Skip to main content
SpringerLink
Log in

Aspects of Topological Superconductivity in 2D Systems: Noncollinear Magnetism, Skyrmions, and Higher-order Topology

  • Review
  • Published:
Journal of Superconductivity and Novel Magnetism Aims and scope Submit manuscript

Abstract

The review is aimed at highlighting the aspects of topological superconductivity in the absence of spin–orbit interaction in two-dimensional systems with long-range noncollinear spin ordering or magnetic skyrmions. Another purpose is to give a brief introduction to the new concept of topological superconductivity, i.e. higher-order topology in two-dimensional systems including spin–orbit coupled structures. The formation of Majorana modes due to magnetic textures is discussed. The role of effective triplet pairings and odd fermion parity of the ground state wave function in different systems is emphasized. We describe the peculiarities of the magnetic skyrmions, leading to the formation of the Majorana modes and defects on which the modes are localized. The problem of braiding in the two-dimensional systems, especially in higher-order topological superconductors, is considered.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22

Similar content being viewed by others

References

  1. Kane, C.L., Mele, E.J.: Phys. Rev. Lett. 95(14), 146802 (2005). https://doi.org/10.1103/PhysRevLett.95.146802.

  2. Volovik, G.E.: J. Exp. Theor. Phys. Lett. 66(7), 522 (1997). https://doi.org/10.1134/1.567563

    Article  Google Scholar 

  3. Read, N., Green, D.: Phys. Rev. B 61(15), 10267 (2000). https://doi.org/10.1103/PhysRevB.61.10267

    Article  ADS  Google Scholar 

  4. Kohmoto, M.: Ann. Phys. 160(2), 343 (1985). https://doi.org/10.1016/0003-4916(85)90148-4

    Article  ADS  Google Scholar 

  5. Anderson, P.W.: Phys. Rev. 112(6), 1900 (1958). https://doi.org/10.1103/PhysRev.112.1900

  6. Ivanov, D.A.: Phys. Rev. Lett. 86(2), 268 (2001). https://doi.org/10.1103/PhysRevLett.86.268

  7. Kitaev, A.Y.: Phys. Usp. 44(10S), 131 (2001). https://doi.org/10.1070/1063-7869/44/10S/S29

    Article  ADS  Google Scholar 

  8. Kaladzhyan, V., Bena, C.: Phys. Rev. B 100(8), 081106 (2019). https://doi.org/10.1103/PhysRevB.100.081106

  9. Björnson, K., Black-Schaffer, A.M.: Phys. Rev. B 88(2), 024501 (2013). https://doi.org/10.1103/PhysRevB.88.024501

  10. Akzyanov, R.S., Rakhmanov, A.L., Rozhkov, A.V., Nori, F.: Phys. Rev. B 94(12), 125428 (2016). https://doi.org/10.1103/PhysRevB.94.125428

  11. Akzyanov, R.S., Rakhmanov, A.L.: J. Supercond. Nov. Magn. 30(11), 3071 (2017). https://doi.org/10.1007/s10948-017-4095-4

    Article  Google Scholar 

  12. Rex, S., Gornyi, I.V., Mirlin, A.D.: Phys. Rev. B 100(6), 064504 (2019). https://doi.org/10.1103/PhysRevB.100.064504

  13. Wang, Q., Liu, C.C., Lu, Y.M., Zhang, F.: Phys. Rev. Lett. 121(18), 186801 (2018). https://doi.org/10.1103/PhysRevLett.121.186801

  14. Nayak, C., Simon, S.H., Stern, A., Freedman, M., Das Sarma, S.: Rev. Mod. Phys. 80(3), 1083 (2008). https://doi.org/10.1103/RevModPhys.80.1083

  15. Alicea, J.: Rep. Prog. Phys. 75(7), 076501 (2012). https://doi.org/10.1088/0034-4885/75/7/076501

  16. Beenakker, C.: Annual Review of Condensed Matter Physics 4(1), 113 (2013). https://doi.org/10.1146/annurev-conmatphys-030212-184337

    Article  ADS  Google Scholar 

  17. Elliott, S.R., Franz, M.: Rev. Mod. Phys. 87(1), 137 (2015). https://doi.org/10.1103/RevModPhys.87.137

    Article  ADS  Google Scholar 

  18. Sato, M., Ando, Y.: Rep. Prog. Phys. 80(7), 076501 (2017). https://doi.org/10.1088/1361-6633/aa6ac7

  19. Val’kov, V., Shustin, M., Aksenov, S., Zlotnikov, A., Fedoseev, A., Mitskan, V., Kagan, M.: Phys. Usp. (2021). https://doi.org/10.3367/UFNe.2021.03.038950

    Article  Google Scholar 

  20. Samokhvalov, A.V., Kopasov, A.A., Kutlin, A.G., Mironov, S.V., Buzdin, A.I., Mel’nikov, A.S.: JETP Lett. 113(1), 34 (2021). https://doi.org/10.1134/S0021364021010070

    Article  ADS  Google Scholar 

  21. Mackenzie, A.P., Maeno, Y.: Rev. Mod. Phys. 75(2), 657 (2003). https://doi.org/10.1103/RevModPhys.75.657

    Article  ADS  Google Scholar 

  22. Das Sarma, S., Nayak, C., Tewari, S.: Phys. Rev. B 73(22), 220502 (2006). https://doi.org/10.1103/PhysRevB.73.220502

  23. Sau, J.D., Tewari, S.: Phys. Rev. B 86(10), 104509 (2012). https://doi.org/10.1103/PhysRevB.86.104509

  24. Mineev, V.P.: Phys. Usp. 60(2), 121 (2017). https://doi.org/10.3367/UFNe.2016.04.037771

    Article  ADS  Google Scholar 

  25. Pustogow, A., Luo, Y., Chronister, A., Su, Y.S., Sokolov, D.A., Jerzembeck, F., Mackenzie, A.P., Hicks, C.W., Kikugawa, N., Raghu, S., Bauer, E.D., Brown, S.E.: Nature 574(7776), 72 (2019). https://doi.org/10.1038/s41586-019-1596-2

    Article  ADS  Google Scholar 

  26. Suzuki, S.I., Sato, M., Tanaka, Y.: Phys. Rev. B 101(5), 054505 (2020). https://doi.org/10.1103/PhysRevB.101.054505

  27. Sau, J.D., Lutchyn, R.M., Tewari, S., Das Sarma, S.: Phys. Rev. Lett. 104(4), 040502 (2010). https://doi.org/10.1103/PhysRevLett.104.040502

  28. R.M. Lutchyn, J.D. Sau, S. Das Sarma, Physical Review Letters 105(7), 077001 (2010). https://doi.org/10.1103/PhysRevLett.105.077001

