Boundary criticality at the Anderson transition between a metal and a quantum spin Hall insulator in two dimensions

Static disorder in a noninteracting gas of electrons confined to two dimensions can drive a continuous quantum (Anderson) transition between a metallic and an insulating state when time-reversal symmetry is preserved but spin-rotation symmetry is broken. The critical exponent $\nu$ that characterizes the diverging localization length and the bulk multifractal scaling exponents that characterize the amplitudes of the critical wave functions at the metal-insulator transition do not depend on the topological nature of the insulating state, i.e., whether it is topologically trivial (ordinary insulator) or nontrivial (a ${\mathbb{Z}}_2$ insulator supporting a quantum spin Hall effect). This is not true of the boundary multifractal scaling exponents, which we show (numerically) to depend on whether the insulating state is topologically trivial or not.

Figure 1

(Color online) Elementary building blocks of the network model. A square Bravais lattice with nearest-neighbor sites connected by bonds underlies the construction of the network model. Any red site of one of the two sublattices of the square lattice is replaced by a red circle $\mathsf{S}$ (a node of type $\mathsf{S}$) with the four bonds meeting at the site replaced by four pairs of directed links numbered according to the rule shown in (a). Any blue site of the complementary sublattice of the square lattice is replaced by a blue circle ${\mathsf{S}}^{\prime }$ (a node of type ${\mathsf{S}}^{\prime }$) with the four bonds meeting at the site replaced by four pairs of directed links numbered according to the rule shown in (b). Observe that a clockwise rotation by $\pi /2$ turns (a) into (b). The directed links represent incoming or outgoing plane waves with a well-defined projection of the spin-1/2 quantum number along the quantization axis. Each pair of links replacing a bond represents a Kramers doublet of plane waves. Each node $\mathsf{S}$ or ${\mathsf{S}}^{\prime }$ depicts a scattering process represented by a $4\times 4$ unitary matrix defined in Eq. (2.1) that preserves TRS but breaks SRS.

Figure 2

(Color online) Phase diagram for the network model after Ref. 12.

Figure 3

(Color online) Quasi-one-dimensional network model with reflecting boundaries. (a) The upper and lower boundaries pass through the nodes ${\mathsf{S}}^{\prime }$. (b) The upper and lower boundaries pass through the nodes $\mathsf{S}$. (c) The upper and lower boundaries pass through the nodes $\mathsf{S}$ and ${\mathsf{S}}^{\prime }$, respectively. Transverse periodic boundary conditions are possible only in geometries (a) and (b), for which the transverse width is here $M=2$.

Figure 4

(a) The network model on a square with $L^2$ nodes of type $\mathsf{S}$ and $L^2$ nodes of type ${\mathsf{S}}^{\prime }$ (a node on a boundary is counted as 1/2) is wrapped around a cylinder by imposing RBCs along the $x$ direction and PBCs along the $y$ direction. (b) It is wrapped around a torus by imposing PBCs in the $x$ and $y$ directions.

Figure 5

Phase diagram of the network model of the quasi-one-dimensional geometries in Figs. 3a, 3b, 3c at $X\gtrsim X_l$. The choice of the boundary decides whether the insulating phase at $X>X_l$ has helical edge states.

Figure 6

(Color online) Scattering in the reflective limit $X\to \infty$ for nodes (a) $\mathsf{S}$ and (b) ${\mathsf{S}}^{\prime }$.

Figure 7

(Color online) Large $X$ limit of the quasi-one-dimensional network model with the reflecting boundaries from Fig. 3.

Figure 8

(Color online) The dependence on $X$ of ${\Lambda }^{\left(1\right)}\left(X,M\right)$ in the geometry of Fig. 3a with transverse RBCs is shown for $M=4$ $(\odot )$, 8 (◼), and 16 (●). The dependence on $X$ of ${\Lambda }^{\left(2\right)}\left(X,M\right)$ with transverse RBCs is shown for $M=4$ $(\times )$, 8 $(+)$, and 16 $(\ast )$. The dependence on $X$ of ${\Lambda }^{\left(1\right)}\left(X,M\right)$ with transverse PBCs is shown for $M=4$ (blue dashed curve), 8 (green dotted curve), and 16 (red solid curve). The vertical dashed lines identify the critical points $X_s$ and $X_l$ deduced in Ref. 12 when transverse PBCs are imposed. The inset displays the same data points with a logarithmic vertical scale.

Figure 9

(Color online) (a) Dependence on $X$ of the normalized localization length ${\Lambda }^{\left(1\right)}\left(X,M\right)\equiv \Lambda \left(X,M\right)$ for the network model in the quasi-one-dimensional geometry of Fig. 3b with transverse RBCs and $M=4,8,16,32,64$. (b) Dependence on the dimensionless disorder strength $W$ of the normalized localization length ${\Lambda }^{\left(1\right)}\left(W,M\right)\equiv \Lambda \left(W,M\right)$ for the SU(2) model in the quasi-one-dimensional geometry with transverse FBCs and $M=4,8,16,32,64$. The solid curves are computed from the finite-size scaling functions that follow. (c) Finite-size scaling analysis for the data shown in (a) and (b): The solid blue curves represent the scaling functions for the network model with RBCs on node $\mathsf{S}$ in the vicinity of the critical point $X_l$, at which we find ${\Lambda }^{\left(\text{RBC}\right)}\left(X_l\right)=1.49\pm 0.02$ and $\nu =2.88\pm 0.04$. The red dashed curves represent the scaling functions for the SU(2) model with FBCs in the vicinity of the critical point $W_c$, at which we find ${\Lambda }^{\left(\text{RBC}\right)}\left(W_c\right)=1.50\pm 0.03$ and $\nu =2.85\pm 0.06$. The scaling functions are obtained from finite-size scaling analysis incorporating corrections from a leading irrelevant scaling variable (Refs. 12, 21). The normalized localization length ${\Lambda }^{\prime }$ is obtained from $\Lambda$ by subtracting these corrections, as defined in Eqs. (3.7) and (3.8) of Ref. 12. The distance to the critical point $\left|X-X_l\right|$ is rescaled by a factor $c\simeq 1.7$ in the scaling function for the network model with RBCs so that it coincides with that for the SU(2) model with FBCs.

Figure 10

(Color online) Dependence on the position $x$ along the cylinder axis of Fig. 4a of ${⟨\text{ln}{\left|\Psi \right|}^2⟩}_{x,L}$ defined by Eq. (5.1) for the values of $L=20,25,30,35,40,50,60,80$ from top to bottom with (a) $\mathsf{S}$ boundaries or with (b) ${\mathsf{S}}^{\prime }$ boundaries. The blue dashed curves in (a) represent the normalized critical wave functions ${⟨\text{ln}{\left|\Psi \right|}^2⟩}_{x,L}$ for $L=36,48,60,72,96,120$ obtained from the SU(2) model defined in Ref. 2 using the cylinder geometry of Fig. 4a.

Figure 11

(Color online) The dependences on $q$ of ${\Delta }_q^{\left(\kappa \right)}/\left[q\left(1-q\right)\right]$ (●) when $X=X_l$ for (a) the bulk: $\left(\kappa \right)=\left(2\right)$; (b) the $\mathsf{S}$ boundary: $\left(\kappa \right)=\left(1,\text{O}\right)$; and (c) the ${\mathsf{S}}^{\prime }$ boundary: $\left(\kappa \right)=\left(1,{\mathbb{Z}}_2\right)$. The solid lines represent ${\alpha }^{\left(\kappa \right)}\left(q\right)-2$ evaluated at $q=0$ according to Eq. (5.4a) with its error bars indicated by dashed lines. The dependences on $q$ of ${\Delta }_q^{\left(\kappa \right)}/\left[q\left(1-q\right)\right]$ for the SU(2) model in the bulk and on the boundary are shown with ◯ in (a) and (b), respectively. The blue open circles in (c) show the $q$ dependence of ${\Delta }_{1-q}^{\left(1,{\mathbb{Z}}_2\right)}/\left[q\left(1-q\right)\right]$.

Figure 12

(Color online) (a) Dependences on ${\alpha }^{\left(\kappa \right)}$ of the multifractal spectra $f^{\left(\kappa \right)}$ in the bulk $\left(\kappa \right)=\left(2\right)$ (▲), at the $\mathsf{S}$ boundaries $\left(\kappa \right)=\left(1,\text{O}\right)$ $(\odot )$, and at the ${\mathsf{S}}^{\prime }$ boundaries $\left(\kappa \right)=\left(1,{\mathbb{Z}}_2\right)$ (●), at $X=X_l$. Dependences on $q$ of ${\alpha }_q$ and $f_q$ are shown in (b) and (c), respectively, with the same symbols as in (a). The bulk and boundary multifractal spectra for the SU(2) model defined in Ref. 2 are shown as solid and dashed curves, respectively.

Figure 13

(Color online) (a) A semi-infinite geometry with two-point contacts (green dot-dashed curves) attached at the two interfaces between different types of boundaries. Each line or curve represents a Kramers doublet. (b) Closed network with mixed boundaries. The thick wavy and solid lines on the edges represent two different types of boundaries, with and without a helical edge mode at $X>X_l$, respectively.