Quantum Matter Seminar

We show effectively non-interacting topological phases can be associated with mappings from the Brillouin zone to the space of observables associated with the occupied states. We specifically consider topological phases of the spin degree of freedom of the ground state, which can be independent of the topological phases associated with mappings to the projectors onto occupied states for systems with a spin degree of freedom and other degrees of freedom. We show these phases realize distinctive momentum-space skyrmionic spin textures, and exhibit three types of bulk-boundary correspondence. We present recipes for constructing toy models, and also explore consequences of this physics in models for transition metal oxides. When spin is not conserved, 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 and keeping occupancy of bands fixed, due to the minimum spin magnitude passing through zero when atomic spin-orbit coupling is non-negligible and there is an orbital degree of freedom: the spin goes to zero magnitude in this case when the total angular momentum becomes entirely orbital in character. If there is time, we will also briefly summarize two other sets of topological phases we have recently introduced, the multiplicative topological phases and the finite-size topological phases.