Short-ranged interaction effects on $Z_2$ topological phase transitions

Using a combined perturbative and self-consistent mean-field approach that we directly compare with quantum Monte Carlo calculations, we study the effects of short-ranged interactions on the $Z_2$ topological insulator phase, also known as the quantum spin Hall phase, in two generalized versions of the Kane-Mele model at half-filling on the honeycomb lattice. For interactions weaker than the critical value for magnetic instability, we find that the interactions can stabilize the quantum spin Hall phase against third neighbor hoppings, which preserve $C_3$ lattice rotation symmetry, but destabilize it for a dimerization that explicitly breaks the $C_3$ symmetry. Consistent with quantum Monte Carlo calculations, we show the phase boundary shifts are linearly proportional to the square of the interaction strength, but with opposite sign--a result that cannot be reproduced with a perturbative treatment that does not also include a self-consistent treatment of the perturbed Hamiltonian. Our results emphasize that short-range interactions can have subtle effects on the stability of topological phases, and may need to be treated by methods analogous to those we use here.