Many-body chaos and energy dynamics in holography

TL;DR

The paper demonstrates that pole-skipping in the energy-density Green's function is a universal consequence of chaos in holographic theories governed by Einstein gravity with matter. By performing a general near-horizon analysis in ingoing coordinates and employing a solvable axion model, the authors show that at the chaos point $\omega = i\lambda$, $k = i\lambda/v_B$, one horizon equation becomes degenerate, yielding an extra ingoing mode and a line of poles and zeros through that point. A gauge-invariant master field $\psi$ analysis plus UV/IR matching yields an explicit slope for the pole line, which is confirmed numerically across the momentum-relaxation parameter $m/T$ and also at the special $SL(2,R)\times SL(2,R)$ symmetric point. These results link quantum chaos indicators (OTOCs) to energy transport and hydrodynamics, reinforcing pole-skipping as a sensitive, gravity-based probe of chaotic dynamics in strongly coupled systems.

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 $ω= i λ$, $k = i λ/v_B$ in the complex frequency and momentum plane. Here $λ$ and $v_B$ 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.

Figures (3)

• Figure 1: Moving slightly away from \ref{['location']} changes the near-horizon behaviour of $\psi$. To calculate the Green's function we solve separately for the solution in the UV region $(r - r_0) \gg \epsilon/b$ and an IR region $(r - r_0) \sim \epsilon/b$. The solutions can then be matched by comparing them in their overlapping regime of validity.
• Figure 2: These plots show the dispersion relation $\omega(k)$ of the hydrodynamic pole in \ref{['eq:retardedgreenspsi']} as a function of imaginary $k$ for the choices $m/T =1/100$ and $m/T = 100$. The blue lines are the hydrodynamic approximations \ref{['ads4sound']} (left panel) and \ref{['ads4diffusion']} (right panel) to the small $k$ hydrodynamic behaviour. The black dots correspond to the exact dispersion relation extracted from our numerics. Despite the qualitatively different small $k$ behaviour, in all cases we find this dispersion relation passes through the special point \ref{['location']} such that $\omega(i k_0) = i \lambda$ for $k_0 = \lambda/v_B$.
• Figure 3: (a) The red line plots the butterfly velocity $v_B = \lambda/k_0$ as determined from the analytic formula \ref{['locationexplicit']}. The black dots correspond to $\lambda/k_1$ where $k_1$ is extracted as the wavevector for which the numerical dispersion relation for the hydrodynamic pole satisfies $\omega(i k_1) = i \lambda$. If the pole passes through \ref{['location']} then we should have $\lambda/k_1 = v_B$ which indeed holds for all $m/T$. (b) The red line plots the analytic formula \ref{['eq:explicitpredict']} for the slope of the line of poles in the energy density correlator as it passes through the point \ref{['location']}. The dots correspond to the values of the slope extracted from the numerical calculations of the dispersion relation. Note that $m/T\rightarrow\infty$ corresponds to $m=\sqrt{6}r_0$, which is the upper limit shown on each plot.