  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 $t_L$ 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 $t_L$-driven
topological transition. Under the condition, critical values of $t_L$ at which
the topological transition occurs are obtain 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.
