Comparing numerical methods for hydrodynamics in a one-dimensional lattice spin model

TL;DR

This paper compares various numerical methods for simulating hydrodynamics in quantum spin chains at infinite temperature, assessing their accuracy, convergence, and ability to capture long-time behavior and oscillatory phenomena.

Contribution

It evaluates the effectiveness of TEBD, DMT, R-UOG, and OST methods in simulating hydrodynamics, introducing the 'hot band second sound' phenomenon and a toy model for it.

Findings

DMT and OST provide consistent correlations up to t=60/J with 1% accuracy.

TEBD converges for t≈20 and yields accurate diffusion coefficients.

No evidence of long-time tails in current-current correlators; power-law corrections observed.

Abstract

In ergodic quantum spin chains, locally conserved quantities such as energy or particle number generically evolve according to hydrodynamic equations as they relax to equilibrium. We investigate the complexity of simulating hydrodynamics at infinite temperature with multiple methods: time evolving block decimation (TEBD), TEBD with density matrix truncation (DMT), the recursion method with a universal operator growth hypothesis (R-UOG), and operator-size truncated (OST) dynamics. Density matrix truncation and the OST dynamics give consistent dynamical correlations to $t=60/J$ and diffusion coefficients agreeing within 1%. TEBD only converges for $t\lesssim 20$, but still produces diffusion coefficients accurate within 1%. The universal operator growth hypothesis fails to converge and only matches other methods on short times. We see no evidence of long-time tails in either DMT or OST calculations of the current-current correlator, although we cannot rule out that they appear at longer times. We do observe power-law corrections to the energy density correlator. At finite wavelength, we observe a crossover from purely diffusive, overdamped decay of the energy density, to underdamped oscillatory behavior similar to that of cold atom experiments. We dub this behavior "hot band second sound" and offer a microscopically-motivated toy model.