Spin-mixing-tunneling network model for Anderson transitions in two-dimensional disordered spinful electrons

TL;DR

This paper introduces a symplectic quantum network model for two-dimensional disordered spinful electrons, revealing a phase diagram with a normal-metal phase influenced by spin-mixing tunneling, and explores transitions between quantum phases.

Contribution

The paper develops a novel SMT-QNM model that treats SMT and SPT as independent channels and links the phase diagram to disordered weak topological insulators.

Findings

Normal-metal phase appears between QSH and NI phases with SMT.

Phase diagram closely related to disordered 3D weak topological insulators.

SMT-driven transition from QSH to NI when TRS is broken.

Abstract

We consider Anderson transitions in two-dimensional spinful electron gases subject to random scalar potentials with time-reversal-symmetric spin-mixing tunneling (SMT) and spin-preserving tunneling (SPT) at potential saddle points (PSPs). A symplectic quantum network model, named as SMT-QNM, is constructed in which SMT and SPT have the same status and contribute independent tunneling channels rather than sharing a total-probability-fixed one. Two-dimensional continuous Dirac Hamiltonian is then extracted out from this discrete network model as the generator of certain unitary transformation. With the help of high-accuracy numerics based on transfer matrix technique, finite-size analysis on two-terminal conductance and normalized localization length provides a phase diagram drawn in the SMT-SPT plane. As a manifestation of symplectic ensembles, a normal-metal (NM) phase emerges between the quantum spin Hall (QSH) and normal-insulator (NI) phases when SMT appears. We systematically analyze the quantum phases on the boundary and in the interior of the phase space. Particularly, the phase diagram is closely related to that of disordered three-dimensional weak topological insulators by appropriate parameter mapping. At last, if time-reversal symmetry in electron trajectories between PSPs is destroyed, the system falls into unitary class with no more NM phase. A direct SMT-driven transition from QSH to NI phases exists and can be explained by spin-flip backscattering between the degenerate doublets at the same sample edge.

Equations

$$\left(\begin{array}{c}{Z}_2^{\mathrm{o}}\\ Z_4^{\mathrm{o}}\end{array}\right)=s_{\mathbf{R}}^{\mathrm{CC}}\left(\begin{array}{c}{Z}_1^{\mathrm{i}}\\ Z_3^{\mathrm{i}}\end{array}\right),s_{\mathbf{R}}^{\mathrm{CC}}={\Psi }_{\mathbf{R}}^{24}{S}_{\mathrm{CC}}{\Psi }_{\mathbf{R}}^{13}.$$

$$S_{\mathrm{CC}}=\left(\begin{array}{cc}r& t\\ {\eta }_{t}t& {\eta }_{r}r\end{array}\right),$$

$$\left(\mathrm{i}{\sigma }_y\right)S_{\mathrm{CC}}^{\ast }\left(-\mathrm{i}{\sigma }_y\right)={\eta }_r^{-1}{S}_{\mathrm{CC}}\ne S_{\mathrm{CC}}^{†},$$

$$\left(\begin{array}{c}{Z}_{2\uparrow }^{\mathrm{o}}\\ Z_{1\downarrow }^{\mathrm{o}}\\ Z_{4\uparrow }^{\mathrm{o}}\\ Z_{3\downarrow }^{\mathrm{o}}\end{array}\right)=s_{\mathbf{R}}^{\mathrm{SMT}}\left(\begin{array}{c}{Z}_{1\uparrow }^{\mathrm{i}}\\ Z_{2\downarrow }^{\mathrm{i}}\\ Z_{3\uparrow }^{\mathrm{i}}\\ Z_{4\downarrow }^{\mathrm{i}}\end{array}\right),s_{\mathbf{R}}^{\mathrm{SMT}}={\Psi }_{\mathbf{R}}^{2143}{S}_{\mathrm{SMT}}{\Psi }_{\mathbf{R}}^{1234}.$$

$$S_{\mathrm{SMT}}=\left(\begin{array}{cc}\left(\begin{array}{cc}{r}_1& 0\\ 0& r_1^{\ast }\end{array}\right)& Q\\ Q^{†}& \left(\begin{array}{cc}-r_1& 0\\ 0& -r_1^{\ast }\end{array}\right)\end{array}\right),$$

$$S_{\mathrm{SMT}}^{†}{S}_{\mathrm{SMT}}=S_{\mathrm{SMT}}{S}_{\mathrm{SMT}}^{†}={\sigma }_0\otimes {\sigma }_0,$$

$$Q{Q}^{†}=Q^{†}Q=\left(t^2+r^2{\sin }^2\theta \right){\sigma }_0,$$

$$\left(\begin{array}{cc}\mathrm{i}{\sigma }_y& 0\\ 0& \mathrm{i}{\sigma }_y\end{array}\right)S_{\mathrm{SMT}}^{\ast }\left(\begin{array}{cc}-\mathrm{i}{\sigma }_y& 0\\ 0& -\mathrm{i}{\sigma }_y\end{array}\right)=S_{\mathrm{SMT}}^{†},$$

$$Q={\sigma }_y{Q}^{\ast }{\sigma }_y.$$

$$\begin{array}[]{c}Q=a_{0}\sigma_{0}+\sum_{k}a_{k}\sigma_{k},\quad a_{0},a_{k=x,y,z}\in\mathbb{C},\end{array}$$

$$\begin{array}[]{ccc}\sum_{\alpha}|a_{\alpha}|^{2}&=&t^{2}+r^{2}\sin^{2}\theta,\quad\alpha=0,x,y,z,\\ \mathrm{Re}\left(a_{0}^{*}a_{k}\right)&=&\mathrm{Im}\left(\epsilon_{klm}a_{l}^{*}a_{m}\right),\quad k,l,m=x,y,z,\\ \end{array}$$

$$a_0=a_0^{\ast },a_k=-a_k^{\ast },k=x,y,z.$$

$$\begin{array}[]{cc}a_{0}=t\cos\phi_{1},&a_{x}=\mathrm{i}r\sin\theta\sin\phi_{2},\\ a_{z}=\mathrm{i}t\sin\phi_{1},&a_{y}=\mathrm{i}r\sin\theta\cos\phi_{2},\\ \end{array}$$

$$Q=\left(\begin{array}{cc}t{e}^{\mathrm{i}{\varphi }_1}& r{e}^{\mathrm{i}{\varphi }_2}\sin \theta \\ -r{e}^{-\mathrm{i}{\varphi }_2}\sin \theta & t{e}^{-\mathrm{i}{\varphi }_1}\end{array}\right).$$

$${\mathbf{R}}_{x,y}=x{\mathbf{e}}_x+y{\mathbf{e}}_y,x,y\in \mathbb{Z},\mathrm{mod}(x+y,2)=0.$$

$$\left(\begin{array}{c}{Z}_{2\uparrow }(x,y)\\ Z_{1\downarrow }(x,y)\\ Z_{4\uparrow }(x+1,y+1)\\ Z_{3\downarrow }(x+1,y+1)\end{array}\right)=S_{x,y}^{\mathrm{urc}}\left(\begin{array}{c}{Z}_{1\uparrow }(x,y)\\ Z_{2\downarrow }(x,y)\\ Z_{3\uparrow }(x+1,y+1)\\ Z_{4\downarrow }(x+1,y+1)\end{array}\right),$$

$$\begin{array}[]{l}S_{x,y}^{\mathrm{urc}}\equiv U_{x,y}S_{\mathrm{SMT}}V_{x,y},\\ U_{x,y}=\mathrm{diag}(e^{\mathrm{i}\psi_{2}(x,y)},e^{\mathrm{i}\psi_{1}(x,y)},e^{\mathrm{i}\psi_{4}(x+1,y+1)},e^{\mathrm{i}\psi_{3}(x+1,y+1)}),\\ V_{x,y}=\mathrm{diag}(e^{\mathrm{i}\psi_{1}(x,y)},e^{\mathrm{i}\psi_{2}(x,y)},e^{\mathrm{i}\psi_{3}(x+1,y+1)},e^{\mathrm{i}\psi_{4}(x+1,y+1)}),\\ \end{array}$$

$$\left(\begin{array}{c}{Z}_{3\uparrow }(x,y)\\ Z_{2\downarrow }(x,y)\\ Z_{1\uparrow }(x+1,y-1)\\ Z_{4\downarrow }(x+1,y-1)\end{array}\right)=S_{x,y}^{\mathrm{lrc}}\left(\begin{array}{c}{Z}_{2\uparrow }(x,y)\\ Z_{3\downarrow }(x,y)\\ Z_{4\uparrow }(x+1,y-1)\\ Z_{1\downarrow }(x+1,y-1)\end{array}\right),$$

$$\begin{array}[]{l}S_{x,y}^{\mathrm{lrc}}\equiv U^{\prime}_{x,y}S_{\mathrm{SMT}}V^{\prime}_{x,y}\\ U^{\prime}_{x,y}=\mathrm{diag}(e^{\mathrm{i}\psi_{3}(x,y)},e^{\mathrm{i}\psi_{2}(x,y)},e^{\mathrm{i}\psi_{1}(x+1,y-1)},e^{\mathrm{i}\psi_{4}(x+1,y-1)})\\ V^{\prime}_{x,y}=\mathrm{diag}(e^{\mathrm{i}\psi_{2}(x,y)},e^{\mathrm{i}\psi_{3}(x,y)},e^{\mathrm{i}\psi_{4}(x+1,y-1)},e^{\mathrm{i}\psi_{1}(x+1,y-1)})\\ \end{array}.$$

