INVERTIBLE PHASES AND FERMIONIC CRYSTALLINE EQUIVALENCE PRINCIPLE (International Journal of Food and Nutritional Sciences, 2021)
The classification of topological phases of matter has been the subject of intensive research in condensedmatter physics and nearby areas of mathematics for the last decade, but difficult problems still remain: for example, there is not yet an accepted mathematical definition of a topological phase of matter, so researchers must study these systems using ansatzes or heuristic definitions of phases. Restricting to invertible phases, also known as symmetry-protected topological (SPT) phases, simplifies the classification question, but defining these phases precisely is also still an open problem. This paper presents Invertible Phases and Fermionic Crystalline Equivalence Principle. We provide a model for the classification of invertible phases on a G-space. The aim of this paper is to formulate and prove such a fermionic crystalline equivalence principle (FCEP). To do so, we provide an ansatz expressing groups of invertible phases on a G-space Y in which the symmetry type can be merely locally constant over space and can mix with G, including as a special case spatial symmetries mixing with fermion parity. The computation of phase homology groups that represent groups of pointgroup-equivariant fermionic phases reduces to computations of bordism groups. For symmetry types, that has been studied before by other methods, our computations agree with the literature, bolstering our ansatz.