Entanglement of Purification in Many Body Systems and Symmetry Breaking

We study the entanglement of purification (EOP), a measure of total correlation between two subsystems A and B, for free scalar field theory on a lattice and the transverse-field Ising model by numerical methods. In both of these models, we find that the EOP becomes a nonmonotonic function of the distance between A and B when the total number of lattice sites is small. When it is large, the EOP becomes monotonic and shows a plateaulike behavior. Moreover, we also show that the original reflection symmetry which exchanges A and B can get broken in optimally purified systems. We provide an interpretation of our results in terms of the interplay between classical and quantum correlations.