Bosonic entanglement crossover from groundstate scaling to volume laws

The crossover behavior of eigenstate entanglement entropies from an area law or log-area law for low energies and small subsystem sizes to volume laws for high energies and large subsystems can be described by scaling functions. We demonstrate this for two bosonic systems. The harmonic lattice model describes a system of coupled harmonic oscillators and is a lattice regularization for free scalar field theories. For dimensions d ≥ 2, the ground state of this model displays an entanglement area law, even at criticality, because excitation energies vanish only at a single point in momentum space. In contrast, Bose metals feature a finite Bose surface with zero excitation energy. One hence finds log-area laws for the groundstate entanglement. For both models, we sample excited states. The distributions of their entanglement entropies are sharply peaked around subsystem entropies of corresponding thermodynamic ensembles in accordance with the eigenstate thermalization hypothesis. In this way, we determine the scaling functions numerically. Eigenstates for quasi-free bosonic systems are not Gaussian. We resolve this problem by considering appropriate squeezed states instead, for which entanglement entropies can be evaluated efficiently.