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Dissipation-induced Majarona 0- and π-modes in a driven Rashba nanowire

Koustabh Gogoi1, Tanay Nag1,*, and Arnob Kumar Ghosh2,†

  • *Contact author: tanay.nag@hyderabad.bits-pilani.ac.in
  • †Contact author: arnob.ghosh@physics.uu.se
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Phys. Rev. B 113, 054301 – Published 4 February, 2026

DOI: https://doi.org/10.1103/fxs7-wdqp

Abstract

Periodic drive is an intriguing way of creating topological phases in a nontopological setup. However, most systems are often studied as a closed system, despite being always in contact with the environment, which induces dissipation. Here, we investigate a periodically driven Rashba nanowire in proximity to an s-wave superconductor in a dissipative background. The system's dynamics is governed by a periodic Liouvillian operator, from which we construct the Liouvillian time-evolution operator and use the third-quantization method to obtain the ‘‘Floquet damping matrix’’, which captures the spectral and topological properties of the system. We show that the system exhibits edge-localized topological Majorana 0-modes (MZMs) and π-modes (MPMs). Additionally, the system also supports a trivial 0-modes (TZMs) and π-modes (TPMs), which are also localized at the edges of the system. The MZMs and the MPMs are connected to the bulk topology and carry a bulk topological invariant, while the emergence of TZMs and TPMs is primarily tied to exceptional points and is topologically trivial. We show that both the topological (MZMs and MPMs) and trivial (TZMs and TPMs) edge modes are robust against onsite disorder. We study the topological phase diagrams in terms of the topological invariants and show that the dissipation can modify the topological phase diagram substantially and even induce topological phases in the system. Our work extends the understanding of a driven-dissipative topological superconductor.

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References (104)

  1. A. Y. Kitaev, Unpaired Majorana fermions in quantum wires, Phys. Usp. 44, 131 (2001).
  2. X.-L. Qi and S.-C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys. 83, 1057 (2011).
  3. M. Leijnse and K. Flensberg, Introduction to topological superconductivity and Majorana fermions, Semicond. Sci. Technol. 27, 124003 (2012).
  4. J. Alicea, New directions in the pursuit of Majorana fermions in solid state systems, Rep. Prog. Phys. 75, 076501 (2012).
  5. C. Beenakker, Search for Majorana fermions in superconductors, Annu. Rev. Condens. Matter Phys. 4, 113 (2013).
  6. R. Aguado, Majorana quasiparticles in condensed matter, Riv. Nuovo Cimento 40, 523 (2017).
  7. Y. Tanaka, S. Tamura, and J. Cayao, Theory of Majorana zero modes in unconventional superconductors, Prog. Theor. Exp. Phys. 2024, 08C105 (2024).
  8. Y. Oreg, G. Refael, and F. von Oppen, Helical liquids and Majorana bound states in quantum wires, Phys. Rev. Lett. 105, 177002 (2010).
  9. R. M. Lutchyn, J. D. Sau, and S. Das Sarma, Majorana fermions and a topological phase transition in semiconductor-superconductor heterostructures, Phys. Rev. Lett. 105, 077001 (2010).
  10. V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P. A. M. Bakkers, and L. P. Kouwenhoven, Signatures of Majorana fermions in hybrid superconductor-semiconductor nanowire devices, Science 336, 1003 (2012).
  11. 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).
  12. M. T. Deng, C. L. Yu, G. Y. Huang, M. Larsson, P. Caroff, and H. Q. Xu, Anomalous zero-bias conductance peak in a Nb–InSb nanowire–Nb hybrid device, Nano Lett. 12, 6414 (2012).
  13. L. P. Rokhinson, X. Liu, and J. K. Furdyna, The fractional a.c. Josephson effect in a semiconductor–superconductor nanowire as a signature of Majorana particles, Nat. Phys. 8, 795 (2012).
  14. A. D. K. Finck, D. J. Van Harlingen, P. K. Mohseni, K. Jung, and X. Li, Anomalous modulation of a zero-bias peak in a hybrid nanowire-superconductor device, Phys. Rev. Lett. 110, 126406 (2013).
  15. S. M. Albrecht, A. P. Higginbotham, M. Madsen, F. Kuemmeth, T. S. Jespersen, J. Nygård, P. Krogstrup, and C. M. Marcus, Exponential protection of zero modes in Majorana islands, Nature (London) 531, 206 (2016).
