Higher-Order Hydrodynamics in 1D: a Promising Direction and a Null Result

TL;DR

This paper investigates the applicability of higher-order hydrodynamics to inhomogeneous 1D spin chains, finding it effective near light cones in free models but failing in more general cases.

Contribution

The authors derive a Moyal dynamical equation for exact evolution and analyze the limits of hydrodynamics in capturing large-time corrections in 1D spin chains.

Findings

Hydrodynamics captures universal scaling near light cones in free fermionic models.

Hydrodynamics fails to describe dynamics in more general, non-free models.

Numerical analysis indicates limitations of hydrodynamics beyond specific conditions.

Abstract

We derive a Moyal dynamical equation that describes exact time evolution in generic (inhomogeneous) noninteracting spin-chain models. Assuming quasistationarity, we develop a hydrodynamic theory. The question at hand is whether some large-time corrections are captured by higher-order hydrodynamics. We consider in particular the dynamics after that two chains, prepared in different conditions, are joined together. In these situations a light cone, separating regions with macroscopically different properties, emerges from the junction. In free fermionic systems some observables close to the light cone follow a universal behavior, known as Tracy-Widom scaling. Universality means weak dependence on the system's details, so this is the perfect setting where hydrodynamics could emerge. For the transverse-field Ising chain and the XX model, we show that hydrodynamics captures the scaling behavior close to the light cone. On the other hand, our numerical analysis suggests that hydrodynamics fails in more general models, whenever a condition is not satisfied.