Eigenstate Structure and Quantum Chaos in the Bose-Hubbard Hamiltonian

Interacting bosons in an optical lattice, conveniently described by the Bose-Hubbard Hamiltonian, constitute an archetypical example of a complex quantum many-body system. When the strengths of nearest-neighbour tunneling and on-site interaction are of comparable magnitude, the Bose-Hubbard model develops a chaotic phase, in which statistical features of the system are characterized by universal random-matrix ensembles and are hence no longer system-specific.

We investigate the dependence of this chaotic phase on particle number N, system size L and particle density n=N/L in terms of spectral and eigenvector statistics, and benchmark these statistical features against 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 delocalisation in the computational basis and are well described by the two random matrix ensembles. 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 N, L, or n constant, the fastest convergence of the Bose-Hubbard chaotic phase to the universal random-matrix benchmarks is achieved at fixed density n≲1. 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. These results provide evidence of a way to discriminate among different many-body Hamiltonians even in the chaotic regime.