Universality versus certification of distinctive features in the Bose-Hubbard Hamiltonian

Understanding the dynamics of complex interacting quantum many-body systems is crucial to exploit them as platforms for quantum-information tasks such as quantum computations and quantum simulations. An archetypical example system for complex quantum many-body dynamics is the Bose-Hubbard model of interacting bosons on a lattice. When the strengths of nearest-neighbour tunneling and on-site interaction are of comparable magnitude, the Bose-Hubbard Hamiltonian develops a chaotic phase, in which statistical features of the system are characterized by universal random-matrix ensembles and are hence no longer distinctive for the system. However, e.g., in a quantum-simulation scenario, one wishes to certify the implementation of a specific Hamiltonian.

We investigate the Bose-Hubbard Hamiltonian's chaotic phase in terms of spectral and eigenvector statistics, benchmarking them 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 Bose-Hubbard eigenstate fractal dimensions, which are quantifiers of the eigenstate structure and related to Rényi information entropies, are well described by the two random-matrix models. As the limit of infinite Hilbert space dimension is approached along different directions, the fastest convergence of the Bose-Hubbard chaotic phase to the universal random-matrix predictions is achieved at fixed particle density $\lesssim 1$. 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.