Connection of Edge States to Bulk Topological Invariance in a Quantum Spin Hall State

We propose a topological understanding of general characteristics of edge states in a quantum spin Hall phase without considering any symmetries. It follows from the requirement of gauge invariance that either the energy gap or the gap in the spectrum of the projected spin operator needs to close on the sample edges. Based upon the Kane-Mele model with a uniform Zeeman field and a smooth confining potential near the sample boundaries, we demonstrate the existence of gapless edge states in the absence of time-reversal symmetry and their robust properties against impurities. These gapless edge states are protected by the band topology alone, rather than any symmetries.

Figure 1

(a) Illustration of the flow of electron states in a looped ribbon of the QHE system, with adiabatically increasing the magnetic flux $\varphi$. The opposite movements of the bulk states below and above $E_F$ are connected through gapless edge states, forming a closed loop. (b) In the first scenario for the QSH system, gapless edge modes appear on the edges, so that states in each spin sector behave like those in the QHE system. (c) In the second scenario, the spin spectrum gap closes on the edges. The flow of states in two spin sectors joins together on the edges within the valence (conduction) band.

Figure 2

(a) A schematic of an armchair honeycomb lattice ribbon with atom sites in two sublattices being labeled by solid dots and hollow dots. (b) Profiles of $|V_i|$ as functions of $x_i$, for an abrupt confining potential (dashed line) and a relatively smooth confining potential (solid line).

Figure 3

The energy spectrum (a),(b) and the spectrum of the projected spin $P{\sigma }_{z}P$ (c),(d) of an armchair ribbon for $\xi =0.1$ (a),(c) and $\xi =4$ (b),(d), in which $L$ ($R$) stands for states on the left (right) edge, and $\uparrow$ ($\downarrow$) for the up (down) spin polarization. The horizontal arrows in (a) point to the small energy gaps in the energy spectrum.

Figure 4

Eigenenergies of edge states as a function of the magnetic flux $\varphi$ for some different values of $l_0$ and $U_0$. The other parameters are the same as in Fig. 3b. Arrows in (a) indicate some of the relatively large energy gaps.