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Reviews of topical problems


Topological superconductivity and Majorana states in low-dimensional systems

 a,  a,  a,  a,  a,  a, b,  c, d
a Kirensky Institute of Physics, Federal Research Center Krasnoyarsk Science Center of the Siberian Branch of the Russian Academy of Sciences, Academgorodok 50, str. 38, Krasnoyarsk, 660036, Russian Federation
b Reshetnev Siberian State Aerospace University, prosp. Gazety Krasnoyarskii rabochii 31, Krasnoyarsk, 660014, Russian Federation
c Kapitza Institute of Physical Problems, Russian Academy of Sciences, ul. Kosygina 2, Moscow, 117334, Russian Federation
d HSE University, ul. Myasnitskaya 20, Moscow, 101000, Russian Federation

We discuss the properties of topologically nontrivial superconducting phases and the conditions for their realization in condensed media, the criteria for the appearance of elementary Majorana-type excitations in solids, and the corresponding principles and experimental methods for identifying Majorana bound states (MBSs). Along with the well-known Kitaev chain and superconducting nanowire (SNW) models with spin--orbit coupling in an external magnetic field, we discuss models of quasi-two-dimensional materials in which MBSs are realized in the presence of noncollinear spin ordering. For finite-length SNWs, we demonstrate a cascade of quantum transitions occurring with a change in the magnetic field, accompanied by a change in the fermion parity of the ground state. The corresponding anomalous behavior of the magnetocaloric effect can be used as a tool for identifying MBSs. We devote considerable attention to the analysis of the transport characteristics of devices that contain topologically nontrivial materials. The results of studying the conductivity of an Aharonov--Bohm ring whose arms are connected by an SNW are discussed in detail. An important feature of this device is the appearance of Fano resonances in the dependence of conductance on the magnetic field when the SNW is in a topologically nontrivial phase. We establish a relation between the characteristics of such resonances and the spatial structure of the lowest-energy SNW state. The conditions for the occurrence of an MBS in the phase of the coexistence of chiral d+id superconductivity and 120-degree spin ordering are determined in the framework of the t-J-V model on a triangular lattice. We take electron--electron interactions into account in discussing the topological invariants of low-dimensional superconducting materials with noncollinear spin ordering. The formation of Majorana modes in regions with an odd value of a topological ℤ invariant is demonstrated. The spatial structure of these excitations in the Hubbard fermion ensemble is determined.

Fulltext pdf (2.3 MB)
Fulltext is also available at DOI: 10.3367/UFNe.2021.03.038950