$$\begin{array}[]{ccc}\tau_{x}^{\pm}f_{n\sigma}(x,y)&=&f_{n\sigma}(x\pm 1,y),\\ \tau_{y}^{\pm}f_{n\sigma}(x,y)&=&f_{n\sigma}(x,y\pm 1),\\ \end{array}$$

$${\left[{\tau }_{x(y)}^{\pm }\right]}^{-1}={\tau }_{x(y)}^{\mp }.$$

$$\begin{array}[]{rcl}\left(\begin{matrix}Z_{2\uparrow}(x,y)\\ Z_{4\uparrow}(x,y)\\ Z_{1\downarrow}(x,y)\\ Z_{3\downarrow}(x,y)\\ \end{matrix}\right)&=&\Omega_{S}\left(\begin{matrix}Z_{2\downarrow}(x,y)\\ Z_{4\downarrow}(x,y)\\ Z_{1\uparrow}(x,y)\\ Z_{3\uparrow}(x,y)\\ \end{matrix}\right),\\ \Omega_{S}&=&O_{x,y}\cdot M_{S}\cdot O_{x,y},\\ \end{array}$$

$$O_{x,y}=\mathrm{diag}(e^{\mathrm{i}{\psi }_2(x,y)},e^{\mathrm{i}{\psi }_4(x,y)},e^{\mathrm{i}{\psi }_1(x,y)},e^{\mathrm{i}{\psi }_3(x,y)})$$

$$\begin{array}[]{rcl}M_{S}&=&\left(\begin{matrix}\Phi_{2}M_{\mathrm{d}}\Phi_{2}&\Phi_{1}^{\dagger}M_{\mathrm{od}}\Phi_{1}\\ \Phi_{1}M_{\mathrm{od}}\Phi_{1}^{\dagger}&-\Phi_{2}^{\dagger}M_{\mathrm{d}}\Phi_{2}^{\dagger}\\ \end{matrix}\right),\\ \Phi_{1}&=&\mathrm{diag}(e^{-\mathrm{i}\frac{\phi_{1}}{2}},e^{\mathrm{i}\frac{\phi_{1}}{2}}),\quad\Phi_{2}=e^{\mathrm{i}\frac{\phi_{2}}{2}}\sigma_{0},\\ M_{\mathrm{d}}&=&\left(\begin{matrix}0&r\sin\theta\tau_{x}^{+}\tau_{y}^{+}\\ -r\sin\theta\tau_{x}^{-}\tau_{y}^{-}&0\\ \end{matrix}\right),\\ M_{\mathrm{od}}&=&\left(\begin{matrix}r\cos\theta&t\tau_{x}^{+}\tau_{y}^{+}\\ t\tau_{x}^{-}\tau_{y}^{-}&-r\cos\theta\\ \end{matrix}\right),\\ \end{array}$$

$$\begin{array}[]{rcl}\left(\begin{matrix}Z_{2\downarrow}(x,y)\\ Z_{4\downarrow}(x,y)\\ Z_{1\uparrow}(x,y)\\ Z_{3\uparrow}(x,y)\\ \end{matrix}\right)&=&\Omega_{S^{\prime}}\left(\begin{matrix}Z_{2\uparrow}(x,y)\\ Z_{4\uparrow}(x,y)\\ Z_{1\downarrow}(x,y)\\ Z_{3\downarrow}(x,y)\\ \end{matrix}\right),\\ \Omega_{S^{\prime}}&=&O_{x,y}\cdot M_{S^{\prime}}\cdot O_{x,y},\\ \end{array}$$

$$\begin{array}[]{rcl}M_{S^{\prime}}&=&\left(\begin{matrix}\Phi_{2}^{\dagger}M^{\prime}_{\mathrm{d}}\Phi_{2}^{\dagger}&\Phi_{1}M^{\prime}_{\mathrm{od}}\Phi_{1}\\ \Phi_{1}\sigma_{x}M^{\prime}_{\mathrm{od}}\sigma_{x}\Phi_{1}&-\Phi_{2}\sigma_{x}M^{\prime}_{\mathrm{d}}\sigma_{x}\Phi_{2}\\ \end{matrix}\right),\\ M^{\prime}_{\mathrm{d}}&=&\left(\begin{matrix}0&-r\sin\theta\tau_{x}^{+}\tau_{y}^{-}\\ r\sin\theta\tau_{x}^{-}\tau_{y}^{+}&0\\ \end{matrix}\right),\\ M^{\prime}_{\mathrm{od}}&=&\left(\begin{matrix}t\tau_{x}^{+}\tau_{y}^{-}&r\cos\theta\\ -r\cos\theta&t\tau_{x}^{-}\tau_{y}^{+}\\ \end{matrix}\right).\\ \end{array}$$

$${\mathbf{Z}}_{x,y}\equiv {\left(Z_{2\uparrow }{Z}_{4\uparrow }{Z}_{1\downarrow }{Z}_{3\downarrow }{Z}_{2\downarrow }{Z}_{4\downarrow }{Z}_{1\uparrow }{Z}_{3\uparrow }\right)}^{\mathrm{T}}$$

$${\left({\mathbf{Z}}_{x,y}\right)}_{\mu +1}=\left(\begin{array}{cc}0& {\Omega }_S\\ {\Omega }_{S^{\prime }}& 0\end{array}\right){\left({\mathbf{Z}}_{x,y}\right)}_{\mu }.$$

$${\left({\mathbf{Z}}_{x,y}\right)}_{\mu +2}=\left(\begin{array}{cc}{\Omega }_S{\Omega }_{S^{\prime }}& 0\\ 0& {\Omega }_{S^{\prime }}{\Omega }_S\end{array}\right){\left({\mathbf{Z}}_{x,y}\right)}_{\mu }$$

$${\psi }_4\to {\psi }_4+\frac{\pi }{2}$$

Full text

I I. Introduction

Anderson transitions (ATs), i.e., transitions between localized and delocalized quantum phases in disordered electronic systems, have attracted intense and continuous attention since its proposalAnderson_1958 due to its fundamental significance in condensed matter physicsLee_1985 ; Kramer_1993 ; Huckestein_1995 ; Evers_Mirlin_2008 . In 1970s and 1980s, scaling-theory and field-theory approaches revealed the connections between Anderson transition and conventional second-order phase transitionsLee_1985 ; Kramer_1993 ; Huckestein_1995 . In 1990s, the symmetry classification of disordered systems was achieved based on its relation to the classical symmetric spacesZirnbauer_1996 ; Zirnbauer_1997_prb ; Caselle_2004 . Later, the completeness of this classification is proved in 2005Heinzner_2005 . Now we know there are totally ten symmetry classes according to how many discrete symmetries are obeyed by the underlying physical system. When a system only has symmetries translationally invariant in energy, such as the time-reversal symmetry (TRS) and spin-rotation symmetry (SRS), it falls into one of the three traditional Wigner-Dyson classes (unitary, orthogonal and symplectic)Wigner_1951 ; Dyson_1962 . However, if we focus on some particular value of energy, extra discrete symmetries could arise and lead to novel symmetry classes. In condensed matter systems described by tight-binding models on a bipartite lattice with randomness only residing in hopping terms, three chiral classes are identifiedZirnbauer_1996 . The remaining four were discovered in superconducting systems and known as the Bogoliubov-de Gennes classesZirnbauer_1997_prb . In the past decades, ATs in these ten classes have been investigated intensively and considerable progress has been made in various directions, such as their scaling-theory and field-theory descriptionsLee_1985 ; Kramer_1993 ; Huckestein_1995 , multifractality in critical wave functionsJanssen_1994 ; Mudry_1996 ; Evers_Mirlin_2000 ; Evers_2001 ; Obuse_2004_prb ; Obuse_2007_prl ; Obuse_2010_prb_1 and level statistics at criticalityMirlin_2000 ; XiongGang_2001_PRL ; XiongGang_2006_JPCM ; Obuse_2005_prb ; GarciaGarcia_2007 , etc.

