain is considered in the form given by the fermionic Gaussian path integrals. A part of the discrete anti-commuting integration variables is subjected to ``automorphic'' boundary conditions in respect of imaginary time. The situation when only a part of the integration variables is subjected to the unusual boundary conditions generalizes more conventional ones when ``automorphic'' boundary conditions appear for all sites in the lattice spin models. The results of the functional integration are expressed as determinants of the matrix operators. The generating function, as well as the partition function of the model, are calculated by means of zeta-regularization. Certain correlation functions at nonzero temperature are obtained explicitly.