$Z_2$ Topological Order and the Quantum Spin Hall Effect

The quantum spin Hall (QSH) phase is a time reversal invariant electronic state with a bulk electronic band gap that supports the transport of charge and spin in gapless edge states. We show that this phase is associated with a novel $Z_2$ topological invariant, which distinguishes it from an ordinary insulator. The $Z_2$ classification, which is defined for time reversal invariant Hamiltonians, is analogous to the Chern number classification of the quantum Hall effect. We establish the $Z_2$ order of the QSH phase in the two band model of graphene and propose a generalization of the formalism applicable to multiband and interacting systems.

Figure 1

Energy bands for a one-dimensional “zigzag” strip in the (a) QSH phase ${\lambda }_v=0.1t$ and (b) the insulating phase ${\lambda }_v=0.4t$. In both cases ${\lambda }_{\mathrm{SO}}=.06t$ and ${\lambda }_R=.05t$. The edge states on a given edge cross at $ka=\pi$. The inset shows the phase diagram as a function of ${\lambda }_v$ and ${\lambda }_R$ for $0<{\lambda }_{\mathrm{SO}}\ll t$.

Figure 2

The zeros of $P(\mathbf{k})$ in the QSH phase occur at points $\pm {\mathbf{k}}^{*}$ for (a) ${\lambda }_v\ne 0$ and on the oval for (b) ${\lambda }_v=0$. (c) $|P(0,{\alpha }_2)|$ in the QSH (solid line) and insulating (dashed line) phases for a $2\times 2$ supercell using parameters in Fig. 1. (d) Point (${\lambda }_v\ne 0$) and line (${\lambda }_v=0$) zeros of $P(\overrightarrow{\alpha })$ for the $2\times 2$ supercell. In (a), (b), and (d) the solid dots are $\mathcal{T}$-symmetric points, which cannot be zeros of $P$, and $C$ is the contour of integration for Eq. (5).