Eigenstate thermalization scaling in approaching the classical limit

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 similar to 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 midspectrum 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.

Automated summary

The study explores the effect of increasing the particle number in fixed lattice topologies in the classical or semiclassical limit, focusing on the behavior of the Bose-Hubbard system. It is found that for larger lattices, the ETH scaling of physical midspectrum eigenstates follows the ideal (Gaussian) expectation, while for smaller lattices, anomalous scaling occurs with a different exponent. Various plausible mechanisms for this anomaly are examined.