Recently, the spin-orbit-induced topological materials, named as topological insulators (TIs), have received intensive attentionRMP_2010_Hasan_Kane ; RMP_2011_XLQi_SCZhang ; RMP_2015_Beenakker ; RPP_2016_ZHQiao_QNiu ; RMP_2016_Bansil ; RMP_2016_Witten ; RMP_2016_Ryu . In TIs, the interplay between topology and symmetry greatly enriches our knowledge of quantum statesRMP_2016_Ryu ; Fu_2012_prl ; Fulga_2012_prb ; Ryu_2007_prl ; Ryu_2010_njp . First, the TRS is crucial for their realization and stabilization. Second, the spin-orbit coupling (SOC) destroys the SRS, thus makes TIs belong to the Wigner-Dyson symplectic class. In two dimensions (2D), they are the well-known quantum spin Hall (QSH) ensembles. In disordered QSH systems, ATs can be extended from traditional metal-insulator transitions to a broader sense which includes transition between topologically trivial and nontrivial phasesEvers_Mirlin_2008 . In the past decade, great efforts have been devoted into this issueRyu_2010_njp ; Obuse_2007_prb ; Obuse_2008_prb ; Obuse_2010_prb_2 ; Obuse_2014_prb ; Slevin_2012 ; Ryu_2012_prb ; Slevin_2014_njp ; XRWang_2015_PRL ; CWang_2017_PRB . The widely-used framework is to construct a quantum network model which consists of two copies of Chalker-Coddington random network model (CC-RNM) describing up and down spins, as well as certain coupling describing spin-flip process. If spin flip occurs in electron trajectories between potential saddle points (PSPs), it is the well-known spin-orbit coupling (SOC). While if it takes place at the PSPs, it is the spin-mixing tunneling (SMT) which is the main focus in this work. Recently, a $Z_{2}$ quantum network model ($Z_{2}$-QNM) is proposedRyu_2010_njp ; Obuse_2007_prb ; Obuse_2008_prb ; Obuse_2010_prb_2 ; Obuse_2014_prb ; Slevin_2012 in which SMT at PSPs are considered. It belongs to the Wigner-Dyson symplectic class and a series of work declare that it provides a good description of ATs in 2D disordered spinful electron gases(2D-DSEGs). In $Z_{2}$-QNM, at PSPs the total tunneling probability are fixed, which means SMT takes away part of the probability from the spin-preserving tunneling (SPT) process. However, from the basic principles of quantum tunneling SMT provides an additional channel and should not affect the existing SPT. In this work, we treat the SMT as an independent quantum tunneling channel and build a new network model, namely the “SMT-QNM”, to provide an alternative perspective to understand ATs in 2D-DSEGs.

This paper is organized as follows. In Sec. II the SMT-QNM is systematically built up based on probability conservation and TRS at PSPs. Then the 2D continuous Dirac Hamiltonian with “valley” degree of freedom is extracted out. In Sec. III numerical algorithms using transfer matrix technique for finite-size analysis on two-terminal conductance and normalized localization length are reviewed. Based on them, in Sec. IV the quantum phases of SMT-QNM are investigated and a phase diagram is then obtained. We discuss its close connection with that of the disordered 3D weak TIs. In Sec. V, we consider the case when TRS in electron trajectories between PSPs is destroyed. The system then falls into unitary class. We briefly summarize the quantum phases and phase transitions therein. Finally, the concluding remarks are provided in the last section.

II II. The SMT-QNM

II.1 II.A Brief review of CC-RNM

Under a strong magnetic field $\vec{B}=B\hat{z}$, the motion of an electron in a smooth enough 2D random scalar potential $V(\vec{r})$ can be decomposed into a rapid cyclotron gyration and a slow drift of the guiding center along an equipotential contour which is generally composed of numerous loops around potential valleys or peaksChalker_Coddington_1988 ; Kramer_2005 . The drifting direction of electrons in each loop is uni-directional (chiral): $\vec{v}(\vec{r})=\nabla V(\vec{r})\times\vec{B}/(eB^{2})$. At PSPs, electrons’ reflecting along equipotential lines and their mutual tunneling are the essential physical ingredients for constructing a network model describing quantum criticality in disordered 2D systems. For modelization, the PSPs are arranged to form a 2D square lattice with the interconnected links representing electron flows along equipotential lines. The potential peaks and valleys distribute alternatively in the square plaquettes enclosed by the links. This endues definite propagating direction of electron flows on the links and then divides the PSPs into two subgroups: the S- and S’-types (see Fig. 1a and 1b). At each PSP, two incoming and two outgoing electron flows intersect hence lead to a $2\times 2$ scattering matrix. Quantum tunneling only occurs at PSPs and in the simplest case can be assumed identical. At last, disorder is introduced by random phases along links. This is the basic framework of CC-RNM. In all illustration figures in this paper, we adopt the following sketch rules: if $r>t$, the reflecting (tunneling) routes are depicted by solid (dash) curves and vice versa.

For a S-type PSP at $\mathbf{R}$, its scattering matrix is,

[TABLE]

where $Z_{j}^{\mathrm{o(i)}}$ is the outgoing (incoming) electron flow amplitude at link $j$, $\Psi_{\mathbf{R}}^{jk}\equiv\mathrm{diag}(e^{\mathrm{i}\psi_{\mathbf{R}}^{j}},e^{\mathrm{i}\psi_{\mathbf{R}}^{k}})$ is a diagonal matrix, with $\psi_{\mathbf{R}}^{j}$ being the dynamical phase an electron acquires when propagating on link $j$ between the observation point and the PSP at $\mathbf{R}$. The kernel matrix $S_{\mathrm{CC}}$ has the general form,

[TABLE]

where $r=\sqrt{p}$ ($t=\sqrt{1-p}$) measuring the reflecting (tunneling) amplitude at a PSP, and $p$ is related to the Fermi level of the systemKramer_2005 . $\eta_{t(r)}$ are undetermined coefficients. In steady states, probability conservation at any PSP requires $\eta_{t(r)}=e^{\mathrm{i}\phi_{t(r)}}$ and $|\phi_{t}-\phi_{r}|=(2n+1)\pi$. Clearly,

[TABLE]

which means TRS is broken thus the CC-RNM belongs to the unitary class. Throughout this work, $\eta_{r}=-\eta_{t}=-1$ which is also the choice in most literatures.

II.2 II.B Scatter matrices of SMT-QNM

To describe ATs in 2D-DSEGs, the CC-RNM should be generalized to include spins, providing the following hypotheses. First, the potential profile is identical for any spin orientation. Second, the absence of external magnetic fields makes TRS possible which turns the original uni-directed electron flow on each link to a Kramers doublet. Opposite spin components then “feel” opposite effective magnetic fields, forming two copies of CC-RNM with opposite chirality. Third, appropriate coupling should be introduced between the two copies of CC-RNM to describe spin-flip process. Generally, spin flip can occur anywhere. In real modelization, two strategies are most common: (a) it only occurs on the links between PSPs; (b) it only occurs at the PSPs. The first strategy reflects the SOC while the second one is the SMT.

The $Z_{2}$-QNM proposed in Refs.Ryu_2010_njp ; Obuse_2007_prb ; Obuse_2008_prb ; Obuse_2010_prb_2 ; Obuse_2014_prb follows the second strategy, however views SMT and SPT as two competing processes sharing a fixed probability “$t^{2}$”. In this work, the SPT channel remains unperturbed. Meantime we treat SMT as an independent quantum tunneling channel and construct the SMT-QNM to understand ATs in 2D-DSEGs. For S-type PSPs (See Fig. 1c), the scattering matrix at position $\mathbf{R}$ reads,

[TABLE]

where $Z_{j\sigma}^{\mathrm{o(i)}}$ is the outgoing (incoming) electron flow amplitude at link $j$ with spin $\sigma(\uparrow\mathrm{or}\downarrow)$, $\Psi_{\mathbf{R}}^{jklm}\equiv\mathrm{diag}(e^{\mathrm{i}\psi_{\mathbf{R}}^{j}},e^{\mathrm{i}\psi_{\mathbf{R}}^{k}},e^{\mathrm{i}\psi_{\mathbf{R}}^{l}},e^{\mathrm{i}\psi_{\mathbf{R}}^{m}})$ with $\psi_{\mathbf{R}}^{j}$ representing the phase an electron acquires when propagating on link $j$ between the observation point and the PSP at $\mathbf{R}$. We have neglected the spin index since the Kramers pair of electron flows have the same accumulated phase on the same link. To mimic the randomness in PSP distribution, these phases are distributed uniformly and independently in the region $[0,2\pi)$. If we focus on the very point where a PSP locates, $\Psi_{\mathbf{R}}^{jklm}$ then becomes unity. The kernel matrix $S_{\mathrm{SMT}}$ describes the reflecting and tunneling at a general S-type PSP and has the following structure,

[TABLE]

where “$\dagger$” means matrix complex conjugate. For this scattering matrix, several points need to be clarified. First, it is hermitian due to TRS. Second, $|r_{1}|\leq r$ since SMT is an additional tunneling channel hence takes probability away from reflecting rather than SPT process. For simplicity, $r_{1}$ can be defined as $r_{1}=r\cos\theta$ (thus is real), with $\theta\in\left[0,\pi/2\right]$ describing the strength of SMT. Third, probability conservation in steady states at any PSP requires the scattering matrix to be unitary,

[TABLE]

which gives

[TABLE]

where $\sigma_{0}$ is the $2\times 2$ unit matrix. Fourth, TRS requires

[TABLE]

where $\sigma_{x,y,z}$ are the Pauli matrices. This gives,

[TABLE]

By writing $Q$ as

[TABLE]

Eq. (7) turns to

[TABLE]

in which $\epsilon_{klm}$ is the 3D Levi-Civita symbol. In addition Eq. (9) gives

[TABLE]

Summarizing these two conditions, a reasonable solution to $a_{\alpha}$ is

[TABLE]

leading to a physical realization of $Q$ as

[TABLE]

Obviously $\phi_{1}$ and $\phi_{2}$ are the phase shifts associated with SPT and SMT processes, respectively. At last, by rotating S-type PSPs 90 degrees clockwise, we get S’-type PSPs and their scattering matrix can be easily obtained from Eq. (4).

