Type-II topological phase transitions of topological skyrmion phases

We present minimal toy models for topological skyrmion phases of matter, which generically realize type-II topological phase transitions in effectively non-interacting systems, those which occur without closing of the minimum direct bulk energy gap. We study the bulk-boundary correspondence in detail to show that a non-trivial skyrmion number yields a rich bulk-boundary correspondence. We observe gapless edge states, which are robust against disorder, due to non-trivial skyrmion number. Edge states corresponds to bands, which do not traverse the bulk gap, instead yielding gaplessness due to their overlap in energy and exponential localization on opposite edges of the system. These gapless boundary modes can occur for total Chern number zero, and furthermore correspond to rich real-space spin textures with strong polarization of spin along the real-space edge. By introducing toy models generically exhibiting type-II topological phase transitions and characterizing the bulk-boundary correspondence due to non-trivial skyrmion number in these models, we lay the groundwork for understanding consequences of the quantum skyrmion Hall effect.