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  • Letter
  • Open Access

Non-Hermitian higher-order topological superconductors in two dimensions: Statics and dynamics

Arnob Kumar Ghosh1,2,* and Tanay Nag3,†

  • 1Institute of Physics, Sachivalaya Marg, Bhubaneswar 751005, India
  • 2Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400094, India
  • 3Department of Physics and Astronomy, Uppsala University, Box 516, 75120 Uppsala, Sweden

  • *arnob@iopb.res.in
  • †tanay.nag@physics.uu.se
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Phys. Rev. B 106, L140303 – Published 19 October, 2022

DOI: https://doi.org/10.1103/PhysRevB.106.L140303

Abstract

Being motivated by intriguing phenomena, such as the breakdown of conventional bulk boundary correspondence and the emergence of skin modes in the context of non-Hermitian (NH) topological insulators, we here propose a NH second-order topological superconductor (SOTSC) model that hosts Majorana zero modes (MZMs). Employing the non-Bloch form of the NH Hamiltonian, we topologically characterize the above modes by biorthogonal nested polarization and resolve the apparent breakdown of the bulk boundary correspondence. Unlike the Hermitian SOTSC, we note that the MZMs inhabit only one corner out of four in the two-dimensional NH SOTSCs. Such a localization profile of MZMs is protected by mirror rotation symmetry and remains robust under on-site random disorder. We extend the static MZMs into the realm of the Floquet drive. We find the anomalous π mode following low-frequency mass kick in addition to the regular 0 mode that is usually engineered in a high-frequency regime. We further characterize the regular 0 mode with biorthogonal Floquet nested polarization. Our proposal is not limited to the d-wave superconductivity only and can be realized in the experiment with strongly correlated optical lattice platforms.

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References (132)

  1. M. Z. Hasan and C. L. Kane, Colloquium: Topological insulators, Rev. Mod. Phys. 82, 3045 (2010).
  2. F. D. M. Haldane, Model for a Quantum Hall Effect without Landau Levels: Condensed-Matter Realization of the Parity Anomaly, Phys. Rev. Lett. 61, 2015 (1988).
  3. C. L. Kane and E. J. Mele, Z2 Topological Order and the Quantum Spin Hall Effect, Phys. Rev. Lett. 95, 146802 (2005).
  4. B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Quantum spin hall effect and topological phase transition in hgte quantum wells, Science 314, 1757 (2006).
  5. J. Alicea, New directions in the pursuit of majorana fermions in solid state systems, Rep. Prog. Phys. 75, 076501 (2012).
  6. C. Beenakker, Search for majorana fermions in superconductors, Annu. Rev. Condens. Matter Phys. 4, 113 (2013).
  7. G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger, D. Greif, and T. Esslinger, Experimental realization of the topological haldane model with ultracold fermions, Nature 515, 237 (2014).
  8. A. Das, Y. Ronen, Y. Most, Y. Oreg, M. Heiblum, and H. Shtrikman, Zero-bias peaks and splitting in an al–inas nanowire topological superconductor as a signature of majorana fermions, Nat. Phys. 8, 887 (2012).
  9. W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, Quantized electric multipole insulators, Science 357, 61 (2017).
  10. W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, Electric multipole moments, topological multipole moment pumping, and chiral hinge states in crystalline insulators, Phys. Rev. B 96, 245115 (2017).
  11. Z. Song, Z. Fang, and C. Fang, (d−2)-Dimensional Edge States of Rotation Symmetry Protected Topological States, Phys. Rev. Lett. 119, 246402 (2017).
  12. J. Langbehn, Y. Peng, L. Trifunovic, F. von Oppen, and P. W. Brouwer, Reflection-Symmetric Second-Order Topological Insulators and Superconductors, Phys. Rev. Lett. 119, 246401 (2017).
  13. F. Schindler, A. M. Cook, M. G. Vergniory, Z. Wang, S. S. Parkin, B. A. Bernevig, and T. Neupert, Higher-order topological insulators, Sci. Adv. 4, eaat0346 (2018).
  14. S. Franca, J. van den Brink, and I. C. Fulga, An anomalous higher-order topological insulator, Phys. Rev. B 98, 201114(R) (2018).
  15. Z. Wang, B. J. Wieder, J. Li, B. Yan, and B. A. Bernevig, Higher-Order Topology, Monopole Nodal Lines, and the Origin of Large Fermi Arcs in Transition Metal Dichalcogenides xte2 (x=Mo,W), Phys. Rev. Lett. 123, 186401 (2019).
