A note on the high temperature expansion of the density matrix for the isotropic Heisenberg chain

TL;DR

This paper applies the high temperature expansion method to the isotropic Heisenberg chain's density matrix, deriving coefficients for short-range correlations, including third neighbor correlations, extending previous work on emptiness formation probability.

Contribution

The paper introduces a high temperature expansion approach for the isotropic Heisenberg chain's density matrix, providing new coefficients for various correlation functions.

Findings

Coefficients for third neighbor correlation function obtained up to order 25.

Extended previous results on emptiness formation probability to more general correlations.

Demonstrated applicability of HTE to finite magnetic field cases.

Abstract

G\"ohmann, Kl\"umper and Seel derived the multiple integral formula of the density matrix of the $XXZ$ Heisenberg chain at finite temperatures. We have applied the high temperature expansion (HTE) method to isotropic case of their formula in a finite magnetic field and obtained coefficients for several short-range correlation functions. For example, we have succeeded to obtain the coefficients of the HTE of the 3rd neighbor correlation function $<\sigma_{j}^{z}\sigma_{j+3}^{z}>$ for zero magnetic field up to the order of 25. These results expand our previous results on the emptiness formation probability [Z.Tsuboi, M.Shiroishi, J. Phys.A: Math. Gen. 38(2005) L363; cond-mat/0502569] to more general correlation functions.