CM In the literature: open

Can a lattice with few states per site saturate the chaos bound

In plain words

Theory says scrambling at temperature $T$ cannot grow faster than the rate $2\pi k_{\mathrm{B}} T/\hbar$, and black holes and SYK reach this limit. Whether any laboratory material or ordinary lattice model in which each site holds only a few quantum states reaches it is unknown.

Precise statement

The Maldacena-Shenker-Stanford bound $\lambda_L \le 2\pi k_B T/\hbar$ holds for regularized thermal OTOCs, and large-N lattice models (SYK chains, Gu, Qi, Stanford 2017) saturate it. For a lattice model with fixed, finite local Hilbert-space dimension (no large-N or semiclassical parameter), determine whether a parametrically wide exponential-growth window exists as $T \to 0$ and, if so, whether $\lambda_L \hbar/(2\pi k_B T) \to 1$ can be reached; or prove a bound strictly below 1. Answer: an explicit model or a bound.

What would settle it

A controlled computation of $\lambda_{\mathrm{L}}(T)$ in a microscopic lattice model with finite local dimension, or direct OTOC measurement in a quantum simulator at temperatures where quantum effects dominate.

Status in the literature

Large-N lattice models saturate the bound; no finite-local-dimension example is known.