Operator size at finite temperature and Planckian bounds on quantum dynamics

TL;DR

This paper proposes a universal bound on the time scale for operators to grow in quantum systems at finite temperature, linking it to Planckian bounds and explaining their applicability across different coupling regimes.

Contribution

It introduces the concept of operator size at finite temperature and conjectures a universal time scale for operator growth, unifying understanding of Planckian bounds in quantum dynamics.

Findings

The conjectured operator growth time scale is consistent with all known many-body theories.

The bound explains the limited applicability of previous Planckian bounds to weakly coupled systems.

It connects Planckian bounds to both transport phenomena and quantum chaos.

Abstract

It has long been believed that dissipative time scales $\tau$ obey a "Planckian" bound $\tau \gtrsim \frac{\hbar}{k_{\mathrm{B}}T}$ in strongly coupled quantum systems. Despite much circumstantial evidence, however, there is no known $\tau$ for which this bound is universal. Here we define operator size at finite temperature, and conjecture such a $\tau$: the time scale over which small operators become large. All known many-body theories are consistent with this conjecture. This proposed bound explains why previously conjectured Planckian bounds do not always apply to weakly coupled theories, and how Planckian time scales can be relevant to both transport and chaos.