# Phases of scrambling in eigenstates

## TL;DR

This paper investigates the behavior of light operator expectation values in heavy eigenstates of holographic 2D CFTs, revealing thermalization above a certain threshold and oscillatory behavior below, with implications for quantum chaos.

## Contribution

It introduces a monodromy method to compute light operator expectation values in heavy eigenstates and characterizes their scrambling behavior across the threshold.

## Key findings

- Eigenstates above the BTZ threshold exhibit thermal behavior with an effective temperature.
- Eigenstates below the threshold show persistent oscillations and non-decaying correlations.
- Heavy eigenstates above the threshold display maximal scrambling with Lyapunov exponent $2 extpi T_{
m eff}$.

## Abstract

We use the monodromy method to compute expectation values of an arbitrary number of light operators in finitely excited ("heavy") eigenstates of holographic 2D CFT. For eigenstates with scaling dimensions above the BTZ threshold, these behave thermally up to small corrections, with an effective temperature determined by the heavy state. Below the threshold we find oscillatory and not decaying behavior. As an application of these results we compute the expectation of the out-of-time order arrangement of four light operators in a heavy eigenstate, i.e. a six-point function. Above the threshold we find maximally scrambling behavior with Lyapunov exponent $2\pi T_{\rm eff}$. Below threshold we find that the eigenstate OTOC shows persistent harmonic oscillations.