Keywords: low-dimensional system, Majorana bound state, spin—orbit coupling, topological invariant, Coulomb correlations, noncollinear magnetism, topological superconductivity, quantum transport
PACS: 71.20.Nr, 71.20.Ps, 71.23.An, 71.70.Ej, 73.23.−b, 74.20.Rp, 74.25.F−, 74.90.+n (all)
DOI: 10.3367/UFNe.2021.03.038950
URL: https://ufn.ru/en/articles/2022/1/b/
000788597000002
2-s2.0-85127103681
2022PhyU...65....2V
Citation: Val’kov V V, Shustin M S, Aksenov S V, Zlotnikov A O, Fedoseev A D, Mitskan V A, Kagan M Yu "Topological superconductivity and Majorana states in low-dimensional systems" Phys. Usp. 65 2–39 (2022)
BibTexBibNote ® (generic)BibNote ® (RIS)MedlineRefWorks

Received: 26th, May 2020, revised: 18th, December 2020, 11th, March 2021

Оригинал: Вальков В В, Шустин М С, Аксенов С В, Злотников А О, Федосеев А Д, Мицкан В А, Каган М Ю «Топологическая сверхпроводимость и майорановские состояния в низкоразмерных системах» УФН 192 3–44 (2022); DOI: 10.3367/UFNr.2021.03.038950

References (285) ↓ Cited by (21) Similar articles (20)

  1. Haldane F D M Phys. Rev. Lett. 47 1840 (1981)
  2. Haldane F D M J. Phys. C 14 2585 (1981)
  3. Niu Q, Thouless D J, Wu Y-S Phys. Rev. B 31 3372 (1985)
  4. Volovik G E The Universe In A Helium Droplet (Intern. Ser. of Monographs on Physics) Vol. 117 (New York: Oxford Univ. Press, 2009)
  5. Hasan M Z, Kane C L Rev. Mod. Phys. 82 3045 (2010)
  6. Lozovik Yu E Usp. Fiz. Nauk 182 1111 (2012); Lozovik Yu E Phys. Usp. 55 1035 (2012)
  7. Tarasenko S A Usp. Fiz. Nauk 188 1129 (2018); Tarasenko S A Phys. Usp. 61 1026 (2018)
  8. Pankratov O A Usp. Fiz. Nauk 188 1226 (2018); Pankratov O A Phys. Usp. 61 1116 (2018)
  9. Schnyder A P et al Phys. Rev. B 78 195125 (2008)
  10. Nayak Ch et al Rev. Mod. Phys. 80 1083 (2008)
  11. Read N, Green D Phys. Rev. B 61 10267 (2000)
  12. Kitaev A Yu Usp. Fiz. Nauk 171 (Suppl. 10) 131 (2001); Kitaev A Yu Phys. Usp. 44 (10S) 131 (2001)
  13. Kagan M Yu, Turlapov A V Usp. Fiz. Nauk 189 225 (2019); Kagan M Yu, Turlapov A V Phys. Usp. 62 215 (2019)
  14. Volovik G E Pis’ma ZhETF 90 440 (2009); Volovik G E JETP Lett. 90 398 (2009)
  15. Laughloin R B Physica C 234 280 (1994)
  16. Volovik G E Pis’ma ZhETF 66 492 (1997); Volovik G E JETP Lett. 66 522 (1997)
  17. Balatsky A V Phys. Rev. Lett. 80 1972 (1998)
  18. Baskaran G Phys. Rev. Lett. 91 097003 (2003)
  19. Nandkishore R, Levitov L S, Chubukov A V Nat. Phys. 8 158 (2012)
  20. Nandkishore R, Thomale R, Chubukov A V Phys. Rev. B 89 144501 (2014)
  21. Kagan M Yu, Mitskan V A, Korovushkin M M Usp. Fiz. Nauk 185 785 (2015); Kagan M Yu, Mitskan V A, Korovushkin M M Phys. Usp. 58 733 (2015)
  22. Mackenzie A P, Maeno Y Rev. Mod. Phys. 75 657 (2003)
  23. Das Sarma S, Nayak C, Tewari S Phys. Rev. B 73 220502 (2006)
  24. Pustogow A et al Nature 574 72 (2019)
