Oscillating terms in the Renyi entropy of Fermi liquids

In this work we compute subleading oscillating terms in the Renyi entropy of Fermi gases and Fermi liquids corresponding to $2k_F$-like oscillations. Our theoretical tools are the one dimensional formulation of Fermi liquid entanglement familiar from discussions of the logarithmic violation of the area law and quantum Monte Carlo calculations. The main result is a formula for the oscillating term for any region geometry and a spherical Fermi surface. We compare this term to numerical calculations of entanglement using the correlation function method and find excellent agreement. We also compare with quantum Monte Carlo data on interacting Fermi liquids where we also find excellent agreement up to moderate interaction strengths.