# Kinetic theory of spin diffusion and superdiffusion in XXZ spin chains

## TL;DR

This paper develops a kinetic theory approach to analyze spin transport in the XXZ spin chain, revealing superdiffusive behavior with a time-dependent diffusion constant scaling as t^{1/3}, and provides explicit formulas for diffusion at all anisotropies.

## Contribution

It introduces a self-consistent kinetic framework combining generalized hydrodynamics and Gaussian fluctuations to explain superdiffusion in the XXZ model, including explicit diffusion constants for all anisotropies.

## Key findings

- Superdiffusion with D(t) ~ t^{1/3} in the isotropic limit.
- Explicit expressions for D in the large-temperature limit for all Δ > 1.
- D saturates at large anisotropy and diverges near the Heisenberg point as (Δ - 1)^{-1/2}.

## Abstract

We address the nature of spin transport in the integrable XXZ spin chain, focusing on the isotropic Heisenberg limit. We calculate the diffusion constant using a kinetic picture based on generalized hydrodynamics combined with Gaussian fluctuations: we find that it diverges, and show that a self-consistent treatment of this divergence gives superdiffusion, with an effective time-dependent diffusion constant that scales as $D(t) \sim t^{1/3}$. This exponent had previously been observed in large-scale numerical simulations, but had not been theoretically explained. We briefly discuss XXZ models with easy-axis anisotropy $\Delta > 1$. Our method gives closed-form expressions for the diffusion constant $D$ in the infinite-temperature limit for all $\Delta > 1$. We find that $D$ saturates at large anisotropy, and diverges as the Heisenberg limit is approached, as $D \sim (\Delta - 1)^{-1/2}$.