  16. D. Călugăru, V. Juričić, and B. Roy, Higher-order topological phases: A general principle of construction, Phys. Rev. B 99, 041301(R) (2019).
  17. P. Szumniak, D. Loss, and J. Klinovaja, Hinge modes and surface states in second-order topological three-dimensional quantum hall systems induced by charge density modulation, Phys. Rev. B 102, 125126 (2020).
  18. X. Ni, M. Li, M. Weiner, A. Alù, and A. B. Khanikaev, Demonstration of a quantized acoustic octupole topological insulator, Nat. Commun. 11, 2108 (2020).
  19. B. Xie, H. Wang, X. Zhang, P. Zhan, J. Jiang, M. Lu, and Y. Chen, Higher-order band topology, Nat. Rev. Phys. 3, 520 (2021).
  20. S. Saha, T. Nag, and S. Mandal, Dipolar quantum spin hall insulator phase in extended haldane model, arXiv:2204.06641.
  21. X. Zhu, Tunable majorana corner states in a two-dimensional second-order topological superconductor induced by magnetic fields, Phys. Rev. B 97, 205134 (2018).
  22. T. Liu, J. J. He, and F. Nori, Majorana corner states in a two-dimensional magnetic topological insulator on a high-temperature superconductor, Phys. Rev. B 98, 245413 (2018).
  23. Z. Yan, F. Song, and Z. Wang, Majorana Corner Modes in a High-Temperature Platform, Phys. Rev. Lett. 121, 096803 (2018).
  24. Y. Wang, M. Lin, and T. L. Hughes, Weak-pairing higher order topological superconductors, Phys. Rev. B 98, 165144 (2018).
  25. R.-X. Zhang, W. S. Cole, and S. Das Sarma, Helical Hinge Majorana Modes in Iron-Based Superconductors, Phys. Rev. Lett. 122, 187001 (2019).
  26. R.-X. Zhang, W. S. Cole, X. Wu, and S. Das Sarma, Higher-Order Topology and Nodal Topological Superconductivity in Fe(Se,Te) Heterostructures, Phys. Rev. Lett. 123, 167001 (2019).
  27. Y. Volpez, D. Loss, and J. Klinovaja, Second-Order Topological Superconductivity in π-Junction Rashba Layers, Phys. Rev. Lett. 122, 126402 (2019).
  28. Z. Yan, Majorana corner and hinge modes in second-order topological insulator/superconductor heterostructures, Phys. Rev. B 100, 205406 (2019).
  29. S. A. A. Ghorashi, X. Hu, T. L. Hughes, and E. Rossi, Second-order dirac superconductors and magnetic field induced majorana hinge modes, Phys. Rev. B 100, 020509(R) (2019).
  30. Y.-J. Wu, J. Hou, Y.-M. Li, X.-W. Luo, X. Shi, and C. Zhang, In-Plane Zeeman-Field-Induced Majorana Corner and Hinge Modes in an s-Wave Superconductor Heterostructure, Phys. Rev. Lett. 124, 227001 (2020).
  31. K. Laubscher, D. Chughtai, D. Loss, and J. Klinovaja, Kramers pairs of majorana corner states in a topological insulator bilayer, Phys. Rev. B 102, 195401 (2020).
  32. B. Roy, Higher-order topological superconductors in P-, T-odd quadrupolar dirac materials, Phys. Rev. B 101, 220506(R) (2020).
  33. S.-B. Zhang, W. B. Rui, A. Calzona, S.-J. Choi, A. P. Schnyder, and B. Trauzettel, Topological and holonomic quantum computation based on second-order topological superconductors, Phys. Rev. Res. 2, 043025 (2020).
  34. M. Kheirkhah, Z. Yan, and F. Marsiglio, Vortex-line topology in iron-based superconductors with and without second-order topology, Phys. Rev. B 103, L140502 (2021).
  35. Z. Yan, Higher-Order Topological Odd-Parity Superconductors, Phys. Rev. Lett. 123, 177001 (2019).
  36. J. Ahn and B.-J. Yang, Higher-order topological superconductivity of spin-polarized fermions, Phys. Rev. Res. 2, 012060(R) (2020).
  37. X.-J. Luo, X.-H. Pan, and X. Liu, Higher-order topological superconductors based on weak topological insulators, Phys. Rev. B 104, 104510 (2021).
  38. Q. Wang, C.-C. Liu, Y.-M. Lu, and F. Zhang, High-Temperature Majorana Corner States, Phys. Rev. Lett. 121, 186801 (2018).
