Quantum Epidemiology: Operator Growth, Thermal Effects, and SYK

TL;DR

This paper develops a comprehensive method to analyze operator growth in fermionic systems at finite temperature, applying it to the SYK model to reveal temperature-dependent scrambling dynamics and modeling these effects with a modified epidemic analogy.

Contribution

It introduces a new methodology for deriving the full operator growth structure at finite temperature, specifically applied to the SYK model in the large q limit.

Findings

Operator growth slows down as temperature decreases.

The temperature dependence of scrambling is simple and consistent.

Finite-temperature effects can be modeled by a vaccinated epidemic model.

Abstract

In many-body chaotic systems, the size of an operator generically grows in Heisenberg evolution, which can be measured by certain out-of-time-ordered four-point functions. However, these only provide a coarse probe of the full underlying operator growth structure. In this article we develop a methodology to derive the full growth structure of fermionic systems, that also naturally introduces the effect of finite temperature. We then apply our methodology to the SYK model, which features all-to-all $q$-body interactions. We derive the full operator growth structure in the large $q$ limit at all temperatures. We see that its temperature dependence has a remarkably simple form consistent with the slowing down of scrambling as temperature is decreased. Furthermore, our finite-temperature scrambling results can be modeled by a modified epidemic model, where the thermal state serves as a vaccinated population, thereby slowing the overall rate of infection.