Entanglement Entropy of Interacting Fermions using Large-$N$ Approach

The R\'enyi entropy $S^{(n)}$ of interacting fermions can be expressed in terms of the normalized partition function of the theory in the presence of an additional vertex $S_{ent}(t_0)$, quadratic in the fields within the subsystem $A$ at the time of measurement $t_0$. For the ground state or in thermal equilibrium, $S_{ent}(t_0)$ couples successive replicas together with a self-replica coupling. This breaks both spatial and temporal translational invariance due to the entanglement cut at the boundary of the subsystem $A$ at time $t_0$. We use this to calculate the entanglement entropy of generic interacting fermionic systems via a large-$N$ approach employing the saddle point approximation. We show that, in the $n \to 1$ limit relevant to the von Neumann entropy $S^{\mathrm{vN}}$, the system can be described by the usual translationally invariant saddle point of the original interaction theory, as the contribution to the saddle point from $S_{ent}$ vanishes in this limit. As applications, we compute the large-$N$ entanglement entropy of interacting systems, including the Fermi gas, superconductors (s-wave and d-wave), the heavy Fermi liquid in the Kondo lattice, and the charge-density-wave (CDW) lattice model.