We show how the two-dimensional (2D) topological insulator evolves, by stacking, into a strong or weak topological insulator with different topological indices, proposing a new conjecture that goes beyond an intuitive picture of the crossover from qu
antum spin Hall to the weak topological insulator. Studying the conductance under different boundary conditions, we demonstrate the existence of two conduction regimes in which conduction happens through either surface- or edge-conduction channels. We show that the two conduction regimes are complementary and exclusive. Conductance maps in the presence and absence of disorder are introduced, together with 2D $mathbb{Z}_2$-index maps, describing the dimensional crossover of the conductance from the 2D to the 3D limit. Stacking layers is an effective way to invert the gap, an alternative to controlling the strength of spin-orbit coupling. The emerging quantum spin Hall insulator phase is not restricted to the case of odd numbers of layers.
We propose classification schemes for characterizing two-dimensional topological phases with nontrivial weak indices. Here, weak implies that the Chern number in the corresponding phase is trivial, while the system shows edge states along specific bo
undaries. As concrete examples, we analyze differen
As a model for describing finite-size effects in topological insulator thin films, we study a one-dimensional (1D) effective model of a topological insulator (TI). Using this effective 1D model, we reveal the precise correspondence between the spatia
l profile of the surface wave function, and the dependence of the finite-size energy gap on the thickness (Lx) of the film. We solve the boundary problem both in the semi-infinite and slab geometries to show that the Lx-dependence of the size gap is a direct measure of the amplitude of the surface wave function at the depth of x=Lx+1 [here, the boundary condition is chosen such that the wave function vanishes at x=0]. Depending on the parameters, the edge state function shows either a damped oscillation (in the TI-oscillatory region of FIG. 2, or becomes overdamped (ibid., in the TI-overdamped phase). In the original 3D bulk TI, an asymmetry in the spectrum of valence and conduction bands is omnipresent. Here, we demonstrate by tuning this asymmetry one can drive a crossover from the TI-oscillatory to the TI-overdamped phase.
The quantum phase transition between the three dimensional Dirac semimetal and the diffusive metal can be induced by increasing disorder. Taking the system of disordered $mathbb{Z}_2$ topological insulator as an important example, we compute the sing
le particle density of states by the kernel polynomial method. We focus on three regions: the Dirac semimetal at the phase boundary between two topologically distinct phases, the tricritical point of the two topological insulator phases and the diffusive metal, and the diffusive metal lying at strong disorder. The density of states obeys a novel single parameter scaling, collapsing onto two branches of a universal scaling function, which correspond to the Dirac semimetal and the diffusive metal. The diverging length scale critical exponent $ u$ and the dynamical critical exponent $z$ are estimated, and found to differ significantly from those for the conventional Anderson transition. Critical behavior of experimentally observable quantities near and at the tricritical point is also discussed.
Existence of a protected surface state described by a massless Dirac equation is a defining property of the topological insulator. Though this statement can be explicitly verified on an idealized flat surface, it remains to be addressed to what exten
t it could be general. On a curved surface, the surface Dirac equation is modified by the spin connection terms. Here, in the light of the differential geometry, we give a general framework for constructing the surface Dirac equation starting from the Hamiltonian for bulk topological insulators. The obtained unified description clarifies the physical meaning of the spin connection.
The metallic surface state of a topological insulator (TI) is not only topologically protected, but exhibits a remarkable property of inducing an effective vector potential on curved surfaces. For an electron in the surface state of a spherical or a
cylindrical TI (TI nanoparticle or nanowire) a pseudo-magnetic monopole or a fictitious solenoid is effectively induced, encoding the geometry of the system. Here, by taking an example of a hyperbolic surface we demonstrate that as a consequence of this property stemming from its active spin degree of freedom, the surface state is by itself topologically protected.
The non-trivialness of a topological insulator (TI) is characterized either by a bulk topological invariant or by the existence of a protected metallic surface state. Yet, in realistic samples of finite size this non-trivialness does not necessarily
guarantee the gaplessness of the surface state. Depending on the geometry and on the topological indices, a finite-size energy gap of different nature can appear, and correspondingly, exhibits various scaling behaviors of the gap. The spin-to-surface locking provides one of such gap-opening mechanisms, resulting in a power-law scaling of the energy gap. Weak and strong TIs show different degrees of sensitivity to the geometry of the sample. As a noteworthy example, a strong TI nanowire of a rectangular prism shape is shown to be more gapped than that of a weak TI of precisely the same geometry.
We report our recent numerical study on the effects of dephasing on a perfectly conducting channel (PCC), its presence believed to be dominant in the transport characteristics of a zigzag graphene nanoribbons (GNR) and of a metallic carbon nanotubes
(CNT). Our data confirms an earlier prediction that a PCC in GNR exhibits a peculiar robustness against dephasing, in contrast to that of the CNT. By studying the behavior of the conductance as a function of the systems length we show that dephasing destroys the PCC in CNT, whereas it stabilizes the PCC in GNR. Such opposing responses of the PCC against dephasing stem from a different nature of the PCC in these systems.
The electronic spectrum on the spherical surface of a topological insulator reflects an active property of the helical surface state that stems from a constraint on its spin on a curved surface. The induced effective vector potential (spin connection
) can be interpreted as an effective vector potential associated with a fictitious magnetic monopole induced at the center of the sphere. The strength of the induced magnetic monopole is found to be g=2pi, -2pi, being the smallest finite (absolute) value compatible with the Dirac quantization condition. We have established an explicit correspondence between the bulk Hamiltonian and the effective Dirac operator on the curved spherical surface. An explicit construction of the surface spinor wave functions implies a rich spin texture possibly realized on the surface of topological insulator nanoparticles. The electronic spectrum inferred by the obtained effective surface Dirac theory, confirmed also by the bulk tight-binding calculation, suggests a specific photo absorption/emission spectrum of such nanoparticles.
We study a Majorana zero-energy state bound to a hedgehog-like point defect in a topological superconductor described by a Bogoliubov-de Gennes (BdG)-Dirac type effective Hamiltonian. We first give an explicit wave function of a Majorana state by sol
ving the BdG equation directly, from which an analytical index can be obtained. Next, by calculating the corresponding topological index, we show a precise equivalence between both indices to confirm the index theorem. Finally, we apply this observation to reexamine the role of another topological invariant, i.e., the Chern number associated with the Berry curvature proposed in the study of protected zero modes along the lines of topological classification of insulators and superconductors. We show that the Chern number is equivalent to the topological index, implying that it indeed reflects the number of zero-energy states. Our theoretical model belongs to the BDI class from the viewpoint of symmetry, whereas the spatial dimension of the system is left arbitrary throughout the paper.
