Incipient quantum spin Hall insulator under strong correlations

To assess prior mean-field claims that the interacting Kane-Mele model hosts a novel $z-$antiferromagnetic (AFM) Chern insulating phase for a wide range of sub-lattice potentials, we analyze the Kane-Mele-Hubbard model in the presence of a sub-lattice potential using non-perturbative determinant quantum Monte Carlo simulations. We find instead that the true low-temperature state is a quantum spin Hall insulator for intermediate values of the sub-lattice potential $\lambda_v$ and large on-site repulsion. Two kinds of magnetic fluctuations are found to compete: $z$- and $xy$-AFM. The latter dominates at low temperature leading to a stabilization of the quantum spin Hall state as opposed to $z-$AFM Chern insulator. Our work is consistent with the robust quantum spin Hall effects which are consistently observed at even-integer fillings over a wide range of parameters in twisted bilayer MoTe$_2$ and WSe$_2$ as well as AB stacked MoTe$_2$/WSe$_2$.