From ETH to algebraic relaxation of OTOCs in systems with conserved quantities

TL;DR

This paper demonstrates that in quantum systems with local conserved quantities, OTOCs typically relax algebraically, especially in local, thermalizing systems, regardless of whether the Hamiltonian is time-dependent or has power-law interactions.

Contribution

It establishes that algebraic relaxation of OTOCs occurs under broad conditions, extending previous understanding to time-dependent and power-law interacting systems.

Findings

OTOCs relax algebraically in systems with conserved quantities

Time-independence of Hamiltonian is not required for algebraic relaxation

Results apply to systems with power-law interactions

Abstract

The relaxation of out-of-time-ordered correlators (OTOCs) has been studied as a mean to characterize the scrambling properties of a quantum system. We show that the presence of local conserved quantities typically results in, at the fastest, an algebraic relaxation of the OTOC provided (i) the dynamics is local and (ii) the system follows the eigenstate thermalization hypothesis. Our result relies on the algebraic scaling of the infinite-time value of OTOCs with system size, which is typical in thermalizing systems with local conserved quantities, and on the existence of finite speed of propagation of correlations for finite-range-interaction systems. We show that time-independence of the Hamiltonian is not necessary as the above conditions (i) and (ii) can occur in time-dependent systems, both periodic or aperiodic. We also remark that our result can be extended to systems with power-law interactions.