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

We study a variety of questions related to entanglement in symmetry-protected phases. 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 further consider the question of coupling these models to gauge fields which we show gives an alternative way to construct different kinds of long-range entangled states without symmetry. We also 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.