A derivation of the late-time volume law for local operator entanglement

TL;DR

This paper derives an analytical expression for late-time local operator entanglement in chaotic quantum systems, demonstrating volume-law scaling and validating it through numerical simulations of the 1D mixed-field Ising model.

Contribution

It provides the first explicit analytical formula for late-time LOE in chaotic systems, based on assumptions like ETH and random state replacement.

Findings

LOE exhibits volume-law scaling at late times in chaotic systems

The derived formula matches numerical simulations in the 1D mixed-field Ising model

Assumptions like ETH and non-resonance are validated within certain regimes

Abstract

Local Operator Entanglement (LOE) has emerged an indicator of quantum chaos in many-body systems. Numerical studies have shown that, in chaotic systems, LOE grows linearly in time and displays a volume-law behavior at late times, scaling proportionally with the number of local degrees of freedom. Despite extensive numerical evidence, complemented by analytical studies in integrable systems, a fully analytical understanding of the emergence of the volume law remains incomplete. In this paper, we contribute toward this goal by deriving a late-time expression for LOE in chaotic systems that exhibits volume-law scaling. Our derivation proceeds by expressing the late-time LOE in the Liouville eigenstate basis and relies on three main assumptions: a higher-order non-resonance condition for the Hamiltonian eigenenergies, the Eigenstate Thermalization Hypothesis (ETH) ansatz for the matrix elements of the initial local operator, and the replacement of Hamiltonian eigenstates with random states in the final expression for LOE. Under these assumptions, we obtain an explicit formula displaying volume-law scaling. Finally, we complement our analytical derivation with numerical simulations of the 1D mixed-field Ising model, testing the resulting formula and exploring the regime of validity of our assumptions.