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We present a new method for calculating Renyi entanglement entropies forfermionic field-theories originating from microscopic Hamiltonians. The methodbuilds on an operator identity which we discover for the first time. Theidentity leads to the representation of traces of operator products, and thusRenyi entropies of a subsystem, in terms of fermionic-displacement operators.This allows for a very transparent path-integral formulation, both in andout-of-equilibrium, having a simple boundary condition on the fermionic fields.The method is validated by reproducing well known expressions for entanglemententropy in terms of the correlation matrix for non-interacting fermions. Wedemonstrate the effectiveness of the method by explicitly formulating the fieldtheory for Renyi entropy in a few zero and higher-dimensional large-$N$interacting models based on the Sachdev-Ye-Kitaev (SYK) model, and for theHubbard model within dynamical mean-field theory (DMFT) approximation. We usethe formulation to compute Renyi entanglement entropy of interacting Fermiliquid (FL) and non-Fermi liquid (NFL) states in the large-$N$ models andcompare successfully with the results obtained via exact diagonalization forfinite $N$. We elucidate the connection between entanglement entropy andresidual entropy of the NFL ground state in the SYK model and extract sharpsignatures of quantum phase transition in the entanglement entropy across anNFL to FL transition. Furthermore, we employ the method to obtain nontrivialsystem-size scaling of entanglement in an interacting diffusive metal describedby a chain of SYK dots.

Renyi entanglement entropy of Fermi liquids and non-Fermi liquids: Sachdev-Ye-Kitaev model and dynamical mean field theories