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On CFT and Quantum Chaos

Gustavo Turiaci, Herman Verlinde

TL;DR

The paper argues that 1+1D CFTs with non-linear conformal realization exhibit chaos signatures, notably maximal Lyapunov growth and a spectrum of Ruelle resonances. It introduces a parafermionic Y-system lattice model that, in the continuum limit, flows to Liouville CFT and shows chaotic behavior, offering a UV-complete framework. The work further connects Ruelle resonances to the analytic structure of OPE coefficients, with Liouville DOZZ data reproducing BTZ quasi-normal-mode poles, supporting the view that Liouville theory captures universal high-energy CFT behavior in holographic contexts. Together, these results illuminate the deep ties between near-horizon gravity, chaotic dynamics, and high-energy CFT data, and provide practical models to study scrambling and ergodicity.

Abstract

We make three observations that help clarify the relation between CFT and quantum chaos. We show that any 1+1-D system in which conformal symmetry is non-linearly realized exhibits two main characteristics of chaos: maximal Lyapunov behavior and a spectrum of Ruelle resonances. We use this insight to identify a lattice model for quantum chaos, built from parafermionic spin variables with an equation of motion given by a Y-system. Finally we point to a relation between the spectrum of Ruelle resonances of a CFT and the analytic properties of OPE coefficients between light and heavy operators. In our model, this spectrum agrees with the quasi-normal modes of the BTZ black hole.

On CFT and Quantum Chaos

TL;DR

The paper argues that 1+1D CFTs with non-linear conformal realization exhibit chaos signatures, notably maximal Lyapunov growth and a spectrum of Ruelle resonances. It introduces a parafermionic Y-system lattice model that, in the continuum limit, flows to Liouville CFT and shows chaotic behavior, offering a UV-complete framework. The work further connects Ruelle resonances to the analytic structure of OPE coefficients, with Liouville DOZZ data reproducing BTZ quasi-normal-mode poles, supporting the view that Liouville theory captures universal high-energy CFT behavior in holographic contexts. Together, these results illuminate the deep ties between near-horizon gravity, chaotic dynamics, and high-energy CFT data, and provide practical models to study scrambling and ergodicity.

Abstract

We make three observations that help clarify the relation between CFT and quantum chaos. We show that any 1+1-D system in which conformal symmetry is non-linearly realized exhibits two main characteristics of chaos: maximal Lyapunov behavior and a spectrum of Ruelle resonances. We use this insight to identify a lattice model for quantum chaos, built from parafermionic spin variables with an equation of motion given by a Y-system. Finally we point to a relation between the spectrum of Ruelle resonances of a CFT and the analytic properties of OPE coefficients between light and heavy operators. In our model, this spectrum agrees with the quasi-normal modes of the BTZ black hole.

Paper Structure

This paper contains 6 sections, 66 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Lyapunov from Goldstone
  3. A Chaotic Lattice Model
  4. Ruelle Resonances
  5. Conclusions
  6. DOZZ three point function

Figures (2)

  • Figure 1: The discrete model is defined on a rhombic lattice. We indicated the center $(\sigma,\tau)$ of the diamond $(\sigma\pm1,\tau\pm1)$. The equation of motion (\ref{['ysystem']}) expresses the variable at the top of the diamond in terms of the other three.
  • Figure 2: Diagrammatical representation of crossing symmetry (given by the first identity) and the exchange algebra (given by the second) of the CFT correlation function of two heavy operators, labeled by $M$, and two light ones, labeled by $a$ and $b$.