  For quantum matter, eigenstate entanglement entropies obey an area law or
log-area law at low energies and small subsystem sizes and cross over to volume
laws for high energies and large subsystems. This transition is captured by
crossover functions, which assume a universal scaling form in quantum critical
regimes. We demonstrate this for the harmonic lattice model, which describes
quantized lattice vibrations and is a regularization for free scalar field
theories, modeling, e.g., spin-0 bosonic particles. In one dimension, the
groundstate entanglement obeys a log-area law. For dimensions $d\geq 2$, it
displays area laws, even at criticality. The distribution of excited-state
entanglement entropies is found to be sharply peaked around subsystem entropies
of corresponding thermodynamic ensembles in accordance with the eigenstate
thermalization hypothesis. Numerically, we determine crossover scaling
functions for the quantum critical regime of the model and do a large-deviation
analysis. We show how infrared singularities of the system can be handled and
how to access the thermodynamic limit using a perturbative trick for the
covariance matrix. Eigenstates for quasi-free bosonic systems are not Gaussian.
We resolve this problem by considering appropriate squeezed states instead. For
these, entanglement entropies can be evaluated efficiently.