  16. M. T. Deng, S. Vaitiekenas, E. B. Hansen, J. Danon, M. Leijnse, K. Flensberg, J. Nygård, P. Krogstrup, and C. M. Marcus, Majorana bound state in a coupled quantum-dot hybrid-nanowire system, Science 354, 1557 (2016).
  17. F. Nichele, A. C. C. Drachmann, A. M. Whiticar, E. C. T. O'Farrell, H. J. Suominen, A. Fornieri, T. Wang, G. C. Gardner, C. Thomas, A. T. Hatke, P. Krogstrup, M. J. Manfra, K. Flensberg, and C. M. Marcus, Scaling of Majorana zero-bias conductance peaks, Phys. Rev. Lett. 119, 136803 (2017).
  18. J. Chen, P. Yu, J. Stenger, M. Hocevar, D. Car, S. R. Plissard, E. P. A. M. Bakkers, T. D. Stanescu, and S. M. Frolov, Experimental phase diagram of zero-bias conductance peaks in superconductor/semiconductor nanowire devices, Sci. Adv. 3, e1701476 (2017).
  19. H. Zhang, Ö. Gül, S. Conesa-Boj, M. P. Nowak, M. Wimmer, K. Zuo, V. Mourik, F. K. de Vries, J. van Veen, M. W. A. de Moor, et al., Ballistic superconductivity in semiconductor nanowires, Nat. Commun. 8, 16025 (2017).
  20. Ö. Gül, H. Zhang, J. D. S. Bommer, M. W. A. de Moor, D. Car, S. R. Plissard, E. P. A. M. Bakkers, A. Geresdi, K. Watanabe, T. Taniguchi, and L. P. Kouwenhoven, Ballistic Majorana nanowire devices, Nat. Nanotechnol. 13, 192 (2018).
  21. A. Grivnin, E. Bor, M. Heiblum, Y. Oreg, and H. Shtrikman, Concomitant opening of a bulk-gap with an emerging possible Majorana zero mode, Nat. Commun. 10, 1940 (2019).
  22. J. Chen, B. D. Woods, P. Yu, M. Hocevar, D. Car, S. R. Plissard, E. P. A. M. Bakkers, T. D. Stanescu, and S. M. Frolov, Ubiquitous non-Majorana zero-bias conductance peaks in nanowire devices, Phys. Rev. Lett. 123, 107703 (2019).
  23. M. Leijnse and K. Flensberg, Parity qubits and poor man's Majorana bound states in double quantum dots, Phys. Rev. B 86, 134528 (2012).
  24. T. Dvir, G. Wang, N. van Loo, C.-X. Liu, G. P. Mazur, A. Bordin, S. L. D. ten Haaf, J.-Y. Wang, D. van Driel, F. Zatelli, et al., Realization of a minimal Kitaev chain in coupled quantum dots, Nature (London) 614, 445 (2023).
  25. L. Jiang, T. Kitagawa, J. Alicea, A. R. Akhmerov, D. Pekker, G. Refael, J. I. Cirac, E. Demler, M. D. Lukin, and P. Zoller, Majorana fermions in equilibrium and in driven cold-atom quantum wires, Phys. Rev. Lett. 106, 220402 (2011).
  26. M. Thakurathi, A. A. Patel, D. Sen, and A. Dutta, Floquet generation of Majorana end modes and topological invariants, Phys. Rev. B 88, 155133 (2013).
  27. D. E. Liu, A. Levchenko, and H. U. Baranger, Floquet Majorana fermions for topological qubits in superconducting devices and cold-atom systems, Phys. Rev. Lett. 111, 047002 (2013).
  28. M. Benito, A. Gómez-León, V. M. Bastidas, T. Brandes, and G. Platero, Floquet engineering of long-range p-wave superconductivity, Phys. Rev. B 90, 205127 (2014).
  29. A. C. Potter, T. Morimoto, and A. Vishwanath, Classification of interacting topological Floquet phases in one dimension, Phys. Rev. X 6, 041001 (2016).
  30. M. Thakurathi, D. Loss, and J. Klinovaja, Floquet Majorana fermions and parafermions in driven Rashba nanowires, Phys. Rev. B 95, 155407 (2017).
  31. Z. Yang, Q. Yang, J. Hu, and D. E. Liu, Dissipative Floquet Majorana modes in proximity-induced topological superconductors, Phys. Rev. Lett. 126, 086801 (2021).