  39. A. K. Ghosh, T. Nag, and A. Saha, Hierarchy of higher-order topological superconductors in three dimensions, Phys. Rev. B 104, 134508 (2021).
  40. B. Roy and V. Juričić, Mixed-parity octupolar pairing and corner majorana modes in three dimensions, Phys. Rev. B 104, L180503 (2021).
  41. T. Li, M. Geier, J. Ingham, and H. D. Scammell, Higher-order topological superconductivity from repulsive interactions in kagome and honeycomb systems, 2D Mater. 9, 015031 (2022).
  42. R. W. Bomantara, L. Zhou, J. Pan, and J. Gong, Coupled-wire construction of static and floquet second-order topological insulators, Phys. Rev. B 99, 045441 (2019).
  43. T. Nag, V. Juričić, and B. Roy, Out of equilibrium higher-order topological insulator: Floquet engineering and quench dynamics, Phys. Rev. Res. 1, 032045(R) (2019).
  44. Y. Peng and G. Refael, Floquet Second-Order Topological Insulators from Nonsymmorphic Space-Time Symmetries, Phys. Rev. Lett. 123, 016806 (2019).
  45. R. Seshadri, A. Dutta, and D. Sen, Generating a second-order topological insulator with multiple corner states by periodic driving, Phys. Rev. B 100, 115403 (2019).
  46. M. Rodriguez-Vega, A. Kumar, and B. Seradjeh, Higher-order floquet topological phases with corner and bulk bound states, Phys. Rev. B 100, 085138 (2019).
  47. A. K. Ghosh, G. C. Paul, and A. Saha, Higher order topological insulator via periodic driving, Phys. Rev. B 101, 235403 (2020).
  48. B. Huang and W. V. Liu, Floquet Higher-Order Topological Insulators with Anomalous Dynamical Polarization, Phys. Rev. Lett. 124, 216601 (2020).
  49. H. Hu, B. Huang, E. Zhao, and W. V. Liu, Dynamical Singularities of Floquet Higher-Order Topological Insulators, Phys. Rev. Lett. 124, 057001 (2020).
  50. Y. Peng, Floquet higher-order topological insulators and superconductors with space-time symmetries, Phys. Rev. Res. 2, 013124 (2020).
  51. T. Nag, V. Juričić, and B. Roy, Hierarchy of higher-order floquet topological phases in three dimensions, Phys. Rev. B 103, 115308 (2021).
  52. R.-X. Zhang and Z.-C. Yang, Tunable fragile topology in floquet systems, Phys. Rev. B 103, L121115 (2021).
  53. R. V. Bhat and S. Bera, Out of equilibrium chiral higher order topological insulator on a π-flux square lattice, J. Phys.: Condens. Matter 33, 164005 (2021).
  54. W. Zhu, Y. D. Chong, and J. Gong, Floquet higher-order topological insulator in a periodically driven bipartite lattice, Phys. Rev. B 103, L041402 (2021).
  55. J. Yu, R.-X. Zhang, and Z.-D. Song, Dynamical symmetry indicators for floquet crystals, Nat. Commun. 12, 5985 (2021).
  56. D. Vu, Dynamic bulk-boundary correspondence for anomalous floquet topology, Phys. Rev. B 105, 064304 (2022).
  57. A. K. Ghosh, T. Nag, and A. Saha, Systematic generation of the cascade of anomalous dynamical first- and higher-order modes in floquet topological insulators, Phys. Rev. B 105, 115418 (2022).
  58. X.-L. Du, R. Chen, R. Wang, and D.-H. Xu, Weyl nodes with higher-order topology in an optically driven nodal-line semimetal, Phys. Rev. B 105, L081102 (2022).
  59. Z. Ning, B. Fu, D.-H. Xu, and R. Wang, Tailoring quadrupole topological insulators with periodic driving and disorder, Phys. Rev. B 105, L201114 (2022).
  60. K. Plekhanov, M. Thakurathi, D. Loss, and J. Klinovaja, Floquet second-order topological superconductor driven via ferromagnetic resonance, Phys. Rev. Res. 1, 032013 (2019).
  61. R. W. Bomantara and J. Gong, Measurement-only quantum computation with floquet majorana corner modes, Phys. Rev. B 101, 085401 (2020).
  62. R. W. Bomantara, Time-induced second-order topological superconductors, Phys. Rev. Res. 2, 033495 (2020).
  63. A. K. Ghosh, T. Nag, and A. Saha, Floquet generation of a second-order topological superconductor, Phys. Rev. B 103, 045424 (2021).
