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

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 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

- Bernevig, B.A.; Hughes, T.L.; Zhang, S.-C. Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells. Science
**2006**, 314, 1757–1761. [Google Scholar] [CrossRef] [PubMed][Green Version] - König, M.; Wiedmann, S.; Brüune, C.; Roth, A.; Buhmann, H.; Molenkamp, L.W.; Qi, X.-L.; Zhang, S.-C. Quantum Spin Hall Insulator State in HgTe Quantum Wells. Science
**2007**, 318, 766–770. [Google Scholar] [CrossRef] [PubMed][Green Version] - König, M.; Buhmann, H.; Molenkamp, L.W.; Hughes, T.; Liu, C.-X.; Qi, X.-L.; Zhang, S.-C. The Quantum Spin Hall Effect: Theory and Experiment. J. Phys. Soc. Jpn.
**2008**, 77, 031007. [Google Scholar] [CrossRef][Green Version] - Roth, A.; Brüne, C.; Buhmann, H.; Molenkamp, L.W.; Maciejko, J.; Qi, X.-L.; Zhang, S.-C. Nonlocal Transport in the Quantum Spin Hall State. Science
**2009**, 325, 294–297. [Google Scholar] [CrossRef] [PubMed][Green Version] - Xia, Y.; Qian, D.; Hsieh, D.; Wray, L.; Pal, A.; Lin, H.; Bansil, A.; Grauer, D.; Hor, Y.S.; Cava, R.J.; Hasan, M.Z. Observation of a large-gap topological-insulator class with a single Dirac cone on the surface. Nat. Phys.
**2009**, 5, 398–402. [Google Scholar] [CrossRef][Green Version] - Zhang, H.; Liu, C.-X.; Qi, X.-L.; Dai, X.; Fang, Z.; Zhang, S.-C. Topological insulators in Bi
_{2}Se_{3}, Bi_{2}Te_{3}and Sb_{2}Te_{3}with a single Dirac cone on the surface. Nat. Phys.**2009**, 5, 438–442. [Google Scholar] [CrossRef] - Jiang, H.; Cheng, S.G.; Sun, Q.F.; Xie, X.C. Topological Insulator: A New Quantized Spin Hall Resistance Robust to Dephasing. Phys. Rev. Lett.
**2009**, 103, 036803. [Google Scholar] [CrossRef] [PubMed][Green Version] - Hasan, M.Z.; Kane, C.L. Colloquium: Topological insulators. Rev. Mod. Phys.
**2010**, 82, 3045–3067. [Google Scholar] [CrossRef] - Qi, X.-L.; Zhang, S.-C. Topological insulators and superconductors. Rev. Mod. Phys.
**2011**, 83, 1057–1110. [Google Scholar] [CrossRef][Green Version] - Qiao, Z.H.; Tse, W.K.; Jiang, H.; Yao, Y.G.; Niu, Q. Two-Dimensional Topological Insulator State and Topological Phase Transition in Bilayer Graphene. Phys. Rev. Lett.
**2011**, 107, 256801. [Google Scholar] [CrossRef] - Chang, C.Z.; Zhang, J.S.; Feng, X.; Shen, J.; Zhang, Z.C.; Guo, M.H.; Li, K.; Ou, Y.B.; Wei, P.; Wang, L.L.; et al. Experimental Observation of the Quantum Anomalous Hall Effect in a Magnetic Topological Insulator. Science
**2013**, 340, 167–170. [Google Scholar] [CrossRef] [PubMed][Green Version] - Wang, Z.F.; Liu, Z.; Liu, F. Quantum anomalous Hall effect in 2D organic topological insulators. Phys. Rev. Lett.
**2013**, 110, 196801. [Google Scholar] [CrossRef] [PubMed] - Thouless, D.J.; Kohmoto, M.; Nightingale, M.P.; den Nijs, M. Quantized Hall Conductance in a Two-Dimensional Periodic Potential. Phys. Rev. Lett.
**1982**, 49, 405–408. [Google Scholar] [CrossRef] - Kane, C.L.; Mele, E.J. Z
_{2}Topological Order and the Quantum Spin Hall Effect. Phys. Rev. Lett.**2005**, 95, 146802. [Google Scholar] [CrossRef] [PubMed] - Sheng, D.N.; Weng, Z.Y.; Sheng, L.; Haldane, F.D.M. Quantum Spin-Hall Effect and Topologically Invariant Chern Numbers. Phys. Rev. Lett.
**2006**, 97, 036808. [Google Scholar] [CrossRef] [PubMed] - Li, J.; Chu, R.L.; Jain, J.K.; Shen, S.Q. Topological Anderson Insulator. Phys. Rev. Lett.
**2009**, 102, 136806. [Google Scholar] [CrossRef] [PubMed] - Jiang, Z.F.; Chu, R.L.; Shen, S.Q. Electric-field modulation of the number of helical edge states in thin-film semiconductors. Phys. Rev. B
**2010**, 81, 115322. [Google Scholar] [CrossRef] - Kim, M.; Kim, C.H.; Kim, H.S.; Ihm, J. Topological quantum phase transitions driven by external electric fields in Sb
_{2}Te_{3}thin films. Proc. Natl. Acad. Sci. USA**2012**, 109, 671–674. [Google Scholar] [CrossRef] - Bahramy, M.; Yang, B.J.; Arita, R.; Nagaosa, N. Emergent quantum confinement at topological insulator surfaces. Nat. Commun.
