Renyi Entropy of Chaotic Eigenstates

TL;DR

This paper derives a universal analytical expression for Renyi entanglement entropies in chaotic many-body eigenstates, revealing non-trivial subsystem size dependence and exceeding thermal entropy in certain regimes, supported by numerical validation.

Contribution

It provides the first analytical formula for Renyi entropies in finite-energy density eigenstates of chaotic systems, extending understanding beyond von Neumann entropy.

Findings

Renyi entropies depend non-linearly on subsystem size.

Renyi entropies for n > 1 exceed thermal entropy at same energy.

Analytical predictions agree with numerical simulations.

Abstract

Using arguments built on ergodicity, we derive an analytical expression for the Renyi entanglement entropies corresponding to the finite-energy density eigenstates of chaotic many-body Hamiltonians. The expression is a universal function of the density of states and is valid even when the subsystem is a finite fraction of the total system - a regime in which the reduced density matrix is not thermal. We find that in the thermodynamic limit, only the von Neumann entropy density is independent of the subsystem to the total system ratio $V_A/V$, while the Renyi entropy densities depend non-linearly on $V_A/V$. Surprisingly, Renyi entropies $S_n$ for $n > 1$ are convex functions of the subsystem size, with a volume law coefficient that depends on $V_A/V$, and exceeds that of a thermal mixed state at the same energy density. We provide two different arguments to support our results: the first one relies on a many-body version of Berry's formula for chaotic quantum mechanical systems, and is closely related to eigenstate thermalization hypothesis. The second argument relies on the assumption that for a fixed energy in a subsystem, all states in its complement allowed by the energy conservation are equally likely. We perform Exact Diagonalization study on quantum spin-chain Hamiltonians to test our analytical predictions, and find good agreement.