# Engineering Topological Nodal Line Semimetals in Rashba Spin-Orbit Coupled Atomic Chains

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## Abstract

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## 1. Introduction

## 2. The Model

## 3. Metal-Insulator Transition Induced by Periodicity

## 4. Symmetry Protection of the Nodal Lines

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Energy spectrum of the Hamiltonian matrix ${\mathcal{H}}_{k}$, for the case $\lambda =4$, as a function of the momentum k (measured in units of $\pi /\tilde{a}$) at (

**a**) ${V}_{0}=0.6t,{\varphi}_{V}=0$, and ${\alpha}_{0}=0$; (

**b**) ${\alpha}_{0}=0.6t,{\varphi}_{\alpha}=0$ and ${V}_{0}=0$ and (

**c**) ${V}_{0}={\alpha}_{0}=0.6t,{\varphi}_{V}=0,{\varphi}_{\alpha}=0$. All energies are measured in the unit of the hopping parameter t.

**Figure 2.**Energy spectrum of Hamiltonian matrix ${\mathcal{H}}_{k}$ for the case $\lambda =4$ (

**a**) at ${V}_{0}=0.6t$ and ${\alpha}_{0}=0$ as a function of the charge potential phase ${\varphi}_{V}$, (

**b**) at ${\alpha}_{0}=0.6t$ and ${V}_{0}=0$ as a function of the RSOC phase ${\varphi}_{\alpha}$, and (

**c**) ${V}_{0}={\alpha}_{0}=0.6t$, ${\varphi}_{V}=0.2\pi $ as a function of the RSOC phase ${\varphi}_{\alpha}$.

**Figure 3.**Density plot of the energy gap between the fourth and fifth energy bands (counted starting from the lowest one in energy) for periodicity $\lambda =4$ at $k=0$ in the synthetic space $\left({\varphi}_{V},{\varphi}_{\alpha}\right)$ at ${\alpha}_{0}=t$ and ${V}_{0}={\alpha}_{0}/2$ in (

**a**), ${V}_{0}=\sqrt{2}{\alpha}_{0}$ in (

**b**) and ${V}_{0}=2{\alpha}_{0}$ in (

**c**).

**Figure 4.**Density plot of the energy gap between the eighth and ninth bands (counted starting from the lowest one in energy) for periodicty $\lambda =8$ at $k=0$ in the synthetic space $\left({\varphi}_{V},{\varphi}_{\alpha}\right)$ at ${\alpha}_{0}=t$ and ${V}_{0}={\alpha}_{0}/2$ in (

**a**), ${V}_{0}=\sqrt{2}{\alpha}_{0}$ in (

**b**) and ${V}_{0}=2{\alpha}_{0}$ in (

**c**).

**Figure 5.**Density plot of the topological invariant and corresponding classification of the different regions as topological (T) and non-topological (NT) at ${\alpha}_{0}=t$ and ${V}_{0}={\alpha}_{0}/2$ in (

**a**), ${V}_{0}=\sqrt{2}{\alpha}_{0}$ in (

**b**), and ${V}_{0}=2{\alpha}_{0}$ in (

**c**).

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**MDPI and ACS Style**

Gentile, P.; Benvenuto, V.; Ortix, C.; Noce, C.; Cuoco, M.
Engineering Topological Nodal Line Semimetals in Rashba Spin-Orbit Coupled Atomic Chains. *Condens. Matter* **2019**, *4*, 25.
https://doi.org/10.3390/condmat4010025

**AMA Style**

Gentile P, Benvenuto V, Ortix C, Noce C, Cuoco M.
Engineering Topological Nodal Line Semimetals in Rashba Spin-Orbit Coupled Atomic Chains. *Condensed Matter*. 2019; 4(1):25.
https://doi.org/10.3390/condmat4010025

**Chicago/Turabian Style**

Gentile, Paola, Vittorio Benvenuto, Carmine Ortix, Canio Noce, and Mario Cuoco.
2019. "Engineering Topological Nodal Line Semimetals in Rashba Spin-Orbit Coupled Atomic Chains" *Condensed Matter* 4, no. 1: 25.
https://doi.org/10.3390/condmat4010025