To summarize, in our SMT-QNM at any PSP (S- and S’-type), for an incoming electron flow with some certain spin orientation and probability 1, it tunnels into an outgoing flow with the same spin orientation via SPT process with probability “$t^{2}$” and also into an outgoing flow with opposite spin orientation via SMT process with probability “$r^{2}\sin^{2}\theta$”, leaving a probability “$r^{2}\cos^{2}\theta$” residing in the original equipotential line.

II.3 II.C 2D Dirac Hamiltonian from SMT-QNM

The mapping from CC-RNM to 2D Dirac Hamiltonian was accomplished in 1996Ho_Chalker_1996 , and the connection between the $Z_{2}$-QNM and 2D Dirac Hamiltonian was established in 2010Ryu_2010_njp . The main strategy of both works is to view the unitary (due to probability conservation) scattering matrices as a unitary time-evolution operation whose infinitesimal generator is the required Hamiltonian, as we all know that a unitary matrix is the exponential of a Hermitian one. In this subsection, we follow this strategy and succeed in extracting the 2D Dirac Hamiltonian from our SMT-QNM and recognizing the roles of phase shifts in SMT and SPT at PSPs. Also, this part of work lays the foundation for understanding the close connection between the phase diagram of our SMT-QNM and that of disordered 3D weak TIs (see Sec. IV.D).

II.3.1 II.C.1 Preparations

We arrange the S-type and S’-type PSPs alternatively in a 2D Cartesian plane to form a bipartite square lattice, as shown in Fig. 2. Then following the sketch rules in Fig. 1c and 1d, a series of closed square plaquettes are obtained, with each edge bearing two opposite-directed links. For $r>t$, the centers of these closed plaquettes are the potential valleys, while the potential peaks reside in the blanks outside. For $r<t$, the situation is just reversed. Quantum tunnelings (SPT and SMT) occur at the plaquette corners, which are the PSPs. We take the $r>t$ case as the framework for our discussion, which does not affect the generality of our results. If one of these plaquettes is assigned with coordinate $(0,0)$, then the position of anyone in this set is

[TABLE]

They form a square lattice and is our main concern. The eight directed links on the edges of a plaquette are labeled by $(n\sigma)$ with $n=1,2,3,4$ and $\sigma=\uparrow$ or $\downarrow$.

For the plaquette with coordinate $(x,y)$, the scattering event at the S-type PSP on its upper-right corner (urc) is as follows

[TABLE]

with

[TABLE]

in which TRS has be invoked in writing $U_{x,y}$ and $V_{x,y}$. While that of the S’-type PSP on the lower-right corner (lrc) reads

[TABLE]

with

[TABLE]

Next the displacement operators $\tau_{x(y)}^{\pm}$ are introduced as

[TABLE]

where $f_{n\sigma}(x,y)$ is an arbitrary function defined at $\mathbf{R}_{x,y}$. By definition, they are commutative and

[TABLE]

By rearranging the amplitudes in the order of “2,4,1,3”, we rewrite Eq. (16) into the form

[TABLE]

with

[TABLE]

and

[TABLE]

in which “d (od)” means diagonal (off-diagonal). Similarly, Eq. (18) is rewritten as

[TABLE]

where

[TABLE]

By defining the total amplitude vector $\mathbf{Z}_{x,y}$ composed of all eight links along the edges of plaquette at $\mathbf{R}_{x,y}$ as

[TABLE]

with the superscript “$\mathrm{T}$” indicating matrix transpose and introducing $\mu\in\mathbb{Z}$, the elementary imaginary discrete-time evolution of $\mathbf{Z}_{x,y}$ is,

[TABLE]

To acquire decoupled equations, the “two-step” time evolution,

[TABLE]

is more convenient since it is diagonal. We will focus on $\Omega_{S}\Omega_{S^{\prime}}\equiv\Omega_{SS^{\prime}}$ in the rest of this work. Also we make the transformation

[TABLE]

to raise the reference point of the total phase flux of each plaquette by $\pi$ [see Eqs. (22) and (25)], which is crucial for the extraction of 2D Dirac Hamiltonian. It can be easily checked that $\Omega_{SS^{\prime}}$ is unitary, thus provide a Hamiltonian as its infinitesimal generator,

[TABLE]

We then demonstrate that in the close vicinity of the CC-RNM critical point

[TABLE]

how $\mathcal{H}_{\mathrm{SMT}}$ is mapped to 2D Dirac Hamiltonian by expanding $\Omega_{SS^{\prime}}$ to the leading-order powers of

[TABLE]

II.3.2 II.C.2 2D Dirac Hamiltonian around $\theta=0$

At $\theta=0$, $M_{\mathrm{d}}=M^{\prime}_{\mathrm{d}}=0$. Eqs. (22) and (25) thus provide

[TABLE]

with

[TABLE]

Then

[TABLE]

Under the following assumptions:

(a) displacement operators act on smooth enough functions thus

[TABLE]

(b) the phases $\psi_{n=1,2,3,4}$ and $\phi_{1}$ are small enough hence

[TABLE]

(c) in the close vicinity of CC-RNM critical point one has

[TABLE]

we get

[TABLE]

with

[TABLE]

acting as a scalar/vector potential, respectively.

Then the system is driven away slightly from the critical point (32) along the $\theta$-line. Hence $\psi_{n}=m=0$ and $\tau_{x(y)}^{\pm}=1$, and to the leading order of $\theta$ one has

[TABLE]

with

[TABLE]

Correspondingly, the SMT Hamiltonian turns to

[TABLE]

After performing a unitary transformation

[TABLE]

we get the final Hamiltonian

[TABLE]

with

[TABLE]

Obviously $\mathcal{H}_{\mathrm{f}}$ describes a pair of Dirac fermions (with mass $\pm m$) subject to the same random scalar potential $A_{0}$ and respective random vector potential $\pm(A_{x},A_{y})$, meantime bearing a mutual coupling $\sqrt{2}\theta\sigma_{0}$. By introducing a “valley” space distinguishing these two Dirac fermions (different locations of Dirac cones in Brillouin zone), the final Hamiltonian can be rewritten as

[TABLE]

where $s_{0}$ and $s_{x,y,z}$ are identity and Pauli matrices in valley space. Therefore our SMT-QNM belongs to the symplectic class and should be an effective model for ATs in QSH ensembles. Also, the above analytics shows that the phase shifts in SPT and SMT processes at PSPs have different roles during the extraction of 2D Dirac Hamiltonian. The former ($\phi_{1}$) enters the vector potentials thus could have impacts on geometric phase accumulated along the plaquette edges. While the latter ($\phi_{2}$) resides in the coupling matrix $J_{\theta}$ between $J_{\pm}$ and then manifests itself in the unitary transformation that changes $\mathcal{H}_{\mathrm{SMT}}$ to $\mathcal{H}_{\mathrm{f}}$, hence acts as a gauge field describing the spin-flip interaction.

III III. Algorithms for finite-size analysis

III.1 III.A Two-terminal conductance $G_{\mathrm{2T}}$

For numerical convenience, by rotating Fig.2 45 degrees clockwise, we obtain a 2D PSP lattice composed of $L$ principal layers (PLs), as shown in Fig. 3. Each PL consists of $W$ S-type and $W$ S’-type PSPs. At a S-type PSP, the “left-ro-right” transfer matrix is obtained from its scattering matrix [see Eq. (4)] as

[TABLE]

in which

[TABLE]

and

[TABLE]

While at a S’-type PSP, the counterpart is

[TABLE]

where

[TABLE]

and

[TABLE]

Then the transfer matrix for the $k-$th PL is

[TABLE]

where the boundary nodes are selected to be S’-type PSPs as an example (see Fig. 3). $V_{1}$ is the transfer matrix of the sub-layer composed merely by S-type PSPs with the following form

[TABLE]

$V_{3}$ is the transfer matrix of the S’-type sub-layer

[TABLE]

where $B_{1,2,3,4}$ are $2\times 2$ matrices and determined by the choice of boundary condition in transverse direction. When we focus on edge modes, the reflecting boundary condition (RBC) is imposed. The Kramers pair is totally reflected without any spin flip at boundary nodes, thus

[TABLE]

If bulk behaviors are the main concern, the periodic boundary condition (PBC) is adopted, which means

[TABLE]

At last, $V_{2}^{(k)}$ and $V_{4}^{(k)}$ are $4W\times 4W$ diagonal matrices,

[TABLE]

describing the left-to-right intra- and inter-PL random phases in $4W$ links connecting S-type and S’-type PSPs in adjacent sub-layers. Note that TRS ensures in any link, spin-up electron flowing in a certain direction acquires the same dynamical phase with that of a spin-down electron in the opposite direction. Thus one has the “phase pairing rule”

[TABLE]

In practice, for certain $\alpha$ the $2W$ phases $\phi_{\alpha,2w-1}^{(k)}$ are independently and uniformly distributed in $[0,2\pi)$.

