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Generalized homology and Atiyah-Hirzebruch spectral sequence in crystalline symmetry protected topological phenomena

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arxiv 1810.00801 v2 pith:IBWDVM4F submitted 2018-10-01 cond-mat.str-el hep-th

Ken Shiozaki , Charles Zhaoxi Xiong , Kiyonori Gomi This is my paper · ORCID
classification cond-mat.str-elhep-th
keywords crystallinesymmetryphasesatiyah-hirzebruchgeneralizedprotectedsequencespectral
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abstract

We propose that symmetry protected topological (SPT) phases with crystalline symmetry are formulated by equivariant generalized homologies $h^G_n(X)$ 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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  1. Weak in the boundary: How weak SPT phases spoil anomaly matching

    cond-mat.str-el 2025-07 conditional novelty 6.0 of 10

    Weak SPT phases with a boundary are not classified by the same cohomology groups as on the infinite periodic system; the difference is a lower-dimensional group of phases that the boundary trivializes.

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