General theorem relating the bulk topological number to edge states in two-dimensional insulators

We prove a general theorem on the relation between the bulk topological quantum number and the edge states in two-dimensional insulators. It is shown that whenever there is a topological order in bulk, characterized by a nonvanishing Chern number, even if it is defined for a nonconserved quantity such as spin in the case of the spin Hall effect, one can always infer the existence of gapless edge states under certain twisted boundary conditions that allow tunneling between edges. This relation is robust against disorder and interactions, and it provides a unified topological classification of both the quantum (charge) Hall effect and the quantum spin Hall effect. In addition, it reconciles the apparent conflict between the stability of bulk topological order and the instability of gapless edge states in systems with open boundaries (a known happening in the spin Hall case). The consequences of time reversal invariance for bulk topological order and edge state dynamics are further studied in the present framework.

Figure 1

(Color online) Schematic picture of the parameter space torus and the definition of coordinates $(r,{\theta }_x,{\theta }_y)$. A monopole singularity is drawn in the solid torus.

Figure 2

(Color online) Schematic picture of twisted boundary conditions (2). The $x$ and $y$ boundaries $L_x$ and $L_y$, respectively, are drawn in red dashed lines. Each hopping across them in the direction indicated by the arrows acquires an additional matrix factor $r{e}^{i{\theta }_x{\Gamma }_x}$ and $e^{i{\theta }_y{\Gamma }_y}$, respectively. The hopping in the opposite direction acquires a conjugate factor. The factor $r$ is interpreted as amplitude reduction due to tunneling between edges.

Figure 3

(Color online) The 1D band structure for (a) $c=0$ and the open-boundary condition $r=0$; (b) $c=0.05$ and the open-boundary condition $r=0$; and (c) $c=0.05$ and the twisted boundary conditions $r=0.18,{\theta }_x=\pm \pi ∕2,{\theta }_y=\pi$.

Figure 4

(Color online) The distribution of (a) charge density and (b) pseudospin ${\Gamma }_{12}$ density of the two edge states with wave vector $k=0.16\pi$ and with $c=0.05,r=0.18,{\theta }_x=\pi ∕2$.

Figure 5

(Color online) Schematic picture of the ${\varphi }_y(r,{\theta }_x)$ configuration for the spin Hall system when (a) $c=0$ and (b) $c\ne 0$. The direction of the arrows stands for the phase ${\varphi }_y$. The crosses indicate the position of singularities and the circle is the unit circle $r=1$.