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Effects of short-ranged interactions on the Kane-Mele model without discrete particle-hole symmetry

We study the effects of short-ranged interactions on the $Z_2$ topological insulator phase, also known as the quantum spin Hall phase, in the Kane-Mele model at half-filling with staggered potentials which explicitly breaks the discrete particle-hole symmetry. Within Hartree-Fock mean-field analysis, we conclude that the on-site repulsive interactions help stabilize the topological phase (quantum spin Hall) against the staggered potentials by enlarging the regime of the topological phase along the axis of the ratio of the staggered potential strength and the spin-orbit coupling. In sharp contrast, the on-site attractive interactions destabilize the topological phase. We also examine the attractive interaction case by means of the unbiased determinant projector quantum Monte Carlo and the results are qualitatively consistent with the Hartree-Fock picture.