Chaos in hyperscaling violating Lifshitz theories

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.