Spin-directed network model for the surface states of weak three-dimensional ${\mathbb{Z}}_2$ topological insulators

A two-dimensional spin-directed ${\mathbb{Z}}_2^{}$ network model is constructed that describes the combined effects of dimerization and disorder for the surface states of a weak three-dimensional ${\mathbb{Z}}_2^{}$ topological insulator. The network model consists of helical edge states of two-dimensional layers of ${\mathbb{Z}}_2^{}$ topological insulators which are coupled by time-reversal-symmetric interlayer tunneling. It is argued that, without dimerization of interlayer couplings, the network model has no insulating phase for any disorder strength. However, a sufficiently strong dimerization induces a transition from a metallic phase to an insulating phase. The critical exponent $\nu$ for the diverging localization length at metal-insulator transition points is obtained by finite-size scaling analysis of numerical data from simulations of this network model. It is shown that the phase transition belongs to the two-dimensional symplectic universality class of Anderson transition.

Figure 1

Cartoon representation of a strong two-dimensional ${\mathbb{Z}}_2^{}$ topological band insulator with one connected boundary. (a) Electrons are noninteracting and confined within an ellipse in two-dimensional position space. A single pair of counterpropagating helical edge states at zero energy is denoted by the thick and dashed lines with arrows along the connected boundary of the ellipse, respectively. They are confined to this one-dimensional boundary. (b) The single-particle spectrum consists of eigenstates separated by an energy gap $\Delta$ and eigenstates crossing the band gap $\Delta$. The former eigenstates are the bulk eigenstates. The latter eigenstates are the edge eigenstates. The momentum $k$ is the momentum along the boundary. The wave functions in position space of bulk eigenstates are supported in the shaded region of the ellipse. The wave functions in position space of edge eigenstates are extended along the edge and labeled by the momentum quantum number $k$ along the edge, while they decay exponentially fast away from the edge.

Figure 2

(a) Cartoon representation of a weak layered ${\mathbb{Z}}_2^{}$ topological band insulator with open boundary conditions along the layering axis. A vertical line represents the amplitude for the Kramers' degenerate helical edge states of a layer consisting of a strong two-dimensional ${\mathbb{Z}}_2^{}$ topological band insulator to hop to one of its adjacent layers. The vertical lines are alternatively drawn with solid and dotted lines in order to indicate that the magnitude of this hopping amplitude takes two distinct values. As a consequence of the breaking of translation invariance by one lattice spacing along the stacking axis, a dimerization gap opens for all helical edge states, i.e., for all surface states of this weak layered ${\mathbb{Z}}_2^{}$ topological band insulator. (b) The topology in (a) is that of a cylinder.

Figure 3

Qualitative beta function $dln\sigma /dlnL$ for the conductivity $\sigma$ of a massless two-component Dirac fermion on the surface with the area $L^2$ of a strong three-dimensional ${\mathbb{Z}}_2^{}$ topological insulator, when subjected to disorder that preserves time-reversal symmetry. The beta function $dln\sigma /dlnL$ is deduced from the numerics of Refs. [33, 34]. It is always positive and the smaller the initial value of the conductivity is the larger the positive value of the beta function.

Figure 4

Phase diagram from Ref. [36] for the surface states of a weak three-dimensional ${\mathbb{Z}}_2^{}$ topological band insulator as a function of the mass that opens a band gap at the Dirac points (horizontal axis) and the disorder strength (vertical axis). It is assumed that the disorder does not mix surface states localized on disconnected boundaries of the three-dimensional ${\mathbb{Z}}_2^{}$ topological band insulator [32]. The origin of the phase diagram is the critical point corresponding to an even number of two-component massless Dirac fermions at their Dirac point.

Figure 5

An elementary scattering event in the spin-directed ${\mathbb{Z}}_2^{}$ network model maps four incoming plane waves into four outgoing plane waves through a unitary $4\times 4$ matrix $S$ compatible with the operation of time reversal. By demanding that the operation of time reversal is an antiunitary operation that squares to minus the $4\times 4$ identity matrix, the scattering matrix $S$ can break spin-rotation symmetry (hence the spin labels on the plane waves), but preserves time-reversal symmetry. In other words, the scattering matrix belongs to the symplectic class of scattering matrices. The matrix $S$ is represented by Eq. (2.3).

