Computation of static Heisenberg-chain correlators: Control over length and temperature dependence

TL;DR

This paper provides explicit formulas for static correlators in the spin-1/2 Heisenberg chain at finite temperature and finite size, enabling precise numerical analysis and revealing asymptotic behaviors.

Contribution

It introduces explicit formulas for short-range correlators in the Heisenberg chain depending on temperature and length, with detailed asymptotic analysis.

Findings

Correlators expressed in terms of a single function ω for up to seven sites.

Asymptotic behavior of correlators shows T^2 and 1/L^2 dependence.

Exact coefficients for leading terms up to eight sites.

Abstract

We communicate results on correlation functions for the spin-1/2 Heisenberg-chain in two particularly important cases: (a) for the infinite chain at arbitrary finite temperature $T$, and (b) for finite chains of arbitrary length $L$ in the ground-state. In both cases we present explicit formulas expressing the short-range correlators in a range of up to seven lattice sites in terms of a single function $\omega$ encoding the dependence of the correlators on $T$ ($L$). These formulas allow us to obtain accurate numerical values for the correlators and derived quantities like the entanglement entropy. By calculating the low $T$ (large $L$) asymptotics of $\omega$ we show that the asymptotics of the static correlation functions at any finite distance are $T^2$ ($1/L^2$) terms. We obtain exact and explicit formulas for the coefficients of the leading order terms for up to eight lattice sites.