Abstract
Recent developments have indicated that in addition to out-of-time ordered correlation functions (OTOCs), quantum chaos also has a sharp manifestation in the thermal energy density two-point functions, at least for maximally chaotic systems. The manifestation, referred to as pole-skipping, concerns the analytic behaviour of energy density two-point functions around a special point ω = iλ, k = iλ/vB in the complex frequency and momentum plane. Here λ and vB are the Lyapunov exponent and butterfly velocity characterising quantum chaos. In this paper we provide an argument that the phenomenon of pole-skipping is universal for general finite temperature systems dual to Einstein gravity coupled to matter. In doing so we uncover a surprising universal feature of the linearised Einstein equations around a static black hole geometry. We also study analytically a holographic axion model where all of the features of our general argument as well as the pole-skipping phenomenon can be verified in detail.
Highlights
The exponential growth of (1.1) is reminiscent of the diverging trajectories of two initially infinitesimally separated particles in classical chaotic systems
In this paper we have shown that in general holographic models dual to Einstein gravity coupled to matter, remarkable signatures of many-body chaos exist in the energy density two-point correlation functions
A key element for the discussion is the observation that one of the Einstein’s equations becomes trivial at the horizon at the special point (1.3) determined by chaos parameters, which leads to a general argument for the phenomenon of pole-skipping [27, 28]
Summary
We will demonstrate a remarkably universal property of the linearised Einstein’s equations coupled to general matter content. We will use this property to argue that in general, the retarded energy density two-point correlator of holographic theories with Einstein gravity exhibits pole-skipping at the location of eq (1.3). This property enables us to make a connection with the gravitational shock wave analysis [2,3,4] of (1.1), establishing that the behaviour of the OTOC and pole-skipping have the same gravitational origin. We wish to study the retarded energy density correlation function GRT 00T 00(ω, k) near (2.4) This correlation function can be extracted from solving the linearised gravitational perturbation equations around (2.2) subject to ingoing boundary conditions at. The fields that couple in this channel are δgvv, δgrr, δgvx, δgvr, δgxixi, δgrx and δφ, where δφ schematically represents any matter fields that couple to these perturbations.
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