Organizing symmetry-protected topological phases by layering and symmetry reduction: a minimalist perspective

It is demonstrated that fermionic/bosonic symmetry-protected topological (SPT) phases across different dimensions and symmetry classes can be organized using geometric constructions that increase dimensions and symmetry-forgetting maps that change symmetry groups. Specifically, it is shown that the interacting classifications of SPT phases with and without glide symmetry fit into a short exact sequence, so that the classification with glide is constrained to be a direct sum of cyclic groups of order 2 or 4. Applied to fermionic SPT phases in the Wigner-Dyson class AII, this implies that the complete interacting classification in the presence of glide is ${\mathbb Z}_4{\oplus}{\mathbb Z}_2{\oplus}{\mathbb Z}_2$ in 3 dimensions. In particular, the hourglass-fermion phase recently realized in the band insulator KHgSb must be robust to interactions. Generalizations to spatiotemporal glide symmetries are discussed.