You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: 'In this Communication, we investigate a toy model of three-dimensional topological insulator surface, coupled homogeneously to a fictitious pseudo spin-$\frac{1}{2}$ particle. We show that this toy model captures the interesting features of topological insulator surface states, which include topological quantum phase transition and quantum spin hall effect. We further incorporate an out-of-plane magnetic field and obtain the Landau levels.'
author:
- 'S. A. Owerre'
- 'J. Nsofini'
title: A toy model for quantum spin Hall effect
---

[*Introduction*]{}.– In recent years, topological insulators (TIs) have captivated considerable attention of researchers [@yu; @hk; @hk1; @hk2; @zhang; @km]. These fascinating materials involve compounds such as Bi$_2$Se$_3$ or Bi$_2$Te$_3$, whose electronic bulk structure is an insulator with a finite gap separating the conduction band and the valence band, but their edges (for 2D TIs) or surfaces (for 3D TIs) have gapless states, which are protected by time-reversal symmetry $(TRS)$. These states are robust to perturbations that do not break this symmetry. In recent years, these gapless states have been the focus of interest due to the simple Dirac-like Hamiltonian which describes them. However, breaking $TRS$ introduces many interesting phenomena; this can be achieved by depositing a ferromagnet or a superconductor on the surface of a TI. Thus, the topologically protected surface state develops a gap, which leads to interesting electronic transport, which include half quantized conductivity, Majorana bound states, etc.[@yu; @bur1; @franz; @sol3; @dim; @kane; @zhang1; @eze]. They also have potential technological applications in the field of spintronics[@zhang2; @take].

In this Communication, we study a toy model that captures some of the interesting features of surface states in topological insulators. Specifically, we study a fictitious pseudo spin-$\frac{1}{2}$ particle interacting homogeneously with the surface of a TI. Assuming anisotropic “in-plane” and “out-of-plane” exchange interactions on the interface, say $\lambda_{\parallel}$ and $\lambda_{\perp}$, the coupled system exhibits a topological quantum phase transition as a function of $\xi=\lambda_{\perp}/\lambda_{\parallel}$. This phase transition is associated with a quantum phase transition point at $\xi=1$, which separates two regions; $\xi<1$, with two topologically protected Dirac points (semimetallic phase) and $\xi>1$, which is fully gapped (quantized spin Hall phase). We further show that when the $z$-component of the pseudospin is conserved, the system decouples into two Hamiltonians which are related by $TRS$. The coupling term opens two gaps at the $\Gamma$ point on the interface, which are degenerate but with opposite spin orientations. Each gap opening gives rise to a half quantized Hall conductivity. Upon applying an electric field, an opposite spin current is induced on each interface leading to a quantized spin Hall conductivity. Furthermore, we introduce a non-conserving term in the quantized spin Hall Hamiltonian by breaking the conservation of the $z$-component of the pseudospins. This leads to the obliteration of the quantized spin Hall conductivity. However, in this non-conserving regime we obtain a nontrivial topological spin Chern number by diagonalizing the projection of the pseudospin onto the occupied valence bands [@pro].

