Invertible phases for mixed spatial symmetries and the fermionic crystalline equivalence principle

TL;DR

This paper extends a mathematical framework to classify gapped invertible phases of matter with complex spatial symmetries, including fermionic cases, and proves a fermionic crystalline equivalence principle, aligning with existing physics results.

Contribution

It generalizes the Freed-Hopkins ansatz to include nontrivial symmetry mixing and establishes a new fermionic crystalline equivalence principle.

Findings

Proves the fermionic crystalline equivalence principle.

Computes classifications of phases with point group symmetry.

Results agree with previous studies where applicable.

Abstract

Freed-Hopkins give a mathematical ansatz for classifying gapped invertible phases of matter with a spatial symmetry in terms of Borel-equivariant generalized homology. We propose a slight generalization of this ansatz to account for cases where the symmetry type mixes nontrivially with the spatial symmetry, such as crystalline phases with spin-1/2 fermions. From this ansatz, we prove as a theorem a "fermionic crystalline equivalence principle," as predicted in the physics literature. Using this and the Adams spectral sequence, we compute classifications of some classes of phases with a point group symmetry; in cases where these phases have been studied by other methods, our results agree with the literature.