Diffusive hydrodynamics of out-of-time-ordered correlators with charge conservation

TL;DR

This paper extends the hydrodynamical description of out-of-time-ordered correlators (OTOCs) to systems with conserved charges, revealing how conservation laws slow information scrambling and lead to diffusive and ballistic spreading behaviors.

Contribution

It introduces a framework for understanding OTOCs in systems with charge conservation, combining analytical and numerical evidence for diffusive and ballistic components in information spreading.

Findings

Conservation laws slow down OTOC relaxation.

OTOCs exhibit diffusive tails and asymmetric wave fronts.

Ballistic fronts in OTOCs develop at exponentially large times for high chemical potential.

Abstract

The scrambling of quantum information in closed many-body systems, as measured by out-of-time-ordered correlation functions (OTOCs), has lately received considerable attention. Recently, a hydrodynamical description of OTOCs has emerged from considering random local circuits, aspects of which are conjectured to be universal to ergodic many-body systems, even without randomness. Here we extend this approach to systems with locally conserved quantities (e.g., energy). We do this by considering local random unitary circuits with a conserved U$(1)$ charge and argue, with numerical and analytical evidence, that the presence of a conservation law slows relaxation in both time ordered {\textit{and}} out-of-time-ordered correlation functions, both can have a diffusively relaxing component or "hydrodynamic tail" at late times. We verify the presence of such tails also in a deterministic, peridocially driven system. We show that for OTOCs, the combination of diffusive and ballistic components leads to a wave front with a specific, asymmetric shape, decaying as a power law behind the front. These results also explain existing numerical investigations in non-noisy ergodic systems with energy conservation. Moreover, we consider OTOCs in Gibbs states, parametrized by a chemical potential $\mu$, and apply perturbative arguments to show that for $\mu\gg 1$ the ballistic front of information-spreading can only develop at times exponentially large in $\mu$ -- with the information traveling diffusively at earlier times. We also develop a new formalism for describing OTOCs and operator spreading, which allows us to interpret the saturation of OTOCs as a form of thermalization on the Hilbert space of operators.