- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Letter
  • Open Access

Gapless superconductivity and its real-space topology in quasicrystals

Kazuma Saito1, Masahiro Hori1,2, Ryo Okugawa1, K. Tanaka2, and Takami Tohyama1

  • 1Department of Applied Physics, Tokyo University of Science, Katsushika, Tokyo 125-8585, Japan
  • 2Department of Physics and Engineering Physics, and Centre for Quantum Topology and Its Applications (quanTA), University of Saskatchewan, 116 Science Place, Saskatoon, Saskatchewan, Canada S7N 5E2

PDF

Phys. Rev. Research 7, L022077 – Published 25 June, 2025

DOI: https://doi.org/10.1103/nb7l-f44r

Abstract

We study superconductivity in Ammann-Beenker quasicrystals under magnetic field. By assuming an intrinsic s-wave pairing interaction and solving for mean-field equations self-consistently, we find gapless superconductivity in the quasicrystals at and near half filling. We show that gapless superconductivity results from the combination of broken translational symmetry and confined states that is characteristic of the quasicrystals. When Rashba spin-orbit coupling is present, the quasicrystalline gapless superconductor can be topologically nontrivial and characterized by a nonzero pseudospectrum invariant given by a spectral localizer. The gapless topological superconducting phase exhibits edge states with near-zero energy. These findings suggest that quasicrystals can be a unique platform for realizing gapless superconductivity with nontrivial topology.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (63)