Multiplying $T^{(k)}$ sequentially, the total transfer matrix $T_{W}^{L}$, which relates the electron flows on the left of the network $(Z_{1\uparrow}^{\mathrm{L,i}},Z_{1\downarrow}^{\mathrm{L,o}},\cdots,Z_{2W\uparrow}^{\mathrm{L,o}},Z_{2W\downarrow}^{\mathrm{L,i}})^{\mathrm{T}}$ and those on the right $(Z_{1\uparrow}^{\mathrm{R,o}},Z_{1\downarrow}^{\mathrm{R,i}},\cdots,Z_{2W\uparrow}^{\mathrm{R,i}},Z_{2W\downarrow}^{\mathrm{R,o}})^{\mathrm{T}}$, is then obtained

[TABLE]

By introducing a unitary matrix $O$ with

[TABLE]

the electron flows on each side of the system are reordered into four subgroups marked by $(\alpha=\mathrm{i/o},\sigma=\uparrow\downarrow)$, i.e.,

[TABLE]

On the other hand, the entire network can be viewed as a whole hence its transport features are provided by a $4W\times 4W$ scattering matrix $S_{\mathrm{t}}$,

[TABLE]

where $T$ and $T^{\prime}$ ($R$ and $R^{\prime}$) are $2W\times 2W$ transmission (reflection) matrices. The Landauer formula tells us that the total two-terminal charge conductance, $G_{\mathrm{2T}}$, is

[TABLE]

Finally, by comparing Eqs. (64) and (65), one has

[TABLE]

which leads to $T^{\prime}=(\widetilde{T}_{22})^{-1}$.

This provides the main algorithm of calculating the two-terminal conductance. Before ending this subsection, a few points need to be addressed. First, the diagonal phases in $\mathcal{U}_{1,2}$ and $\mathcal{U}^{\prime}_{1}$ are already grouped to pairs with opposite signs, thus can be absorbed into random phase matrices $V_{2,4}^{(k)}$. This feature has two consequences: (i) we directly use $T_{0}(T^{\prime}_{0})$ rather than $T_{S}(T_{S^{\prime}})$ to build $V_{1}(V_{3})$, (ii) in real calculations, usually $\phi_{1,2}$ are assumed to be distributed independently and uniformly in $[0,2\pi)$ or even neglected. Second, during the calculation of $\widetilde{T}_{22}$, the numerical instability of multiplying iteratively $T^{(k)},k=1,\cdots,L$ can be fixed by performing QR decompositions where needed.

III.2 III.B Lyapunov exponents and normalized localization length

For a quasi-one-dimensional (Q1D) system ($W$ finite, $L\rightarrow\infty$), generally the Anderson localization effect makes the two-terminal transmission decays exponentially. The corresponding decay length is called the Q1D localization length $\xi_{W}$, which is the function of Fermi level ($p$), SMT ($\theta$) and transverse dimension $W$.

Now we define a real $4W\times 4W$ symmetric matrix

[TABLE]

The TRS makes the $4W$ eigenvalues of $\Xi$ doubly degenerate into $2W$ pairs, and further fall into $W$ groups with opposite signs due to the current conservation request. In other words, the eigenvalues of $\Xi$ can be written as $\pm\omega_{i},i=1,\ldots,2W$ meantime satisfying $0<\omega_{1}=\omega_{2}<\omega_{3}=\omega_{4}<\ldots<\omega_{2W-1}=\omega_{2W}$. The Lyapunov exponents (LEs) associated with this Q1D network system with fixed width $W$ are then defined by the following limit

[TABLE]

and are self-averaging random variables.

The Q1D localization length of electrons is defined as the reciprocal of the smallest positive LE,

[TABLE]

since the decay of transmission should be controlled by the lowest decay rate in this system. Finally, the criticality of the 2D system is determined by the behavior of normalized localization length $\Lambda$,

[TABLE]

as the transverse dimension $W$ increases: the system falls into NM (NI) phase when $\Lambda$ is an increasing (decreasing) function of $W$ for sufficient large $W$.

In practical numerical calculations, the LEs are not obtained by directly diagonalizing $\Xi$, which comes from iterative multiplication of transfer matrices and turns to be numerically unstable. Following Ref.Slevin_2014_njp , we employ the following algorithm to achieve satisfactory estimations for both the LEs and their precision. For simplicity, suppose $L=s\cdot r\cdot m$, where $s,r,m$ are integers. To estimate all $2W$ LEs, a $4W\times 2W$ matrix $K^{(0)}$ with random orthogonal columns is multiplied to $T^{(1)}$. We then perform the following QR decomposition every $m$ steps,

[TABLE]

where $j=1,\ldots,sr$, $K^{(j)}$ are $4W\times 2W$ matrices with orthogonal columns and $M^{(j)}$ are $2W\times 2W$ upper triangular matrices with positive diagonal elements.

The total length $L$ is divided into $s$ segments and each consists of $r\cdot m$ PLs. In each $k-$segment ($1\leq k\leq s$), we calculate,

[TABLE]

The $2W$ LEs are then evaluated as,

[TABLE]

If each segment ($rm$) is long enough, it is reasonable to assume that $\gamma_{i}^{(k)}$($1\leq k\leq s$) are statistically independent. The standard error $\sigma_{i}$ of $\Gamma_{i}$ is given by,

[TABLE]

In most cases, $\epsilon_{1}=\sigma_{1}/\Gamma_{1}=1\%$ is an acceptable criterion for a good estimation of $\Gamma_{1}$ and thus $\Lambda$.

IV IV. Quantum phases and phase transitions in SMT-QNM

The simplest S-type PSP is realized by 2D quadratic potential $V_{\mathrm{S-PSP}}=U_{0}\cdot(y^{2}-x^{2})$ with $U_{0}>0$, which is identical for arbitrary spin orientation. In CC-RNM, the total Hamiltonian of an electron close to a S-type PSP

[TABLE]

is quadratic hence can be diagonalized. Under symmetric gauge of the vector potential $\mathbf{A}=(B/2)(-y,x)$, the reflecting probability isKramer_2005 ; Fertig_1987

[TABLE]

with

[TABLE]

in which $E_{F}$ is the Fermi energy of the system. If $E_{F}$ is well below (above) the saddle point energy, the quantum tunneling probability vanishes (approaches to 1). When $E_{F}$ lies exactly at the PSP energy, $\varepsilon=0$ hence $p_{c}=1/2$ being the CC-RNM critical point.

In SMT-QNM, there are no external magnetic fields. However we preserve the mathematical structure in Eq. (77) and in the simplest case assume $\varepsilon\equiv E_{F}$ without loss of generality. Then the mapping from $E_{F}\in[-\infty,+\infty]$ to $p\in[0,1]$ is bijective with $E_{F}=0$ corresponding to $p=p_{c}$. Hence the “SMT-SPT” phase space is isomorphic to “$\theta-p$” parameter space.

To get the full phase diagram, both bulk and edge behaviors are important. In the first step, PBC is exerted to distinguish insulating (NI and TI) and NM states. Then RBC is adopted to further check whether there are topological non-trivial edge modes. The network layout is depicted in Fig. 3, with S’-type PSPs being the boundary nodes. There is also counterpart with marginal S-type PSPs. However they are equivalent under PBC while differ only in boundary modes under RBC (symmetric about $p=p_{c}$). Throughout the rest of this work, we fix the boundary nodes to be S’-type PSPs under RBC. In this section, the quantum phases and phase transitions in the closed phase space

[TABLE]

are investigated in details.

IV.1 IV.A Phase diagram of SMT-QNM in $\Omega_{1}$

Following the algorithms in Sec. III, $G_{\mathrm{2T}}$ and $\Lambda$ are calculated under PBC and/or RBC. Based on these numerical data, the complete phase diagram of SMT-QNM is obtained, as plotted in Fig. 4a. Several important features are collected and explained as follows.

IV.1.1 IV.A.1 Symmetry about $p=p_{c}$ when $\theta=0$

When SMT is absent ($\theta=0$), the SMT-QNM is nothing but two decoupled copies of CC-RNM with opposite chiralities meantime bearing opposite spin orientations. At all PSPs, When $p\rightarrow 0$ the quantum tunneling $t=\sqrt{1-p}$ defeats the reflecting amplitude $r=\sqrt{p}$ along equipotential lines. Hence all electron current loops around potential peaks become closed. On the contrary, when $p\rightarrow 1$ at PSPs the quantum tunneling gets weak and the reflecting along equipotential lines dominates. All electron current loops around potential valleys then become closed. Under PBC, these two cases are equivalent and both lead to NI phase. Between these two phases, $p_{c}=0.5$ ($P_{\mathrm{CC}}$ point in Fig. 4) is the quantum critical point, which can be obtained from the infinitesimal Migdal-Kadanoff transformation for real-space renormalization of CC-RNMArovas_1997 . While under RBC, different choices of marginal PSP nodes result in different boundary modes on network edges. In Fig. 5 we illustrate the case in which S’-type PSPs reside in boundaries thus a quantum doublet emerges on each edge leading to the QSH state when $p<p_{c}$.

