Midgap states induced by Zeeman field and $p$-wave superconductor pairing (2024)

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Midgap states induced by Zeeman field and $p$ -wave superconductor pairing

Yuanjun Jin, XingYu Yue, Yong Xu, Xiang-Long Yu, and Guoqing Chang

Phys. Rev. B 109, L241101 – Published 3 June 2024

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Abstract

The one-dimensional Su-Schrieffer-Heeger (SSH) model is central to band topology in condensed matter physics, which allows us to understand and design distinct topological states. In this work we find another mechanism to analogize the SSH model in a spinful system, realizing an obstructed atomic insulator by introducing intrinsic spin-orbit coupling and in-plane Zeeman field. In our model the midgap states originate from a quantized hidden polarization with invariant index $Z_{2}$ (0; 01) due to the local inversion symmetry breaking. When the global inversion symmetry is broken, a charge pumping is designed by tuning the polarization. Moreover, by introducing the $p + i p$ superconductor pairing potential, a topological phase dubbed obstructed superconductor (OSC) is identified. This new state is characterized by invariant index $Z_{2}$ (0; 01) and nonchiral midgap states. More interestingly, these nonchiral edge states result in a chiral-like nonlocal conductance, which is different from the traditional chiral topological superconductor. Our findings not only find another strategy to achieve a spinful SSH model but also predict the existence of OSC, providing a promising avenue for further exploration of its transport properties.

Physics Subject Headings (PhySH)

Physical Systems

2-dimensional systemsElectronic structureNode-line semimetalsTopological insulatorsTopological materialsTopological superconductors

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Yuanjun Jin^1,2,*, XingYu Yue³, Yong Xu⁴, Xiang-Long Yu^5,6, and Guoqing Chang^2,†

¹Guangdong Basic Research Center of Excellence for Structure and Fundamental Interactions of Matter, Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, School of Physics, South China Normal University, Guangzhou 510006, China
²Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore
³Physics Department and Guangdong-Hong Kong Joint Laboratory of Quantum Matter, the University of Hong Kong, Pokfulam Road, Hong Kong, China
⁴Institute of Micro/Nano Materials and Devices, Ningbo University of Technology, Ningbo 315016, Zhejiang, China
⁵Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology, Shenzhen 518055, China
⁶International Quantum Academy, Shenzhen 518048, China

^*yuanjunjin@m.scnu.edu.cn
^†guoqing.chang@ntu.edu.sg

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Vol. 109, Iss. 24 — 15 June 2024

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Midgap states induced by Zeeman field and $p$-wave superconductor pairing (8)

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Figure 1
(a)2D rectangle lattice includes two sites, A and B, with Wyckoff positions (0, 0.25) and (0, 0.75), respectively. The red rectangle represents the unit cell, and the curves are marked with the hopping parameters $t_{i}$ ( $i = 1$ , 2, 3) and intrinsic SOC $t_{s o}$ . (b)The 2D BZ and the projected 1D BZ along $y$ axis. (c)Band structure in the presence of SOC. (d)Band structure with in-plane Zeeman field along $x$ direction. The color bar represents the spin direction along the $x$ axis.
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Figure 2
(a)Phase diagram depending on $t_{3}$ and $m$ . (b)WCC and nuclei positions in the unit cell. Blue spheres denote the nuclei, and yellow stars present WCC. (c)Edge states in the ribbon along $x$ axis. The width of this ribbon is about 100 unit cells. (d)The real-space probability distribution at $k_{x} = 0$ . Red and green indicate the states are localized on the two edges of the ribbon.
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Figure 3
(a)Polarization as a function of $t_{s o}^{'}$ , $t_{s o}$ , and $t_{3} = 1$ . The unit is $2 π$ . (b)Polarization as a function of $t_{s o}^{'}$ , $t_{s o}$ , and $t_{3} = - 1$ . The unit is $2 π$ . (c)The edge-state splitting when $I$ symmetry is broken with $t_{s o}^{'} = 0.1$ . (d)Charge pumping as the revolution of $t_{s o}^{'}$ .
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Figure 4
(a)Topological phase diagram. (b)Edge states of obstructed SC, red and green indicate the states that come from two edges of the ribbon. (c)Nonlocal conductance of chiral TSC in our model. The insert shows the schematic three-terminal device with two normal metal leads. The figurecorresponds to the setup for measuring $G_{12}$ and $G_{21}$ . The bias voltage $V_{1}$ is applied to lead 1, while lead 2 and the superconductor are grounded. (d)The nonlocal conductance of OSC in our model.
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