  1. D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, Metallic phase with long-range orientational order and no translational symmetry, Phys. Rev. Lett. 53, 1951 (1984).
  2. D. Levine and P. J. Steinhardt, Quasicrystals. I. Definition and structure, Phys. Rev. B 34, 596 (1986).
  3. R. N. Araújo and E. C. Andrade, Conventional superconductivity in quasicrystals, Phys. Rev. B 100, 014510 (2019).
  4. S. Sakai, N. Takemori, A. Koga, and R. Arita, Superconductivity on a quasiperiodic lattice: Extended-to-localized crossover of Cooper pairs, Phys. Rev. B 95, 024509 (2017).
  5. S. Sakai and R. Arita, Exotic pairing state in quasicrystalline superconductors under a magnetic field, Phys. Rev. Res. 1, 022002(R) (2019).
  6. N. Takemori, R. Arita, and S. Sakai, Physical properties of weak-coupling quasiperiodic superconductors, Phys. Rev. B 102, 115108 (2020).
  7. K. Kamiya, T. Takeuchi, N. Kabeya, N. Wada, T. Ishimasa, A. Ochiai, K. Deguchi, K. Imura, and N. K. Sato, Discovery of superconductivity in quasicrystal, Nat. Commun. 9, 154 (2018).
  8. Y. Tokumoto, K. Hamano, S. Nakagawa, Y. Kamimura, S. Suzuki, R. Tamura, and K. Edagawa, Superconductivity in a van der Waals layered quasicrystal, Nat. Commun. 15, 1529 (2024).
  9. T. Terashima, Y. Tokumoto, K. Hamano, T. Konoike, N. Kikugawa, and K. Edagawa, Anomalous upper critical field in the quasicrystal superconductor Ta1.6Te, npj Quantum Mater. 9, 56 (2024).
  10. J. Fan and H. Huang, Topological states in quasicrystals, Front. Phys. 17, 13203 (2022).
  11. I. C. Fulga, D. I. Pikulin, and T. A. Loring, Aperiodic weak topological superconductors, Phys. Rev. Lett. 116, 257002 (2016).
  12. T. A. Loring, Bulk spectrum and K-theory for infinite-area topological quasicrystals, J. Math. Phys. (NY) 60, 081903 (2019).
  13. D. Varjas, A. Lau, K. Pöyhönen, A. R. Akhmerov, D. I. Pikulin, and I. C. Fulga, Topological phases without crystalline counterparts, Phys. Rev. Lett. 123, 196401 (2019).
  14. Y. Cao, Y. Zhang, Y.-B. Liu, C.-C. Liu, W.-Q. Chen, and F. Yang, Kohn-Luttinger mechanism driven exotic topological superconductivity on the Penrose lattice, Phys. Rev. Lett. 125, 017002 (2020).
  15. C.-B. Hua, Z.-R. Liu, T. Peng, R. Chen, D.-H. Xu, and B. Zhou, Disorder-induced chiral and helical Majorana edge modes in a two-dimensional Ammann-Beenker quasicrystal, Phys. Rev. B 104, 155304 (2021).
  16. R. Ghadimi, T. Sugimoto, K. Tanaka, and T. Tohyama, Topological superconductivity in quasicrystals, Phys. Rev. B 104, 144511 (2021).
  17. C. Wang, F. Liu, and H. Huang, Effective model for fractional topological corner modes in quasicrystals, Phys. Rev. Lett. 129, 056403 (2022).
  18. Y.-B. Liu, Y. Zhang, W.-Q. Chen, and F. Yang, High-angular-momentum topological superconductivities in twisted bilayer quasicrystal systems, Phys. Rev. B 107, 014501 (2023).
  19. S. Manna, S. K. Das, and B. Roy, Noncrystalline topological superconductors, Phys. Rev. B 109, 174512 (2024).
  20. Z. Qi, I. Na, G. Refael, and Y. Peng, Real-space topological invariant for time-quasiperiodic Majorana modes, Phys. Rev. B 110, 014309 (2024).
  21. M. Hori, T. Sugimoto, T. Tohyama, and K. Tanaka, Self-consistent study of topological superconductivity in two-dimensional quasicrystals, Phys. Rev. B 110, 144512 (2024).
  22. M. Hori, R. Okugawa, K. Tanaka, and T. Tohyama, Weyl superconductivity and quasiperiodic Majorana arcs in quasicrystals, Phys. Rev. Res. 6, 033088 (2024).
  23. M. Sato, Y. Takahashi, and S. Fujimoto, Non-Abelian topological order in s-wave superfluids of ultracold fermionic atoms, Phys. Rev. Lett. 103, 020401 (2009).
  24. M. Sato, Y. Takahashi, and S. Fujimoto, Non-Abelian topological orders and Majorana fermions in spin-singlet superconductors, Phys. Rev. B 82, 134521 (2010).
  25. M. Kohmoto and B. Sutherland, Electronic and vibrational modes on a Penrose lattice: Localized states and band structure, Phys. Rev. B 34, 3849 (1986).
  26. M. Arai, T. Tokihiro, T. Fujiwara, and M. Kohmoto, Strictly localized states on a two-dimensional Penrose lattice, Phys. Rev. B 38, 1621 (1988).
  27. T. Rieth and M. Schreiber, Identification of spatially confined states in two-dimensional quasiperiodic lattices, Phys. Rev. B 51, 15827 (1995).
  28. A. Koga and H. Tsunetsugu, Antiferromagnetic order in the Hubbard model on the Penrose lattice, Phys. Rev. B 96, 214402 (2017).
  29. E. Day-Roberts, R. M. Fernandes, and A. Kamenev, Nature of protected zero-energy states in penrose quasicrystals, Phys. Rev. B 102, 064210 (2020).
  30. A. Koga, Superlattice structure in the antiferromagnetically ordered state in the Hubbard model on the Ammann-Beenker tiling, Phys. Rev. B 102, 115125 (2020).
  31. M. Ö. Oktel, Strictly localized states in the octagonal Ammann-Beenker quasicrystal, Phys. Rev. B 104, 014204 (2021).