  29. Oreg, Y., Refael, G., von Oppen, F.: Phys. Rev. Lett. 105(17), 177002 (2010). https://doi.org/10.1103/PhysRevLett.105.177002

  30. Deng, M.T., Yu, C.L., Huang, G.Y., Larsson, M., Caroff, P., Xu, H.Q.: Nano Lett. 12(12), 6414 (2012). https://doi.org/10.1021/nl303758w

    Article  ADS  Google Scholar 

  31. Martin, I., Morpurgo, A.F.: Phys. Rev. B 85(14), 144505 (2012). https://doi.org/10.1103/PhysRevB.85.144505

  32. Lu, Y.M., Wang, Z.: Phys. Rev. Lett. 110(9), 096403 (2013). https://doi.org/10.1103/PhysRevLett.110.096403

  33. Choy, T.P., Edge, J.M., Akhmerov, A.R., Beenakker, C.W.J.: Phys. Rev. B 84(19), 195442 (2011). https://doi.org/10.1103/PhysRevB.84.195442

  34. Chen, W., Schnyder, A.P.: Phys. Rev. B 92(21), 214502 (2015). https://doi.org/10.1103/PhysRevB.92.214502

  35. Yang, G., Stano, P., Klinovaja, J., Loss, D.: Phys. Rev. B 93(22), 224505 (2016). https://doi.org/10.1103/PhysRevB.93.224505

  36. Mascot, E., Bedow, J., Graham, M., Rachel, S., Morr, D.K.: npj Quantum Materials. 6(1), 1 (2021). https://doi.org/10.1038/s41535-020-00299-x

  37. Mohanta, N., Okamoto, S., Dagotto, E.: Commun. Phys. 4(1), 1 (2021). https://doi.org/10.1038/s42005-021-00666-5. https://www.nature.com/articles/s42005-021-00666-5

  38. Zhang, S.B., Rui, W.B., Calzona, A., Choi, S.J., Schnyder, A.P., Trauzettel, B.: Phys. Rev. Research 2, 043025 (2020). https://doi.org/10.1103/PhysRevResearch.2.043025

  39. Zhang, S.B., Calzona, A., Trauzettel, B.: Phys. Rev. B 102, 100503 (2020). https://doi.org/10.1103/PhysRevB.102.100503

  40. Wu, Y.J., Hou, J., Li, Y.M., Luo, X.W., Shi, X., Zhang, C.: Phys. Rev. Lett. 124, 227001 (2020). https://doi.org/10.1103/PhysRevLett.124.227001

  41. Zhang, R.X., Cole, W.S., Wu, X., Das Sarma, S.: Phys. Rev. Lett. 123, 167001 (2019). https://doi.org/10.1103/PhysRevLett.123.167001

  42. Zhu, X.: Phys. Rev. B 97, 205134 (2018). https://doi.org/10.1103/PhysRevB.97.205134

  43. Fu, L., Kane, C.L.: Phys. Rev. Lett. 100(9), 096407 (2008). https://doi.org/10.1103/PhysRevLett.100.096407

  44. Hor, Y.S., Williams, A.J., Checkelsky, J.G., Roushan, P., Seo, J., Xu, Q., Zandbergen, H.W., Yazdani, A., Ong, N.P., Cava, R.J.: Phys. Rev. Lett. 104(5), 057001 (2010). https://doi.org/10.1103/PhysRevLett.104.057001

  45. Hirsch, J.E., Maple, M.B., Marsiglio, F.: Physica C: Superconductivity and its Applications 514, 1 (2015). https://doi.org/10.1016/j.physc.2015.03.002

    Article  ADS  Google Scholar 

  46. Zhang, P., Yaji, K., Hashimoto, T., Ota, Y., Kondo, T., Okazaki, K., Wang, Z., Wen, J., Gu, G.D., Ding, H., Shin, S.: Science 360(6385), 182 (2018). https://doi.org/10.1126/science.aan4596

    Article  ADS  Google Scholar 

  47. Mourik, V., Zuo, K., Frolov, S.M., Plissard, S.R., Bakkers, E.P.A.M., Kouwenhoven, L.P.: Science 336(6084), 1003 (2012). https://doi.org/10.1126/science.1222360

  48. Deng, M.T., Vaitiekėnas, S., Hansen, E.B., Danon, J., Leijnse, M., Flensberg, K., Nygård, J., Krogstrup, P., Marcus, C.M.: Science 354(6319), 1557 (2016). https://doi.org/10.1126/science.aaf3961

    Article  ADS  Google Scholar 

  49. Nichele, F., Drachmann, A.C., Whiticar, A.M., O’Farrell, E.C., Suominen, H.J., Fornieri, A., Wang, T., Gardner, G.C., Thomas, C., Hatke, A.T., Krogstrup, P., Manfra, M.J., Flensberg, K., Marcus, C.M.: Phys. Rev. Lett. 119(13), 136803 (2017). https://doi.org/10.1103/PhysRevLett.119.136803

  50. Vaitiekėnas, S., Winkler, G.W., Heck, B.V., Karzig, T., Deng, M.T., Flensberg, K., Glazman, L.I., Nayak, C., Krogstrup, P., Lutchyn, R.M., Marcus, C.M.: Science 367(6485), (2020). https://doi.org/10.1126/science.aav3392

  51. Shen, J., Winkler, G.W., Borsoi, F., Heedt, S., Levajac, V., Wang, J.Y., van Driel, D., Bouman, D., Gazibegovic, S., Op Het Veld, R.L.M., Car, D., Logan, J.A., Pendharkar, M., Palmstrøm, C.J., Bakkers, E.P.A.M., Kouwenhoven, L.P., van Heck, B.: Phys. Rev. B 104(4), 045422 (2021). https://doi.org/10.1103/PhysRevB.104.045422

  52. Moore, C., Stanescu, T.D., Tewari, S.: Phys. Rev. B 97(16), 165302 (2018). https://doi.org/10.1103/PhysRevB.97.165302

  53. Reeg, C., Dmytruk, O., Chevallier, D., Loss, D., Klinovaja, J.: Phys. Rev. B 98(24), 245407 (2018). https://doi.org/10.1103/PhysRevB.98.245407

  54. Zhang, H., de Moor, M.W.A., Bommer, J.D.S., Xu, D., Wang, G., van Loo, N., Liu, C.X., Gazibegovic, S., Logan, J.A., Car, D., Veld, R.L.M.O.H., van Veldhoven, P.J., Koelling, S., Verheijen, M.A., Pendharkar, M., Pennachio, D.J., Shojaei, B., Lee, J.S., Palmstrøm, C.J., Bakkers, E.P.A.M., Sarma, S.D., Kouwenhoven, L.P.: arXiv p. 2101.11456 (2021)