  32. A. K. Ghosh, T. Nag, and A. Saha, Dynamical construction of quadrupolar and octupolar topological superconductors, Phys. Rev. B 105, 155406 (2022).
  33. D. Mondal, A. K. Ghosh, T. Nag, and A. Saha, Topological characterization and stability of Floquet Majorana modes in Rashba nanowires, Phys. Rev. B 107, 035427 (2023).
  34. D. Mondal, A. K. Ghosh, T. Nag, and A. Saha, Engineering anomalous Floquet Majorana modes and their time evolution in a helical Shiba chain, Phys. Rev. B 108, L081403 (2023).
  35. A. K. Ghosh, T. Nag, and A. Saha, Time evolution of Majorana corner modes in a Floquet second-order topological superconductor, Phys. Rev. B 107, 035419 (2023).
  36. K. Roy and S. Basu, Single and multifrequency driving protocols in a Rashba nanowire proximitized to an s-wave superconductor, Phys. Rev. B 110, 165403 (2024).
  37. E. Ahmed, S. Tamura, Y. Tanaka, and J. Cayao, Odd-frequency superconducting pairing due to multiple Majorana edge modes in driven topological superconductors, Phys. Rev. B 111, 024507 (2025).
  38. R. W. Bomantara and J. Gong, Quantum computation via Floquet topological edge modes, Phys. Rev. B 98, 165421 (2018).
  39. B. Bauer, T. Pereg-Barnea, T. Karzig, M.-T. Rieder, G. Refael, E. Berg, and Y. Oreg, Topologically protected braiding in a single wire using Floquet Majorana modes, Phys. Rev. B 100, 041102 (2019).
  40. L. Zhou, Non-Hermitian Floquet topological superconductors with multiple Majorana edge modes, Phys. Rev. B 101, 014306 (2020).
  41. A. K. Ghosh and T. Nag, Non-Hermitian higher-order topological superconductors in two dimensions: Statics and dynamics, Phys. Rev. B 106, L140303 (2022).
  42. S. Yao and Z. Wang, Edge states and topological invariants of non-Hermitian systems, Phys. Rev. Lett. 121, 086803 (2018).
  43. 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).
  44. E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Exceptional topology of non-Hermitian systems, Rev. Mod. Phys. 93, 015005 (2021).
  45. N. Okuma and M. Sato, Non-Hermitian topological phenomena: A review, Annu. Rev. Condens. Matter Phys. 14, 83 (2023).
  46. C. H. Lee and R. Thomale, Anatomy of skin modes and topology in non-Hermitian systems, Phys. Rev. B 99, 201103 (2019).
  47. L. Li, C. H. Lee, S. Mu, and J. Gong, Critical non-Hermitian skin effect, Nat. Commun. 11, 5491 (2020).
  48. A. Banerjee, R. Sarkar, S. Dey, and A. Narayan, Non-Hermitian topological phases: Principles and prospects, J. Phys.: Condens. Matter 35, 333001 (2023).
  49. T. E. Lee, Anomalous edge state in a non-Hermitian lattice, Phys. Rev. Lett. 116, 133903 (2016).
  50. H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, Enhanced sensitivity at higher-order exceptional points, Nature (London) 548, 187 (2017).
  51. A. Ghatak and T. Das, New topological invariants in non-Hermitian systems, J. Phys.: Condens. Matter 31, 263001 (2019).
  52. Y. Ashida, Z. Gong, and M. Ueda, Non-Hermitian physics, Adv. Phys. 69, 249 (2020).
  53. 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).
  54. S. Sayyad and F. K. Kunst, Realizing exceptional points of any order in the presence of symmetry, Phys. Rev. Res. 4, 023130 (2022).
  55. D. Mondal and T. Nag, Anomaly in the dynamical quantum phase transition in a non-Hermitian system with extended gapless phases, Phys. Rev. B 106, 054308 (2022).
  56. D. Mondal and T. Nag, Finite-temperature dynamical quantum phase transition in a non-Hermitian system, Phys. Rev. B 107, 184311 (2023).
  57. K. Roy, K. Gogoi, and S. Basu, Topological characterization of a non-Hermitian ladder via Floquet non-Bloch theory, Phys. Rev. B 111, 115424 (2025).
  58. 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).
  59. F. Dangel, M. Wagner, H. Cartarius, J. Main, and G. Wunner, Topological invariants in dissipative extensions of the Su-Schrieffer-Heeger model, Phys. Rev. A 98, 013628 (2018).
