Operator front broadening in chaotic and integrable quantum chains

TL;DR

This paper investigates how operator front broadening differs in chaotic versus integrable quantum chains, revealing subtle signatures of chaos through the shape and scaling of the operator front using MPO and analytical methods.

Contribution

It introduces a detailed analysis of operator front broadening in 1D quantum systems, highlighting differences between chaotic and integrable dynamics with a quasiparticle interpretation.

Findings

Operator front broadens diffusively in both systems.

MPO with small bond dimension captures tail but not the front.

Integrable systems show anomalous front shape and scaling.

Abstract

Operator spreading under unitary time evolution has attracted a lot of attention recently, as a way to probe many-body quantum chaos. While quantities such as out-of-time-ordered correlators (OTOC) do distinguish interacting from non-interacting systems, it has remained unclear to what extent they can truly diagnose chaotic {\it vs} integrable dynamics in many-body quantum systems. Here, we analyze operator spreading in generic 1D many-body quantum systems using a combination of matrix product operator (MPO) and analytical techniques, focusing on the operator {\em right-weight}. First, we show that while small bond dimension MPOs allow one to capture the exponentially-decaying tail of the operator front, in agreement with earlier results, they lead to significant quantitative and qualitative errors for the actual front -- defined by the maximum of the right-weight. We find that while the operator front broadens diffusively in both integrable and chaotic interacting spin chains, the precise shape and scaling of the height of the front in integrable systems is anomalous for all accessible times. We interpret these results using a quasiparticle picture. This provides a sharp, though rather subtle signature of many-body quantum chaos in the operator front.