# Spin diffusion from an inhomogeneous quench in an integrable system

## TL;DR

This paper investigates spin transport in the anisotropic Heisenberg XXZ chain starting from symmetric inhomogeneous states, revealing diffusive behavior in the easy-axis regime and super-diffusive transport near the isotropic point, challenging the universality of ballistic transport predictions.

## Contribution

It provides large-scale numerical evidence of non-ballistic spin transport in symmetric initial states of the XXZ chain, highlighting diffusive and super-diffusive regimes.

## Key findings

- Normal diffusion in the easy-axis regime.
- Super-diffusive transport near the isotropic point with exponent close to 2/3.
- Universal scaling dynamics obeying the diffusion equation in nonlinear time.

## Abstract

Generalised hydrodynamics predicts universal ballistic transport in integrable lattice systems when prepared in generic inhomogeneous initial states. However, the ballistic contribution to transport can vanish in systems with additional discrete symmetries. Here we perform large scale numerical simulations of spin dynamics in the anisotropic Heisenberg $XXZ$ spin $1/2$ chain starting from an inhomogeneous mixed initial state which is symmetric with respect to a combination of spin-reversal and spatial reflection. In the isotropic and easy-axis regimes we find non-ballistic spin transport which we analyse in detail in terms of scaling exponents of the transported magnetisation and scaling profiles of the spin density. While in the easy-axis regime we find accurate evidence of normal diffusion, the spin transport in the isotropic case is clearly super-diffusive, with the scaling exponent very close to $2/3$, but with universal scaling dynamics which obeys the diffusion equation in nonlinearly scaled time.