3d symmetry protected phases and surface topological order

Lukasz Fidkowski
SUNY Stony Brook

I will consider gapped Hamiltonians of generalized spin models, which are invariant under a certain unbroken onsite unitary symmetry group. It is well known that such Hamiltonians can realize topologically ordered phases, which in (2+1)d can be studied with modular tensor categories. When a symmetry is included, the corresponding `symmetry enriched' phases correspond to a richer mathematical structure - e.g. braided G-crossed categories in (2+1)d. However, in systematically constructing such braided G-crossed categories by extending ordinary modular ones one sometimes encounters obstructions. Here we give a physical interpretation for such obstruction, and show that the corresponding topologically ordered theory, though it cannot be realized in 2d in a G-symmetric way, can be realized at the surface of a 3d `symmetry protected' phase. I will try to emphasize the physical interpretation of the various mathematical concepts involved, and I will explain a specific example in detail.

Back to Symmetry and Topology in Quantum Matter