**2012**, 3, 1159. [Google Scholar] [CrossRef][Green Version] - Esaki, L.; Tsu, R. Superlattice and negative differential conductivity in semiconductors. IBM J. Res. Dev.
**1970**, 14, 61–65. [Google Scholar] [CrossRef] - Tsu, R. Superlattice to Nanoelectronics; Elsevier: Oxford, UK, 2005. [Google Scholar]
- Schneider, H.; Grahn, H.T.; Klitzing, K.V.; Ploog, K. Sequential resonant tunneling of holes in GaAs-AlAs superlattices. Phys. Rev. B
**1989**, 40, 10040–10043. [Google Scholar] [CrossRef] - Wacker, A.; Moscoso, M.; Kindelan, M.; Bonilla, L.L. Current-voltage characteristic and stability in resonant-tunneling n-dopedsemiconductor superlattices. Phys. Rev. B
**2003**, 55, 2466–2475. [Google Scholar] [CrossRef] - Cheng, Y.-C.; Yang, S.-T.; Yang, J.-N.; Lan, W.-H.; Chang, L.-B.; Hsieh, L.-Z. Fabrication of far-infrared photodetector based on InAs/GaAs quantum dot superlattices. Opt. Eng.
**2003**, 42, 119–123. [Google Scholar] - Zheng, Y.; Yang, S.-J. Topological bands in one-dimensional periodic potentials. Phys. B
**2014**, 454, 93–97. [Google Scholar] [CrossRef][Green Version] - Fu, B.; Zheng, H.; Li, Q.; Shi, Q.; Yang, J. Topological phase transition driven by a spatially periodic potential. Phys. Rev. B
**2014**, 90, 214502. [Google Scholar] [CrossRef] - Gentile, P.; Cuoco, M.; Ortix, C. Edge States and Topological Insulating Phases Generated by Curving a Nanowire with Rashba Spin-Orbit Coupling. Phys. Rev. Lett.
**2015**, 115, 256801. [Google Scholar] [CrossRef] [PubMed] - Pandey, S.; Scopigno, N.; Gentile, P.; Cuoco, M.; Ortix, C. Topological quantum pump in serpentine-shaped semiconducting narrow channels. Phys. Rev. B
**2018**, 97, 241103. [Google Scholar] [CrossRef][Green Version] - Lau, A.; van den Brink, J.; Ortix, C. Topological mirror insulators in one dimension. Phys. Rev. B
**2016**, 94, 165164. [Google Scholar] [CrossRef] - Harper, P.G. Single Band Motion of Conduction Electrons in a Uniform Magnetic Field. Proc. Phys. Soc. Lond. Sect. A
**1955**, 68, 874–878. [Google Scholar] [CrossRef] - Aubry, S.; André, G. Analyticity breaking and Anderson localization in incommensurate lattices. Ann. Isr. Phys. Soc.
**1980**, 3, 133–151. [Google Scholar] - Ganeshan, S.; Sun, K.; Das Sarma, S. Topological zero-energy modes in gapless commensurate Aubry-André-Harper models. Phys. Rev. Lett.
**2013**, 110, 180403. [Google Scholar] [CrossRef] [PubMed] - Matsuura, S.; Chang, P.-Y.; Schnyder, A.P.; Ryu, S. Protected boundary states in gapless topological phases. New J. Phys.
**2013**, 15, 065001. [Google Scholar] [CrossRef][Green Version] - Loring, T.A. K-theory and pseudospectra for topological insulators. Ann. Phys.
**2015**, 356, 383–416. [Google Scholar] [CrossRef] - Koshino, M.; Morimoto, T.; Sato, M. Topological zero modes and Dirac points protected by spatial symmetry and chiral symmetry. Phys. Rev. B
**2014**, 90, 115207. [Google Scholar] [CrossRef][Green Version] - Ying, Z.-J.; Gentile, P.; Ortix, C.; Cuoco, M. Designing electron spin textures and spin interferometers by shape deformations. Phys. Rev. B
**2016**, 94, 081406. [Google Scholar] [CrossRef] - Ying, Z.Y.; Cuoco, M.; Ortix, C.; Gentile, P. Tuning pairing amplitude and spin-triplet texture by curving superconducting nanostructures. Phys. Rev. B
**2017**, 96, 100506. [Google Scholar] [CrossRef] - Ying, Z.-J.; Cuoco, M.; Gentile, P.; Ortix, C. Josephson Current in Rashba-Based Superconducting Nanowires with Geometric Misalignment: Rashba-Based Superconducting Nanowires with Geometric Misalignment. In Proceedings of the 16th International Superconductive Electronics Conference (ISEC), Naples, Italy, 12–16 June 2017. [Google Scholar]

**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 $\left(\right)$ 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 $\left(\right)$ 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**).

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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