Spin diffusion and the spin 1/2 XXZ chain at $T=\infty$ from exact diagonalization

TL;DR

This paper investigates the long-time spin autocorrelation functions in the spin 1/2 XXZ chain at infinite temperature using exact diagonalization, revealing a crossover from ballistic to diffusive dynamics.

Contribution

It provides a detailed numerical analysis of autocorrelation decay and estimates the dynamical exponent as a function of anisotropy, highlighting the crossover behavior.

Findings

Autocorrelation functions fit a specific decay formula with oscillations.

Estimated the dynamical exponent d as a function of anisotropy.

Identified a crossover from ballistic to diffusive spin transport.

Abstract

We study the long time behavior of the zz and xx time dependent autocorrelation function of the spin 1/2 XXZ chain at $T=\infty$ by exact diagonalizations on a chain of 16 sites. We find that the numerical results for the zz correlation are very well fit by the formula $t^{-d}[A+Be^{-\gamma (t-t_0)}\cos \Omega(t-t_0)].$ From this we estimate $d$ as a function of the anisotropy of the chain and study the crossover from ballistic to diffusive behavior.