  55. Vaitiekènas, S., Liu, Y., Krogstrup, P., Marcus, C.M.: Nat. Phys. 17(1), 43 (2021). https://doi.org/10.1038/s41567-020-1017-3

    Article  Google Scholar 

  56. Li, J., Neupert, T., Wang, Z., MacDonald, A.H., Yazdani, A., Bernevig, B.A.: Nat. Commun. 7(1), 12297 (2016). https://doi.org/10.1038/ncomms12297

    Article  ADS  Google Scholar 

  57. Ménard, G.C., Mesaros, A., Brun, C., Debontridder, F., Roditchev, D., Simon, P., Cren, T.: Nat. Commun. 10(1), 2587 (2019). https://doi.org/10.1038/s41467-019-10397-5

    Article  ADS  Google Scholar 

  58. Lutchyn, R.M., Bakkers, E.P.A.M., Kouwenhoven, L.P., Krogstrup, P., Marcus, C.M., Oreg, Y.: Nat. Rev. Mater. 3(5), 52 (2018). https://doi.org/10.1038/s41578-018-0003-1

  59. Alicea, J., Oreg, Y., Rafael, G., von Oppen, F., Fisher, M.P.A.: Nat. Phys. 7, 412 (2011). https://doi.org/10.1038/nphys1915

    Article  Google Scholar 

  60. Braunecker, B., Japaridze, G.I., Klinovaja, J., Loss, D.: Phys. Rev. B 82(4), 045127 (2010). https://doi.org/10.1103/PhysRevB.82.045127

  61. Heimes, A., Kotetes, P., Schön, G.: Phys. Rev. B 90(6), 060507 (2014). https://doi.org/10.1103/PhysRevB.90.060507

  62. Vazifeh, M.M., Franz, M.: Phys. Rev. Lett. 111(20), 206802 (2013). https://doi.org/10.1103/PhysRevLett.111.206802

  63. Nadj-Perge, S., Drozdov, I.K., Bernevig, B.A., Yazdani, A.: Phys. Rev. B 88(2), 020407 (2013). https://doi.org/10.1103/PhysRevB.88.020407

  64. Pientka, F., Glazman, L.I., von Oppen, F.: Phys. Rev. B 88(15), 155420 (2013). https://doi.org/10.1103/PhysRevB.88.155420

  65. Schecter, M., Flensberg, K., Christensen, M.H., Andersen, B.M., Paaske, J.: Phys. Rev. B 93(14), 140503 (2016). https://doi.org/10.1103/PhysRevB.93.140503

  66. Christensen, M.H., Schecter, M., Flensberg, K., Andersen, B.M., Paaske, J.: Phys. Rev. B 94(14), 144509 (2016). https://doi.org/10.1103/PhysRevB.94.144509

  67. Kjaergaard, M., Wölms, K., Flensberg, K.: Phys. Rev. B 85(2), 020503 (2012). https://doi.org/10.1103/PhysRevB.85.020503

  68. Egger, R., Flensberg, K.: Phys. Rev. B 85(23), 235462 (2012). https://doi.org/10.1103/PhysRevB.85.235462

  69. Klinovaja, J., Stano, P., Loss, D.: Phys. Rev. Lett. 109(23), 236801 (2012). https://doi.org/10.1103/PhysRevLett.109.236801

  70. Kornich, V., Vavilov, M.G., Friesen, M., Eriksson, M.A., Coppersmith, S.N.: Phys. Rev. B 101(12), 125414 (2020). https://doi.org/10.1103/PhysRevB.101.125414

  71. Braunecker, B., Simon, P.: Phys. Rev. Lett. 111(14), 147202 (2013). https://doi.org/10.1103/PhysRevLett.111.147202

  72. Klinovaja, J., Stano, P., Yazdani, A., Loss, D.: Phys. Rev. Lett. 111(18), 186805 (2013). https://doi.org/10.1103/PhysRevLett.111.186805

  73. Hsu, C.H., Stano, P., Klinovaja, J., Loss, D.: Phys. Rev. B 92(23), 235435 (2015). https://doi.org/10.1103/PhysRevB.92.235435

  74. Nadj-Perge, S., Drozdov, I.K., Li, J., Chen, H., Jeon, S., Seo, J., MacDonald, A.H., Bernevig, B.A., Yazdani, A.: Science 346(6209), 602 (2014). https://doi.org/10.1126/science.1259327

    Article  ADS  Google Scholar 

  75. Ruby, M., Pientka, F., Peng, Y., von Oppen, F., Heinrich, B.W., Franke, K.J.: Phys. Rev. Lett. 115(19), 197204 (2015). https://doi.org/10.1103/PhysRevLett.115.197204

  76. Feldman, B.E., Randeria, M.T., Li, J., Jeon, S., Xie, Y., Wang, Z., Drozdov, I.K., Andrei Bernevig, B., Yazdani, A.: Nat. Phys. 13(3), 286 (2017). https://doi.org/10.1038/nphys3947

  77. Peng, Y., Pientka, F., Glazman, L.I., von Oppen, F.: Phys. Rev. Lett. 114(10), 106801 (2015). https://doi.org/10.1103/PhysRevLett.114.106801

  78. Kim, Y., Cheng, M., Bauer, B., Lutchyn, R.M., Das Sarma, S.: Phys. Rev. B 90(6), 060401 (2014). https://doi.org/10.1103/PhysRevB.90.060401

  79. Menzel, M., Mokrousov, Y., Wieser, R., Bickel, J.E., Vedmedenko, E., Blügel, S., Heinze, S., von Bergmann, K., Kubetzka, A., Wiesendanger, R.: Phys. Rev. Lett. 108(19), 197204 (2012). https://doi.org/10.1103/PhysRevLett.108.197204

  80. Kim, H., Palacio-Morales, A., Posske, T., Rózsa, L., Palotás, K., Szunyogh, L., Thorwart, M., Wiesendanger, R.: Sci. Adv. 4(5), eaar5251 (2018). https://doi.org/10.1126/sciadv.aar5251

  81. Desjardins, M.M., Contamin, L.C., Delbecq, M.R., Dartiailh, M.C., Bruhat, L.E., Cubaynes, T., Viennot, J.J., Mallet, F., Rohart, S., Thiaville, A., Cottet, A., Kontos, T.: Nat. Mater. 18(10), 1060 (2019). https://doi.org/10.1038/s41563-019-0457-6

    Article  ADS  Google Scholar 

  82. Mohanta, N., Zhou, T., Xu, J.W., Han, J.E., Kent, A.D., Shabani, J., Žutić, I., Matos-Abiague, A.: Phys. Rev. Appl. 12(3), 034048 (2019). https://doi.org/10.1103/PhysRevApplied.12.034048

