Operator Size at Finite Temperature and Planckian Bounds on Quantum Dynamics

It has long been believed that dissipative timescales $\tau$ obey a “Planckian” bound $\tau \gtrsim (\hslash /k_{\mathrm{B}}T)$ 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 a finite temperature, and conjecture such a $\tau$: the timescale 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 timescales can be relevant to both transport and chaos.

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 Eq. (1) only constrains the time for $|B)$ to rotate into the large subspace.