Correlation Function and Simplified TBA Equations for XXZ Chain

TL;DR

This paper derives simplified equations for calculating correlation functions in the XXZ spin chain, providing analytic solutions at specific anisotropy and enabling better understanding of entanglement and thermodynamics in integrable models.

Contribution

It introduces a new simplified TBA equation for the XXZ chain and derives its analytic solution at =0, advancing the analytical study of correlation functions.

Findings

Correlation functions expressed by generalized zeta functions.

Analytic solution of the new TBA equation at =0.

Entanglement entropy for seven-site subsystem calculated.

Abstract

The calculation of the correlation functions of Bethe ansatz solvable models is very difficult problem. Among these solvable models spin 1/2 XXX chain has been investigated for a long time. Even for this model only the nearest neighbor and the second neighbor correlations were known. In 1990's multiple integral formula for the general correlations is derived. But the integration of this formula is also very difficult problem. Recently these integrals are decomposed to products of one dimensional integrals and at zero temperature, zero magnetic field and isotropic case, correlation functions are expressed by $\log 2$ and Riemann's zeta functions with odd integer argument $\zeta(3),\zeta(5),\zeta(7),...$. We can calculate density sub-matrix of successive seven sites. Entanglement entropy of seven sites is calculated. These methods can be extended to XXZ chain up to $n=4$. Correlation functions are expressed by the generalized zeta functions. Several years ago I derived new thermodynamic Bethe ansatz equation for XXZ chain. This is quite different with Yang-Yang type TBA equations and contains only one unknown function. This equation is very useful to get the high temperature expansion. In this paper we get the analytic solution of this equation at $\Delta=0$.