  83. Pöyhönen, K., Westström, A., Röntynen, J., Ojanen, T.: Phys. Rev. B 89(11), 115109 (2014). https://doi.org/10.1103/PhysRevB.89.115109

  84. Xiao, J., An, J.: New J. Phys. 17(11), 113034 (2015). https://doi.org/10.1088/1367-2630/17/11/113034

  85. Mikheenkov, A.V., Valiulin, V.E., Shvartsberg, A.V., Barabanov, A.F.: J. Exp. Theor. Phys. 126(3), 404 (2018). https://doi.org/10.1134/S1063776118030147

    Article  ADS  Google Scholar 

  86. Devizorova, Z., Buzdin, A.: Phys. Rev. B. 100(10), 104523 (2019). https://doi.org/10.1103/PhysRevB.100.104523

  87. Nakosai, S., Tanaka, Y., Nagaosa, N.: Phys. Rev. B. 88(18), 180503 (2013). https://doi.org/10.1103/PhysRevB.88.180503

  88. Igoshev, P.A., Timirgazin, M.A., Katanin, A.A., Arzhnikov, A.K., Irkhin, V.Y.: Phys. Rev. B 81(9), 094407 (2010). https://doi.org/10.1103/PhysRevB.81.094407

  89. Chubukov, A.V., Sachdev, S., Senthil, T.: J. Phys. Condens. Matter 6(42), 8891 (1994). https://doi.org/10.1088/0953-8984/6/42/019

    Article  ADS  Google Scholar 

  90. Steffensen, D., Christensen, M.H., Andersen, B.M., Kotetes, P.: arXiv:2012.09691 [cond-mat] (2020).

  91. Buzdin, A.I., Bulaevskii, L.N., Kulich, M.L., Panyukov, S.V.: Soviet Physics Uspekhi 27(12), 927 (1984). https://doi.org/10.1070/PU1984v027n12ABEH004085

    Article  ADS  Google Scholar 

  92. Val’kov, V.V., Zlotnikov, A.O., Fedoseev, A.D., Shustin, M.S.: J. Magn. Magn. Mater. 440, 37 (2017). https://doi.org/10.1016/j.jmmm.2016.12.093

    Article  ADS  Google Scholar 

  93. Val’kov, V.V., Zlotnikov, A.O., Shustin, M.S.: J. Magn. Magn. Mater. 459, 112 (2018). https://doi.org/10.1016/j.jmmm.2017.11.115

    Article  ADS  Google Scholar 

  94. Zhou, S., Wang, Z.: Phys. Rev. Lett. 100(21), 217002 (2008). https://doi.org/10.1103/PhysRevLett.100.217002

  95. Iida, K., Nagai, Y., Ishida, S., Ishikado, M., Murai, N., Christianson, A.D., Yoshida, H., Inamura, Y., Nakamura, H., Nakao, A., Munakata, K., Kagerbauer, D., Eisterer, M., Kawashima, K., Yoshida, Y., Eisaki, H., Iyo, A.: Phys. Rev. B 100(1), 014506 (2019). https://doi.org/10.1103/PhysRevB.100.014506

  96. Kim, T.K., Pervakov, K.S., Evtushinsky, D.V., Jung, S.W., Poelchen, G., Kummer, K., Vlasenko, V.A., Sadakov, A.V., Usoltsev, A.S., Pudalov, V.M., Roditchev, D., Stolyarov, V.S., Vyalikh, D.V., Borisov, V., Valenti, R., Ernst, A., Eremeev, S.V., Chulkov, E.V.: Phys. Rev. B 103(17), 174517 (2021). https://doi.org/10.1103/PhysRevB.103.174517

  97. Collomb, D., Bending, S., Koshelev, A., Smylie, M., Farrar, L., Bao, J.K., Chung, D., Kanatzidis, M., Kwok, W.K., Welp, U.: Phys. Rev. Lett. 126(15), 157001 (2021). https://doi.org/10.1103/PhysRevLett.126.157001

  98. Klinovaja, J., Loss, D.: Phys. Rev. X 3(1), 011008 (2013). https://doi.org/10.1103/PhysRevX.3.011008

  99. Sedlmayr, N., Aguiar-Hualde, J.M., Bena, C.: Phys. Rev. B 91(11), 115415 (2015). https://doi.org/10.1103/PhysRevB.91.115415

  100. Bedow, J., Mascot, E., Posske, T., Uhrig, G.S., Wiesendanger, R., Rachel, S., Morr, D.K.: Phys. Rev. B 102(18), 180504 (2020). https://doi.org/10.1103/PhysRevB.102.180504

  101. Rex, S., Gornyi, I.V., Mirlin, A.D.: Phys. Rev. B. 102(22), 224501 (2020). https://doi.org/10.1103/PhysRevB.102.224501

  102. Palacio-Morales, A., Kubetzka, A., von Bergmann, K., Wiesendanger, R.: Nano Lett. 16(10), 6252 (2016). https://doi.org/10.1021/acs.nanolett.6b02528

    Article  ADS  Google Scholar 

  103. Palacio-Morales, A., Mascot, E., Cocklin, S., Kim, H., Rachel, S., Morr, D.K., Wiesendanger, R.: Sci. Adv. 5(7), eaav6600 (2019). https://doi.org/10.1126/sciadv.aav6600

  104. Spethmann, J., Meyer, S., von Bergmann, K., Wiesendanger, R., Heinze, S., Kubetzka, A.: Phys. Rev. Lett. 124(22), 227203 (2020). https://doi.org/10.1103/PhysRevLett.124.227203

  105. Fatin, G.L., Matos-Abiague, A., Scharf, B., Žutić, I.: Phys. Rev. Lett. 117(7), 077002 (2016). https://doi.org/10.1103/PhysRevLett.117.077002

  106. Zhou, T., Mohanta, N., Han, J.E., Matos-Abiague, A., Žutić, I.: Phys. Rev. B 99(13),134505 (2019). https://doi.org/10.1103/PhysRevB.99.134505

  107. Psaltakis, G.C., Fenton, E.W.: J. Phys. C Solid State Phys. 16(20), 3913 (1983). https://doi.org/10.1088/0022-3719/16/20/015

    Article  ADS  Google Scholar 

  108. Kyung, B.: Phys. Rev. B 62(13), 9083 (2000). https://doi.org/10.1103/PhysRevB.62.9083

    Article  ADS  Google Scholar 

  109. Rashba, E.I.: Physica E: Low-dimensional Systems and Nanostructures 20(3), 189 (2004). https://doi.org/10.1016/j.physe.2003.08.002

    Article  ADS  Google Scholar 

  110. Ashkenazi, J., Kuper, C.G., Ron, A.: Phys. Rev. B. 28(1), 418 (1983). https://doi.org/10.1103/PhysRevB.28.418

  111. Buzdin, A.I., Bulaevskii, L.N.: Soviet Physics Uspekhi 29(5), 412 (1986). https://doi.org/10.1070/PU1986v029n05ABEH003375

