Eigenstate thermalization scaling in approaching the classical limit

TL;DR

This paper investigates how eigenstate fluctuations in the Bose-Hubbard model scale as the system approaches the classical limit, revealing deviations from ETH predictions in small lattices and confirming expected behavior in larger systems.

Contribution

It derives and tests scaling laws for eigenstate fluctuations in the semiclassical limit, highlighting size-dependent deviations from ETH expectations.

Findings

ETH scaling follows Gaussian predictions in large lattices

Small lattices exhibit anomalous scaling exponents

Numerical analysis supports theoretical scaling laws

Abstract

According to the eigenstate thermalization hypothesis (ETH), the eigenstate-to-eigenstate fluctuations of expectation values of local observables should decrease with increasing system size. In approaching the thermodynamic limit - the number of sites and the particle number increasing at the same rate - the fluctuations should scale as $\sim D^{-1/2}$ with the Hilbert space dimension $D$. Here, we study a different limit - the classical or semiclassical limit - by increasing the particle number in fixed lattice topologies. We focus on the paradigmatic Bose-Hubbard system, which is quantum-chaotic for large lattices and shows mixed behavior for small lattices. We derive expressions for the expected scaling, assuming ideal eigenstates having Gaussian-distributed random components. We show numerically that, for larger lattices, ETH scaling of physical mid-spectrum eigenstates follows the ideal (Gaussian) expectation, but for smaller lattices, the scaling occurs via a different exponent. We examine several plausible mechanisms for this anomalous scaling.