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  • Letter
  • Open Access

Spin-aligned butterfly spectral map in non-Hermitian quasicrystals

Soumya Ranjan Padhi1,2,*, Souvik Roy1,2,†, Debashree Chowdhury3,‡, and Tapan Mishra1,2,§

  • *Contact author: padhisoumyaranjan778@gmail.com
  • †Contact author: souvikroy138@gmail.com
  • ‡Contact author: debashreephys@gmail.com
  • §Contact author: mishratapan@gmail.com
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APS Open Sci. 1, L000079 – Published 3 August, 2026

DOI: https://doi.org/10.1103/bkcp-l8t4

Abstract

The non-Hermitian spinful Aubry-André-Harper model in the presence of Rashba-type spin-orbit coupling and a spatially varying textured magnetic field is studied. Interestingly, our analysis produces a butterfly spectral map due to the nontrivial extent of localization of the states in the spectrum. This spectral map also exhibits an asymmetric spin alignment with respect to the wings of the butterfly. Our analysis also suggests that the onset of such a spectral map is a combined effect of the non-Hermiticity, spin-orbit interaction, and the textured magnetic field.

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References (65)

  1. X. Wang, Y. Li, X. Hu, R. Gu, Y. Ao, P. Jiang, and Q. Gong, Non-Hermitian high-quality-factor topological photonic crystal cavities, Phys. Rev. A 105, 023531 (2022).
  2. F. Koch and J. C. Budich, Quantum non-Hermitian topological sensors, Phys. Rev. Res. 4, 013113 (2022).
  3. J. C. Budich and E. J. Bergholtz, Non-Hermitian topological sensors, Phys. Rev. Lett. 125, 180403 (2020).
  4. E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Exceptional topology of non-Hermitian systems, Rev. Mod. Phys. 93, 015005 (2021).
  5. T. E. Lee, Anomalous edge state in a non-Hermitian lattice, Phys. Rev. Lett. 116, 133903 (2016).
  6. H. Shen, B. Zhen, and L. Fu, Topological band theory for non-Hermitian Hamiltonians, Phys. Rev. Lett. 120, 146402 (2018).
  7. K. Takata and M. Notomi, Photonic topological insulating phase induced solely by gain and loss, Phys. Rev. Lett. 121, 213902 (2018).
  8. S. Yao and Z. Wang, Edge states and topological invariants of non-Hermitian systems, Phys. Rev. Lett. 121, 086803 (2018).
  9. C. Yuce, Non-Hermitian anomalous skin effect, Phys. Lett. A 384, 126094 (2020).
  10. N. Okuma, K. Kawabata, K. Shiozaki, and M. Sato, Topological origin of non-Hermitian skin effects, Phys. Rev. Lett. 124, 086801 (2020).
  11. L. Li, C. H. Lee, S. Mu, and J. Gong, Critical non-Hermitian skin effect, Nat. Commun. 11, 5491 (2020).
  12. N. Okuma and M. Sato, Non-Hermitian topological phenomena: A review, Annu. Rev. Condens. Matter Phys. 14, 83 (2023).
  13. L. Li, C. H. Lee, and J. Gong, Impurity induced scale-free localization, Commun. Phys. 4, 42 (2021).
  14. R. Lin, T. Tai, L. Li, and C. H. Lee, Topological non-Hermitian skin effect, Front. Phys. 18, 53605 (2023).
  15. P. Molignini, O. Arandes, and E. J. Bergholtz, Anomalous skin effects in disordered systems with a single non-Hermitian impurity, Phys. Rev. Res. 5, 033058 (2023).
  16. D. S. Borgnia, A. J. Kruchkov, and R.-J. Slager, Non-Hermitian boundary modes and topology, Phys. Rev. Lett. 124, 056802 (2020).
  17. M.-A. Miri and A. Alù, Exceptional points in optics and photonics, Science 363, eaar7709 (2019).
  18. Ş. K. Özdemir, S. Rotter, F. Nori, and L. Yang, Parity–time symmetry and exceptional points in photonics, Nat. Mater. 18, 783 (2019).
  19. V. M. Martinez Alvarez, J. E. Barrios Vargas, and L. E. F. Foa Torres, Non-Hermitian robust edge states in one dimension: Anomalous localization and eigenspace condensation at exceptional points, Phys. Rev. B 97, 121401(R) (2018).
  20. Z. Gong, Y. Ashida, K. Kawabata, K. Takasan, S. Higashikawa, and M. Ueda, Topological phases of non-Hermitian systems, Phys. Rev. X 8, 031079 (2018).