[*Simplest Single Dirac Point*]{}.– Most of the interesting physics of $2D$ metals are captured by investigating the robustness of the band touching points (nodes). The bulk energy band near these points usually replicates a $2D$ Dirac Hamiltonian; thus these nodes are called Dirac points. The simplest form of Dirac point can be mapped out from this simple Hamiltonian: H=v\_F(), \[8\] where $v_F$ is the Fermi velocity, $\sigma_i, i=x,y,z$ are the Pauli matrices representing the real spins of the surface states. The $z$-component is taken to be perpendicular to the plane of the $2$D sample, and $\bold k=(k_x,k_y)$ is the $2$D surface Brillouin zone momentum. This Hamiltonian describes many known physical systems, such as the surface of a $3$D topological insulator [@zhang], and the Dirac point Hamiltonian in graphene in which the Pauli matrices are pseudospins [@net0]. Evidently, this Hamiltonian is invariant when the spin and momentum go to minus of their original values; in other words, the Hamiltonian possesses $TRS$, where the time reversal operator is given $\Theta= i\sigma_y\mathcal{K}$; $\mathcal{K}$ is a complex conjugation. The energy spectrum is trivial and given by $\mathcal{E}_s=sv_F\sqrt{k_x^2+k_y^2}$; $s=\pm $. It is apparent that the two bands touch each other at $\bold k=0$. The robustness of this Dirac point is guaranteed by any perturbation that preserves the $TRS$ of Eq.. By breaking $TRS$, the degeneracy of the band might be lifted and many interesting phenomena can emerged. There are several terms that break $TRS$; for instance, $\Delta_x\sigma_x$ and $\Delta_y\sigma_y$ do break $TRS$; they correspond to Zeeman magnetic field contributions along the $x$ and $y$ directions respectively. However, addition of these terms to Eq. simply shifts the position of the Dirac points to $\bold k=(0, -\Delta_x/v_F)$ and $\bold k=(\Delta_y/v_F, 0)$ respectively. Thus, the degeneracy is not lifted. The only $TRS$ breaking term that lifts the degeneracy of the bands is the direct coupling of the surface electron spins to the Zeeman magnetic field term perpendicular to the plane, [*i.e.,*]{} $\Delta_z\sigma_z$. This term can also be generated by depositing a ferromagnet on the surface electrons [@franz; @kane; @sol3]. It is evident that this contribution opens a gap of size $2|\Delta_z|$ at $\bold k =0$. Consequently, when the Fermi energy lies between the gap, this leads to a half-quantized Hall conductivity [@res; @zhang]. In graphene, however, there are an even number of Dirac points. In this case, the charge Hall conductivity vanishes in the ordinary insulating state [@neil; @adm; @hk].

[*Toy Model*]{}.– As mentioned in the preceding sections, the surface of a $3$D topological insulator possesses many interesting features when $TRS$ is explicitly broken. In this section, we will consider a $3$D topological insulator whose surface is homogeneously coupled to a fictitious pseudospin-$\frac{1}{2}$ particle; the low-energy effective Hamiltonian can be written as $$\begin{aligned}
\mathcal{H}=v_F(\hat{z}\times \boldsymbol{\sigma})\cdot\bold{k}-\lambda_{\parallel}\tau_x\sigma_x -\lambda_{\perp}\tau_z\sigma_z,
\label{appen1}\end{aligned}$$ where $\sigma_i$ and $\tau_i$, ($i=x,y,z$) are the Pauli matrices acting on the topological insulator space and the fictitious pseudospin-$\frac{1}{2}$ particle space respectively. The coupling constants $\lambda_{\parallel}$ and $\lambda_{\perp}$ are the anisotropic in-plane and out-of-plane homogeneous exchange interactions. In most cases of physical interest, such interaction is usually inhomogeneous, [*i.e.*]{}, the last two terms in Eq. should contain a delta function. Thus, Eq. does not describe any known physical system. However, it is possible that it might be applicable to a pseudo quantum qubit, but we are not interested in any specific system, instead we will consider it as a toy model that captures some of the interesting physics of TI surface states. Although our model does not describe any known physical system, it possesses some features that are similar to other models that describe known systems. This is the main purpose of this Communication. We will consider Eq. as the basis of our investigation in this Communication. It is apparent that Eq. explicitly breaks $TRS$, [*i.e* ]{}, $[\Theta, \mathcal{H}]\neq 0$, where $\Theta= \tau_x\otimes i\sigma_y\mathcal{K}$. Due to the fictitious pseudospin-$\frac{1}{2}$ particle, Eq. is obviously a $4$-band model.