  60. M. van Caspel, S. E. T. Arze, and I. P. Castillo, Dynamical signatures of topological order in the driven-dissipative Kitaev chain, SciPost Phys. 6, 026 (2019).
  61. M. Kawasaki, K. Mochizuki, and H. Obuse, Topological phases protected by shifted sublattice symmetry in dissipative quantum systems, Phys. Rev. B 106, 035408 (2022).
  62. F. Yang, P. Molignini, and E. J. Bergholtz, Dissipative boundary state preparation, Phys. Rev. Res. 5, 043229 (2023).
  63. S. S. Hegde, T. Ehmcke, and T. Meng, Edge-selective extremal damping from topological heritage of dissipative Chern insulators, Phys. Rev. Lett. 131, 256601 (2023).
  64. F. Song, S. Yao, and Z. Wang, Non-Hermitian skin effect and chiral damping in open quantum systems, Phys. Rev. Lett. 123, 170401 (2019).
  65. S. Lieu, M. McGinley, and N. R. Cooper, Tenfold way for quadratic Lindbladians, Phys. Rev. Lett. 124, 040401 (2020).
  66. F. Minganti, A. Miranowicz, R. W. Chhajlany, and F. Nori, Quantum exceptional points of non-Hermitian Hamiltonians and Liouvillians: The effects of quantum jumps, Phys. Rev. A 100, 062131 (2019).
  67. C.-H. Liu, K. Zhang, Z. Yang, and S. Chen, Helical damping and dynamical critical skin effect in open quantum systems, Phys. Rev. Res. 2, 043167 (2020).
  68. T. Haga, M. Nakagawa, R. Hamazaki, and M. Ueda, Liouvillian skin effect: Slowing down of relaxation processes without gap closing, Phys. Rev. Lett. 127, 070402 (2021).
  69. N. Okuma and M. Sato, Quantum anomaly, non-Hermitian skin effects, and entanglement entropy in open systems, Phys. Rev. B 103, 085428 (2021).
  70. Z. Zhou and Z. Yu, Non-Hermitian skin effect in quadratic Lindbladian systems: An adjoint fermion approach, Phys. Rev. A 106, 032216 (2022).
  71. F. Yang, Q.-D. Jiang, and E. J. Bergholtz, Liouvillian skin effect in an exactly solvable model, Phys. Rev. Res. 4, 023160 (2022).
  72. G. Lee, A. McDonald, and A. Clerk, Anomalously large relaxation times in dissipative lattice models beyond the non-Hermitian skin effect, Phys. Rev. B 108, 064311 (2023).
  73. K. Kawabata, T. Numasawa, and S. Ryu, Entanglement phase transition induced by the non-Hermitian skin effect, Phys. Rev. X 13, 021007 (2023).
  74. F. Yang, Z. Wei, X. Tong, K. Cao, and S.-P. Kou, Symmetry classes of dissipative topological insulators with edge dark states, Phys. Rev. B 107, 165139 (2023).
  75. S. Diehl, A. Micheli, A. Kantian, B. Kraus, H. P. Büchler, and P. Zoller, Quantum states and phases in driven open quantum systems with cold atoms, Nat. Phys. 4, 878 (2008).
  76. B. Kraus, H. P. Büchler, S. Diehl, A. Kantian, A. Micheli, and P. Zoller, Preparation of entangled states by quantum Markov processes, Phys. Rev. A 78, 042307 (2008).
  77. F. Verstraete, M. M. Wolf, and J. I. Cirac, Quantum computation and quantum-state engineering driven by dissipation, Nat. Phys. 5, 633 (2009).
  78. H. Weimer, M. Müller, I. Lesanovsky, P. Zoller, and H. P. Büchler, A Rydberg quantum simulator, Nat. Phys. 6, 382 (2010).
  79. S. Diehl, E. Rico, M. A. Baranov, and P. Zoller, Topology by dissipation in atomic quantum wires, Nat. Phys. 7, 971 (2011).
  80. C.-E. Bardyn, M. A. Baranov, E. Rico, A. İmamoğlu, P. Zoller, and S. Diehl, Majorana modes in driven-dissipative atomic superfluids with a zero Chern number, Phys. Rev. Lett. 109, 130402 (2012).
  81. C.-E. Bardyn, M. A. Baranov, C. V. Kraus, E. Rico, A. İmamoğlu, P. Zoller, and S. Diehl, Topology by dissipation, New J. Phys. 15, 085001 (2013).