  21. F. K. Kunst, E. Edvardsson, J. C. Budich, and E. J. Bergholtz, Biorthogonal bulk-boundary correspondence in non-Hermitian systems, Phys. Rev. Lett. 121, 026808 (2018).
  22. R. Koch and J. C. Budich, Bulk-boundary correspondence in non-Hermitian systems: Stability analysis for generalized boundary conditions, Eur. Phys. J. D 74, 70 (2020).
  23. Y. Cao, Y. Li, and X. Yang, Non-Hermitian bulk-boundary correspondence in a periodically driven system, Phys. Rev. B 103, 075126 (2021).
  24. L. Xiao, T. Deng, K. Wang, G. Zhu, Z. Wang, W. Yi, and P. Xue, Non-Hermitian bulk–boundary correspondence in quantum dynamics, Nat. Phys. 16, 761 (2020).
  25. L. Jin and Z. Song, Bulk-boundary correspondence in a non-Hermitian system in one dimension with chiral inversion symmetry, Phys. Rev. B 99, 081103(R) (2019).
  26. S. R. Padhi, S. Roy, B. Paul, S. Banerjee, and T. Mishra, Non-Hermitian comb effect in coupled clean and quasiperiodic chains, Phys. Rev. B 114, 024202 (2026).
  27. S. R. Padhi, S. Banerjee, T. Nag, and T. Mishra, Anomalous spectrum in a non-Hermitian quasiperiodic chain, Phys. Rev. B 112, 174201 (2025).
  28. S. Aubry and G. André, Analyticity breaking and Anderson localization in incommensurate lattices, Ann. Israel Phys. Soc. 3, 133 (1980).
  29. P. G. Harper, Single band motion of conduction electrons in a uniform magnetic field, Proc. Phys. Soc. A 68, 874 (1955).
  30. G. A. Domínguez-Castro and R. Paredes, The Aubry–André model as a hobbyhorse for understanding the localization phenomenon, Eur. J. Phys. 40, 045403 (2019).
  31. G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, Anderson localization of a non-interacting Bose–Einstein condensate, Nature (London) 453, 895 (2008).
  32. S. Longhi, Topological phase transition in non-Hermitian quasicrystals, Phys. Rev. Lett. 122, 237601 (2019).
  33. L.-Z. Tang, G.-Q. Zhang, L.-F. Zhang, and D.-W. Zhang, Localization and topological transitions in non-Hermitian quasiperiodic lattices, Phys. Rev. A 103, 033325 (2021).
  34. X. Cai, Localization transitions and winding numbers for non-Hermitian Aubry-André-Harper models with off-diagonal modulations, Phys. Rev. B 106, 214207 (2022).
  35. Y. Liu, Q. Zhou, and S. Chen, Localization transition, spectrum structure, and winding numbers for one-dimensional non-Hermitian quasicrystals, Phys. Rev. B 104, 024201 (2021).
  36. L. Zhou, Non-Abelian generalization of non-Hermitian quasicrystals: PT-symmetry breaking, localization, entanglement, and topological transitions, Phys. Rev. B 108, 014202 (2023).
  37. H. Wang, X. Zheng, J. Chen, L. Xiao, S. Jia, and L. Zhang, Fate of the reentrant localization phenomenon in the one-dimensional dimerized quasiperiodic chain with long-range hopping, Phys. Rev. B 107, 075128 (2023).
  38. W. Han and L. Zhou, Dimerization-induced mobility edges and multiple reentrant localization transitions in non-Hermitian quasicrystals, Phys. Rev. B 105, 054204 (2022).
  39. C. Wu, J. Fan, G. Chen, and S. Jia, Non-Hermiticity-induced reentrant localization in a quasiperiodic lattice, New J. Phys. 23, 123048 (2021).
  40. Q.-B. Zeng, Y.-B. Yang, and Y. Xu, Topological phases in non-Hermitian Aubry-André-Harper models, Phys. Rev. B 101, 020201(R) (2020).
  41. Y. Liu, Y. Wang, X.-J. Liu, Q. Zhou, and S. Chen, Exact mobility edges, PT-symmetry breaking, and skin effect in one-dimensional non-Hermitian quasicrystals, Phys. Rev. B 103, 014203 (2021).
  42. Y. Liu, Y. Wang, Z. Zheng, and S. Chen, Exact non-Hermitian mobility edges in one-dimensional quasicrystal lattice with exponentially decaying hopping and its dual lattice, Phys. Rev. B 103, 134208 (2021).
  43. Z.-H. Wang, F. Xu, L. Li, D.-H. Xu, and B. Wang, Unconventional real-complex spectral transition and Majorana zero modes in nonreciprocal quasicrystals, Phys. Rev. B 104, 174501 (2021).
