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

We investigate, firstly, the competition between the Rashba spin–orbit coupling (SOC) and the intrinsic SOC in Kane–Mele model. For the small intrinsic SOC, we investigate the effects of the Rashba SOC on the touching point of the valence and conduction bands when the ratio of the Rashba SOC to the intrinsic SOC is greater than classical value 2 3 . For the large intrinsic SOC, we find that the critical ratio of the two SOCs at which the band touching occurs decreases with the increasing intrinsic SOC and the locations of these touching points deviate from points K and K′ of the Brillouin zone. Furthermore, effects of the Rashba SOC on these touching points are discussed in detail when the ratio is greater than the critical value. The Rashba SOC-driven topologically trivial and non-trivial transitions are also obtained in the first part of the work. Secondly, using the slave-rotor mean field method we investigate the influences of the correlation on the Rashba SOC-driven topologically trivial and non-trivial transitions in both the charge condensate and Mott regions. The topological Mott insulator with gapped or gapless spin excitations which arises from the interplay of the Rashba SOC and correlations is obtained in the work.

Abstract. The dimerized Kane-Mele model with/without the strong interaction is studied using analytical methods. The boundary of the topological phase transition of the model without strong interaction is obtained. Our results show that the occurrence of the transition only depends on dimerized parameter  . From the one-particle spectrum, we obtain the completed phase diagram including the quantum spin Hall (QSH) state and the topologically trivial insulator.Then, using different mean-field methods, we investigate the Mott transition and the magnetic transition of the strongly correlated dimerized Kane-Mele model. In the region between the two transitions, the topological Mott insulator (TMI) with characters of Mott insulators and topological phases may be the most interesting phase. In this work, effects of the hopping anisotropy and Hubbard interaction U on boundaries of the two transitions are observed in detail.The completed phase diagram of the dimerized Kane-Mele-Hubbard model is also obtained in this work. Quantum fluctuations have extremely important influences on a quantum system. However, investigations are under the framework of the mean field treatment in this work and the effects of fluctuations in this model will be discussed in the future.

We examine the contribution of electromagnetic field to the atomic spin, by adopting two different, both gauge invariant definitions of the electromagnetic angular momentum: and . Notably, at the classical level, gives an exactly null result while gives a finite value. This suggests that leads to a simpler and more reasonable picture of the atomic spin, therefore qualifies as a more appropriate definition of the electromagnetic angular momentum. Our observation gives important hint on the delicate issue of gluon contribution to the nucleon spin.