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

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 $2πk_B T/\hbar$. Here we introduce a one-parameter family of out-of-time-ordered correlators $F_γ(t)$ ($0\leqγ\leq 1$), 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 $γ=1/2$. We rigorously prove that if $F_γ(t)$ shows a transient exponential growth for all $γ$ in $0\leqγ\leq 1$, that is, if the OTOC shows an exponential growth regardless of the choice of the regularization, then the growth rate $λ$ does not depend on the regularization parameter $γ$, and satisfies the inequality $λ\leq 2πk_B T/\hbar$.