# Bound on the exponential growth rate of out-of-time-ordered correlators

## TL;DR

This paper introduces a family of out-of-time-ordered correlators and proves that if they exhibit exponential growth, the growth rate is universally bounded by a temperature-dependent limit, supporting the conjectured chaos bound.

## Contribution

The authors define a one-parameter family of OTOCs and rigorously prove the universal bound on their exponential growth rate, independent of regularization.

## Key findings

- The growth rate of OTOCs is bounded by 2πk_B T/ħ.
- Exponential growth in all regularizations implies a universal bound.
- The introduced correlators are effective regularizations of quantum chaos diagnostics.

## Abstract

It has been conjectured by Maldacena, Shenker, and Stanford [J. High Energy Phys.~08 (2016) 106] that the exponential growth rate of the out-of-time-ordered correlator (OTOC) $F(t)$ has a universal upper bound $2\pi k_B T/\hbar$. Here we introduce a one-parameter family of out-of-time-ordered correlators $F_\gamma(t)$ ($0\leq\gamma\leq 1$), which has as good properties as $F(t)$ as a regularization of the out-of-time-ordered part of the squared commutator $\langle [A(t), B(0)]^2\rangle$ that diagnoses quantum many-body chaos, and coincides with $F(t)$ at $\gamma=1/2$. We rigorously prove that if $F_\gamma(t)$ shows a transient exponential growth for all $\gamma$ in $0\leq\gamma\leq 1$, that is, if the OTOC shows an exponential growth regardless of the choice of the regularization, then the growth rate $\lambda$ does not depend on the regularization parameter $\gamma$, and satisfies the inequality $\lambda\leq 2\pi k_B T/\hbar$.