  We study entanglement properties of systems with spontaneously broken
continuous symmetry. We find that in addition to the expected area law
behavior, the entanglement entropy contains a subleading contribution which
diverges logarithmically with the subsystem size in agreement with the Monte
Carlo simulations of A. Kallin et. al. (Phys. Rev. B 84, 165134 (2011)). The
coefficient of the logarithm is a universal number given simply by $N_G
(d-1)/2$, where $N_G$ is the number of Goldstone modes and $d$ is the spatial
dimension. This term is present even when the subsystem boundary is straight
and contains no corners, and its origin lies in the interplay of Goldstone
modes and restoration of symmetry in a finite volume. We also compute the
"low-energy" part of the entanglement spectrum and show that it has the same
characteristic "tower of states" form as the physical low-energy spectrum
obtained when a system with spontaneously broken continuous symmetry is placed
in a finite volume.