    Article  ADS  Google Scholar 

  112. Inui, M., Doniach, S., Hirschfeld, P.J., Ruckenstein, A.E.: Phys. Rev. B 37(4), 2320 (1988). https://doi.org/10.1103/PhysRevB.37.2320

    Article  ADS  Google Scholar 

  113. Lichtenstein, A.I., Katsnelson, M.I.: Phys. Rev. B 62(14), R9283 (2000). https://doi.org/10.1103/PhysRevB.62.R9283

    Article  ADS  Google Scholar 

  114. Sacramento, P.D.: J. Phys. Condens. Matter 15(36), 6285 (2003). https://doi.org/10.1088/0953-8984/15/36/315

    Article  ADS  Google Scholar 

  115. Demler, E., Hanke, W., Zhang, S.C.: Rev. Mod. Phys. 76(3), 909 (2004). https://doi.org/10.1103/RevModPhys.76.909

    Article  ADS  Google Scholar 

  116. Capone, M., Kotliar, G.: Phys. Rev. B 74(5), 054513 (2006). https://doi.org/10.1103/PhysRevB.74.054513

  117. Weber, C., Läuchli, A., Mila, F., Giamarchi, T.: Phys. Rev. B. 73(1), 014519 (2006). https://doi.org/10.1103/PhysRevB.73.014519

  118. Pathak, S., Shenoy, V.B., Randeria, M., Trivedi, N.: Phys. Rev. Lett. 102(2), 027002 (2009). https://doi.org/10.1103/PhysRevLett.102.027002

  119. Kuboki, K., Yoneya, M., Yamase, H.: Physica C: Superconductivity and its Applications 470, S163 (2010). https://doi.org/10.1016/j.physc.2009.12.053

    Article  ADS  Google Scholar 

  120. Sahoo, B.K., Panda, B.N.: Physica C: Superconductivity 470(13), 547 (2010). https://doi.org/10.1016/j.physc.2010.05.001

    Article  ADS  Google Scholar 

  121. Kaczmarczyk, J., Spałek, J.: Phys. Rev. B 84(12), 125140 (2011). https://doi.org/10.1103/PhysRevB.84.125140

  122. Val’kov, V.V., Zlotnikov, A.O.: J. Supercond. Nov. Magn. 26(9), 2885 (2013). https://doi.org/10.1007/s10948-013-2222-4

    Article  Google Scholar 

  123. Yamase, H., Eberlein, A., Metzner, W.: Phys. Rev. Lett. 116(9), 096402 (2016). https://doi.org/10.1103/PhysRevLett.116.096402

  124. Val’kov, V.V., Zlotnikov, A.O.: JETP Lett. 104(7), 483 (2016). https://doi.org/10.1134/S0021364016190127

    Article  ADS  Google Scholar 

  125. Foley, A., Verret, S., Tremblay, A.M.S., Sénéchal, D.: Phys. Rev. B 99(18), 184510 (2019). https://doi.org/10.1103/PhysRevB.99.184510

  126. Koshelev, A.E.: Phys. Rev. B 100(22), 224503 (2019). https://doi.org/10.1103/PhysRevB.100.224503

  127. Tewari, S., Sau, J.D.: Phys. Rev. Lett. 109(15), 150408 (2012). https://doi.org/10.1103/PhysRevLett.109.150408

  128. Baskaran, G.: Phys. Rev. Lett. 91(9), 097003 (2003). https://doi.org/10.1103/PhysRevLett.91.097003

  129. Val’kov, V.V., Val’kova, T.A., Mitskan, V.A.: JETP Lett. 102(6), 361 (2015). https://doi.org/10.1134/S0021364015180162

    Article  ADS  Google Scholar 

  130. Val’kov, V.V., Mitskan, V.A., Shustin, M.S.: J. Exp. Theor. Phys. 129(3), 426 (2019). https://doi.org/10.1134/S1063776119080144

    Article  ADS  Google Scholar 

  131. Wong, C.L.M., Law, K.T.: Phys. Rev. B 86(18), 184516 (2012). https://doi.org/10.1103/PhysRevB.86.184516

  132. Aksenov, S.V., Zlotnikov, A.O., Shustin, M.S.: Phys. Rev. B 101(12), 125431 (2020). https://doi.org/10.1103/PhysRevB.101.125431

  133. Crawford, D., Mascot, E., Morr, D.K., Rachel, S.: Phys. Rev. B 101(17), 174510 (2020). https://doi.org/10.1103/PhysRevB.101.174510

  134. Ishikawa, K., Matsuyama, T.: Nucl. Phys. B 280, 523 (1987). https://doi.org/10.1016/0550-3213(87)90160-X

    Article  ADS  Google Scholar 

  135. Volovik, G.E.: The Universe in a Helium Droplet. Oxford University Press (2009)

  136. Val’kov, V.V., Zlotnikov, A.O.: JETP Lett. 109(11), 736 (2019). https://doi.org/10.1134/S0021364019110158

    Article  ADS  Google Scholar 

  137. Ghosh, P., Sau, J.D., Tewari, S., Das Sarma, S.: Phys. Rev. B 82(18), 184525 (2010). https://doi.org/10.1103/PhysRevB.82.184525

  138. Bloch, F.: Z. Phys. 61(3), 206 (1930). https://doi.org/10.1007/BF01339661

    Article  ADS  Google Scholar 

  139. Ter haar, D. (ed.): Collected Papers of L.D. Landau, pp. 101–114. Pergamon (1965). https://doi.org/10.1016/B978-0-08-010586-4.50023-7

  140. Skyrme, T.H.R., Schonland, B.F.J.: Proc. R. Soc. Lond. A Math. Phys. Sci. 260(1300), 127 (1961). https://doi.org/10.1098/rspa.1961.0018

    Article  ADS  Google Scholar 

  141. Skyrme, T.H.R.: Nucl. Phys. 31, 556 (1962). https://doi.org/10.1016/0029-5582(62)90775-7

    Article  MathSciNet  Google Scholar 

  142. Bogdanov, A.N., Yablonskiǐ, D.A.: Sov. Phys. JETP 68(1), 101 (1989). http://www.jetp.ras.ru/cgi-bin/e/index/r/95/1/p178?a=list

  143. Mühlbauer, S., Binz, B., Jonietz, F., Pfleiderer, C., Rosch, A., Neubauer, A., Georgii, R., Böni, P.: Science 323(5916), 915 (2009). https://doi.org/10.1126/science.1166767