  25. Suzuki S-I, Sato M, Tanaka Y Phys. Rev. B 101 054505 (2020)
  26. Sau J D, Tewari S Phys. Rev. B 86 104509 (2012)
  27. Mineev V P Usp. Fiz. Nauk 187 129 (2017); Mineev V P Phys. Usp. 60 121 (2017)
  28. Fu L, Kane C L Phys. Rev. Lett. 100 096407 (2008)
  29. Snelder M et al J. Phys. Condens. Matter 27 315701 (2015)
  30. Sato M, Fujimoto S Phys. Rev. B 79 094504 (2009)
  31. Hor Y S et al Phys. Rev. Lett. 104 057001 (2010)
  32. Sasaki S, Mizushima T Physica C 514 206 (2015)
  33. Zhang P et al Science 360 182 (2018)
  34. Sau J D et al Phys. Rev. Lett. 104 040502 (2010)
  35. Lutchyn R M, Sau J D, Das Sarma S Phys. Rev. Lett. 105 077001 (2010)
  36. Oreg Y, Refael G, von Oppen F Phys. Rev. Lett. 105 177002 (2010)
  37. Stanescu T D, Lutchyn R M, Das Sarma S Phys. Rev. B 84 144522 (2011)
  38. Mourik V et al Science 336 1003 (2012)
  39. Deng M T et al Nano Lett. 12 6414 (2012)
  40. Choy T-P et al Phys. Rev. B 84 195442 (2011)
  41. Martin I, Morpurgo A F Phys. Rev. B 85 144505 (2012)
  42. Alicea J Rep. Prog. Phys. 75 076501 (2012)
  43. Beenakker C W J Annu. Rev. Condens. Matter Phys. 4 113 (2013)
  44. Elliot S R, Franz M Rev. Mod. Phys. 87 137 (2015)
  45. Sato M, Ando Y Rep. Prog. Phys. 80 076501 (2017)
  46. Goldstein G, Chamon C Phys. Rev. B 86 115122 (2012)
  47. Kells G Phys. Rev. B 92 081401 (2015)
  48. Kells G Phys. Rev. B 92 155434 (2015)
  49. Monthus C J. Phys. A 51 265303 (2018)
  50. Niu Y et al Phys. Rev. B 85 035110 (2012)
  51. DeGottardi W et al Phys. Rev. B 88 165111 (2013)
  52. Tong Q-J et al Phys. Rev. B 87 201109 (2013)
  53. Viyuela O et al Phys. Rev. B 94 125121 (2016)
  54. Lepori L, Dell’Anna L New. J. Phys 19 103030 (2017)
  55. Alecce A, Dell’Anna L Phys. Rev. B 95 195160 (2017)
  56. Kitaev A Ann. Physics 321 2 (2006)
  57. Ghosh P et al Phys. Rev. B 82 184525 (2010)
  58. Budich J C, Ardonne E Phys. Rev. B 88 075419 (2013)
  59. Berry M V Proc. R. Soc. Lond. A 392 45 (1984)
  60. Zak J Phys. Rev. Lett. 62 2747 (1989)
  61. Kao H Phys. Rev. B 90 245435 (2014)
  62. Zvyagin A A Fizika Nizkikh Temperatur 41 806 (2015); Zvyagin A A Low Temp. Phys. 41 625 (2015)
  63. Hegde S et al New J. Phys. 17 053036 (2015)
  64. Hegde S, Vishveshwara S Phys. Rev. B 94 115166 (2016)
  65. Kawabata K et al Phys. Rev. B 95 195140 (2017)
  66. Zeng C et al Phys. Rev. B 99 094523 (2019)
  67. Leumer N et al J. Phys. Condens. Matter 32 445502 (2020)
  68. Zyuzin A A et al Phys. Rev. Lett. 111 056802 (2013)
  69. Fedoseev A D Zh. Eksp. Teor. Fiz. 155 138 (2019); Fedoseev A D J. Exp. Theor. Phys. 128 125 (2019)
  70. Thomale R, Rachel S, Schmitteckert P Phys. Rev. B 88 161103 (2013)
  71. Chan Y-H, Chiu C-K, Sun K Phys. Rev. B 92 104514 (2015)
  72. Gergs M, Fritz L, Schuricht D Phys. Rev. B 93 075129 (2016)
  73. Miao J-J et al Sci. Rep. 8 488 (2018)