[*Topological Quantum Phase Transition*]{}.– Most known systems such as graphene and thin film topological insulators possess topological quantum phase transition. This is manifested at the gap closing point which separates an ordinary insulator from a quantized Hall insulator [@phaset; @burr3]. These phases are usually distinguished by a topological invariant quantity called the Chern number. It is generally defined as [@vol] $$\begin{aligned}
 \mathscr{C}= \frac{1}{24\pi^2}\epsilon_{\alpha\beta\gamma}\text{Tr}\int d^3k\mathcal{G}\partial_{k_{\alpha}}\mathcal{G}^{-1}\mathcal{G}\partial_{k_{\beta}}\mathcal{G}^{-1}\mathcal{G}\partial_{k_{\gamma}}\mathcal{G}^{-1},\end{aligned}$$ where $\epsilon_{\alpha\beta\gamma}$ is the totally antisymmetric tensor, $\alpha= 0, x,y,z$, etc., labels the components of a four-vector, and the Matsubara Green’s function can be written as \^[-1]{}(i\_n, )= i\_n\_[44]{}-.

The identity $\mathbf{I}_{4\times 4}$ is a $4\times 4$ matrix and $k_\alpha=(i\omega_n,\bold{k})$ is the momentum four-vector. In principle the Chern number can be computed for any model of interest. One finds that it has a unique value in each phase, which is immutable by any smooth deformation of the system (provided the gap does not close). In general, the emergence of quantum phase transition in electron systems requires the violation of CPT symmetry [@vol], [*i.e.*]{}, charge conjugation, parity, and time-reversal symmetries. It is evident that our model in Eq. explicitly breaks parity and time-reversal symmetries. Thus it can capture quantum phase transitions similar to those predicted in known systems. In order to see this, we diagonalize Eq. and find that the eigenvalues are given by $$\begin{aligned}
\mathcal{E}_{s\eta}=(-1)^{\eta}\sqrt{v_F^2k_x^2 +\lb \sqrt{v_F^2k_y^2 + \lambda_{\perp}^2}+ s\lambda_{\parallel} \rb^2},\end{aligned}$$ where $s=\pm$ and $\eta=0,1$. There are two energy sectors $\mathcal{E}_{+\eta}$ and $\mathcal{E}_{-\eta}$. In each sector there are two bands with $\eta=0$ being the conduction band and $\eta=1$ being the valence band.

![Color online. The phase transition in $(\xi,k_y)$ space with $v_F=1=\lambda_{\parallel}$. The quantum phase transition point $\xi=1$ separates two regions; $\xi<1$, with two Dirac points and $\xi>1$, which is fully gapped; see Fig..[]{data-label="Dirac"}](Dirac.pdf){width="2.5in"}

The $\mathcal{E}_{-\eta}$ bands have two Dirac points ($\mathcal{E}_{+\eta}$ bands are always gapped) at $\pm\bold{K}$, where = 0, ; =\_/\_. \[wav\] The topological quantum phase transition as a function of $\xi$ can be understood as follows. At the point $\xi = 0$, the system is a semimetal with two Dirac points separated by the wave vector $v_Fk_y= 2\lambda_{\parallel}$ along the $y$-direction. The semimetallic phase is controlled by the in-plane exchange interaction. Provided $\xi <1$, these two Dirac points remain intact, separated in momentum space by a wave vector $v_Fk_y= 2\lambda_{\parallel}\sqrt{1-\xi^2}$. They are topologically protected and cannot be eliminated by changing the parameters of the Hamiltonian. However, as $\xi$ is increased, the two Dirac points remain stable and subsequently annihilate each other at the phase transition point $\xi =1$, and emerge as a single Dirac point at the $\Gamma$ point $\bold{K}=(0,0)$. The gap further reopens as one moves to the region $\xi >1$ and the system becomes an insulator with a gap of $2\lambda_{\parallel}(\xi-1)$ at $\bold{K}=(0,0)$. As shown in Fig. and Fig., the quantum phase transition point $\xi=1$ separates two regions; $\xi<1$, with two Dirac points and $\xi>1$, which is fully gapped. The homogeneous in-plane coupling $\lambda_{\parallel}$ plays a similar role as the parallel magnetic field in a thin film topological insulator [@burr3].