    Article  ADS  Google Scholar 

  144. Romming, N., Kubetzka, A., Hanneken, C., von Bergmann, K., Wiesendanger, R.: Phys. Rev. Lett. 114(17), 177203 (2015). https://doi.org/10.1103/PhysRevLett.114.177203

  145. Nayak, A.K., Kumar, V., Ma, T., Werner, P., Pippel, E., Sahoo, R., Damay, F., Rößer, U.K., Felser, C., Parkin, S.S.P.: Nature 548(7669), 561 (2017). https://doi.org/10.1038/nature23466

    Article  ADS  Google Scholar 

  146. Kanazawa, N., Seki, S., Tokura, Y.: Adv. Mater. 29(25), 1603227 (2017). https://doi.org/10.1002/adma.201603227

    Article  Google Scholar 

  147. Grenz, J., Köhler, A., Schwarz, A., Wiesendanger, R.: Phys. Rev. Lett. 119(4), 047205 (2017). https://doi.org/10.1103/PhysRevLett.119.047205

  148. Yang, S.H., Ryu, K.S., Parkin, S.: Nat. Nanotechnol. 10(3), 221 (2015). https://doi.org/10.1038/nnano.2014.324

    Article  ADS  Google Scholar 

  149. Moreau-Luchaire, C., Moutafis, C., Reyren, N., Sampaio, J., Vaz, C.A.F., Van Horne, N., Bouzehouane, K., Garcia, K., Deranlot, C., Warnicke, P., Wohlhüter, P., George, J.M., Weigand, M., Raabe, J., Cros, V., Fert, A.: Nat. Nanotechnol. 11(5), 444 (2016). https://doi.org/10.1038/nnano.2015.313

  150. Everschor-Sitte, K., Masell, J., Reeve, R.M., Kläui, M.: J. Appl. Phys. 124(24), 240901 (2018). https://doi.org/10.1063/1.5048972

  151. Zhang, X., Ezawa, M., Zhou, Y.: Sci. Rep. 5(1), 9400 (2015). https://doi.org/10.1038/srep09400

    Article  ADS  Google Scholar 

  152. Zázvorka, J., Jakobs, F., Heinze, D., Keil, N., Kromin, S., Jaiswal, S., Litzius, K., Jakob, G., Virnau, P., Pinna, D., Everschor-Sitte, K., Rózsa, L., Donges, A., Nowak, U., Kläui, M.: Nat. Nanotechnol. 14(7), 658 (2019). https://doi.org/10.1038/s41565-019-0436-8

    Article  ADS  Google Scholar 

  153. Yu, G., Upadhyaya, P., Li, X., Li, W., Kim, S.K., Fan, Y., Wong, K.L., Tserkovnyak, Y., Amiri, P.K., Wang, K.L.: Nano Lett. 16(3), 1981 (2016). https://doi.org/10.1021/acs.nanolett.5b05257

  154. Miller, H., Anders, J.: New J. Phys. 19, 062001 (2017). https://doi.org/10.1088/1367-2630/aa703f

  155. Nakahara, M.: Geometry, Topology and Physics, 2nd edn. CRC Press, Bristol; Philadelphia (2003)

    MATH  Google Scholar 

  156. Schwarz, A.S.: Quantum Field Theory and Topology. Grundlehren der mathematischen Wissenschaften. Springer-Verlag, Berlin Heidelberg, 1993. https://doi.org/10.1007/978-3-662-02943-5

  157. Uzdin, V.M., Potkina, M.N., Lobanov, I.S., Bessarab, P.F., Jónsson, H.: J. Magn. Magn. Mater. 459, 236 (2018). https://doi.org/10.1016/j.jmmm.2017.10.100

    Article  ADS  Google Scholar 

  158. Göbel, B., Mertig, I., Tretiakov, O.A.: Phys. Rep. 895, 1 (2021). https://doi.org/10.1016/j.physrep.2020.10.001

    Article  MathSciNet  ADS  Google Scholar 

  159. Murakami, S., Nagaosa, N., Zhang, S.C.: Science 301(5638), 1348 (2003). https://doi.org/10.1126/science.1087128

    Article  ADS  Google Scholar 

  160. Sinova, J., Culcer, D., Niu, Q., Sinitsyn, N.A., Jungwirth, T., MacDonald, A.H.: Phys. Rev. Lett. 92(12), 126603 (2004). https://doi.org/10.1103/PhysRevLett.92.126603

  161. Büttner, F., Lemesh, I., Beach, G.S.D.: Sci. Rep. 8(1), 4464 (2018). https://doi.org/10.1038/s41598-018-22242-8

    Article  ADS  Google Scholar 

  162. Wang, X.S., Yuan, H.Y., Wang, X.R.: Commun. Phys. 1(1), 1 (2018). https://doi.org/10.1038/s42005-018-0029-0

    Article  ADS  Google Scholar 

  163. Garnier, M., Mesaros, A., Simon, P.: Commun. Phys. 2(1), 1 (2019). https://doi.org/10.1038/s42005-019-0226-5

    Article  Google Scholar 

  164. Rybakov, F.N., Kiselev, N.S.: Phys. Rev. B 99(6), 064437 (2019). https://doi.org/10.1103/PhysRevB.99.064437

  165. Pershoguba, S.S., Nakosai, S., Balatsky, A.V.: Phys. Rev. B 94(6), 064513 (2016). https://doi.org/10.1103/PhysRevB.94.064513

  166. Pöyhönen, K., Westström, A., Pershoguba, S.S., Ojanen, T., Balatsky, A.V.: Phys. Rev. B 94(21), 214509 (2016). https://doi.org/10.1103/PhysRevB.94.214509

  167. Yokoyama, T., Linder, J.: Phys. Rev. B 92(6), 060503 (2015). https://doi.org/10.1103/PhysRevB.92.060503

  168. Vadimov, V.L., Sapozhnikov, M.V., Mel’nikov, A.S.: Appl. Phys. Lett. 113(3), 032402 (2018). https://doi.org/10.1063/1.5037934

  169. Hals, K.M., Schecter, M., Rudner, M.S.: Phys. Rev. Lett. 117(1), 017001 (2016). https://doi.org/10.1103/PhysRevLett.117.017001

  170. Baumard, J., Cayssol, J., Bergeret, F.S., Buzdin, A.: Phys. Rev. B 99(1), 014511 (2019). https://doi.org/10.1103/PhysRevB.99.014511

  171. Dahir, S.M., Volkov, A.F., Eremin, I.M.: Phys. Rev. Lett. 122(9), 097001 (2019). https://doi.org/10.1103/PhysRevLett.122.097001

  172. Petroviç, A., Raju, M., Tee, X., Louat, A., Maggio-Aprile, I., Menezes, R., Wyszyński, M., Duong, N., Reznikov, M., Renner, C., Miloševiç, M., Panagopoulos, C.: Phys. Rev. Lett. 126(11), 117205 (2021). https://doi.org/10.1103/PhysRevLett.126.117205

