We investigate the transport properties of the surface states of a three-dimensional topological insulator in the presence of a spin-splitting Zeeman field. We propose a picture that the chiral edge state forms on the surface, and is split into two halves that are spatially separated, each carrying one half of the conductance quantum (e 2 /h). This picture is confirmed by numerical simulation in a fourterminal setup. It is demonstrated that the difference between the clockwise and counterclockwise transmission coefficients of the two neighboring terminals is approximately one half, which suggests that the half quantized Hall conductance can be manifested in an appropriate experimental setup. Topological insulators are insulating in the bulk, but have metallic surface states possessing an odd number of Dirac cones of massless fermions [1][2][3]. The band structure and the quantum spin texture of these surfaces states have been well established theoretically and experimentally [4][5][6][7][8][9]. In the presence of a spin-splitting Zeeman field, which could be induced by magnetically doping the samples or putting the samples in the proximity of ferromagnetic materials, the Dirac fermions will gain a mass and the spectrum opens a gap [10,11]. When the Fermi level is located within the energy gap, it was proposed that the Hall conductance of the surface states will be one half of the conductance quantum e 2 /h [12][13][14]. Based on this, Qi et al. proposed the unconventional magnetoelectric effect, which is regarded as one of the characteristic features of the topological insulators [14][15][16].On the other hand, it is not clear whether or not the half quantization of the Hall conductance can be directly observed in the transport measurement. In the usual quantum Hall system, the current-carrying chiral edge states are responsible for the integer quantized Hall conductance measured in the transport experiment. [17,18] It is not immediately clear whether or not the similar chiral edge state will form on the closed surface of a topological insulator, and how the quantized nature of the edge states can be reconciled with the prediction of the half quantization of the Hall conductance. [12][13][14] To get a definite answer to these questions, we investigate the multi-terminal transport properties of a 3D topological insulator in the presence of a uniform spinsplitting Zeeman field. We propose that the the closed surface of the topological insulator will form two insulating domains of the different topologies (i.e., positive vs. negative gaps), separated by a gapless metallic belt. A chiral edge state will form and is concentrated around the boundaries between the insulating domains and the metallic belt. Effectively, the chiral edge state is split into two halves, each of which is circulating around the boundary of one of the domains and carrying one half of the conductance quantum (e 2 /h). Such a picture reconciles the apparent conflict between the half quantization and the index theorem. It also suggests that by at...

It has been proposed that disorder may lead to a new type of topological insulator, called topological Anderson insulator (TAI). Here we examine the physical origin of this phenomenon. We calculate the topological invariants and density of states of disordered model in a super-cell of 2-dimensional HgTe/CdTe quantum well. The topologically non-trivial phase is triggered by a band touching as the disorder strength increases. The TAI is protected by a mobility gap, in contrast to the band gap in conventional quantum spin Hall systems. The mobility gap in the TAI consists of a cluster of non-trivial subgaps separated by almost flat and localized bands.

We study the topologically non-trivial semi-metals by means of the 6-band Kane model. Existence of surface states is explicitly demonstrated by calculating the LDOS on the material surface. In the strain free condition, surface states are divided into two parts in the energy spectrum, one part is in the direct gap, the other part including the crossing point of surface state Dirac cone is submerged in the valence band. We also show how uni-axial strain induces an insulating band gap and raises the crossing point from the valence band into the band gap, making the system a true topological insulator. We predict existence of helical edge states and spin Hall effect in the thin film topological semi-metals, which could be tested with future experiment. Disorder is found to significantly enhance the spin Hall effect in the valence band of the thin films.PACS numbers:

We study the Anderson transition with interactions in one dimension from the perspective of quantum entanglement. Extensive numerical calculations of the entanglement entropy (EE) of the systems are carried out through the density matrix renormalization group algorithm. We demonstrate that the EE can be used for the finite-size scaling (FSS) to characterize the Anderson transition in both noninteracting and interacting systems. From the FSS analysis we can obtain a precise estimate of the critical parameters of the transition. The method can be applied to various one-dimensional models, either interacting or noninteracting, to quantitatively characterize the Anderson transitions.