Operator size at finite temperature and Planckian bounds on quantum dynamics

TL;DR

The paper addresses the lack of a universal Planckian bound on dissipation and chaos in many-body quantum systems by introducing a finite-temperature notion of operator size and a corresponding growth time $\tau$. It formulates a thermal inner product and a size operator $\mathcal{S}$ to categorize operators as small or large, and conjectures that in $k$-local systems there exists $R$ and $\delta$ such that $\tau$—the time for a small operator to become large—obeys the Planckian bound $\tau \gtrsim \frac{\hbar}{k_{\mathrm{B}}T}$ (i.e., $\tau \gtrsim \beta$ when $\hbar=k_B=1$). The authors provide multiple lines of evidence: bounds derived from two-point functions and OTOCs, random-operator and region-based estimates, and consistency with known strongly coupled theories and models like the SYK, while showing how integrability and certain transport regimes can evade universal Planckian saturation. They argue that this framework unifies Planckian transport and chaos bounds, clarifying when such bounds apply and why they fail in weakly coupled or localized systems, and offers experimentally testable predictions through thermal correlators and operator dynamics measurements.

Abstract

It has long been believed that dissipative time scales $τ$ obey a "Planckian" bound $τ\gtrsim \frac{\hbar}{k_{\mathrm{B}}T}$ in strongly coupled quantum systems. Despite much circumstantial evidence, however, there is no known $τ$ for which this bound is universal. Here we define operator size at finite temperature, and conjecture such a $τ$: 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.

Figures (1)

• Figure 1: A cartoon of operator space. The subspace of small operators is shaded. Since operator $|A)$ is mostly small (depicted by dotted lines), we know that $|A(t))$ has a significant small component whenever $(A|A(t))$ is sufficiently large. There is no bound on how quickly the nearly small operator $|B)$ can evolve into an orthogonal operator in the small subspace. Our conjecture is that (\ref{['eq:main']}) only constrains the time for $|B)$ to rotate into the large subspace.