Chaos in hyperscaling violating Lifshitz theories

TL;DR

This paper investigates quantum chaos in hyperscaling-violating Lifshitz theories using holographic methods, computing the butterfly velocity and analyzing its dependence on various physical parameters.

Contribution

It provides the first detailed holographic computation of the butterfly velocity in HVL theories, relating it to boundary thermodynamic variables and exploring its complex parameter dependence.

Findings

Butterfly velocity matches shockwave and entanglement wedge results.

$v_B$ exhibits non-monotonic behavior with respect to $z$ and $ heta$ under certain conditions.

$v_B$ increases with entropy ratio $ ilde{S}$ and decreases with charge ratio $ ilde{Q}$.

Abstract

We holographically study quantum chaos in hyperscaling-violating Lifshitz (HVL) theories (with charge). Specifically, we present a detailed computation of the out-of-time ordered correlator (OTOC) via shockwave analysis in the bulk HVL geometry with a planar horizon topology. We also compute the butterfly velocity ($v_{B}$) using the entanglement wedge reconstruction and find that it matches the result obtained from the shockwave analysis. Using a recently developed thermodynamic dictionary for HVL theories, we express $v_B$ purely in terms of boundary thermodynamic variables. Furthermore, we analyze in detail the behavior of $v_{B}$ with respect to the dynamical critical exponent ($z$), hyperscaling-violating parameter ($\theta$), entropy (more precisely, the ratio of entropy to the central charge, $\tilde{S}$), and charge (more precisely, the ratio of charge to the central charge, $\tilde{Q}$). Interestingly, $v_B$ varies non-monotonically with $z$ for $\tilde{S} < 1$, whereas it increases monotonically with $z$ for $\tilde{S} \geq 1$. Additionally, $v_B$ varies non-monotonically with $\theta$ for non-zero charge. Moreover, $v_B$ monotonically increases with $\tilde{S}$ and decreases with $\tilde{Q}$ for all allowed values of $z$ and $\theta$. All these features are reported for combinations /{$z$, $\theta$, $\tilde{S}$, $\tilde{Q}$/} for which the temperature is positive, the null energy condition is satisfied, and $v_B$ is not superluminal. Unpacking the non-monotonicities in $v_B$ can offer interesting insights into these theories.