Generalized homology and Atiyah-Hirzebruch spectral sequence in crystalline symmetry protected topological phenomena
Abstract
We propose that symmetry protected topological (SPT) phases with crystalline symmetry are formulated by equivariant generalized homologies hGn(X) over a real space manifold X with G a crystalline symmetry group. The Atiyah-Hirzebruch spectral sequence unifies various notions in crystalline SPT phases such as the layer construction, higher-order SPT phases and Lieb-Schultz-Mattis type theorems. Our formulation is applicable to interacting systems with onsite and crystalline symmetries as well as free fermions.
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