  64. A. K. Ghosh, T. Nag, and A. Saha, Floquet second order topological superconductor based on unconventional pairing, Phys. Rev. B 103, 085413 (2021).
  65. D. D. Vu, R.-X. Zhang, Z.-C. Yang, and S. Das Sarma, Superconductors with anomalous floquet higher-order topology, Phys. Rev. B 104, L140502 (2021).
  66. A. K. Ghosh, T. Nag, and A. Saha, Dynamical construction of quadrupolar and octupolar topological superconductors, Phys. Rev. B 105, 155406 (2022).
  67. N. Parappurath, F. Alpeggiani, L. Kuipers, and E. Verhagen, Direct observation of topological edge states in silicon photonic crystals: Spin, dispersion, and chiral routing, Sci. Adv. 6, eaaw4137 (2020).
  68. Y. Yang, Z. Gao, H. Xue, L. Zhang, M. He, Z. Yang, R. Singh, Y. Chong, B. Zhang, and H. Chen, Realization of a three-dimensional photonic topological insulator, Nature 565, 622 (2019).
  69. S. Malzard, C. Poli, and H. Schomerus, Topologically Protected Defect States in Open Photonic Systems with Non-Hermitian Charge-Conjugation and Parity-Time Symmetry, Phys. Rev. Lett. 115, 200402 (2015).
  70. A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, Parity–time synthetic photonic lattices, Nature 488, 167 (2012).
  71. R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, Non-hermitian physics and pt symmetry, Nat. Phys. 14, 11 (2018).
  72. M. M. Denner, A. Skurativska, F. Schindler, M. H. Fischer, R. Thomale, T. Bzdušek, and T. Neupert, Exceptional topological insulators, Nat. Commun. 12, 5681 (2021).
  73. Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, Optical solitons in PT periodic potentials, Phys. Rev. Lett. 100, 030402 (2008).
  74. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Beam dynamics in PT symmetric optical lattices, Phys. Rev. Lett. 100, 103904 (2008).
  75. E. J. Bergholtz and J. C. Budich, Non-hermitian weyl physics in topological insulator ferromagnet junctions, Phys. Rev. Res. 1, 012003(R) (2019).
  76. K. Yang, S. C. Morampudi, and E. J. Bergholtz, Exceptional Spin Liquids from Couplings to the Environment, Phys. Rev. Lett. 126, 077201 (2021).
  77. P. San-Jose, J. Cayao, E. Prada, and R. Aguado, Majorana bound states from exceptional points in non-topological superconductors, Sci. Rep. 6, 21427 (2016).
  78. V. Kozii and L. Fu, Non-hermitian topological theory of finite-lifetime quasiparticles: Prediction of bulk fermi arc due to exceptional point, arXiv:1708.05841.
  79. T. Yoshida, R. Peters, and N. Kawakami, Non-hermitian perspective of the band structure in heavy-fermion systems, Phys. Rev. B 98, 035141 (2018).
  80. H. Shen and L. Fu, Quantum Oscillation from In-Gap States and a Non-Hermitian Landau Level Problem, Phys. Rev. Lett. 121, 026403 (2018).
  81. S. Yao, F. Song, and Z. Wang, Non-Hermitian Chern Bands, Phys. Rev. Lett. 121, 136802 (2018).
  82. K. Kawabata, K. Shiozaki, M. Ueda, and M. Sato, Symmetry and Topology in Non-Hermitian Physics, Phys. Rev. X 9, 041015 (2019).
  83. E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Exceptional topology of non-hermitian systems, Rev. Mod. Phys. 93, 015005 (2021).
  84. K. Sone, Y. Ashida, and T. Sagawa, Exceptional non-hermitian topological edge mode and its application to active matter, Nat. Commun. 11, 5745 (2020).
  85. C. M. Bender, Making sense of non-hermitian hamiltonians, Rep. Prog. Phys. 70, 947 (2007).
  86. W. D. Heiss, The physics of exceptional points, J. Phys. A: Math. Theor. 45, 444016 (2012).
  87. S. Yao and Z. Wang, Edge States and Topological Invariants of Non-Hermitian Systems, Phys. Rev. Lett. 121, 086803 (2018).
  88. 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).
  89. T. Helbig, T. Hofmann, S. Imhof, M. Abdelghany, T. Kiessling, L. W. Molenkamp, C. H. Lee, A. Szameit, M. Greiter, and R. Thomale, Generalized bulk–boundary correspondence in non-hermitian topolectrical circuits, Nat. Phys. 16, 747 (2020).