Figure 6

A two-dimensional spin-directed ${\mathbb{Z}}_2^{}$ network model built out of the elementary scattering processes shown in Fig. 5 in the vertex (a) and the brick-wall representations (b), respectively. The red and blue lines represent the flow of electrons with up and down spins, respectively. The dimensions of both networks in (a) and (b) are $M\times M$ with $M=6$. In both representations, one $M$ counts the number of pairs of up and down spins that move from left to right or right to left, while the other $M$ counts the number of pairs of up and down spins that move from up to down or down to up. In the brick-wall representation, each dotted line represents an elementary scattering shown in Fig. 5 with the transmission amplitude $t_{+}^{}$ or $t_{-}^{}$. This periodic pattern implements one of two dimerization patterns, the other one following from interchanging $t_{+}^{}$ and $t_{-}^{}$.

Figure 7

The two-dimensional ${\mathbb{Z}}_2^{}$ network model studied in Refs. [42, 43, 44] for a strong two-dimensional ${\mathbb{Z}}_2^{}$ topological insulator with $M=6$ in the vertex (a) and the brick-wall representations (b), respectively. Note that the flow direction of the same spin component (up or down) is opposite on adjacent layers in (b), in contrast to Fig. 6.

Figure 8

When ${\delta }^2=t^2=0$, the two-dimensional spin-directed ${\mathbb{Z}}_2^{}$ network model from Fig. 6 decouples into $M$ one-dimensional channels, each of which consists of a single Kramers' degenerate pair of helical edge states that propagate unimpeded by the disorder. Panels (a) and (b) are drawn using the vertex and the brick-wall representations, respectively.

Figure 9

When ${\delta }^2=\theta =0$, the two-dimensional spin-directed ${\mathbb{Z}}_2^{}$ network model from Fig. 6 decouples into two directed CC models that are related by time reversal. Panels (a) and (b) are drawn using the vertex and the brick-wall representations, respectively.

Figure 10

When ${\delta }^2=r^2=0$, the two-dimensional spin-directed ${\mathbb{Z}}_2^{}$ network model from Fig. 6 decouples into two directed CC models that are related by time reversal. Panels (a) and (b) are drawn using the vertex and the brick-wall representations, respectively.

Figure 11

When ${\delta }^2=\theta -\pi /2=0$, the two-dimensional spin-directed ${\mathbb{Z}}_2^{}$ network model from Fig. 6 decouples into two directed CC network models that are related by time reversal. Panels (a) and (b) are drawn using the vertex and the brick-wall representations, respectively. Since all elementary scattering $2\times 2$ blocks in Eq. (2.20) are the same, criticality holds for any $0\le t^2\le 1$. At the isotropic point $t^2=r^2=\frac{1}{2}$, the system exhibits the criticality of the usual isotropic CC model, otherwise, the system exhibits the critical behavior of the anisotropic CC model.

Figure 12

(a) Phase diagram in the two-dimensional parameter space (a square) (2.8) without dimerization of the two-dimensional spin-directed ${\mathbb{Z}}_2^{}$ network model. The boundaries of the square correspond to a two-dimensional chiral metal when $\theta =0$, a two-dimensional chiral metal when $t=1$, the metal-insulator critical point of the (two-dimensional) CC model when $\theta =\pi /2$, and the quasi-one-dimensional symplectic metallic fixed point when $t^2=0$. Numerics of the two-dimensional spin-directed ${\mathbb{Z}}_2^{}$ network model are consistent with a metallic phase in the interior of the square except in the region of $t^2\ll 1$ for which numerics are inconclusive. However, we show in Appendix pp1 that the numerical results from Ref. [36] apply to this region, thereby implying that this region is metallic. (b) Two-dimensional cut of the phase diagram in the three-dimensional parameter space (2.7) with fixed and nonvanishing dimerization of the two-dimensional spin-directed ${\mathbb{Z}}_2^{}$ network model. The green dot is on the boundary $\frac{1}{2}\le t^2=1-{\delta }^2\le 1$ of the three-dimensional parameter space (2.7). The presence of the dimerization improves the quality of the numerics. Dimerization reduces the area of the metallic phase. The divergences of the localization length at the metal-insulator transitions are consistent with a metal-insulator critical point belonging to the two-dimensional symplectic class of Anderson localization. (c) Qualitative phase diagram in the three-dimensional parameter space (2.7) with fixed and nonvanishing dimerization. The phase boundary represented by the green curve is on the boundary $\frac{1}{2}\le t^2=1-{\delta }^2\le 1$ of the three-dimensional parameter space (2.7). The phase diagram that includes the effect of the sign of the dimerization in Eq. (2.7) can be represented by gluing along the negative ${\delta }^2$ axis the mirror image about the plane ${\delta }^2=0$ of the phase diagram for positive ${\delta }^2$. This delivers two insulating phases separated by a metallic phase.

