Ergodic Equilibration of R\'enyi Entropies and Replica Wormholes

TL;DR

This paper investigates the long-time behavior of Rnyi entropies in chaotic many-body quantum systems, revealing exponential suppression of quantum noise and connections to black hole physics via AdS/CFT correspondence.

Contribution

It provides exact long-time averages of Rnyi entropies for pure states, extending the equilibrium approximation to black hole scenarios and analyzing non-planar permutation effects.

Findings

Quantum noise around Rnyi entropy averages is exponentially suppressed.

Long-time averages align with the equilibrium proposal for delocalized states.

Extension of analysis to evaporating black holes and gravitational interpretations.

Abstract

We study the behavior of R\'enyi entropies for pure states from standard assumptions about chaos in the high-energy spectrum of the Hamiltonian of a many-body quantum system. We compute the exact long-time averages of R\'enyi entropies and show that the quantum noise around these values is exponentially suppressed in the microcanonical entropy. For delocalized states over the microcanonical band, the long-time average approximately reproduces the equilibration proposal of H. Liu and S. Vardhan, with extra structure arising at the order of non-planar permutations. We analyze the equilibrium approximation for AdS/CFT systems describing black holes in equilibrium in a box. We extend our analysis to the situation of an evaporating black hole, and comment on the possible gravitational description of the new terms in our approximation.