![Color online. The energy bands along the $k_y$ direction for different ranges of the parameters; see text for explanation.[]{data-label="phase"}](phase){width="3.5in"}

[*Quantum Spin Hall Effect*]{}.– Quantum spin Hall effect is known to be manifested in many known systems [@km; @And; @bat; @hai1; @hui1]. In this section we will show how the toy model in Eq. captures this interesting physics. Let us consider the limit $\lambda_{\parallel}\to 0$. In this limit there are no Dirac points, the system is fully gapped. The Hamiltonian decouples into two copies which are related by $TRS$: $$\begin{aligned}
\mathcal{H} =
 \begin{pmatrix}
  \mathcal{H}_{\uparrow} & 0  \\
 0 & \mathcal{H}_{\downarrow}
 \end{pmatrix},
 \label{eqn7}
 \end{aligned}$$ where \_[,]{}= v\_F(k\_y\_x-k\_x\_y) \_z, \[ham\] and $\lambda=\lambda_\perp$. The eigenvalues are pair degenerate ($\tau_z$ is conserved), given by

$$\begin{aligned}
\mathcal{E}_{s}&=s\sqrt{ v_F^2k^2 +\lambda^2}=s\epsilon_k,
\label{12} \end{aligned}$$

![Color online. Schematic diagram of the two Kramers energy bands on the interface with a gap of $2\lambda$ at the $\Gamma$ point.[]{data-label="cone"}](cone.pdf){width="2.5in"}

where $k=\sqrt{k_x^2+k_y^2}$. The corresponding eigenspinors are found to be $$\begin{aligned}
 \psi_s^{\uparrow}&=  {\chi_s(\lambda) \choose 0} \quad\text{and} \quad \psi_s^{\downarrow}=  {0\choose \chi_s(-\lambda)},
  \end{aligned}$$ where $\chi_s(\lambda) = \frac{1}{\sqrt{2}}\lb\sqrt{1+s\frac{\lambda}{\epsilon_k}},-is e^{i\theta_k}\sqrt{1-s\frac{\lambda}{\epsilon_k}}\rb^T$ and $\theta_k =\text{tan}^{-1}\lb k_y/k_x \rb$. Fig. shows the Kramers pairs of the energy bands for spin-up and spin-down, with an energy gap of $2\lambda$ at the $\Gamma$ point $(k_x=0, k_y=0)$. At zero temperature, and when the Fermi energy lies between the gap, the conduction band is empty and the valence band is full; the Hall conductivity is given by [@thou]

\_[xy]{}\^[\_z]{}= \^[\_z]{}, \[thou0\] where $$\mathscr{C}^{\tau_z}=\frac{1}{2\pi}\int d^2k\mathcal{F}_{\tau_z},
\label{thou1}$$ is a topological invariant Chern number, $\mathcal{F}^{\tau_z}=\lb\boldsymbol{\nabla}\times \bold{A}^{\tau_z}\rb_z$ is the Berry curvature and $\bold{A}^{\tau_z}=i\braket{\psi_{-}^{\tau_z}|\boldsymbol{\nabla}|\psi_{-}^{\tau_z}}$ is the Berry connection. Expressing the differential operator in polar coordinates yields: \_[\_z]{}= -, \[becu\] where $A_{\theta_k}=\frac{1}{k}i\braket{\psi^{\tau_z}_-|\partial_{\theta_k}\psi^{\tau_z}_-}$ and $A_{k}=i\braket{\psi^{\tau_z}_-|\partial_{k}\psi^{\tau_z}_-}$. Since $\theta_k$ appears as a $U(1)$ phase in the eigenspinors, $A_{\theta_k}$ and $A_k$ are independent of $\theta_k$, thus the second term in Eq. vanishes and the resulting expression becomes [@hui]: $$\mathcal{F}^{\tau_z}= \frac{1}{k}\frac{\partial}{\partial k}\mathcal{M}^{\tau_z}(k); \quad \mathcal{M}^{\tau_z}(k)=i\braket{\psi^{\tau_z}_-|\partial_{\theta_k}\psi^{\tau_z}_-},
\label{thou2}$$

Integrating Eq. using Eq. yields a half quantized Hall conductivity at each gap opening: \_[xy]{}\^[\_z]{}= ()(). \[hallc\] Evidently, the Hall conductivity is coupled to the fictitious pseudospin orientations; see Fig.. Thus, an application of an electric field will induce an opposite fictitious spin current. The charge-Hall conductivity vanishes due to $TRS$: \_[H]{}\^c=\_[xy]{}\^+\_[xy]{}\^=0, while the resulting half quantized spin Hall conductivity is nonzero: \_H\^s=\_[xy]{}\^-\_[xy]{}\^= -. \[shf\] It should be noted that this spin Hall conductivity is not measurable. Physical known systems in which spin Hall effect has been realized include HgTe/CdTe semiconductor quantum wells [@bat], and the ultrathin film topological insulator [@hai1]. Thus our coupling constant $\lambda$ plays a similar role as the coupling constant in these systems.

