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Topological superconductivity in two-dimensional altermagnetic metals

Di Zhu, Zheng-Yang Zhuang, Zhigang Wu, and Zhongbo Yan
Phys. Rev. B 108, 184505 – Published 3 November 2023
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  • INTRODUCTION
  • MODEL AND RESULTS
  • DISCUSSIONS AND CONCLUSIONS
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    Bringing magnetic metals into superconducting states represents an important approach for realizing unconventional superconductors and potentially even topological superconductors. Altermagnetism, classified as a third basic collinear magnetic phase, gives rise to intriguing momentum-dependent spin-splitting of the band structure and results in an even number of spin-polarized Fermi surfaces due to the symmetry-enforced zero net magnetization. In this work, we investigate the effect of this new magnetic order on the superconductivity of a two-dimensional metal with d-wave altermagnetism and Rashba spin-orbital coupling. Specifically we consider an extended attractive Hubbard interaction and determine the types of superconducting pairing that can occur in this system and ascertain whether they possess topological properties. Through self-consistent mean-field calculations, we find that the system in general favors a mixture of spin-singlet s-wave and spin-triplet p-wave pairings and that the altermagnetism is beneficial to the latter. Using symmetry arguments supported by detailed calculations, we show that a number of topological superconductors, including both first-order and second-order ones, can emerge when the p-wave pairing dominates. In particular, we find that the second-order topological superconductor is enforced by a C4zT symmetry, which renders the spin polarization of Majorana corner modes into a unique entangled structure. Our study demonstrates that altermagnetic metals are fascinating platforms for the exploration of intrinsic unconventional superconductivity and topological superconductivity.

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    • Received 22 May 2023
    • Revised 19 October 2023
    • Accepted 19 October 2023

    DOI:https://doi.org/10.1103/PhysRevB.108.184505

    ©2023 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Superconducting order parameterTopological superconductors
    1. Physical Systems
    Two-dimensional electron systemUnconventional superconductors
    1. Techniques
    BCS theoryBogoliubov-de Gennes equationsDiscrete symmetries in condensed matterMean field theoryTight-binding model
    Condensed Matter, Materials & Applied Physics

    Authors & Affiliations

    Di Zhu1, Zheng-Yang Zhuang1, Zhigang Wu2,3,4,*, and Zhongbo Yan1,†

    • 1Guangdong Provincial Key Laboratory of Magnetoelectric Physics and Devices, School of Physics, Sun Yat-sen University, Guangzhou 510275, People's Republic of China
    • 2Shenzhen Institute for Quantum Science and Engineering (SIQSE), Southern University of Science and Technology, Shenzhen, People's Republic of China
    • 3International Quantum Academy, Shenzhen 518048, People's Republic of China
    • 4Guangdong Provincial Key Laboratory of Quantum Science and Engineering, Southern University of Science and Technology, Shenzhen 518055, People's Republic of China

    • *wuzg@sustech.edu.cn
    • †yanzhb5@mail.sysu.edu.cn

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    Issue

    Vol. 108, Iss. 18 — 1 November 2023

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    Images

    • Figure 1
      Figure 1

      Band structures (left) and Fermi surfaces (right) of the AM metal with and without RSOC. [(a) and (b)] λ=0 and [(c) and (d)] λ=0.2. The color of the Fermi surfaces indicates the magnitude of 〈σz〉, and the length and direction of the arrows indicate the magnitude and direction of the in-plane spin polarization. The spin textures clearly display the C4zT symmetry. Common parameters are t=0.5, tAM=0.1, and μ=−1.

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    • Figure 2
      Figure 2

      A representative pairing phase diagram. The white solid line divides the pairing phase diagram into two regions, with one favoring the mixed s+helicalp-wave pairing and the other favoring the mixed s+chiralp-wave pairing. The former preserves the C4zT symmetry, while the latter does not. Below the black dashed line, along which |Δ↑↓s|=|Δ↑↑p+|, is a sizable region with dominant helical p-wave pairing. Here the parameters are t=0.5, tAM=0.1, μ=−1, and Vp=1.5.

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    • Figure 3
      Figure 3

      (a) A chiral TSC with Chern number C=−2. The blue and red solid lines refer to chiral Majorana modes on two opposite edges. (b) A helical TSC with a pair of helical Majorana modes. The blue and red solid lines are doubly degenerate. [(c) and (d)] A second-order TSC with gapped edge spectra and four Majorana corner modes. The four red dots in the inset of (d) refers to four Majorana zero modes, with their probability density profiles concentrating around the four corners. Common parameters are t=0.5, tAM=0.1, and μ=−1. The rest of parameters are as follows: (a) λ=0.2, Δ↑↓s=0.05, and Δ↑↑p+=Δ↓↓p+=0.25 for the chiral p-wave pairing amplitude; (b) λ=0, Δ↑↓s=0; and [(c) and (d)] λ=0.2, Δ↑↓s=0.05. In (b)–(d), the helical p-wave pairing amplitude is chosen to be Δ↑↑p+=0.25.

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    • Figure 4
      Figure 4

      The s+ chiral p-wave solution of the gap equations. Here t=0.5, tAM=0.1, Vp=1.5, and μ=−1. Left: The pairing amplitudes. Right: The condensation energy.

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    • Figure 5
      Figure 5

      The s+ helical p-wave solution of the gap equations. Here t=0.5, tAM=0.1, Vp=1.5, and μ=−1. Left: The pairing amplitudes. Right: The condensation energy.

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    • Figure 6
      Figure 6

      Left: The difference between the p-wave pairing amplitude and the s-wave pairing amplitude for the s+ helical p solutions (only positive differences are shown). Right: The difference of condensation energies for the s+ chiral p-wave and the s+ helical p-wave solutions (only positive differences are shown).

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    • Figure 7
      Figure 7

      Common parameters are t=0.5, tAM=0.1, λ=0.2, μ=−0.0, and Δp=0.25. (a) Δ↑↓s=0.0 and (b) Δ↑↓s=0.2. The red and blue solid lines refer to midgap states on opposite edges. The edge-state energy spectra tangentially touch with the bulk energy spectra when Δ↑↓s=0, and become floating bands when Δ↑↓s becomes nonzero.

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