Diffusion and operator entanglement spreading

TL;DR

This paper investigates the growth of operator entanglement in integrable quantum systems, revealing a universal logarithmic spreading linked to quasiparticle diffusion, with implications for understanding out-of-equilibrium dynamics.

Contribution

It establishes a universal logarithmic bound for operator entanglement entropy in integrable models and explores the effects of integrability breaking.

Findings

Logarithmic growth of OSEE as 1/2 ln(t) in integrable models

Bound saturation observed in rule 54 and Heisenberg XXZ chains

Finite-time effects hinder asymptotic behavior analysis

Abstract

Understanding the spreading of the operator space entanglement entropy ($OSEE$) is key in order to explore out-of-equilibrium quantum many-body systems. Here we argue that for integrable models the dynamics of the $OSEE$ is related to the diffusion of the underlying quasiparticles. We derive the logarithmic bound $1/2\ln(t)$ for the $OSEE$ of some simple, i.e., low-rank, diagonal local operators. We numerically check that the bound is saturated in the rule $54$ chain, which is representative of interacting integrable systems. Remarkably, the same bound is saturated in the spin-1/2 Heisenberg $XXZ$ chain. Away from the isotropic point and from the free-fermion point, the $OSEE$ grows as $1/2\ln(t)$, irrespective of the chain anisotropy, suggesting universality. Finally, we discuss the effect of integrability breaking. We show that strong finite-time effects are present, which prevent from probing the asymptotic behavior of the $OSEE$.