  82. J. C. Budich, P. Zoller, and S. Diehl, Dissipative preparation of Chern insulators, Phys. Rev. A 91, 042117 (2015).
  83. F. Iemini, D. Rossini, R. Fazio, S. Diehl, and L. Mazza, Dissipative topological superconductors in number-conserving systems, Phys. Rev. B 93, 115113 (2016).
  84. A. K. Ghosh and A. M. Black-Schaffer, Majorana zero-modes in a dissipative Rashba nanowire, SciPost Phys. 17, 036 (2024).
  85. H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, New York, 2002).
  86. G. Lindblad, On the generators of quantum dynamical semigroups, Commun. Math. Phys. 48, 119 (1976).
  87. V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, Completely positive dynamical semigroups of N–level systems, J. Math. Phys. 17, 821 (1976).
  88. T. Prosen, Third quantization: A general method to solve master equations for quadratic open Fermi systems, New J. Phys. 10, 043026 (2008).
  89. T. Prosen, Spectral theorem for the Lindblad equation for quadratic open fermionic systems, J. Stat. Mech. (2010) P07020.
  90. T. Prosen and B. Žunkovič, Exact solution of Markovian master equations for quadratic Fermi systems: Thermal baths, open XY spin chains and non-equilibrium phase transition, New J. Phys. 12, 025016 (2010).
  91. G. T. Landi, D. Poletti, and G. Schaller, Nonequilibrium boundary-driven quantum systems: Models, methods, and properties, Rev. Mod. Phys. 94, 045006 (2022).
  92. A. K. Ghosh, T. Nag, and A. Saha, Generation of higher-order topological insulators using periodic driving, J. Phys.: Condens. Matter 36, 093001 (2024).
  93. A. Schnell, A. Eckardt, and S. Denisov, Is there a Floquet Lindbladian? Phys. Rev. B 101, 100301 (2020).
  94. K. Kawabata, K. Shiozaki, M. Ueda, and M. Sato, Symmetry and topology in non-Hermitian physics, Phys. Rev. X 9, 041015 (2019).
  95. J. K. Asbóth, B. Tarasinski, and P. Delplace, Chiral symmetry and bulk-boundary correspondence in periodically driven one-dimensional systems, Phys. Rev. B 90, 125143 (2014).
  96. K. Esaki, M. Sato, K. Hasebe, and M. Kohmoto, Edge states and topological phases in non-Hermitian systems, Phys. Rev. B 84, 205128 (2011).
  97. L. Lin, Y. Ke, and C. Lee, Real-space representation of the winding number for a one-dimensional chiral-symmetric topological insulator, Phys. Rev. B 103, 224208 (2021).
  98. J. Reuther, J. Alicea, and A. Yacoby, Gate-defined wires in HgTe quantum wells: From Majorana fermions to spintronics, Phys. Rev. X 3, 031011 (2013).
  99. Z. Wang, D.-K. Ki, H. Chen, H. Berger, A. H. MacDonald, and A. F. Morpurgo, Strong interface-induced spin–orbit interaction in graphene on WS2, Nat. Commun. 6, 8339 (2015).
  100. E. Khosravi, G. Stefanucci, S. Kurth, and E. Gross, Bound states in time-dependent quantum transport: Oscillations and memory effects in current and density, Phys. Chem. Chem. Phys. 11, 4535 (2009).
  101. J. Dubois, T. Jullien, F. Portier, P. Roche, A. Cavanna, Y. Jin, W. Wegscheider, P. Roulleau, and D. Glattli, Minimal-excitation states for electron quantum optics using levitons, Nature (London) 502, 659 (2013).
  102. J. Gabelli and B. Reulet, Shaping a time-dependent excitation to minimize the shot noise in a tunnel junction, Phys. Rev. B 87, 075403 (2013).
  103. M. Misiorny, G. Fève, and J. Splettstoesser, Shaping charge excitations in chiral edge states with a time-dependent gate voltage, Phys. Rev. B 97, 075426 (2018).
  104. K. Gogoi, T. Nag, and A. K. Ghosh, Dissipation induced Majarona 0- and π-modes in a driven Rashba nanowire (Version v2) [Data set], Zenodo (2025), doi: 10.5281/zenodo.17983359.

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Phys. Rev. B 113, 054301– Published 4 February, 2026
  • Received 15 July 2025
  • Revised 19 December 2025
  • Accepted 21 January 2026
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