Figure 13

Two examples in the brick-wall representation of two-dimensional spin-directed ${\mathbb{Z}}_2^{}$ network models built out of the elementary scattering processes shown in Fig. 5. The periodic choice of the transmission amplitudes $t_{+}^{}$ and $t_{-}^{}$ implements a dimerization pattern. The red and blue lines represent the flow of electron with up and down spins, respectively. The continuous (dotted) black line represents the scattering with transmission amplitude $t_{+}^{}$ $\left(t_{-}^{}\right)$ between them. Periodic boundary conditions are imposed along the horizontal and vertical directions in panels (a) and (b), respectively. The dimensions of these two-dimensional spin-directed ${\mathbb{Z}}_2^{}$ network models are $M\times L=8\times 4$ and $L\times M=8\times 4$ for examples (a) and (b), respectively.

Figure 14

One and two-dimensional cuts in the parameter space (2.7) studied in this paper. The one-dimensional cuts, A defined in Eq. (3.17), B in (3.20), C in (3.21), D in (3.23), E in (3.26), F in (3.27), and G in (3.28), as well as the two-dimensional cuts, I–III, defined in Eq. (3.25) are shown. The red triangle where the three one-dimensional cuts A,D, and E cross represents the critical point (3.19).

Figure 15

The two-dimensional spin-directed ${\mathbb{Z}}_2^{}$ network model is solved numerically along the one-dimensional cut (3.17) in the three-dimensional parameter space (2.7). Panel (a-1) shows the ${\delta }^2$ dependence of the normalized localization length ${\Lambda }_{\perp }^{}$ corresponding to the geometry of Fig. 13 for several values of $M$. Panel (b-1) shows the ${\delta }^2$ dependence of the normalized localization lengths ${\Lambda }_{\parallel }^{}$ corresponding to the geometry of Fig. 13 for several values of $M$. A finite-size scaling analysis of panels (a-1) and (b-1) is performed in panels (a-2) and (b-2), respectively. The horizontal axis is $M^{1/\nu }|{\delta }^2-{\delta }_{\mathrm{c}}^2|$ with $\nu$ and ${\delta }_{\mathrm{c}}^2$ given in Tables 2 and 3 in Appendix pp3. The vertical axis ${\Lambda }_{\mathrm{x}}^{\prime }$ with $\mathrm{x}=\perp ,\parallel$ is defined by subtracting from the normalized localization length ${\Lambda }_{\mathrm{x}}^{}$ its finite-size correction from the leading irrelevant exponent $y$ given in Tables S-I and S-II from Ref. [45]. The red solid curve demonstrates the quality of the data collapse onto a one-parameter scaling function.

Figure 16

The $\theta$ dependence of the normalized localization lengths ${\Lambda }_{\perp }^{}$ in panel (a) and ${\Lambda }_{\parallel }^{}$ in panel (b) of the two-dimensional spin-directed ${\mathbb{Z}}_2^{}$ network model with dimerization along the one-dimensional cut (3.23). The strength of the spin-orbit coupling $\theta$ is varied given the values $t^2=0.5$ and ${\delta }^2=0.234$ for fixed $M$ ranging from 16 to 256. The vertical and horizontal dashed lines represent ${\theta }_{\mathrm{c}}^{}$ and ${\Lambda }_{\mathrm{x},\mathrm{c}}^{}$, respectively, as deduced from the finite-size scaling analysis summarized in Table 1.

