Chaos in the Bose‐Hubbard model versus Gaussian orthogonal and embedded random matrix ensembles

We benchmark spectral and eigenvector statistics of the Bose‐Hubbard Hamiltonian against those of the Gaussian orthogonal and the bosonic two‐body embedded random‐matrix ensembles. The latter, in contrast to the Gaussian ensemble, mirrors the few‐body nature of interactions and is therefore expected to better describe chaotic quantum many‐particle systems. Within the energy and parameter range where chaos fully unfolds, the expectation value and the eigenstate‐to‐eigenstate fluctuations of the fractal dimensions of Bose‐Hubbard eigenstates show clear signatures of ergodicity and are well described by the two random‐matrix ensembles [1,2]. On top, the bosonic embedded ensemble reproduces the energy dependence of the chaotic domain. As the limit of infinite Hilbert space dimension is approached along different directions, keeping either particle number N, system size L, or particle density n = N/L constant, the limit N → ∞ at constant n leads to a faster convergence of the chaotic phase towards the random‐matrix benchmarks than the same limit at constant L. The fastest route to chaos is found at fixed density n 1 [3]. Despite the agreement of the three models on the level of the fractal dimensions' lowest‐order statistical moments, the models are ever more distinguishable from each other in terms of their full fractal dimension distributions as Hilbert space grows, even along the fastest route to chaos. These results provide evidence of a way to discriminate among different many‐body Hamiltonians in the chaotic regime.
[1] L. Pausch et al., Phys. Rev. Lett. 126, 150601 (2021) [2] L. Pausch et al., New J. Phys. 23, 123036 (2021) [3] L. Pausch et al., J. Phys. A 55, 324002 (2022)