Entanglement Entropy and Mutual Information in Bose-Einstein Condensates

We study the entanglement properties of free non-relativistic Bose gases. At zero temperature, we calculate the bipartite block entanglement entropy of the system, and find that it diverges logarithmically ($\frac{1}{2} \ln N$) with the particle number in the subsystem. For finite temperatures, we study the mutual information between the two blocks. We first analytically study an infinite-range hopping model, then numerically study a set of long-range hopping models in one-dimension that exhibit Bose-Einstein condensation. In both cases we find that a Bose-Einstein condensate, if present, makes a divergent contribution to the mutual information which is proportional to the logarithm of the number of particles in the condensate in the subsystem. Below $T_C$ the prefactor of the logarithmic divergent term is 1/2 for the infinite-range hopping model, and model dependent ($<$1/2) for the long-range hopping models.