  We present a new method for calculating Renyi entanglement entropies for
fermionic field-theories originating from microscopic Hamiltonians. The method
builds on an operator identity which we discover for the first time. The
identity leads to the representation of traces of operator products, and thus
Renyi entropies of a subsystem, in terms of fermionic-displacement operators.
This allows for a very transparent path-integral formulation, both in and
out-of-equilibrium, having a simple boundary condition on the fermionic fields.
The method is validated by reproducing well known expressions for entanglement
entropy in terms of the correlation matrix for non-interacting fermions. We
demonstrate the effectiveness of the method by explicitly formulating the field
theory for Renyi entropy in a few zero and higher-dimensional large-$N$
interacting models based on the Sachdev-Ye-Kitaev (SYK) model, and for the
Hubbard model within dynamical mean-field theory (DMFT) approximation. We use
the formulation to compute Renyi entanglement entropy of interacting Fermi
liquid (FL) and non-Fermi liquid (NFL) states in the large-$N$ models and
compare successfully with the results obtained via exact diagonalization for
finite $N$. We elucidate the connection between entanglement entropy and
residual entropy of the NFL ground state in the SYK model and extract sharp
signatures of quantum phase transition in the entanglement entropy across an
NFL to FL transition. Furthermore, we employ the method to obtain nontrivial
system-size scaling of entanglement in an interacting diffusive metal described
by a chain of SYK dots.
