Narrow Results

We investigate the topological phases of a three-dimensional (3D) topological insulator (TI) without the top–bottom inversion symmetry. We calculate the momentum depended spin Chern number to extract the phase diagram. Various phases are found and we address the dependence of phase boundaries on the strength of inversion asymmetry. Opposite to the quasi-two-dimensional thin film TI, in our 3D system the TI state is stabilized by the structure inversion asymmetry (SIA). With a strong SIA the 3D TI phase can exist even under a large Zeeman field. In a tight-binding form, the surface modes are discussed to confirm with the phase diagram. Particularly we find that the SIA cannot destroy the surface states but open a gap on its spectrum.

In this paper, we demonstrate an anomalous topological phase transition without closing of bulk energy gap. We find such an effect in a model of three-dimensional (3D) topological insulator (TI) subjected to the in-plane exchange field. The energy spectrum, spin spectrum and momentum-dependent spin Chern numbers are calculated. It is shown that our system realizes both the 3D TI phase and the integer quantum Hall (QH) phase. By varying the strength of exchange field, a series of topological phase transitions takes place and in the mean time the energy gap remains open. However, the spin spectrum is closed at the transition and various topological phases are characterized with different number of nodes in spin spectrum. In a tight-binding form, the surface modes are discussed to confirm with the phase diagram. Particularly for a strong field, we find the flat band edge modes which may provide an opportunity for realizing the two-dimensional (2D) fractional QH effect on the boundary of our 3D system.

Topological phase transitions of a two-dimensional topologically nontrivial polaritonic system are studied. A generic model of semiconductor excitons strongly coupled with tailored photonic modes is considered. We introduce a pseudospin operator, measuring the polariton polarization between photonic-like and excitonic-like. The associated pseudospin spectrum and pseudospin Chern numbers are calculated. It is shown that the pseudospin Chern number phase diagram exhibits certain features resembling the topological phase of quantum-spin-Hall-like. Moreover, a series of topological phase transitions may occur with the closing of the bulk energy gap or the pseudospin spectrum gap. In a tight-binding form, the edge-mode simulation is done numerically to confirm the analytically results.

A semiclassical formulation of the spin Hall effect for physical systems satisfying Dirac-like equation is introduced. We demonstrate that the main contribution to the spin Hall conductivity is given by the spin Chern number whether the spin is conserved or not at the quantum level. We illustrated the formulation within the Kane–Mele model of graphene in the absence and in the presence of the Rashba spin-orbit coupling term.