  90. D. S. Borgnia, A. J. Kruchkov, and R.-J. Slager, Non-Hermitian Boundary Modes and Topology, Phys. Rev. Lett. 124, 056802 (2020).
  91. 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).
  92. H.-G. Zirnstein, G. Refael, and B. Rosenow, Bulk-Boundary Correspondence for Non-Hermitian Hamiltonians via Green Functions, Phys. Rev. Lett. 126, 216407 (2021).
  93. Y. Takane, Bulk-boundary correspondence in a non-hermitian chern insulator, J. Phys. Soc. Jpn. 90, 033704 (2021).
  94. K. Kawabata, M. Sato, and K. Shiozaki, Higher-order non-hermitian skin effect, Phys. Rev. B 102, 205118 (2020).
  95. K. Zhang, Z. Yang, and C. Fang, Correspondence between Winding Numbers and Skin Modes in Non-Hermitian Systems, Phys. Rev. Lett. 125, 126402 (2020).
  96. 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).
  97. C. Yin, H. Jiang, L. Li, R. Lü, and S. Chen, Geometrical meaning of winding number and its characterization of topological phases in one-dimensional chiral non-hermitian systems, Phys. Rev. A 97, 052115 (2018).
  98. A. Banerjee and A. Narayan, Controlling exceptional points with light, Phys. Rev. B 102, 205423 (2020).
  99. D. Chowdhury, A. Banerjee, and A. Narayan, Light-driven lifshitz transitions in non-hermitian multi-weyl semimetals, Phys. Rev. A 103, L051101 (2021).
  100. T. Liu, Y.-R. Zhang, Q. Ai, Z. Gong, K. Kawabata, M. Ueda, and F. Nori, Second-Order Topological Phases in Non-Hermitian Systems, Phys. Rev. Lett. 122, 076801 (2019).
  101. X.-W. Luo and C. Zhang, Higher-Order Topological Corner States Induced by Gain and Loss, Phys. Rev. Lett. 123, 073601 (2019).
  102. E. Edvardsson, F. K. Kunst, and E. J. Bergholtz, Non-hermitian extensions of higher-order topological phases and their biorthogonal bulk-boundary correspondence, Phys. Rev. B 99, 081302(R) (2019).
  103. Z. Zhang, M. Rosendo López, Y. Cheng, X. Liu, and J. Christensen, Non-Hermitian Sonic Second-Order Topological Insulator, Phys. Rev. Lett. 122, 195501 (2019).
  104. C. H. Lee, L. Li, and J. Gong, Hybrid Higher-Order Skin-Topological Modes in Nonreciprocal Systems, Phys. Rev. Lett. 123, 016805 (2019).
  105. Y.-J. Wu, C.-C. Liu, and J. Hou, Wannier-type photonic higher-order topological corner states induced solely by gain and loss, Phys. Rev. A 101, 043833 (2020).
  106. R. Okugawa, R. Takahashi, and K. Yokomizo, Second-order topological non-hermitian skin effects, Phys. Rev. B 102, 241202(R) (2020).
  107. R. Okugawa, R. Takahashi, and K. Yokomizo, Non-hermitian band topology with generalized inversion symmetry, Phys. Rev. B 103, 205205 (2021).
  108. K. Shiozaki and S. Ono, Symmetry indicator in non-hermitian systems, Phys. Rev. B 104, 035424 (2021).
  109. Y. Yu, M. Jung, and G. Shvets, Zero-energy corner states in a non-hermitian quadrupole insulator, Phys. Rev. B 103, L041102 (2021).
  110. N. Okuma and M. Sato, Topological Phase Transition Driven by Infinitesimal Instability: Majorana Fermions in Non-Hermitian Spintronics, Phys. Rev. Lett. 123, 097701 (2019).
  111. S. Lieu, Non-hermitian majorana modes protect degenerate steady states, Phys. Rev. B 100, 085110 (2019).
  112. J. Avila, F. Peñaranda, E. Prada, P. San-Jose, and R. Aguado, Non-hermitian topology as a unifying framework for the andreev versus majorana states controversy, Commun. Phys. 2, 133 (2019).
  113. X.-M. Zhao, C.-X. Guo, M.-L. Yang, H. Wang, W.-M. Liu, and S.-P. Kou, Anomalous non-abelian statistics for non-hermitian generalization of majorana zero modes, Phys. Rev. B 104, 214502 (2021).
