  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.
