Renyi entropy, mutual information, and fluctuation properties of Fermi liquids

We compute the leading contribution to the ground-state Renyi entropy S-alpha for a region of linear size L in a Fermi liquid in d dimensions. The result contains a universal boundary law violating term simply related to the more familiar entanglement entropy. We also obtain a universal crossover function that smoothly interpolates between the zero-temperature result and the ordinary thermal Renyi entropy of a Fermi liquid. Formulas for the entanglement entropy of more complicated regions, including nonconvex and disconnected regions, are obtained from the conformal field theory formulation of Fermi surface dynamics. These results permit an evaluation of the quantum mutual information between different regions in a Fermi liquid. We also study the number fluctuations in a Fermi liquid. Taken together, these results give a reasonably complete characterization of the low-energy quantum information content of Fermi liquids.