  74. Val’kov V V, Mitskan V A, Shustin M S Pis’ma ZhETF 106 762 (2017); Val’kov V V, Mitskan V A, Shustin M S JETP Lett. 106 798 (2017)
  75. Val’kov V V, Kagan M Yu, Aksenov S V J. Phys. Condens. Matter 31 225301 (2019)
  76. Val’kov V V, Aksenov S V J. Magn. Magn. Mater. 465 88 (2018)
  77. Karcher J F, Sonner M, Mirlin A D Phys. Rev. B 100 134207 (2019)
  78. Zhang H et al Nature 556 74 (2018)
  79. Gangadharaiah S et al Phys. Rev. Lett. 107 036801 (2011)
  80. Fidkowski L, Kitaev A Phys. Rev. B 81 134509 (2010)
  81. Fidkowski L, Kitaev A Phys. Rev. B 83 075103 (2011)
  82. Turner A M, Pollmann F, Berg E Phys. Rev. B 83 075102 (2011)
  83. Katsura H, Schuricht D, Takahashi M Phys. Rev. B 92 115137 (2015)
  84. Miao F-C et al Phys. Rev. Lett. 118 267701 (2017)
  85. Iemini F et al Phys. Rev. Lett. 115 156402 (2015)
  86. Rahmani A et al Phys. Rev. B 92 235123 (2015)
  87. Hassler F, Schuricht D New. J. Phys. 14 125018 (2012)
  88. Tao Z et al Phys. Rev. B 101 035109 (2020)
  89. Brouwer P W et al Phys. Rev. Lett. 107 196804 (2011)
  90. Kaladzhyan V, Bena C Phys. Rev. B 100 081106 (2019)
  91. Tinyukova T S, Chuburin Yu P Teor. Mat. Fiz. 200 137 (2019); Tinyukova T S, Chuburin Yu P Theor. Math. Phys. 200 1043 (2019)
  92. Thakurathi M, Deb O, Sen D J. Phys. Condens. Matter 27 275702 (2015)
  93. Sedlmayr N, Bena C Phys. Rev. B 92 115115 (2015)
  94. Sedlmayr N, Aguiar-Hualde J M, Bena C Phys. Rev. B 93 155425 (2016)
  95. Zhang G, Song Z Phys. Rev. Lett. 115 177204 (2015)
  96. Val’kov V V, Mitskan V A, Shustin M S Zh. Eksp. Teor. Fiz. 157 281 (2019); Val’kov V V, Mitskan V A, Shustin M S J. Exp. Theor. Phys. 130 235 (2019)
  97. Val’kov V V i dr Pis’ma ZhETF 126 126 (2019); Val’kov V V et al JETP Lett. 110 140 (2019)
  98. Das A et al Nat. Phys. 8 887 (2012)
  99. Andreev A F Zh. Eksp. Teor. Fiz. 46 1823 (1964); Andreev A F Sov. Phys. JETP 19 1228 (1964)
  100. Blonder G E, Tinkham M, Klapwijk T M Phys. Rev. B 25 4515 (1982)
  101. Law K T, Lee P A, Ng T K Phys. Rev. Lett. 103 237001 (2009)
  102. Flensberg K Phys. Rev. B 82 180516 (2010)
  103. Wu B H, Cao J C Phys. Rev. B 85 085415 (2012)
  104. Das Sarma S, Sau J D, Stanescu T D Phys. Rev. B 86 220506 (2012)
  105. Rainis D et al Phys. Rev. B 87 024515 (2013)
  106. Deng M T et al Science 354 1557 (2016)
  107. Nichele F et al Phys. Rev. Lett. 119 136803 (2017)
  108. Takei S et al Phys. Rev. Lett. 110 186803 (2013)
  109. Liu C-X et al Phys. Rev. B 96 054520 (2017)
  110. Cheng M, Lutchyn R M, Das Sarma S Phys. Rev. B 85 165124 (2012)
  111. Rainis D, Loss D Phys. Rev. B 85 174533 (2012)
  112. Liu J et al Phys. Rev. Lett. 109 267002 (2012)
  113. Bagrets D, Altland A Phys. Rev. Lett. 109 227005 (2012)
  114. Rieder M-T et al Phys. Rev. B 86 125423 (2012)
  115. Pan H et al Phys. Rev. B 101 024506 (2020)
  116. Kells G, Meidan D, Brouwer P W Phys. Rev. B 85 060507 (2012)
