Quantum Spin-Hall Effect and Topologically Invariant Chern Numbers

We present a topological description of the quantum spin-Hall effect (QSHE) in a two-dimensional electron system on a honeycomb lattice with both intrinsic and Rashba spin-orbit couplings. We show that the topology of the band insulator can be characterized by a $2\times 2$ matrix of first Chern integers. The nontrivial QSHE phase is identified by the nonzero diagonal matrix elements of the Chern number matrix (CNM). A spin Chern number is derived from the CNM, which is conserved in the presence of finite disorder scattering and spin nonconserving Rashba coupling. By using the Laughlin gedanken experiment, we numerically calculate the spin polarization and spin transfer rate of the conducting edge states and determine a phase diagram for the QSHE.

Figure 1

The evolution of edge levels inside the bulk gap with twist boundary phase ${\theta }_y$ in a cylindric 2D sample of $N=480\times 240$, $V_{\mathrm{SO}}=0.05t$, $V_R=0.05t$, and $W=1.0t$.

Figure 2

Spin polarization $P_m^z$ defined in the text (in units of $\hslash /2$) as a function of eigenenergy $E_m$ for two random disorder configurations (we have checked over 100 configurations) of strength $W=1.0t$ at different system sizes $N=120\times 120–480\times 480$ (with $N_x=N_y=L$). $V_{\mathrm{SO}}$ and $V_R$ are the same as in Fig. 1.

Figure 3

(a) Spin transfer rate $\gamma =\frac{\Delta ⟨S^z{⟩}_{\mathrm{edge}}}{\Delta \Phi }$ (in units of $e/4\pi$) vs disorder strength $W$ at $V_{\mathrm{SO}}=0.05t$ and $V_R=0.05t$ averaged over 500 disorder configurations. (b) The phase boundary for the QSHE determined from the spin transfer, where the transition happens with a change of the numbers of edge channels from $N_{\mathrm{edge}}=2$ to $N_{\mathrm{edge}}=0$.

Figure 4

(a) Level crossing below $W_c$ at $W=2.2t$ in the energy region $0.0–0.07t$. (b) The avoided level crossing (pointed by arrows) between edgelike states above $W_c$ at $W=2.8t$ with changing ${\theta }_y$. Here, $N=240\times 120$, $V_{\mathrm{SO}}$, and $V_R$ are the same as in Fig. 3a with $W_c=2.4t$.

Figure 5

Solid angle ${\Omega }_j$ vs spin twist ${\theta }_x$ and momentum $k_y$, which is meshed into $N_{\mathrm{mesh}}=120\times 120$ points for a pure system with $N=240\times 240$ at $V_{\mathrm{SO}}=0.1t$ and $V_R=0.3t$. The total Berry phase $\sum _{j=1}^{N_{\mathrm{mesh}}}{\Omega }_j=4\pi$, and thus $C_{\mathrm{sc}}=2$.