Dorokhov-Mello-Pereyra-Kumar equation for the edge transport of a quantum spin Hall insulator

Using the random matrix theory, we investigate the ensemble statistics of edge transport of a quantum spin Hall insulator with multiple edge states in the presence of quenched disorder. Dorokhov-Mello-Pereyra-Kumar equation applicable for such a system is established. It is found that a two-dimensional quantum spin Hall insulator is effectively a new type of one-dimensional (1D) quantum conductor with the different ensemble statistics from that of the ordinary 1D quantum conductor or the insulator with an even number of Kramers edge pairs. The ensemble statistics provides a physical manifestation of the $Z_2$ classification for the time-reversal invariant insulators.

Figure 1

(Color online) (a) The geometry of the system: an insulator with multiple edge conducting channels (represented by the color shaded bands at the two edges) is connected to the left and right measurement leads. (b) The transport along each of edges can be characterized by a $S$ matrix. $a_i^{+}$ $\left(b_i^{+}\right)$ and $a_i^{-}$ $\left(b_i^{-}\right)$ denote the right-going and left-going wave amplitudes, respectively. $a_i^{\pm }$ (or $b_i^{\pm }$) with the same index $i$ are related by the TR and form a Kramers pair. $N$ denotes the total number of Kramers pairs at each edge, and can be odd (QSHI) or even (ordinary insulator).

Figure 2

(Color online) The weak localization correction to the conductance $\left(\delta \sigma \right)$ and the variance of conductance $\left[\text{var}\left(\sigma \right)\right]$. Left: the results obtained from the numerical solutions of DMPK equation for different values of $N$ (the numbers by the curves). The $N\to \infty$ asymptotic behaviors [Eqs. (16, 17)] are indicated by the red curves. Right: the comparison between the results from the DMPK equation and that from the simulation of the 2D model [Eq. (18)]. The results for $N=4$ and $N=5$ are shown. In the 2D simulation, the parameters for the Kane-Mele model are $t=1$, ${\lambda }_{\text{so}}=0.2$, ${\lambda }_R=0.2$, and ${\lambda }_v=0$ (see Ref. 3 for the definitions of these parameters), and the other parameters are $W=0.6$ and $t_1=0.2$. The stripe has a width of 48 lattice sites (LSs) along the zig-zag direction. We have determined the mean-free paths to be $l=2240$ LS $\left(N=4\right)$ and $l=1680$ LS $\left(N=5\right)$, respectively. Two hundred disorder configurations are averaged.

Figure 3

The distribution of conductance for $N=4$ (lower) and 5 (upper). The drop lines show the results from the DMPK equation. The solid lines indicate the distributions predicted by the asymptotic formulas (a)–(c) Eq. (14) and (d) Eq. (13). $\Delta \sigma \equiv \sigma \left(s\right)-\sigma \left(\infty \right)$.