  We characterize non-perturbatively the R\'enyi entropies of degree n=2,3,4,
and 5 of three-dimensional, strongly coupled many-fermion systems in the
scale-invariant regime of short interaction range and large scattering length,
i.e. in the unitary limit. We carry out our calculations using lattice methods
devised recently by us. Our results show the effect of strong pairing
correlations on the entanglement entropy, which modify the sub-leading behavior
for large subsystem sizes (as characterized by the dimensionless parameter x=kF
L_A, where kF is the Fermi momentum and L_A the linear subsystem size), but
leave the leading order unchanged relative to the non-interacting case.
Moreover, we find that the onset of the sub-leading asymptotic regime is at
surprisingly small x=2-4. We provide further insight into the entanglement
properties of this system by analyzing the spectrum of the entanglement
Hamiltonian of the two-body problem from weak to strong coupling. The low-lying
entanglement spectrum displays clear features as the strength of the coupling
is varied, such as eigenvalue crossing, a sharp change in the Schmidt gap, and
scale invariance at unitarity. Beyond the low-lying component, the spectrum
appears as a quasi-continuum distribution, for which we present a statistical
characterization; we find, in particular, that the mean shifts to infinity as
the coupling is turned off, which indicates that that part of the spectrum
represents non-perturbative contributions to the entanglement Hamiltonian. In
contrast, the low-lying entanglement spectrum evolves to finite values in the
noninteracting limit. The scale invariance of the unitary regime guarantees
that our results are universal features intrinsic to 3D quantum mechanics and
represent a well-defined prediction for ultracold atom experiments, which were
recently shown to have direct access to the entanglement entropy.
