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

TL;DR

This paper introduces a unified mathematical framework using equivariant generalized homologies and the Atiyah-Hirzebruch spectral sequence to analyze crystalline symmetry protected topological phases, applicable to both interacting and free fermion systems.

Contribution

It formulates crystalline SPT phases via equivariant generalized homologies and unifies various concepts like layer construction and higher-order phases through spectral sequences.

Findings

Unified framework for crystalline SPT phases.

Applicable to interacting and free fermion systems.

Connections to layer construction and higher-order phases.

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.