Figure 17

The phase diagrams obtained from the $\theta$ dependence of the normalized localization length ${\Lambda }_{\parallel }^{}$ on the two-dimensional cuts (a) I (${\delta }^2=0.1$), (b) II (${\delta }^2=0.234$), and (c) III (${\delta }^2=0.35$). The critical points obtained by the finite-size scaling analysis are marked by red dots. The connecting green lines are guides to the eyes. The point $t^2={\delta }^2$ and $\theta =0$ is critical by construction. The two vertical dashed lines in each panel indicate the allowed minimum and maximum values of $t^2$. The horizontal dashed line in panel (a) represents the one-dimensional cut “F” ($\theta =19\pi /48$) defined in Eq. (3.27).

Figure 18

The $t^2$ dependence of the normalized localization lengths ${\Lambda }_{\perp }^{}$ in panel (a) and ${\Lambda }_{\parallel }^{}$ in panel (b) of the two-dimensional spin-directed ${\mathbb{Z}}_2^{}$ network model with dimerization along the one-dimensional cut (3.26) at $\theta =\pi /4$ and ${\delta }^2=0.234$ for fixed $M$ ranging from 16 to 128. The vertical and horizontal dashed lines represent $t_{\mathrm{c}}^2$ and ${\Lambda }_{\mathrm{x},\mathrm{c}}^{}$ as deduced from the finite-size scaling analysis summarized in Table 1, respectively.

Figure 19

The $t^2$ dependence of the normalized localization lengths ${\Lambda }_{\perp }^{}$ in panel (a) and ${\Lambda }_{\parallel }^{}$ in panel (b) of the two-dimensional spin-directed ${\mathbb{Z}}_2^{}$ network model with dimerization along the one-dimensional cut (3.27) at $\theta =19\pi /48$ and ${\delta }^2=0.1$ for fixed $M$ ranging from 16 to 128. Inset: Dependence on $t^2$ near $t^2\approx 0.4$ for panel (a) and near $t^2\approx 0.8$ for panel (b).

Figure 20

The $t^2$ dependence of the normalized localization lengths ${\Lambda }_{\perp }^{}$ in panel (a) and ${\Lambda }_{\parallel }^{}$ in panel (b) of the two-dimensional spin-directed ${\mathbb{Z}}_2^{}$ network model without dimerization, at $\theta =\pi /4$, and for fixed $M$ ranging from 16 to 256. Inset: Dependence on $t^2$ near $t^2=0$ for panel (a) and near $t^2=1$ for panel (b).

Figure 21

The $\delta \varphi$ dependence of the normalized localization lengths ${\Lambda }_{\perp }^{}$ in panel (a) and ${\Lambda }_{\parallel }^{}$ in panel (b) of the two-dimensional spin-directed ${\mathbb{Z}}_2^{}$ network model at $t^2=0.5$, $\theta =\pi /4$, and ${\delta }^2=0.234$ for fixed $M$ ranging from 16 to 128. The horizontal dashed lines in panels (a) and (b) represent ${\Lambda }_{\perp }^{\mathrm{c}}=0.935$ and ${\Lambda }_{\parallel }^{\mathrm{c}}=3.657$, respectively.

Figure 22

A two-dimensional spin-directed ${\mathbb{Z}}_2^{}$ network model with the dimensions $(2M^{\prime }+1)\times 2M^{\prime }$ where $M^{\prime }=3$ in the vertex (a) and the brick-wall representations (b), respectively. The difference with the two-dimensional spin-directed ${\mathbb{Z}}_2^{}$ network model shown in Fig. 6 is the addition of a bottom and a top row of vertices represented by empty circles at which perfectly reflecting boundary conditions must be imposed in panel (a).

