Optimal route to quantum chaos in the Bose-Hubbard model

The dependence of the chaotic phase of the Bose-Hubbard Hamiltonian on particle number $N$, system size $L$ and particle density is investigated in terms of spectral and eigenstate features. We analyze the development of the chaotic phase as the limit of infinite Hilbert space dimension is approached along different directions, and show that the fastest route to chaos is the path at fixed density $n \lesssim 1$. The limit $N \to \infty$ at constant $L$ leads to a slower convergence of the chaotic phase towards the random matrix theory benchmarks. In this case, from the distribution of the eigenstate generalized fractal dimensions, the ergodic phase becomes more distinguishable from random matrix theory for larger $N$, in a similar way as along trajectories at fixed density.