# Topological Properties of the 2D 2-Band System with Generalized W-Shaped Band Inversion

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## Abstract

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## 1. Introduction

## 2. Methods

#### 2.1. The Model

#### 2.2. The Berry Phase

#### 2.3. The Anomalous QHE

## 3. Results

## 4. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Schematics of the energy bands (3), conduction ${\mathcal{E}}_{+}$ and valence ${\mathcal{E}}_{-}$ as a function of $\mathbf{k}$ assuming the rotational symmetry for several choices of system parameters. The vertical distance of the cone tips, along the energy axis, is $2\delta $ ($\delta =1$ in all four panels). The other parameters are: (

**a**) $n=m=1$, $\gamma =0$ (the gap is closed); (

**b**) $n=m=1$, $\gamma =0.5$; (

**c**) $n=m=2$, $\gamma =0.5$; (

**d**) $n=4$, $m=7$, $\gamma =0.5$.

**Figure 2.**The energy band ${\mathcal{E}}_{+}\left(k\right)$, assuming the angular rotational symmetry, for several choices of system parameters. The other energy band is simply ${\mathcal{E}}_{-}\left(k\right)=-{\mathcal{E}}_{+}\left(k\right)$. In both panels, $\delta =0.5$, the energy gap between ${\mathcal{E}}_{+}\left(k\right)$ and ${\mathcal{E}}_{-}\left(k\right)$ closes for $\gamma =0$ and remains opened for finite $\gamma $ varied between values zero and 20. The other parameters are: (

**a**) $n=m=1$; (

**b**) $n=m=2$. The minimum of ${\mathcal{E}}_{+}\left(k\right)$ determines the radius of the nodal ring (see Figure 1) ${k}_{0}$ which shrinks with increasing $\gamma $ and becomes zero in the limit $\gamma \to \infty $, turning the picture of two intersecting cones (with the gap opened along the intersection) into the picture of two distant cones, which are separated “vertically” by $2\delta $ from each other.

**Figure 3.**The Berry phase $\mathcal{B}$ (19) as a function of parameter $\gamma $ for the case $n=m$. It attains a value of $\mathcal{B}=-2\pi n$ for $\gamma =0$ and tends to the value $\mathcal{B}\to -\pi n$ as $\gamma \to \infty $.

**Figure 4.**Topological phase diagram of the system described by the Hamiltonian (1) characterized by the “winding number” $\mathcal{C}$, given by Equation (20), depending on parameters $(n,m)\in \{0,1,2,3,\dots \}$ and $\gamma \in {\mathbf{R}}^{+}$. (0) $n=0$, $\forall m$ (orange): the trivial phase, $\mathcal{C}=0$; (I) $m<n$ (green): $\mathcal{C}=-n/2$; (II) $m>n$ (blue): $\mathcal{C}=-n$; (III) $m=n\ne 0$ (red): $\mathcal{C}=-n\left[1+1/\sqrt{1+{\gamma}^{2}}\right]/2$. Along the $m=n>0$ line, $\mathcal{C}$ is a function of $\gamma $ which interpolates between values of $\mathcal{C}$ in the regions I and II, being equal to the value in I in the limit $\gamma \to \infty $, and equal to the value in II in the limit $\gamma =0$.

**Figure 5.**The vector field representing the expected value of $({\sigma}_{x},{\sigma}_{z})$ operator with respect to the state ${\mathsf{\Psi}}_{-}({k}_{x},{k}_{y})$ (4) on the $({k}_{x},{k}_{y})$ domain. The horizontal labels (1), (2), (3) represent values of parameter $\gamma =$ 0.1, 1, 10, respectively, while the vertical ones are: (

**a**) $(m,n)=(1,1)$, $\mathcal{C}=$−0.998, −0.854, and −0.651, respectively; (

**b**) $(m,n)=(1,2)$, $\mathcal{C}=-1/2$, (

**c**) $(m,n)=(2,1)$, $\mathcal{C}=-1$. The topologically trivial case for which the bands touch $(\gamma =0,\mathcal{C}=0)$ would have all “spins” directed upwards.

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**MDPI and ACS Style**

Rukelj, Z.; Radić, D.
Topological Properties of the 2D 2-Band System with Generalized W-Shaped Band Inversion. *Quantum Rep.* **2022**, *4*, 476-485.
https://doi.org/10.3390/quantum4040034

**AMA Style**

Rukelj Z, Radić D.
Topological Properties of the 2D 2-Band System with Generalized W-Shaped Band Inversion. *Quantum Reports*. 2022; 4(4):476-485.
https://doi.org/10.3390/quantum4040034

**Chicago/Turabian Style**

Rukelj, Zoran, and Danko Radić.
2022. "Topological Properties of the 2D 2-Band System with Generalized W-Shaped Band Inversion" *Quantum Reports* 4, no. 4: 476-485.
https://doi.org/10.3390/quantum4040034