Phase transitions of the Kane–Mele–Hubbard model with a long-range hopping

Abstract

The interacting Kane–Mele model with a long-range hopping is studied using analytical method. The original Kane–Mele model is defined on a honeycomb lattice. In the work, we introduce a four-lattice-constant range hopping and the on-site Hubbard interaction into the model and keep its lattice structure unchanged. From the single-particle energy spectrum, we obtain the critical strength of the long-range hopping tL at which the topological transition occurs in the non-interacting limit of the model and our results show that it is independent of the spin–orbit coupling. After introducing the Hubbard interaction, we investigate the Mott transition and the magnetic transition of the generalized strongly correlated Kane–Mele model using the slave-rotor mean field theory and Hartree–Fock mean field theory respectively. In the small long-range hopping region, it is a correlated quantum spin Hall state below the Mott transition, while a topological Mott insulator above the Mott transition. By comparing the energy band of spin degree of freedom with the one of electrons in non-interacting limit, we find a condition for the tL-driven topological transition. Under the condition, critical values of tL at which the topological transition occurs are obtained numerically from seven self-consistency equations in both regions below and above the Mott transition. Influences of the interaction and the spin–orbit coupling on the topological transition are discussed in this work. Finally, we show complete phase diagrams of the generalized interacting topological model at some strength of spin–orbital coupling.