# Electric Polarization in Magnetic Topological Nodal Semimetal Thin Films

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## Abstract

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## 1. Introduction

## 2. Model Hamiltonian

## 3. Electric Polarization Induced by the Screened Potential

## 4. Numerical Results

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Schematic picture of the slab system. M represents magnetization set on the x-z plane, and $\theta $ is an angle between the z axis and M. We assume the open boundary condition for the x direction, and the periodic boundary condition for the y and z directions. Topological phase diagram of bulk system for: (

**b**) $\theta =0$; and (

**c**) $\theta =\pi /2$. The topological nodal semimetal phases emerge once the exchange coupling overcomes the band gap.

**Figure 2.**Band structure of nodal line semimetals: (

**a**) in the bulk system; (

**b**) in the slab system with no E-field; and (

**c**) in the slab system with E-field. We set the parameters as ${m}_{2}/t=1,{m}_{0}/t=-0.5,{J}_{0}/t=1,{N}_{x}=30,$ and $\alpha =0.1$.

**Figure 3.**(

**a**) The black closed curve represents nodal line in the Brillouin zone. Two paths ${C}_{1}$ and ${C}_{2}$ are depicted by red arrows and give the Zak phase $\pi $ and 0, respectively. (

**b**) The Zak phase calculated at fixed (${k}_{y},{k}_{z}$). The Zak phase is quantized as $\pi $ in the area enclosed by the nodal line projectoed on ${k}_{y}$-${k}_{z}$ plane and it is zero in the other area.

**Figure 4.**Band structure of nodal point (Weyl) semimetals: (

**a**) in the bulk system; (

**b**) in the slab system with no external potential; and (

**c**) in the slab system with an external potential. We set the parameters as ${m}_{2}/t=1,{m}_{0}/t=-0.5,{J}_{0}/t=1,{N}_{x}=30,$ and $\alpha =0.1$.

**Figure 5.**Polarization as a function of the external electric field for several screening strength $\alpha $. There are the kink structures only in the nodal line semimetals.

**Figure 6.**(

**a**) The total polarization; and (

**b**) the total energy as a function of the angle $\theta $. We set ${E}_{0}=0.01t/\left(ea\right)$, $\alpha =0.1$, and $\mu =0$.

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**MDPI and ACS Style**

Ominato, Y.; Yamakage, A.; Nomura, K.
Electric Polarization in Magnetic Topological Nodal Semimetal Thin Films. *Condens. Matter* **2018**, *3*, 43.
https://doi.org/10.3390/condmat3040043

**AMA Style**

Ominato Y, Yamakage A, Nomura K.
Electric Polarization in Magnetic Topological Nodal Semimetal Thin Films. *Condensed Matter*. 2018; 3(4):43.
https://doi.org/10.3390/condmat3040043

**Chicago/Turabian Style**

Ominato, Yuya, Ai Yamakage, and Kentaro Nomura.
2018. "Electric Polarization in Magnetic Topological Nodal Semimetal Thin Films" *Condensed Matter* 3, no. 4: 43.
https://doi.org/10.3390/condmat3040043