Phase diagram of the Kane-Mele-Hubbard model

Motivated by recent numerical results, we study the phase diagram of the Kane-Mele-Hubbard model, especially the nature of its quantum critical points. The phase diagram of the Kane-Mele-Hubbard model can be understood by breaking the SO(4) symmetry of our previous work down to U(1)ₛₚᵢₙ×U(1)charge×PH symmetry. The vortices of the in-plane N\'eel phase carry charge, and the proliferation of the charged magnetic vortex drives the transition between the in-plane N\'eel phase and the quantum spin Hall (QSH) insulator phase; this transition belongs to the three-dimensional XY universality class. The transition between the liquid phase and the in-plane N\'eel phase is an anisotropic O(4) transition, which eventually becomes first order due to quantum fluctuation. The liquid-QSH transition is predicted to be first order based on a $1/N$ calculation.