Figure 23

The ${\delta }^2$ dependence of the normalized localization length ${\Lambda }_{\parallel }^{}$ calculated from the second smallest positive Lyapunov exponent $X_{\parallel ,1}^{}$ for the spin-directed ${\mathbb{Z}}_2^{}$ network model with an odd number channels. The parameter $t^2$ and $\theta$ are fixed to $t^2=0.5$ and $\theta =\pi /4$, so as to facilitate comparison with Fig. 15 when the number of channels is even, while all other parameters are the same. The vertical dashed line represents ${\delta }_{\mathrm{c}}^2=0.234$, as deduced from the finite-size scaling analysis summarized in Table 1.

Figure 24

Two-dimensional spin-directed ${\mathbb{Z}}_2^{}$ network model with $M=6M^{\prime }$ an even integer multiple of 3. The periodic pattern for the transmission amplitudes implements a trimerization, as is indicated by the enlarged unit cell.

Figure 25

Combined effects of trimerization and disorder for a two-dimensional spin-directed ${\mathbb{Z}}_2^{}$ network model. The ${\delta }^2$ dependence of the normalized localization length at $t^2=0.5$ and $\theta =\pi /4$ for fixed $M$ ranging from 24 to 192 is shown in panel (a) for ${\Lambda }_{\perp }^{}$ and in panel (b) for ${\Lambda }_{\parallel }^{}$. Inset: Dependence on ${\delta }^2$ near ${\delta }^2=0.5$ for panel (a) and for panel (b).

Figure 26

The two-dimensional spin-directed ${\mathbb{Z}}_2^{}$ network model is solved numerically along the one-dimensional cut (3.20) in the three-dimensional parameter space (2.7). Panel (a-1) shows the ${\delta }^2$ dependence of the normalized localization length ${\Lambda }_{\perp }^{}$ corresponding to the geometry of Fig. 13 for several values of $M$. Panel (b-1) shows the ${\delta }^2$ dependence of the normalized localization lengths ${\Lambda }_{\parallel }^{}$ corresponding to the geometry of Fig. 13 for several values of $M$. A finite-size scaling analysis of panels (a-1) and (b-1) is performed in panels (a-2) and (b-2), respectively. The horizontal axis is $M^{1/\nu }|{\delta }^2-{\delta }_{\mathrm{c}}^2|$ with $\nu$ and ${\delta }_{\mathrm{c}}^2$ given in Tables 4 and 5. The vertical axis ${\Lambda }_{\mathrm{x}}^{\prime }$ with $\mathrm{x}=\perp ,\parallel$ is defined by subtracting from the normalized localization length ${\Lambda }_{\mathrm{x}}^{}$ its finite-size correction from the leading irrelevant exponent $y$ given in Tables 4 and 5. The red solid curve demonstrates the quality of the data collapse onto a one-parameter scaling function.

Figure 27

The two-dimensional spin-directed ${\mathbb{Z}}_2^{}$ network model is solved numerically along the one-dimensional cut (3.21) in the three-dimensional parameter space (2.7). Panel (a-1) shows the ${\delta }^2$ dependence of the normalized localization length ${\Lambda }_{\perp }^{}$ corresponding to the geometry of Fig. 13 for several values of $M$. Panel (b-1) shows the ${\delta }^2$ dependence of the normalized localization lengths ${\Lambda }_{\parallel }^{}$ corresponding to the geometry of Fig. 13 for several values of $M$. A finite-size scaling analysis of panels (a-1) and (b-1) is performed in panels (a-2) and (b-2), respectively. The horizontal axis is $M^{1/\nu }|{\delta }^2-{\delta }_{\mathrm{c}}^2|$ with $\nu$ and ${\delta }_{\mathrm{c}}^2$ given in Tables 6 and 7. The vertical axis ${\Lambda }_{\mathrm{x}}^{\prime }$ with $\mathrm{x}=\perp ,\parallel$ is defined by subtracting from the normalized localization length ${\Lambda }_{\mathrm{x}}^{}$ its finite-size correction from the leading irrelevant exponent $y$ given in Tables 6 and 7. The red solid curve demonstrates the quality of the data collapse onto a one-parameter scaling function.