  173. Andriyakhina, E.S., Burmistrov, I.S.: Phys. Rev. B 103(17), 174519 (2021). https://doi.org/10.1103/PhysRevB.103.174519

  174. Díaz, S.A., Klinovaja, J., Loss, D., Hoffman, S.: arXiv:2102.03423 [cond-mat] (2021)

  175. Lobanov, I., Potkina, M., Jónsson, H., Uzdin, V.: Nanosystems: Physics, Chemistry, Mathematics pp. 586–595 (2017). https://doi.org/10.17586/2220-8054-2017-8-5-586-595

  176. E, W., Ren, W., Vanden-Eijnden, E.: Phys. Rev. B 66(5), 052301 (2002). https://doi.org/10.1103/PhysRevB.66.052301

  177. Suess, D., Vogler, C., Bruckner, F., Heistracher, P., Slanovc, F., Abert, C.: Sci. Rep. 9(1), 4827 (2019). https://doi.org/10.1038/s41598-019-41062-y

    Article  ADS  Google Scholar 

  178. Olsen, R.A., Kroes, G.J., Henkelman, G., Arnaldsson, A., Jónsson, H.: J. Chem. Phys. 121(20), 9776 (2004). https://doi.org/10.1063/1.1809574

    Article  ADS  Google Scholar 

  179. Müller, G.P., Bessarab, P.F., Vlasov, S.M., Lux, F., Kiselev, N.S., Blügel, S., Uzdin, V.M., Jónsson, H.: Phys. Rev. Lett. 121(19), 197202 (2018). https://doi.org/10.1103/PhysRevLett.121.197202

  180. Foster, D., Kind, C., Ackerman, P.J., Tai, J.S.B., Dennis, M.R., Smalyukh, I.I.: Nat. Phys. 15(7), 655 (2019). https://doi.org/10.1038/s41567-019-0476-x

    Article  Google Scholar 

  181. Kuchkin, V.M., Barton-Singer, B., Rybakov, F.N., Blügel, S., Schroers, B.J., Kiselev, N.S.: Phys. Rev. B 102(14), 144422 (2020). https://doi.org/10.1103/PhysRevB.102.144422

  182. Kind, C., Foster, D.: Physical Review B 103(10), L100413 (2021). https://doi.org/10.1103/PhysRevB.103.L100413

    Article  ADS  Google Scholar 

  183. Kuchkin, V.M., Chichay, K., Barton-Singer, B., Rybakov, F.N., Blügel, S., Schroers, B.J., Kiselev, N.S.: arXiv:2104.02342 [cond-mat, physics:hep-th] (2021)

  184. Lin, S.Z., Bulaevskii, L.N.: Physical Review B 88(6), 060404 (2013). https://doi.org/10.1103/PhysRevB.88.060404

  185. Roldán-Molina, A., Santander, M.J., Nunez, A.S., Fernández-Rossier, J.: Phys. Rev. B 92(24), 245436 (2015). https://doi.org/10.1103/PhysRevB.92.245436

  186. Takashima, R., Ishizuka, H., Balents, L.: Phys. Rev. B 94(13), 134415 (2016). https://doi.org/10.1103/PhysRevB.94.134415

  187. Aristov, D.N., Matveeva, P.G.: Phys. Rev. B 94(21), 214425 (2016). https://doi.org/10.1103/PhysRevB.94.214425

  188. Díaz, S., Arovas, D.: Memorial Volume for Shoucheng Zhang, p. 19-33 (2021). https://doi.org/10.1142/9789811231711_0004

  189. Psaroudaki, C., Hoffman, S., Klinovaja, J., Loss, D.: Phys. Rev. X 7(4), 041045 (2017). https://doi.org/10.1103/PhysRevX.7.041045

  190. Douçot, B., Kovrizhin, D.L., Moessner, R.: Ann. Physics 399, 239 (2018). https://doi.org/10.1016/j.aop.2018.10.008

    Article  MathSciNet  ADS  Google Scholar 

  191. Derras-Chouk, A., Chudnovsky, E.M., Garanin, D.A.: Phys. Rev. B 98(2), 024423 (2018). https://doi.org/10.1103/PhysRevB.98.024423

  192. Psaroudaki, C., Loss, D.: Phys. Rev. Lett. 120(23), 237203 (2018). https://doi.org/10.1103/PhysRevLett.120.237203

  193. Psaroudaki, C., Aseev, P., Loss, D.: Phys. Rev. B 100(13), 134404 (2019). https://doi.org/10.1103/PhysRevB.100.134404

  194. Vlasov, S.M., Bessarab, P.F., Lobanov, I.S., Potkina, M.N., Uzdin, V.M., Jónsson, H.: New J. Phys. 22(8), 083013 (2020). https://doi.org/10.1088/1367-2630/ab9f6d

  195. Mook, A., Klinovaja, J., Loss, D.: Phys. Rev. Research 2(3), 033491 (2020). https://doi.org/10.1103/PhysRevResearch.2.033491

  196. Benalcazar, W.A., Bernevig, B.A., Hughes, T.L.: Science 357(6346), 61 (2017). https://doi.org/10.1126/science.aah6442

    Article  ADS  Google Scholar 

  197. Cheng, Q.B., He, J., Kou, S.P.: Phys. Lett. A 380(5), 779 (2016). https://doi.org/10.1016/j.physleta.2015.11.030

    Article  ADS  Google Scholar 

  198. Zhou, T., Dartiailh, M.C., Mayer, W., Han, J.E., Matos-Abiague, A., Shabani, J., Žutić, I.: Phys. Rev. Lett. 124, 137001 (2020). https://doi.org/10.1103/PhysRevLett.124.137001

  199. Harper, F., Pushp, A., Roy, R.: Phys. Rev. Research 1, 033207 (2019). https://doi.org/10.1103/PhysRevResearch.1.033207

  200. Potter, A.C., Lee, P.A.: Phys. Rev. Lett. 105, 227003 (2010). https://doi.org/10.1103/PhysRevLett.105.227003

  201. Sedlmayr, N., Aguiar-Hualde, J.M., Bena, C.: Phys. Rev. B 93, 155425 (2016). https://doi.org/10.1103/PhysRevB.93.155425

  202. Liu, T., He, J.J., Nori, F.: Phys. Rev. B 98, 245413 (2018). https://doi.org/10.1103/PhysRevB.98.245413

  203. Yan, Z., Song, F., Wang, Z.: Phys. Rev. Lett. 121, 096803 (2018). https://doi.org/10.1103/PhysRevLett.121.096803

  204. Wang, Q., Liu, C.C., Lu, Y.M., Zhang, F.: Phys. Rev. Lett. 121, 186801 (2018). https://doi.org/10.1103/PhysRevLett.121.186801

