Chaos and pole skipping in CFT$_2$

TL;DR

This work analyzes the connection between quantum chaos and pole skipping in 2D CFTs by computing stress-tensor retarded Green's functions on a torus. It shows that on the cylinder the pole-skipping frequency is $\omega_* = k_* = 2\pi i T$, while on the torus the location depends on the spectrum via $\omega_*^2 = k_*^2 = \langle T\rangle_{\mathrm{T}^2}/(c/24)$ and, for energy density, $\omega_*^2 = -\frac{12}{c}\frac{E}{R}$; these results lead to the bound $\lambda \leq 2\pi T$, analogous to the MSS chaos bound. In the large-$c$ regime with a sparse light spectrum, the bound is saturated above the Hawking-Page transition, connecting modular invariance, BTZ thermodynamics, and maximal chaos in holographic CFT$_2$. The findings highlight the spectrum-dependence of pole skipping on compact spaces and suggest broader implications for OTOCs and effective field theories of chaos in two dimensions.

Abstract

Recent work has suggested an intriguing relation between quantum chaos and energy density correlations, known as pole skipping. We investigate this relationship in two dimensional conformal field theories on a finite size spatial circle by studying the thermal energy density retarded two-point function on a torus. We find that the location $ω_* = i λ$ of pole skipping in the complex frequency plane is determined by the central charge and the stress energy one-point function $\langle T\rangle$ on the torus. In addition, we find a bound on $λ$ in $c>1$ compact, unitary CFT$_2$s identical to the chaos bound, $λ\leq 2πT$. This bound is saturated in large $c$ CFT$_2$s with a sparse light spectrum, as quantified by arXiv:1405.5137, for all temperatures above the dual Hawking-Page transition temperature.

Figures (2)

• Figure 1: Contour ${\cal C}$ in the complex (Lorentzian) time plane used in evaluating the Fourier transform of $G^R_{\mathbb{R}\times \mathrm{S}^1}(t,x)$, shown with $x>0$. The two branches, infinitessimally close to the real line, correspond to the two terms in the commutator defining $G^R$. The Heaviside function in the definition of $G^R$ restricts the contour to positive real times, and hence the singularity at $t=x$ is not enclosed by the contour of integration for $x<0$.
• Figure 2: Contour used in evaluating the Fourier transform of $G^R_{\mathrm{T}^2}$. Here the singularity at $t=x$ has doubly periodic images, separated by $\mathrm{i} \beta$ and $2\pi R$, as required by the torus geometry.