Topological phases characterized by spin Chern number and skyrmion number in triangular Bose-Hubbard model

Topological phases are important topics in condensed matter physics. Here, we investigate a spin-orbit coupling Bose-Hubbard model in triangular lattice. In the strong coupling limit, we obtained the single particle Green's function and constructed the phase diagram for ground states. We found two types of nontrivial topological ground phases characterized by spin Chern number and skyrmion number, respectively. The spin Chern numbers characterize the spin Chern insulators. While the skyrmion numbers characterize the skyrmion textures. We show that the phase transitions between different spin Chern insulators take place with gap closing and reopening. While the phase transitions between different skyrmion textures occur without gap closing.