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Notes on Scrambling in Conformal Field Theory

Chang Liu, David A. Lowe

TL;DR

This work probes quantum scrambling in 2d conformal field theories at finite temperature by examining out-of-time-order correlators and the role of conformal blocks. It shows that when the identity block dominates, a scrambling timescale scales as $t_* \sim (β/(2π))\log(c/h_w)$, but smearing primary operators into wavepackets reveals a richer behavior with a late-time onset where higher-weight intermediate states can dominate. The authors demonstrate that intermediate states with $0<h_p\ll c$ can destroy or drastically modify scrambling, implying UV data influence observables typically thought to be captured by bulk EFT, and raising questions about the existence of fast-scrambling 2d CFTs at finite temperature. The findings suggest that bulk effective field theory may not predict the fate of certain commutators, highlighting a nontrivial link between UV structure and IR scrambling in holographic contexts, and pointing to a potential class of theories where fast scrambling could persist at finite temperature.

Abstract

The onset of quantum chaos in quantum field theory may be studied using out-of-time-order correlators at finite temperature. Recent work argued that a timescale logarithmic in the central charge emerged in the context of two-dimensional conformal field theories, provided the intermediate channel was dominated by the Virasoro identity block. This suggests a wide class of conformal field theories exhibit a version of fast scrambling. In the present work we study this idea in more detail. We begin by clarifying to what extent correlators of wavepackets built out of superpositions of primary operators may be used to quantify quantum scrambling. Subject to certain caveats, these results concur with previous work. We then go on to study the contribution of intermediate states beyond the Virasoro identity block. We find that at late times, time-ordered correlators exhibit a familiar decoupling theorem, suppressing the contribution of higher dimension operators. However this is no longer true of the out-of-time-order correlators relevant for the discussion of quantum chaos. We compute the contributions of these conformal blocks to the relevant correlators, and find they are able to dominate in many interesting limits. Interpreting these results in the context of holographic models of quantum gravity, sheds new light on the black hole information problem by exhibiting a class of correlators where bulk effective field theory does not predict its own demise.

Notes on Scrambling in Conformal Field Theory

TL;DR

This work probes quantum scrambling in 2d conformal field theories at finite temperature by examining out-of-time-order correlators and the role of conformal blocks. It shows that when the identity block dominates, a scrambling timescale scales as , but smearing primary operators into wavepackets reveals a richer behavior with a late-time onset where higher-weight intermediate states can dominate. The authors demonstrate that intermediate states with can destroy or drastically modify scrambling, implying UV data influence observables typically thought to be captured by bulk EFT, and raising questions about the existence of fast-scrambling 2d CFTs at finite temperature. The findings suggest that bulk effective field theory may not predict the fate of certain commutators, highlighting a nontrivial link between UV structure and IR scrambling in holographic contexts, and pointing to a potential class of theories where fast scrambling could persist at finite temperature.

Abstract

The onset of quantum chaos in quantum field theory may be studied using out-of-time-order correlators at finite temperature. Recent work argued that a timescale logarithmic in the central charge emerged in the context of two-dimensional conformal field theories, provided the intermediate channel was dominated by the Virasoro identity block. This suggests a wide class of conformal field theories exhibit a version of fast scrambling. In the present work we study this idea in more detail. We begin by clarifying to what extent correlators of wavepackets built out of superpositions of primary operators may be used to quantify quantum scrambling. Subject to certain caveats, these results concur with previous work. We then go on to study the contribution of intermediate states beyond the Virasoro identity block. We find that at late times, time-ordered correlators exhibit a familiar decoupling theorem, suppressing the contribution of higher dimension operators. However this is no longer true of the out-of-time-order correlators relevant for the discussion of quantum chaos. We compute the contributions of these conformal blocks to the relevant correlators, and find they are able to dominate in many interesting limits. Interpreting these results in the context of holographic models of quantum gravity, sheds new light on the black hole information problem by exhibiting a class of correlators where bulk effective field theory does not predict its own demise.

Paper Structure

This paper contains 6 sections, 37 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Scrambling and CFT Correlators
  3. Higher Weight Intermediate States
  4. Intermediate channels with $h_{p}\gg c$
  5. Conclusions
  6. Correlators and Conformal Blocks

Figures (2)

  • Figure 1: The configuration of four pair-wise identical local operators, themselves separated by large $t$.
  • Figure 2: Plot of function $|z^{2h_{v}}\mathcal{F}(z)|=|F(z(t))|=1/\left|1-12\pi ih_{w}\exp\left(2\pi/\beta\left(t-\log c-x\right)\right)\right|^{2h_{v}}$ where $c=10^{7}$, $h_{v}=100$, $h_{w}=10$, $\beta=2\pi$ and $x=0$. Here $t_{*}=7.7$ according to \ref{['eq:scrambletime']}. In the right panel, a plot of $\mathrm{Re}\,F(z(t))$ is shown.