Generalized homology and Atiyah–Hirzebruch spectral sequence in crystalline symmetry protected topological phenomena
Ken Shiozaki, Charles Zhaoxi Xiong, Kiyonori Gomi
Abstract
Abstract We propose that symmetry-protected topological (SPT) phases with crystalline symmetry are formulated by an equivariant generalized homology $h^G_n(X)$ 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. This formulation is applicable to not only free fermionic systems but also interacting systems with arbitrary onsite and crystal symmetries.
Topics & Concepts
Equivariant mapPhysicsHomogeneous spaceSymmetry (geometry)Spectral sequenceSymmetry groupManifold (fluid mechanics)Homology (biology)Mathematical physicsTopology (electrical circuits)Theoretical physicsPure mathematicsGeometryMathematicsCombinatoricsChemistryEngineeringCohomologyGeneBiochemistryMechanical engineeringTopological Materials and PhenomenaAlgebraic structures and combinatorial modelsNoncommutative and Quantum Gravity Theories