In addition, under PBC the quantum phases on $\theta=0$ line are symmetric about $p=p_{c}$. There are two strategies to understand this symmetry. The first one comes from global considerations. To begin with, a given arbitrary random scalar potential profile (with statistical average being zero) is denoted as $\Sigma$. Then, following our sketch rules, we define $\mathbb{A}_{\Sigma\ominus}^{\uparrow(\downarrow)}(p)$ as the network composed of all solid closed loops around potential valleys with up(down) spin. They are inter-connected by dashed SPTs for $p>p_{c}(E_{F}<0)$, as shown in Fig. 1c. Since $\theta=0$, $\mathbb{A}_{\Sigma\ominus}^{\uparrow}(p)$ is decoupled from $\mathbb{A}_{\Sigma\ominus}^{\downarrow}(p)$, although they coincide with each other in real space. Similarly the network including all SPT-interconnected closed loops around potential peaks with up(down) spin for $p^{\prime}<p_{c}(E_{F}>0)$ are defined as $\mathbb{B}_{\Sigma\oplus}^{\uparrow(\downarrow)}(p^{\prime})$ (not shown in Fig. 1). Also $\mathbb{B}_{\Sigma\oplus}^{\uparrow}(p^{\prime})$ is unrelated to $\mathbb{B}_{\Sigma\oplus}^{\downarrow}(p^{\prime})$ in the absence of SMT. For any $p_{1}(>p_{c})$, by definition we have the following mappings under PBC,

[TABLE]

On the other hand, we define $-\Sigma\equiv\overline{\Sigma}$. Obviously, peaks (valleys) of $\overline{\Sigma}$ are valleys (peaks) of $\Sigma$, hence S-type (S’-type) PSPs of $\overline{\Sigma}$ are S’-type (S-type) PSPs of $\Sigma$. By symmetry, under PBC one has

[TABLE]

Then the mappings in Eq. (80) becomes

[TABLE]

Note that both $\Sigma$ and $\overline{\Sigma}$ are examples of “random scalar potential with zero statistical average”. Then naturally $G_{\mathrm{2T}}^{\mathrm{PBC}}$ and $\Lambda^{\mathrm{PBC}}$ are both statistically symmetric about $p=p_{c}$.

The second strategy focuses locally on each PSP, in which $T_{0}$ and $T^{\prime}_{0}$ (kernels of transfer matrices $T_{S}$ and $T_{S^{\prime}}$) are the main concern. At $\theta=0$, for an arbitrary $p(0<p<1)$, Eqs. (51) and (54) provide

[TABLE]

Then the following connections

[TABLE]

hold. A possible misunderstanding must be clarified here. The “$p\leftrightarrow 1-p$” mapping does not change the random scalar potential profile. S-type (S’-type) PSPs are always S-type (S’-type). What it really changes is the Fermi level of this system, i.e. from “$E_{F}(p)$” to “$-E_{F}(p)$” due to Eq. (77), since we have fixed the energy reference point to be zero. Under our sketch rules, at a S-type PSP, for “$p(>p_{c})$”, the electron flows are shown in Fig. 1c. For “$1-p$”, the valley-peak-distribution is unchanged but the electron flows change to those depicted in Fig. 1d. Now the PSP is still S-type and only its scattering matrix takes a similar mathematical format as a S’-type PSP. Bearing this in mind, the connection (84) actually means at a certain PSP, the transfer matrix at “$p$” in an original closed equipotential loop surrounding a potential valley (peak) is mathematically related to the transfer matrix at “$1-p$” in a mapped loop around a potential peak (valley). This is exactly what Eq. (80) tells us. Therefore mathematically S-type and S’-type PSPs exchange their roles in constructing the total transfer matrix. Hence under PBC, the total transfer matrix is unchanged, resulting in the symmetry about $p=p_{c}$.

IV.1.2 IV.A.2 Asymmetry about $p=p_{c}$ when $\theta>0$

When $\theta>0$ (SMT appears), an intermediate NM phase emerges between the two NI phases (PBC) or “QSH+NI” phases (RBC), as a manifestation of Wigner-Dyson symplectic ensembles. Numerical data in Fig. 4a clearly show that the phase diagram is asymmetric about $p=p_{c}$ line. This can also be explained by the global and local strategies introduced in the above subsection.

From the global strategy, the definitions of $\mathbb{A}_{\Sigma\ominus}^{\uparrow(\downarrow)}(p)$ and $\mathbb{B}_{\Sigma\oplus}^{\uparrow(\downarrow)}(p^{\prime})$ are unchanged. However, now $\mathbb{A}_{\Sigma\ominus}^{\uparrow}(p)$ is coupled with $\mathbb{A}_{\Sigma\ominus}^{\downarrow}(p)$ via SMT. The situation is similar for $\mathbb{B}_{\Sigma\oplus}^{\uparrow(\downarrow)}(p^{\prime})$. The mappings in Eq. (80) still hold. But the symmetry in Eq. (81) fails due to the SMT terms. Hence the final mappings in Eq. (82) do not exist, leading to the asymmetry about $p=p_{c}$ line when $\theta>0$.

From the local strategy, the general form of $T_{0}(p,\theta)$ and $T^{\prime}_{0}(p,\theta)$ are given in Eqs. (51) and (54). For $0<\theta\leq\pi/2$, generally $T_{0}(1-p,\theta)$ and $T^{\prime}_{0}(1-p,\theta)$ have no explicit connections with $T^{\prime}_{0}(p,\theta)$ and $T_{0}(p,\theta)$ as in Eq. (84). This also explains the asymmetry about $p=p_{c}$.

IV.1.3 IV.A.3 QSH phase on $p=0$

On the vertical $p=0$ line in Fig. 4a, Eq. (51) becomes

[TABLE]

which is irrelevant to $\theta$, meaning that the SMT has no effects on “left-to-right” transfer of electron flows. However from Eq. (54), $T^{\prime}_{0}(0,\theta)$ provides singularity since $r=\sqrt{p}=0$. This is due to the fact that when $p=0$, at S’-type PSPs in the bulk, terminals on the left-hand side are decoupled from those on the right-hand side, thus have no contributions to left-to-right transfer. All electron current loops around potential peaks then become completely closed. Under RBC, at boundary S’-type PSPs, the completely reflecting of electron flows results in dissipationless edge modes thus make the system fall into QSH phase.

IV.1.4 IV.A.4 One-to-one mapping between $p=1$ and $\theta=\frac{\pi}{2}$ lines

On $p=1$ line, one has

[TABLE]

While on $\theta=\pi/2$ line, the counterparts are

[TABLE]

If we perform the bijection

[TABLE]

between the two line segments $\left\{p=1,\theta\in(0,\frac{\pi}{2}]\right\}$ and $\left\{\theta=\frac{\pi}{2},p\in(0,1]\right\}$, then the following connections

[TABLE]

hold. Note although the unitary matrix “$-\sigma_{z}\otimes\sigma_{0}$” lies on different sides, its $\pi$-phases (originated from diagonal “$-1$” elements) can be absorbed into phase matrices $\mathcal{U}_{1}$ and $\mathcal{U}^{\prime}_{1}$, thus do not affect the mathematical role-reversal of S- and S’-type PSPs under bijection (88). Then $p=1$ and $\theta=\pi/2$ lines are equivalent and both fall into NI phase under PBC. Under RBC, on $\theta=\pi/2$ line dissipationless edge modes appear at boundary S’-type PSPs thus make the system fall into QSH phase. While for $p=1$ line, similar to Fig. 5b, closed electron-flow loops around potential valleys turn the system to NI state.

At last, at the phase point $(p,\theta)=(1,\frac{\pi}{2})$, which is the cross point of the above two line segments, one has

[TABLE]

The completely diagonal transfer matrices fully mix the up and down spins and meantime greatly enhance the itinerant range of electrons. Thus at this very point, the system becomes metallic.

IV.2 IV.B Mapping to phase diagram of $Z_{2}$-QNM

In fact, we can map our phase diagram (Fig. 4a) to a more symmetric one. However, before do that, it is interesting to point out that our phase diagram has close connection with that from the existing $Z_{2}$-QNM [see Fig. 8 and Fig. 11 in Ref. Obuse_2007_prb ]: under PBC, they are symmetric about the vertical $p=p_{c}$ line.

The reason is straightforward. By mapping the horizontal axis “$x$” in Ref. Obuse_2007_prb to the counterpart in this work “$p$” through $p=\tanh^{2}x$, we rewrite their Eq. (2.3) in terms of “$r$” and “$t$” as

[TABLE]

and

[TABLE]

By comparing them with the transfer matrix kernels $T_{0}$ and $T^{\prime}_{0}$ of our SMT-QNM, we have the following connections

[TABLE]

which are quite similar to Eq. (84). Therefore, similar discussions as in the end of Sec. IV.A.1 can be performed.

In Fig. 6 we illustrate a typical mapping starting from the SMT-QNM with $p>p_{c}(r>t)$. The main procedure is: (s1) $p\to 1-p$, or equivalently exchange $t$ and $r$; (s2) following our sketch rules, the electron flows are redrawn; (s3) by exchanging up and down spins, the S-type (S’-type) PSPs in SMT-QNM has the same electron-flow structure as the S’-type (S-type) PSPs in $Z_{2}$-QNM. Then it is understandable that under PBC by performing a mirror-symmetry operation on our phase diagram (Fig. 4a) about $p=p_{c}$ line, one gets the phase diagram of the $Z_{2}$-QNM. Note that in this mapping only the electron-flows are converted. The potential valleys and peaks are unchanged.

