Projection operator approach to spin diffusion in the anisotropic Heisenberg chain at high temperatures

TL;DR

This paper uses a projection operator method to analyze spin diffusion in the anisotropic Heisenberg chain at high temperatures, deriving diffusion constants that depend on anisotropy and spin quantum number, and validating results with numerical methods.

Contribution

It introduces a perturbative projection operator approach to calculate spin diffusion constants in the anisotropic Heisenberg chain, revealing their dependence on anisotropy and spin quantum number.

Findings

Diffusion constants scale as 1/Δ^2 for small Δ (s=1/2).

Diffusion constants scale as 1/Δ for large Δ (arbitrary s).

Results agree well with exact diagonalization and non-equilibrium methods.

Abstract

We investigate spin transport in the anisotropic Heisenberg chain in the limit of high temperatures ({\beta} \to 0). We particularly focus on diffusion and the quantitative evaluation of diffusion constants from current autocorrelations as a function of the anisotropy parameter {\Delta} and the spin quantum number s. Our approach is essentially based on an application of the time-convolutionless (TCL) projection operator technique. Within this perturbative approach the projection onto the current yields the decay of autocorrelations to lowest order of {\Delta}. The resulting diffusion constants scale as 1/{\Delta}^2 in the Markovian regime {\Delta}<<1 (s=1/2) and as 1/{\Delta} in the highly non-Markovian regime above {\Delta} \sim 1 (arbitrary s). In the latter regime the dependence on s appears approximately as an overall scaling factor \sqrt{s(s+1)} only. These results are in remarkably good agreement with diffusion constants for {\Delta}>1 which are obtained directly from the exact diagonalization of autocorrelations or have been obtained from non-equilibrium bath scenarios.