  32. H. Ha and B.-J. Yang, Macroscopically degenerate localized zero-energy states of quasicrystalline bilayer systems in the strong coupling limit, Phys. Rev. B 104, 165112 (2021).
  33. A. Koga, Antiferromagnetically ordered state in the half-filled Hubbard model on the Socolar dodecagonal tiling, Mater. Trans. 62, 360 (2021).
  34. M. A. Keskiner and M. O. Oktel, Strictly localized states on the Socolar dodecagonal lattice, Phys. Rev. B 106, 064207 (2022).
  35. R. Ghadimi, M. Hori, T. Sugimoto, and T. Tohyama, Confined states and topological phases in two-dimensional quasicrystalline π-flux model, Phys. Rev. B 108, 125104 (2023).
  36. T. Matsubara, A. Koga, and S. Coates, Ferromagnetically ordered states in the Hubbard model on the H00 hexagonal golden-mean tiling, Phys. Rev. B 109, 014413 (2024).
  37. J. E. S. Socolar, Simple octagonal and dodecagonal quasicrystals, Phys. Rev. B 39, 10519 (1989).
  38. A. A. Abrikosov and L. P. Gor'kov, Contribution to the theory of superconducting alloys with paramagnetic impurities, Zh. Eksp. Teor. Fiz. 39, 1781 (1960).
  39. M. A. Woolf and F. Reif, Effect of magnetic impurities on the density of states of superconductors, Phys. Rev. 137, A557 (1965).
  40. A. V. Balatsky, I. Vekhter, and J.-X. Zhu, Impurity-induced states in conventional and unconventional superconductors, Rev. Mod. Phys. 78, 373 (2006).
  41. L. Yu, Bound state in superconductors with paramagnetic impurities, Acta Phys. Sin. 21, 75 (1965).
  42. H. Shiba, Classical Spins in Superconductors, Prog. Theor. Phys. 40, 435 (1968).
  43. A. I. Rusinov, On the theory of gapless superconductivity in alloys containing paramagnetic impurities, Sov. Phys. JETP 29, 1101 (1969).
  44. R. Nanguneri, M. Jiang, T. Cary, G. G. Batrouni, and R. T. Scalettar, Interplay of superconductivity and spin-dependent disorder, Phys. Rev. B 85, 134506 (2012).
  45. M. Jiang, R. Nanguneri, N. Trivedi, G. G. Batrouni, and R. T. Scalettar, Gapless inhomogeneous superfluid phase with spin-dependent disorder, New J. Phys. 15, 023023 (2013).
  46. S. S. Babkin, A. P. Higginbotham, and M. Serbyn, Proximity-induced gapless superconductivity in two-dimensional Rashba semiconductor in magnetic field, SciPost Phys. 16, 115 (2024).
  47. T. A. Loring, K-theory and pseudospectra for topological insulators, Ann. Phys. 356, 383 (2015).
  48. A. Cerjan and T. A. Loring, Local invariants identify topology in metals and gapless systems, Phys. Rev. B 106, 064109 (2022).
  49. C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu, Classification of topological quantum matter with symmetries, Rev. Mod. Phys. 88, 035005 (2016).
  50. See Supplemental Material at http://link.aps.org/supplemental/10.1103/nb7l-f44r for details on the system and the mechanism of gapless superconductivity, and topological phase transitions.
  51. S. L. Goertzen, K. Tanaka, and Y. Nagai, Self-consistent study of Abelian and non-Abelian order in a two-dimensional topological superconductor, Phys. Rev. B 95, 064509 (2017).
  52. A. Ghosal, M. Randeria, and N. Trivedi, Inhomogeneous pairing in highly disordered s-wave superconductors, Phys. Rev. B 65, 014501 (2001).
  53. J. J. Kinnunen, J. E. Baarsma, J.-P. Martikainen, and P. Törmä, The Fulde–Ferrell–Larkin–Ovchinnikov state for ultracold fermions in lattice and harmonic potentials: a review, Rep. Prog. Phys. 81, 046401 (2018).
  54. B. S. Chandrasekhar, A note on the maximum critical field of high-field superconductors, Appl. Phys. Lett. 1, 7 (1962).
  55. A. M. Clogston, Upper limit for the critical field in hard superconductors, Phys. Rev. Lett. 9, 266 (1962).
  56. A. Ghosal, M. Randeria, and N. Trivedi, Role of spatial amplitude fluctuations in highly disordered s-wave superconductors, Phys. Rev. Lett. 81, 3940 (1998).
  57. M. Z. Hasan and C. L. Kane, Colloquium: Topological insulators, Rev. Mod. Phys. 82, 3045 (2010).
  58. X.-L. Qi and S.-C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys. 83, 1057 (2011).
  59. A. Cerjan, T. A. Loring, and F. Vides, Quadratic pseudospectrum for identifying localized states, J. Math. Phys. (NY) 64, 023501 (2023).
  60. A. Altland and M. R. Zirnbauer, Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures, Phys. Rev. B 55, 1142 (1997).
  61. T. A. Loring and H. Schulz-Baldes, The spectral localizer for even index pairings, J. Noncommut. Geom. 14, 1 (2020).
  62. E. Lozano Viesca, J. Schober, and H. Schulz-Baldes, Chern numbers as half-signature of the spectral localizer, J. Math. Phys. (NY) 60, 072101 (2019).
  63. alliancecan.ca

Outline

Information

Phys. Rev. Research 7, L022077– Published 25 June, 2025
  • Received 1 October 2024
  • Revised 19 February 2025
  • Accepted 13 June 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation