  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.
