Topological aspects of the quantum spin-Hall effect in graphene: ${\mathrm{Z}}_2$ topological order and spin Chern number

For generic time-reversal-invariant systems with spin-orbit couplings, we clarify a close relationship between the ${\mathrm{Z}}_2$ topological order and the spin Chern number (SChN) in the quantum spin-Hall effect. It turns out that a global gauge transformation connects sectors with different SChNs (even integers) modulo 4, which implies that the SChN and ${\mathrm{Z}}_2$ topological orders yield the same classification. We present a method of computing the SChN and demonstrate it in single and double planes of graphene.

Figure 1

Spectrum for ${\theta }_1=0$ as a function of $k_2$. Parameters used are $V_{\mathrm{so}}=0.1t$, $v_{\mathrm{s}}=0.3t$, and $V_{\mathrm{R}}=0.1t$ (left), $V_{\mathrm{R}}=0.225207t$ (middle), and $V_{\mathrm{R}}=0.3t$ (right).

Figure 2

Upper left: spectrum for nonzero $\phi =\pi ∕4$ as a function of $k_2$ at ${\theta }_1=\pi ∕2$. Other parameters are the same as those of the left in Fig. 1. Upper right: the amplitude ${\mid {\psi }_{j_1}\mid }^2$ of the wave functions at $k_2=0.8\pi$ indicated by arrow. The solid, dotted, and dashed lines correspond to the first, second, and third states from the zero energy into negative energy, respectively. Lower left and right: the $n$-field configuration corresponding to the left in Fig. 1. The left (right) is at $\phi =0$ $(\pi ∕2)$. The white (black) circle denotes $n=1(-1)$, while the blank means $n=0$. We used the meshes $N_1=N_2=10$.

Figure 3

The $n$-field configuration for ${\gamma }_1=0.1t$. Other parameters are same as those of the left in Fig. 1. Left: ${\phi }_1=0$ and ${\phi }_2=0$ $(c_{\mathrm{s}}=4)$. Middle: ${\phi }_1=\pi ∕2$ and ${\phi }_2=0$ $(c_{\mathrm{s}}=0)$. Right: ${\phi }_1=\pi ∕4$ and ${\phi }_2=-\pi ∕4$ $(c_{\mathrm{s}}=0)$. In the case ${\phi }_1=\pi ∕2$ and ${\phi }_2=\pi ∕2$ we have the same figure as the left but with black circles $(c_{\mathrm{s}}=-4)$. We used the meshes $N_1=N_2=10$.