[*Effects of Non-Conserving Terms and Magnetic Field*]{}.– Our analysis for the quantum spin Hall effect in the previous section relied on the fact that the $z$-component of the fictitious spin $1/2$ system is conserved. We will now investigate the effect of adding a non-conserving term to the Hamiltonian. There are many ways to do this, but we will choose the most plausible way. We assume that the fictitious particle is a two-level system with degenerate minima. Tunneling obviously breaks this degeneracy, which introduces dynamics into the system and thus breaks the conservation of the $z$-component of the spin. This dynamics effectively corresponds to a term of the form $\Delta_x\tau_x$. Thus, the model in Eq. then becomes: $$\begin{aligned}
\mathcal{H} =
 \begin{pmatrix}
  \mathcal{H}_{\uparrow} & \Delta_x  \\
 \Delta_x & \mathcal{H}_{\downarrow}
 \end{pmatrix}.
 \label{11a}
 \end{aligned}$$ In this Hamiltonian, the off-diagonal term is akin to structure inversion asymmetry potential in a thin film topological insulator [@hui1]. Diagonalizing this Hamiltonian we find that the eigenvalues are given by $$\begin{aligned}
\mathcal{E}_{s\eta}&=(-1)^{\eta}\sqrt{\lb v_Fk +s\Delta_x\rb^2 +\lambda^2}.
\label{12a} \end{aligned}$$

The corresponding eigenspinors for the conduction band ($\eta=0$) and the valence band ($\eta=1$) are given respectively by $$\begin{aligned}
\chi_{s}^{\eta=0}=\lb -is e^{-i\theta_k} u_{s\eta +}, -su_{s\eta-},-ie^{-i\theta_k}u_{s\eta-}, u_{s\eta+}\rb^{T},\end{aligned}$$ $$\begin{aligned}
\chi_{s}^{\eta=1}=\lb -is e^{-i\theta_k} u_{s\eta -}, -su_{s\eta+},ie^{-i\theta_k}u_{s\eta+}, u_{s\eta-}\rb^{T},
\label{vale}\end{aligned}$$ where u\_[s]{}= .

Since $\tau_z$ is no longer conserved, the Hall conductivity in Eq. is obliterated. However, in the regime $\Delta_x/\lambda \ll 1$, one can still define a pseudo spin Chern number by diagonalizing the operator $\mathcal{P}\tau_z\mathcal{P}$ [@pro; @hui], where $\mathcal{P}=\sum_{s=\pm}\chi_{s}\chi^{\dagger}_s$ is the projection operator onto the occupied valence bands in Eq.. The matrix elements are expressed as $\braket{\chi_{s}|\tau_z\otimes\mathbf{I}_\sigma|\chi_{s^{\prime}}}$; $s,s^{\prime}=\pm$, and the eigenvalues of this matrix are given by $\omega_{\pm}(k)=\pm|\rho(k)|$, where (k)=2u\_[++]{}u\_[-+]{}-u\_[–]{}u\_[+-]{}, and $\eta=1$ always. The corresponding eigenfunctions of $\tau_z$ are given by \_=\_[+]{}\_[-]{}. The pseudo spin Chern number is now given by Eqs. and by replacing $\psi^{\tau_z}_-$ with $\Psi_{\pm}$ in $\mathcal{M}(k)$. Computing the corresponding inner products yields: \_(k) = +1. Integrating Eq. one obtains the Chern number: $$\mathscr{C}^{\pm}=\mathcal{M}^{\pm}(\infty)-\mathcal{M}^{\pm}(0)=\pm\frac{1}{2}\lb \rho(\infty)-\rho(0)\rb,
\label{thou3}$$ which defines a nontrivial topological spin Hall phase provided that $\Delta_x/\lambda \ll 1$. The function $\rho(k)$ is depicted in Fig. for a specific value of $\Delta_x/\lambda$. It is analogous to the [*Pfaffian*]{} [@km; @hui1], which is obtained by diagonalizing the $TRS$ operator $\Theta$ in the occupied valence bands basis.