  205. Bernevig, B.A., Hughes, T.L., Zhang, S.C.: Science 314(5806), 1757 (2006). https://doi.org/10.1126/science.1133734

    Article  ADS  Google Scholar 

  206. Zhu, X.: Phys. Rev. Lett. 122, 236401 (2019). https://doi.org/10.1103/PhysRevLett.122.236401

  207. Huang, S.J., Hsu, Y.T.: Phys. Rev. Research 3, 013243 (2021). https://doi.org/10.1103/PhysRevResearch.3.013243

  208. Hsu, Y.T., Cole, W.S., Zhang, R.X., Sau, J.D.: Phys. Rev. Lett. 125, 097001 (2020). https://doi.org/10.1103/PhysRevLett.125.097001

  209. Volpez, Y., Loss, D., Klinovaja, J.: Phys. Rev. Lett. 122, 126402 (2019). https://doi.org/10.1103/PhysRevLett.122.126402

  210. Plekhanov, K., Müller, N., Volpez, Y., Kennes, D.M., Schoeller, H., Loss, D., Klinovaja, J.: Phys. Rev. B 103, L041401 (2021). https://doi.org/10.1103/PhysRevB.103.L041401

    Article  ADS  Google Scholar 

  211. Tiwari, A., Jahin, A., Wang, Y.: Phys. Rev. Research 2, 043300 (2020). https://doi.org/10.1103/PhysRevResearch.2.043300

  212. Ahn, J., Yang, B.J.: Phys. Rev. Research 2, 012060 (2020). https://doi.org/10.1103/PhysRevResearch.2.012060

  213. Roy, B.: Phys. Rev. B 101, 220506 (2020). https://doi.org/10.1103/PhysRevB.101.220506

  214. Wang, Y., Lin, M., Hughes, T.L.: Phys. Rev. B 98, 165144 (2018). https://doi.org/10.1103/PhysRevB.98.165144

  215. Laubscher, K., Loss, D., Klinovaja, J.: Phys. Rev. Research 1, 032017 (2019). https://doi.org/10.1103/PhysRevResearch.1.032017

  216. Franca, S., Efremov, D.V., Fulga, I.C.: Phys. Rev. B 100, 075415 (2019). https://doi.org/10.1103/PhysRevB.100.075415

  217. Pahomi, T.E., Sigrist, M., Soluyanov, A.A.: Phys. Rev. Research 2, 032068 (2020). https://doi.org/10.1103/PhysRevResearch.2.032068

  218. Li, Y.X., Zhou, T.: Phys. Rev. B 103, 024517 (2021). https://doi.org/10.1103/PhysRevB.103.024517

  219. Kheirkhah, M., Yan, Z., Nagai, Y., Marsiglio, F.: Phys. Rev. Lett. 125, 017001 (2020). https://doi.org/10.1103/PhysRevLett.125.017001

  220. Roberts, E., Behrends, J., Béri, B.: Phys. Rev. B 101, 155133 (2020). https://doi.org/10.1103/PhysRevB.101.155133

  221. van Miert, G., Ortix, C.: npj Quantum Materials 5, 63 (2020). https://doi.org/10.1038/s41535-020-00265-7

  222. Fedoseev, A.D.: J. Phys. Condens. Matter 32(40), 405302 (2020). https://doi.org/10.1088/1361-648x/ab9e2f

  223. Ikegaya, S., Rui, W.B., Manske, D., Schnyder, A.P.: Phys. Rev. Research 3, 023007 (2021). https://doi.org/10.1103/PhysRevResearch.3.023007

  224. El Hassan, A., Kunst, F.K., Moritz, A., Andler, G., Bergholtz, E.J., Bourennane, M.: Nat. Photonics 13, 697 (2019). https://doi.org/10.1038/s41566-019-0519-y

    Article  ADS  Google Scholar 

  225. Chen, X.D., Deng, W.M., Shi, F.L., Zhao, F.L., Chen, M., Dong, J.W.: Phys. Rev. Lett. 122, 233902 (2019). https://doi.org/10.1103/PhysRevLett.122.233902

  226. Ni, X., Weiner, M., Alu, A., Khanikaev, A.B.: Nat. Mater. 18, 113 (2019). https://doi.org/10.1038/s41563-018-0252-9

    Article  Google Scholar 

  227. Xue, H., Yang, Y., Liu, G., Gao, F., Chong, Y., Zhang, B.: Phys. Rev. Lett. 122, 244301 (2019). https://doi.org/10.1103/PhysRevLett.122.244301

  228. Imhof, S., Berger, C., Bayer, F., Brehm, J., Molenkamp, L.W., Kiessling, T., Schindler, F., Lee, C.H., Greiter, M., Neupert, T., Thomalem, R.: Nat. Phys. 14, 925 (2018). https://doi.org/10.1038/s41567-018-0246-1

    Article  Google Scholar 

  229. Serra-Garcia, M., Süsstrunk, R., Huber, S.D.: Phys. Rev. B 99, 020304 (2019). https://doi.org/10.1103/PhysRevB.99.020304

  230. Bao, J., Zou, D., Zhang, W., He, W., Sun, H., Zhang, X.: Phys. Rev. B 100, 201406 (2019). https://doi.org/10.1103/PhysRevB.100.201406

  231. Volovik, G.E.: JETP Letters 91, 201 (2010). https://doi.org/10.1134/S0021364010040090

Download references

Acknowledgements

We acknowledge fruitful discussions with V.V. Val’kov and S.V. Aksenov. M.S.S. also thanks M.N. Potkina for the valuable discussions.

Funding

The study was funded by the Russian Foundation for Basic Research (Project No. 19-02-00348), Government of Krasnoyarsk Territory, Krasnoyarsk Regional Fund of Science (Grants No. 19-42-240011, 20-42-243001). A.O.Z. and M.S.S. are grateful to the support of the Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS”.

Author information

Authors and Affiliations

  1. Kirensky Institute of Physics, Federal Research Center KSC SB RAS, 660036, Krasnoyarsk, Russia

    A. O. Zlotnikov, M. S. Shustin & A. D. Fedoseev

Authors
  1. A. O. Zlotnikov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. S. Shustin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. D. Fedoseev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. O. Zlotnikov.

Ethics declarations

Conflicts of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zlotnikov, A.O., Shustin, M.S. & Fedoseev, A.D. Aspects of Topological Superconductivity in 2D Systems: Noncollinear Magnetism, Skyrmions, and Higher-order Topology. J Supercond Nov Magn 34, 3053–3088 (2021). https://doi.org/10.1007/s10948-021-06029-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10948-021-06029-z

Keywords

PACS

Search

Navigation