Interplay between short and long-range entanglement in symmetry protected phases

We study a variety of questions related to entanglement in symmetry protected phases, especially those introduced in arXiv:1106.4772 (Chen et al., 2011). These phases are analogous to topological insulators in that they are short range entangled states with symmetry protected edge or surface states. We show that the now standard bulk-edge correspondence relating the entanglement spectrum to the gapless edge spectrum holds for these phases as well. We also consider the question of coupling these models to gauge fields or equivalently of introducing long range entanglement. We argue that this procedure yields models with perturbatively stable edge or surface states at a variety of interface types. The non-onsite nature of the edge symmetry plays an important role in our considerations.