![Color online. Plot of $\rho(k)$ vs. $k$ with $\Delta_x/\lambda =0.04$.[]{data-label="smm_split"}](m_split){width="3.5in"}

Interesting features also arise in the presence of a magnetic field. Now let us apply a constant magnetic field along the out-of-plane direction in the system described by Eq.. The resulting Hamiltonian becomes: $$\begin{aligned}
\mathcal{H}&=v_F(\hat{z}\times \boldsymbol{\sigma})\cdot \bold{k}_{\bold A}+(\Delta_z-\lambda\tau_z)\sigma_z-\Delta_x\tau_x,
\label{16}\end{aligned}$$ where $\bold k_{\bold A}=(-i\boldsymbol{\nabla}+\frac{e}{c}\bold{A})$ is the magnetic field experienced by the surface electrons through the Peierls substitution, also known as the orbital effects on the electrons. Here $\bold{A}=-yB\hat{x}$ is the vector potential for the Landau gauge. Diagonalization of Eq. is achieved by introducing the creation and annihilation operators: $$\begin{aligned}
b= \frac{l_B}{\sqrt{2}}(\Pi_x+i\Pi_y); \quad b^{\dagger}= \frac{l_B}{\sqrt{2}}(\Pi_x-i\Pi_y),\end{aligned}$$ where $l_B^2=c/eB$ is the magnetic length and $\boldsymbol{\Pi}=-i\boldsymbol{\nabla} +\frac{e}{c}\bold{A}$. Thus, Eq. becomes: $$\begin{aligned}
\mathcal{H}=i\omega_B\sqrt{2}(\sigma^+b-\sigma^-b^{\dagger})+(\Delta_z-\lambda\tau_z)\sigma_z-\Delta_x\tau_x,
\label{18}\end{aligned}$$ with $\omega_B=v_F/l_B$ being the characteristic frequency analogous to cyclotron frequency. The corresponding landau levels for $n\neq0$ are obtained as: $$\begin{aligned}
\mathcal{E}^2_{n\pm}&=2\omega_B^2n+\Delta_x^2 +\Delta_z^2+\lambda^2\nonumber\\&\pm 2\sqrt{2\omega_B^2n\Delta_x^2+\Delta_z^2\lb\lambda^2+\Delta_x^2\rb},
\label{19a} \end{aligned}$$ There are two energy sectors for each Landau level which are controlled by several parameters. Both sectors behave quite differently as a function of the magnetic field. The Hall conductivity can as well be computed using these Landau levels [@bur2; @bur3].

[*Conclusion*]{}.– In conclusion, we have studied a toy model that captures the interesting physics of quantum spin Hall effect and topological quantum phase transition on the surface of a 3D topological insulator. Our model is comprised of a homogeneous exchange coupling between the surface electrons and the fictitious spin-$\frac{1}{2}$ particle; we showed that this model exhibits both quantum phase transition and half quantized spin Hall effect. We presented a lucid exposition of the quantum phase transition and the topological invariant quantity that characterizes the Dirac points. Introducing dynamics in the fictitious particle leads to a non-conserving term in the Hamiltonian, which breaks the conservation of $z$-component; thus obliterates the spin Hall effect. We obtained a nontrivial topological spin Chern number by diagonalizing the projection of the pseudo spin onto the occupied valence bands. Furthermore, we obtained the Landau levels in the presence of a magnetic field.

[*[ Acknowledgments]{}.–*]{} The authors would like to thank Tami Pereg-Barnea for useful discussions, and Manu Paranjape for his continual supports. This work commenced at Université de Montréal, and was completed at Perimeter Institute for Theoretical Physics.

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