Analytic solution of the Domain Wall non-equilibrium stationary state

TL;DR

This paper derives an exact analytic description of the non-equilibrium stationary state in a quantum spin chain after joining two domains with opposite magnetizations, revealing a discontinuous dependence on interaction strength.

Contribution

It provides a simple, exact formula for the space-time profile of spin current and magnetization in the gapless XXZ model using generalized hydrodynamics, highlighting interaction effects.

Findings

Stationary state exhibits a discontinuous dependence on interaction strength.

Derived an exact formula for spin current and magnetization profiles.

Interactions prevent Tracy-Widom scaling at the edge, unlike free fermions.

Abstract

We consider the out-of-equilibrium dynamics generated by joining two domains with arbitrary opposite magnetisations. We study the stationary state which emerges by the unitary evolution via the spin $1/2$ XXZ Hamiltonian, in the gapless regime, where the system develops a stationary spin current. Using the generalized hydrodynamic approach, we present a simple formula for the space-time profile of the spin current and the magnetisation exact in the limit of large times. As a remarkable effect, we show that the stationary state has a strongly discontinuous dependence on the strength of interaction. This feature allows us to give a qualitative estimation for the transient behavior of the current which is compared with numerical simulations. Moreover, we analyse the behavior around the edge of the magnetisation profile and we argue that, unlike the XX free-fermionic point, interactions always prevent the emergence of a Tracy-Widom scaling.