Abstract

Symmetry-protected topological phases of matter have long been classified based on mappings from the Brillouin zone to the space of projectors, or flat band Hamiltonians. This is the basis of the ten fold way classification scheme. Here, we show effectively non-interacting topological phases are associated with mappings from the Brillouin zone to the space of other observables. We specifically consider topological phases of the spin degree of freedom of the ground state, protected by a generalized particle-hole symmetry, which are independent of the topological phases of the full set of degrees of freedom of the ground state. We show these phases realize distinctive momentum-space skyrmionic spin textures, and exhibit two types of bulk-boundary correspondence. We present recipes for constructing toy models, and also explore consequences of this physics in models for transition metal oxide superconductors as the generalized particle-hole symmetry occurs in centrosymmetric superconductors. When spin is not conserved, we find two kinds of topological phase transitions are possible. The second kind occurs without the closing of an energy gap while respecting the symmetry protecting the topological phase, due to the minimum spin magnitude becoming zero. This contradicts the flat band limit assumption used to construct the ten fold way classification scheme.