  114. Z.-H. Wang, F. Xu, L. Li, D.-H. Xu, W.-Q. Chen, and B. Wang, Majorana polarization in non-hermitian topological superconductors, Phys. Rev. B 103, 134507 (2021).
  115. H. Liu, M. Lu, Y. Wu, J. Liu, and X. C. Xie, Non-hermiticity stabilized majorana zero modes in semiconductor-superconductor nanowires, Phys. Rev. B 106, 064505 (2022).
  116. L. Zhou, Non-hermitian floquet topological superconductors with multiple majorana edge modes, Phys. Rev. B 101, 014306 (2020).
  117. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevB.106.L140303 for discussions on underlying symmetry of the model, localization of MZMs for the asymmetry in the non-Hermiticity, symmetry constraint of the Wannier band, effect of disorder, and tuning the number of Majorana corner modes dynamically.
  118. O. Viyuela, D. Vodola, G. Pupillo, and M. A. Martin-Delgado, Topological massive dirac edge modes and long-range superconducting hamiltonians, Phys. Rev. B 94, 125121 (2016).
  119. H. Shen, B. Zhen, and L. Fu, Topological Band Theory for Non-Hermitian Hamiltonians, Phys. Rev. Lett. 120, 146402 (2018).
  120. K. Esaki, M. Sato, K. Hasebe, and M. Kohmoto, Edge states and topological phases in non-hermitian systems, Phys. Rev. B 84, 205128 (2011).
  121. W. DeGottardi, M. Thakurathi, S. Vishveshwara, and D. Sen, Majorana fermions in superconducting wires: Effects of long-range hopping, broken time-reversal symmetry, and potential landscapes, Phys. Rev. B 88, 165111 (2013).
  122. W. A. Benalcazar and A. Cerjan, Chiral-Symmetric Higher-Order Topological Phases of Matter, Phys. Rev. Lett. 128, 127601 (2022).
  123. W. Chen, M. Legner, A. Rüegg, and M. Sigrist, Correlation length, universality classes, and scaling laws associated with topological phase transitions, Phys. Rev. B 95, 075116 (2017).
  124. R. Arouca, C. H. Lee, and C. Morais Smith, Unconventional scaling at non-hermitian critical points, Phys. Rev. B 102, 245145 (2020).
  125. L. Huang, Z. Meng, P. Wang, P. Peng, S.-L. Zhang, L. Chen, D. Li, Q. Zhou, and J. Zhang, Experimental realization of two-dimensional synthetic spin–orbit coupling in ultracold fermi gases, Nat. Phys. 12, 540 (2016).
  126. Z. Wu, L. Zhang, W. Sun, X.-T. Xu, B.-Z. Wang, S.-C. Ji, Y. Deng, S. Chen, X.-J. Liu, and J.-W. Pan, Realization of two-dimensional spin-orbit coupling for bose-einstein condensates, Science 354, 83 (2016).
  127. J. Li, A. K. Harter, J. Liu, L. de Melo, Y. N. Joglekar, and L. Luo, Observation of parity-time symmetry breaking transitions in a dissipative floquet system of ultracold atoms, Nat. Commun. 10, 855 (2019).
  128. W. Song, W. Sun, C. Chen, Q. Song, S. Xiao, S. Zhu, and T. Li, Breakup and Recovery of Topological Zero Modes in Finite Non-Hermitian Optical Lattices, Phys. Rev. Lett. 123, 165701 (2019).
  129. W. Jia, Z.-H. Huang, X. Wei, Q. Zhao, and X.-J. Liu, Topological superfluids for spin-orbit coupled ultracold fermi gases, Phys. Rev. B 99, 094520 (2019).
  130. A. Bühler, N. Lang, C. V. Kraus, G. Möller, S. D. Huber, and H.-P. Büchler, Majorana modes and p-wave superfluids for fermionic atoms in optical lattices, Nat. Commun. 5, 4504 (2014).
  131. H. Miyake, G. A. Siviloglou, C. J. Kennedy, W. C. Burton, and W. Ketterle, Realizing the Harper Hamiltonian with Laser-Assisted Tunneling in Optical Lattices, Phys. Rev. Lett. 111, 185302 (2013).
  132. D. McKay and B. DeMarco, Cooling in strongly correlated optical lattices: Prospects and challenges, Rep. Prog. Phys. 74, 054401 (2011).

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Phys. Rev. B 106, L140303– Published 19 October, 2022
  • Received 1 June 2022
  • Accepted 11 October 2022
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