Surfaces of 3d symmetry protected phases

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Surfaces of 3d symmetry protected phases

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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.


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Fidkowski, Lukasz. (2015). Surfaces of 3d symmetry protected phases. Retrieved from the University Digital Conservancy,

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