Topological skyrmion phases of matter

Abstract:

Symmetry-protected topological phases of matter have long been classified based on mappings from the full Brillouin zone to the space of projectors onto occupied states. 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 distinguished by winding of the ground state spin rather than winding of the projector onto the ground state, in scenarios where these are distinct quantities. We show these phases realize momentum-space skyrmionic spin textures, and exhibit exotic bulk-boundary correspondence, which we characterize in detail by introducing the symmetry-enriched partial trace and symmetry-enriched slab entanglement spectrum. Non-trivial spin skyrmion number corresponds to additional spin-momentum-locking at the edge in slab geometries, and topologically-protected gapless boundary modes even when the projector topological invariant is trivial. We present recipes for constructing myriad toy models of these phases, and also explore consequences of this physics in transition metal oxide superconductors as the generalized particle-hole symmetry occurs in centrosymmetric superconductors. When spin is not conserved due to non-negligible atomic spin-orbit coupling, we find two kinds of topological phase transitions are possible. The second kind occurs without the closing of the minimum direct bulk energy gap while respecting the symmetry protecting the topological phase, due to the minimum spin magnitude going to zero somewhere in the Brillouin zone. This type-II topological phase transition serves as the first-known contradiction of the flat-band limit assumption, widely-used since the early 1980's.