  44. S. Longhi, Non-Hermitian Maryland model, Phys. Rev. B 103, 224206 (2021).
  45. L. Zhou, Floquet engineering of topological localization transitions and mobility edges in one-dimensional non-Hermitian quasicrystals, Phys. Rev. Res. 3, 033184 (2021).
  46. L. Zhou and Y. Gu, Topological delocalization transitions and mobility edges in the nonreciprocal Maryland model, J. Phys.: Condens. Matter 34, 115402 (2022).
  47. Q.-B. Zeng and Y. Xu, Winding numbers and generalized mobility edges in non-Hermitian systems, Phys. Rev. Res. 2, 033052 (2020).
  48. S.-Z. Li and Z. Li, Ring structure in the complex plane: A fingerprint of a non-Hermitian mobility edge, Phys. Rev. B 110, L041102 (2024).
  49. A. Padhan, S. R. Padhi, and T. Mishra, Complete delocalization and reentrant topological transition in a non-Hermitian quasiperiodic lattice, Phys. Rev. B 109, L020203 (2024).
  50. S. Gandhi and J. N. Bandyopadhyay, Non-Hermitian Aubry-André-Harper model with short- and long-range p-wave pairing, Phys. Rev. B 110, 094203 (2024).
  51. D. Peng, S. Cheng, and G. Xianlong, Power law hopping of single particles in one-dimensional non-Hermitian quasicrystals, Phys. Rev. B 107, 174205 (2023).
  52. Z. Xu, X. Xia, and S. Chen, Non-Hermitian Aubry-André model with power-law hopping, Phys. Rev. B 104, 224204 (2021).
  53. T. Liu, H. Guo, Y. Pu, and S. Longhi, Generalized Aubry-André self-duality and mobility edges in non-Hermitian quasiperiodic lattices, Phys. Rev. B 102, 024205 (2020).
  54. X.-P. Jiang, W. Zeng, Y. Hu, and L. Pan, Exact anomalous mobility edges in one-dimensional non-Hermitian quasicrystals, arXiv:2409.03591 [cond-mat.dis-nn].
  55. A. P. Acharya, A. Chakrabarty, D. K. Sahu, and S. Datta, Localization, PT symmetry breaking, and topological transitions in non-Hermitian quasicrystals, Phys. Rev. B 105, 014202 (2022).
  56. Y.-P. Wang, C.-K. Chang, R. Okugawa, and C.-H. Hsu, Quasiperiodicity-induced bulk localization with self-similarity in non-Hermitian systems, Phys. Rev. Res. 7, 043353 (2025).
  57. L. Du et al., Experimental observation of interconversion of non-Hermitian skin effect and topological edge states, Phys. Rev. B 112, 115417 (2025).
  58. Q. Lin, T. Li, L. Xiao, K. Wang, W. Yi, and P. Xue, Topological phase transitions and mobility edges in non-Hermitian quasicrystals, Phys. Rev. Lett. 129, 113601 (2022).
  59. A. Ghatak, D. H. Kaltsas, M. Kulkarni, and K. G. Makris, Diffraction and pseudospectra in non-Hermitian quasiperiodic lattices, Phys. Rev. E 110, 064228 (2024).
  60. S. Weidemann, M. Kremer, S. Longhi, and A. Szameit, Topological triple phase transition in non-Hermitian Floquet quasicrystals, Nature (London) 601, 354 (2022).
  61. R. K. Malla and M. E. Raikh, Spinful aubry-andré model in a magnetic field: Delocalization facilitated by a weak spin-orbit coupling, Phys. Rev. B 97, 214209 (2018).
  62. D. Kumar Sahu, A. P. Acharya, D. Choudhuri, and S. Datta, Self-duality of one-dimensional quasicrystals with spin-orbit interaction, Phys. Rev. B 104, 054202 (2021).
  63. E. Zhao, Y.-J. Liu, K. K. Pak, P. Ren, M. Guo, C. He, and G.-B. Jo, Localization behavior in a Hermitian and non-Hermitian Raman lattice, Phys. Rev. A 113, 053310 (2026).
  64. Z. W. Lin Wang and S. Chen, Non-Hermitian butterfly spectra in a family of quasiperiodic lattices, Phys. Rev. B 110, L060201 (2024).
  65. T. Mishra, D. Chowdhury, S. Roy, and S. R. Padhi, Spin-aligned butterfly spectral map in non-Hermitian quasicrystals [Data set], Zenodo, 2026, http://doi.org/10.5281/zenodo.18465080

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APS Open Sci. 1, L000079– Published 3 August, 2026
Vol. 1
  • Received 11 February 2026
  • Revised 20 July 2026
  • Accepted 23 July 2026
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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.

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