Based on this result, the critical exponent and normalized localization length at phase transitions should be the same as those from $Z_{2}$-QNM. This is confirmed by numerical calculations within error permissibility. To save space, we do not show this part of our data here. However, this close connection should not downgrade the significance of SMT-QNM constructed in this work. First, in our SMT-QNM, SMT process is an additional tunneling channel and does not take probability away from the existing SPT channel, which is a more physical assumption. Second, the symmetry about $p=p_{c}$ line between these two phase diagrams indicates a possible way to check which network model provides better description to real 2D-DSEGs. From Eq. (77), $p$ is directly related to system Fermi level. By sweeping the Fermi level and check out the quantum phase a 2D-DSEG falls in, experimentally one can make reasonable judgment. Third, as will be shown next, the phase diagram of SMT-QNM can be topologically transformed to a symmetric one which is highly similar to the phase diagram of disordered 3D weak TIs. This enriches the possible applications of our 2D SMT-QNM.

IV.3 IV.C Mapping to a symmetric phase diagram

The asymmetry of phase diagram in the original $(p,\theta)$ phase space is unfavorable for a deep understanding of ATs in 2D-DSEGs. Fortunately, its features summarized in Sec.VI.A provide us enough information to topologically transform it to a completely symmetric one. Mathematically, the following mapping

[TABLE]

perfectly achieves this target:

(a) The original phase space “$\Omega_{1}$” [see Eq. (79)] is mapped to a new phase space

[TABLE]

(b) The original $\theta=0$ line is mapped to $Y=0$ line with the one-to-one correspondence $p\leftrightarrow X$, hence $Y=0$ line is symmetric about the vertical line $X=X_{c}(\equiv p_{c})$;

(c) The original $p=0$ line shrinks to a single point $(X,Y)=(0,0)$;

(d) The metallic phase-point $(p,\theta)=(1,\frac{\pi}{2})$ stretches itself to $Y=1$ line, which is indeed a singularity of the mapping in Eq. (94);

(e) The $\theta=\frac{\pi}{2}$ and $p=1$ lines are mapped to $X=0$ and $X=1$ lines, respectively. The combination of mappings in Eqs. (88) and (94) generates a one-to-one correspondence of phase points on $X=0$ and $X=1$ lines with the same $Y$. Therefore these two lines are completely symmetric about $X=X_{c}$ line.

(f) For an arbitrary point $(X,Y)$ in the interior region of $\Omega_{2}$, one has

[TABLE]

and

[TABLE]

Then the following connections

[TABLE]

indicate the symmetry of mapped phase diagram about $X=X_{c}$ line in $\Omega_{2}$.

Following the mapping in Eq. (94), we transform the phase diagram in $\Omega_{1}$ into the one in $\Omega_{2}$ which is plotted in Fig. 4b. Obviously, the new phase diagram looks better. However, it is not completely symmetric about $X=X_{c}$ due to the finite-size effect during our calculation, since we only perform calculations on normalized localization length to $W=2^{5}$ limited by our existing computing capability. It is expected that when $W$ is sufficient large, the symmetry in $\Omega_{2}$ should be more apparent.

The results in this subsection have several potential applications. First, phase boundaries in $\Omega_{1}$ can be double-checked through the mapping (94) and its inversion, since in $\Omega_{2}$ phase boundaries should be symmetric about $X=X_{c}$. Second, the narrow and long NI (or QSH) phase in the close vicinity of $(p,\theta)=(1,\frac{\pi}{2})$, which is hard to precisely determined due to strong symmetry-crossover effects, is enlarged a bit in $\Omega_{2}$. This should be helpful for better determination of NM-NI (QSH) boundaries.

IV.4 IV.D Connection with disordered 3D weak TIs

In addition, the new phase diagram (Fig. 4b) shows apparent similarity with that of disordered 3D weak TIs (see Fig. 1 in Ref.Moore_PRL_2012 ), indicating a close connection between 2D-DSEGs described by our SMT-QNM and the helical surface modes of 3D weak TIs under scalar disorder potentials respecting TRS. Comparing our 2D Dirac Hamiltonian [see Eq. (48) in this work] and the effective Hamiltonian in Ref.Moore_PRL_2012 [see Eqs. (1)-(3) therein], the energy gap of the system is $2|m|$ with $m=1-\frac{p}{p_{c}}=1-\frac{X}{X_{c}}$. For clean limit, $V_{00}=A_{0}=0$ meaning on $Y=0$ line in Fig. 4b, the intermediate metallic region shrinks to a single critical point $X=X_{c}$. In the presence of disorder which couples the two Dirac fermions (with mass $\pm m$) with strength $V_{x0}=\sqrt{2}\theta$, direct transitions between the insulating phases (NI and QSH) are forbidden due to the stability of the symplectic metal, which results in the finite width of intermediate metallic phase. In addition, the disorder strength

[TABLE]

in the vicinity of the critical point $(X,Y)=(X_{c},0)$. All these correspondences confirm the close connection we mentioned at the beginning of this section. This implies the possible application of our SMT-QNM on investigations of disordered helical surface modes of 3D weak TIs. For $Z_{2}$-QNM, similar works have been done systematicallyObuse_2014_prb . For our SMT-QNM, this is an interesting direction but out of the scope of this work.

V V. Quantum phases and phase transitions in TRS-breaking SMT-QNM

The TRS-preserving SMT-QNM introduced above can be downgraded to the counterpart which still preserves TRS at PSPs but breaks it in the links between PSPs. Physically, this corresponds to 2D-DSEGs with TRS-breaking (usually called magnetic) isotropic impurities. These impurities inevitably affect the random potential profile, however will not create PSPs at their very locations due to the isotropic nature, thus can be described by the TRS-breaking SMT-QNM. In these systems, spin-flip backscattering on each link between PSPs emerges thus destroys the original Krammer’s doublet. For modelization, this can be simply realized by neglecting the “phase pairing rule” in Eq. (61), meanwhile leaving the rest unchanged. Here we briefly summarize our data and provide reasonable explanations.

V.1 V.A Phase diagram

Now the system falls into Wigner-Dyson unitary class (TRS fails, regardless of SRS) and generally no intermediate NM phase exists. This is confirmed by finite-size analysis on $G_{\mathrm{2T}}$ and $\Lambda$. The resulting phase diagram is plotted in Fig. 7 and totally different from the TRS-preserving SMT-QNM. Our data show that QSH state only survives on the line segment $\{0\leq p<0.5,\theta=0\}$ if we choose S’-type PSPs as boundary nodes. When SMT emerges, there is no intermediate NM phase. In the entire phase space $\Omega_{1}$, NI phase dominates.

V.2 V.B The NI phase

The TRS-breaking in links connecting PSPs will turns both NM and QSH phases (except for the segment on $\theta=0$ line) into NI phase, which is the typical behavior of unitary ensembles. To check for this, first we perform numerical calculations of $G_{\mathrm{2T}}$ under RBC for enough dense grid of the phase space $\Omega_{1}$. For all phase points, the network size $W(=L)$ increases from $2^{2}$ to $2^{9}$. Further enlargement of $W$ is out of our computing capability. The sample number is always 128, which is enough to provide sufficient small error. To save space, we summarize the main features and present typical data, if necessary. First, for all phase points in $\Omega_{1}$, $\left\langle G_{\mathrm{2T}}^{\mathrm{RBC}}\right\rangle$ is smaller than 1 and decrease with $W$ for sufficient large $W$ without sign of convergence. Obviously this can not be QSH state. In addition, we have known that in NM phase (if exists) of systems with unitary symmetry, $\left\langle G_{\mathrm{2T}}^{\mathrm{RBC}}\right\rangle$ should converge to the Boltzmann conductance $G_{0}(\gg 1)$Hikami_1980 . Hence our data clearly show that the system falls into neither QSH nor unitary metallic phase. The only possibility is the NI phase.

Next, the finite-size calculations of the normalized localization length $\Lambda$ for $\theta=0.1\pi$ and $\theta=0.3\pi$ are performed and the data are plotted in Fig. 8. In these calculates, RBC in transverse direction is imposed and the sample number is chosen to be 32. In addition, the relative standard error of the first LE is set to be $1\%$. The numerics clearly shows that for both SMT strengths, $\Lambda$ always decreases with the network width $W$. This confirms that along $\theta=0.1\pi$ and $\theta=0.3\pi$ lines in $\Omega_{1}$, the system falls into NI phase. Similar calculations have been performed for other nonzero $\theta$ values. All results support our conclusion that NI phase fills up $\Omega_{1}$.

V.3 V.C Direct transition from QSH to NI phases

The QSH phase on line segment $\{0\leq p<0.5,\theta=0\}$ is absolutely unstable to SMT. This means no matter how small the SMT is, the QSH state will be destroyed completely. To see it, the point X ($p=0.3,\theta=0$) in Fig. 7 is selected as an example. We set $W=L$ and vary $W$ from $2^{1}$ to $2^{9}$. In the close vicinity of point X, $\left\langle G_{\mathrm{2T}}^{\mathrm{RBC}}\right\rangle$ and the corresponding error of 128 independent configurations are calculated and plotted in Fig. 9. For point X, numerical data (hollow squares in Fig. 9) show that when the system size increases to $W=2^{9}$, $\left\langle G_{\mathrm{2T}}^{\mathrm{RBC}}\right\rangle$ approaches the quantized value 2, with the standard error as small as $5.6\times 10^{-9}$. This validates that point X belongs to the QSH phase. Next we perform calculations for $\theta=0.01$. The result is shown in Fig. 9 by solid magenta squares. As the system size gets larger, $\left\langle G_{\mathrm{2T}}^{\mathrm{RBC}}\right\rangle$ falls to $10^{-3}$ or even smaller. We then gradually approach the point X by decreasing $\theta$ by an order of magnitude and calculate the corresponding $\left\langle G_{\mathrm{2T}}^{\mathrm{RBC}}\right\rangle$ until $\theta$ reaches $1.0\times 10^{-6}$. The results are plotted in Fig. 9, showing that as $\theta$ decreases, the deviation of $\left\langle G_{\mathrm{2T}}^{\mathrm{RBC}}\right\rangle$ from quantized value 2 gets weaker at $W=L=2^{9}$. However, it always exists and has no sign of convergence. Even for $q=1.0\times 10^{-6}$ (solid black squares), if the system size is further increased to $2^{10}$, $\left\langle G_{\mathrm{2T}}^{\mathrm{RBC}}\right\rangle$ deviates from 2 evidently. Limited by computing capability, we can not perform calculations to the system size at which $\left\langle G_{\mathrm{2T}}^{\mathrm{RBC}}\right\rangle$ falls to zero. However, the data in Fig. 9 clearly imply that QSH state can not survive when SMT emerges, no matter how small it is.

For cross validation, we also perform finite-size calculations for the normalized localization length $\Lambda$, in which RBC is adopted and the sample number is 16. In addition, the relative standard error of the first LE is $1\%$, leading to the network length $L\sim 10^{6}$. The network width $W$ increases from $2^{2}$ to $2^{5}$ and the SMT strength $\theta$ varies from [math] to $0.01$ with the step $\mathrm{d}\theta=0.001$. The resulting data are plotted in Fig. 10. It is clear that $\Lambda$ for $\theta=0$ always increases with $W$. This comes from the dissipationless edge modes and confirms the fact that point X belongs to QSH phase. On the other hand, when $\theta\geq\mathrm{d}\theta$, $\Lambda$ eventually decreases as $W$ increases to $2^{5}$. This validates the conclusion based on data from $\left\langle G_{\mathrm{2T}}^{\mathrm{RBC}}\right\rangle$ that $(p=0.3,\theta\geq\mathrm{d}\theta)$ falls into NI phase. Further increase of $W$ and decrease of $\mathrm{d}\theta$ are beyond our present computing capability. However, the data in Fig. 10 already provide enough cross-validation evidences of the absolute instability of QSH state.

This result can be understood by the physical process sketched in Fig. 11. Initially the system is in QSH phase, i.e. the situation depicted in Fig. 5a. For simplicity, we take the part close to the upper edge as an example and redraw it in Fig. 11. We focus on an electron with up spin propagating in the dissipationless left-to-right edge channel. Suppose at some moment, the electron is at point A which is set as the starting point. When SMT is absent, the electron at most tunnels into the closed loops with up spins via SPT and can not fall into trajectories associated with down spins. Hence the electron will never be backscattered into the right-to-left edge channel with down spin at the upper edge. On the other hand, the backscattering into the right-to-left edge channel with up spin at the lower edge (not depicted in Fig. 11) by means of multi-SPTs through closed spin-up loops will be suppressed when the network is wide enough since this is a high-order process. When SMT emerges, the situation is completely different. When a spin-up electron propagates from point A and reaches point B, SMT at this S-type PSP allows it to tunnel into the closed loop associated with down spins (point C on the red loop). After circling this loop (C to D to E), the electron comes back to this PSP and tunnels into the right-to-left edge channel (point F) with down spin via SPT and then go to point G and even leftward. Now we realize a spin-flip backscattering event (A$\rightarrow$B$\rightarrow$C$\rightarrow$D$\rightarrow$E$\rightarrow$F$\rightarrow$G) which includes only one step of SMT. Therefore this process is not a high-order one and should take effect as long as SMT appears. Combing with the fact that a number of S-type PSPs distribute along the upper edge, it is understandable that the QSH state should be absolutely unstable with respect to SMT. Therefore,* the QSH line segment acts as a critical line rather than a phase boundary*. Hence the critical exponent for this direct transition can hardly be extracted out using the standard finite-size scaling procedureSlevin_prl_1999 .

V.4 V.D Revisit of the CC-RNM critical point $(p_{c},0)$

In the end of this section, we turn back to the CC critical point $(p,\theta)=(p_{c},0)$, where the network decouples into two copies of CC-RNM with opposite chiralities. As mentioned above, the infinitesimal Migdal-Kadanoff transformation for real-space renormalization of CC-RNM provides that it is the quantum critical point that separates two insulating (NI and QSH) phases in the bulk. However, “whether or not a NM phase exists in a finite range around the quantum critical point $p=p_{c}$ along $\theta=0$ line” is still a controversial issue. Here we present numerical data within error permissibility and within the scope of our computing capability to give a reasonable estimation about the width of this NM phase, if exists.

To begin with, we calculate $\left\langle G_{\mathrm{2T}}\right\rangle$ under both RBC and PBC in the vicinity of $p=p_{c}$ along $\theta=0$ line. The results are shown in Fig. 12a and 12b, respectively. In these calculations, $W(=L)$ varies from $2^{3}$ to $W_{\mathrm{max}}=2^{7}$ and the step of $p$ is $\mathrm{d}p=0.01$. Meantime, $N=10^{5}$ independent samples are generated for acceptable averages. Generally, $\mathrm{d}p$ is limited by our computing capability ($W_{\mathrm{max}}$ and $N$) hence can not be arbitrarily small. In this sense, we can only detect the existence of NM phase at the level of $\mathrm{d}p$. For RBC, $\left\langle G_{\mathrm{2T}}^{\mathrm{RBC}}\right\rangle$ is an increasing function of $W$ when $p\leq p_{c}$. When $p\geq p_{c}+\mathrm{d}p$, $\left\langle G_{\mathrm{2T}}^{\mathrm{RBC}}\right\rangle$ decreases for sufficient large $W$. For PBC, $\left\langle G_{\mathrm{2T}}^{\mathrm{PBC}}\right\rangle$ is an increasing function of $W$ when $|p-p_{c}|\leq\mathrm{d}p$, at least for $W\leq W_{\mathrm{max}}$. While if $|p-p_{c}|\geq 2\mathrm{d}p$, $\left\langle G_{\mathrm{2T}}^{\mathrm{PBC}}\right\rangle$ eventually decreases at sufficient large $W$. These results imply that the NM phase at most appears in $(p_{c}-\mathrm{d}p,p_{c}+\mathrm{d}p)$, if exists.

To further check this conclusion, we calculate the normalized localization length $\Lambda$ and the data are plotted in Fig. 12c. In these calculations, several points should be clarified. First, PBC is adopted to eliminate the effect of dissipationless edge states in QSH phase. Second, the relative standard error $\epsilon_{1}$ of the first LE is set to be $0.5\%$, which results in the strip length $L\sim 10^{7}\gg\xi_{W}$ for $W=2^{7}$. Third, the sample number is 16 and proved to be enough for sufficiently small error bars. The data in Fig. 12c clear show that only at $p=p_{c}$, $\Lambda$ is an increasing function of the width $W$ (at least for $W\leq W_{\mathrm{max}}$). For other $p$ satisfying $|p-p_{c}|\geq\mathrm{d}p$, $\Lambda$ decreases for sufficient large strip width. These results validate the fact that NM-phase is only possible to reside in $|p-p_{c}|<\mathrm{d}p$, if exists, in the absence of SMT. Further verifications need smaller $\mathrm{d}p$, greater network width $M_{\mathrm{max}}$ and sample number $N$, which are out of our present computing capability.

VI IV. Conclusions

In this work, we have constructed the symplectic SMT-QNM by recognizing the SMT as an independent tunneling channel. By leading-order expansion method, the 2D Dirac Hamiltonian is extracted out from SMT-QNM in the close vicinity of CC-RNM critical point, with the SMT strength associating with the spin-flip coupling. A sandwiched (QSH-NM-NI) phase diagram in original phase space $\Omega_{1}$ is then obtained by finite-size analysis of two-terminal conductance and normalized localization length. It is first mapped to the phase diagram of the existing $Z_{2}$-QNM, and then closely related to the counterpart of disordered 3D weak TIs. In the end, the TRS-breaking (in the links between PSPs) version of SMT-QNM is considered and turns out to fall into unitary class. Its phase diagram is filled by NI phase except for a marginal line segment still hosting QSH phase. A direct transition from QSH to NI phases exists and is explained by the SMT-induced spin-flip backscattering.

VII Acknowledgement

We thank Prof. X. R. Wang and Prof. G. Xiong for fruitful discussions. This work is supported by the Science Foundation for The Excellent Youth Scholars of Educational Commission of Hebei Province, China (Grant No. Y2012027) and the Natural Science Foundation of Hebei Province, China (Grant No. A2014205080). J.L also acknowledges the support from the National Natural Science Foundation of China (Grant Nos. 11374088 and 11104060). B.Xi acknowledges the support from the National Natural Science Foundation of China (Grant No. 11774300).