# Bethe-Boltzmann Hydrodynamics and Spin Transport in the XXZ Chain

## TL;DR

This paper develops a numerical hydrodynamic approach to study spin and heat transport in the XXZ quantum spin chain, providing new insights into non-equilibrium steady states and spin Drude weight calculations.

## Contribution

It introduces a numerical scheme for hydrodynamic equations in integrable systems and applies it to accurately compute spin Drude weight in the XXZ chain.

## Key findings

- Hydrodynamic methods agree with DMRG results for spin Drude weight.
- The approach describes non-equilibrium steady states with ballistic transport.
- Numerical scheme enables finite-time analysis of integrable quantum systems.

## Abstract

Quantum integrable systems, such as the interacting Bose gas in one dimension and the XXZ quantum spin chain, have an extensive number of local conserved quantities that endow them with exotic thermalization and transport properties. We discuss recently introduced hydrodynamic approaches for such integrable systems from the viewpoint of kinetic theory and extend the previous works by proposing a numerical scheme to solve the hydrodynamic equations for finite times and arbitrary locally equilibrated initial conditions. We then discuss how such methods can be applied to describe non-equilibrium steady states involving ballistic heat and spin currents. In particular, we show that the spin Drude weight in the XXZ chain, previously accessible only by rigorous techniques of limited scope or controversial thermodynamic Bethe ansatz arguments, may be evaluated from hydrodynamics in very good agreement with density-matrix renormalization group calculations.