Spin conductivity of the XXZ chain in the antiferromagnetic massive regime

TL;DR

This paper develops a high-precision series representation for the spin current correlation function in the XXZ chain's antiferromagnetic massive regime, enabling detailed analysis of spin conductivity and excitation contributions.

Contribution

It introduces a novel series expansion approach for the dynamical two-point function, allowing accurate long-time and large-distance computations and explicit expressions for key excitation contributions.

Findings

The lowest term determines the asymptotic behaviour.

Two-spinon contribution carries most spectral weight at high anisotropy.

Numerical estimates of optical spin conductivity are highly accurate.

Abstract

We present a series representation for the dynamical two-point function of the local spin current for the XXZ chain in the antiferromagnetic massive regime at zero temperature. From this series we can compute the correlation function with very high accuracy up to very long times and large distances. Each term in the series corresponds to the contribution of all scattering states of an even number of excitations. These excitations can be interpreted in terms of an equal number of particles and holes. The lowest term in the series comprises all scattering states of one hole and one particle. This term determines the long-time large-distance asymptotic behaviour which can be obtained explicitly from a saddle-point analysis. The space-time Fourier transform of the two-point function of currents at zero momentum gives the optical spin conductivity of the model. We obtain highly accurate numerical estimates for this quantity by numerically Fourier transforming our data. For the one-particle, one-hole contribution, equivalently interpreted as a two-spinon contribution, we obtain an exact and explicit expression in terms of known special functions. For large enough anisotropy, the two-spinon contribution carries most of the spectral weight, as can be seen by calculating the f-sum rule.