  117. Lee E J H et al Phys. Rev. Lett. 109 186802 (2012)
  118. Pikulin D I et al New J. Phys. 14 125011 (2012)
  119. Krogstrup P et al Nat. Mater. 14 400 (2015)
  120. Chang W et al Nat. Nanotechnol. 10 232 (2015)
  121. Gül Ö et al Nano Lett. 17 2690 (2017)
  122. Gazibegovic S et al Nature 548 434 (2017)
  123. Zhang H et al Nat. Commun. 8 16025 (2017)
  124. Franz M, Pikulin D I Nat. Phys. 14 334 (2018)
  125. Zhang H et al Nature 581 E4 (2020)
  126. Moore C, Stanescu T D, Tewari S Phys. Rev. B 97 165302 (2018)
  127. Kells G, Meidan D, Brouwer P Phys. Rev. B 86 100503 (2012)
  128. Stanescu T D, Lutchyn R M, Das Sarma S Phys. Rev. B 87 094518 (2013)
  129. Haim A et al Phys. Rev. Lett. 114 166406 (2015)
  130. Chen J et al Phys. Rev. Lett. 123 107703 (2019)
  131. Woods B D et al Phys. Rev. B 100 125407 (2019)
  132. Liu C-X et al Phys. Rev. B 96 075161 (2017)
  133. Moore C et al Phys. Rev. B 98 155314 (2018)
  134. Lee E J H et al Nat. Nanotechnol. 9 79 (2014)
  135. Liu C-X, Sau J D, Das Sarma S Phys. Rev. B 97 214502 (2018)
  136. Reeg C et al Phys. Rev. B 98 245407 (2018)
  137. Deng M-T et al Phys. Rev. B 98 085125 (2018)
  138. Yu P et al Nat. Phys. 17 482 (2021)
  139. Li J et al Sci. Rep. 4 4930 (2014)
  140. Prada E, Aguado R, San-Jose P Phys. Rev. B 96 085418 (2017)
  141. Clarke D J Phys. Rev. B 96 201109 (2017)
  142. Ricco L S et al Phys. Rev. B 99 155159 (2019)
  143. Lee M, Lim J S, López R Phys. Rev. B 87 241402 (2013)
  144. Nilsson J, Akhmerov A R, Beenakker C W J Phys. Rev. Lett. 101 120403 (2008)
  145. Liu J, Zhang F-C, Law K T Phys. Rev. B 88 064509 (2013)
  146. Bolech C J, Demler E Phys. Rev. Lett. 98 237002 (2007)
  147. Liu D E, Cheng M, Lutchyn R M Phys. Rev. B 91 081405 (2015)
  148. Haim A et al Phys. Rev. B 92 245112 (2015)
  149. Valentini S et al Physica E 82 254 (2016)
  150. Smirnov S New J. Phys. 19 063020 (2017)
  151. Smirnov S Phys. Rev. B 97 165434 (2018)
  152. Leijnse M, Flensberg K Phys. Rev. Lett. 107 210502 (2011)
  153. Sticlet D, Bena C, Simon P Phys. Rev. Lett. 108 096802 (2012)
  154. Nagai Y, Nakamura H, Machida M J. Phys. Soc. Jpn. 83 064703 (2014)
  155. Guigou M et al Europhys. Lett. 115 47005 (2016)
  156. Serina M, Loss D, Klinovaja J Phys. Rev. B 98 035419 (2018)
  157. Val’kov V V, Aksenov S V FNT 43 546 (2017); Val’kov V V, Aksenov S V Low Temp. Phys. 43 436 (2017)
  158. Val’kov V V, Aksenov S V J. Magn. Magn. Mater. 440 112 (2017)
  159. He J J et al Phys. Rev. Lett. 112 037001 (2014)
  160. He J J et al Nat. Commun. 5 3232 (2014)
  161. Wu B H et al Phys. Rev. B 90 205435 (2014)
  162. Albrecht S M et al Nature 531 206 (2016)
  163. Hekking F W J et al Phys. Rev. Lett. 70 4138 (1993)
  164. Hergenrother J M, Tuominen M T, Tinkham M Phys. Rev. Lett. 72 1742 (1994)
  165. Averin D V, Nazarov Yu V Phys. Rev. Lett. 69 1993 (1992)
  166. Fu L Phys. Rev. Lett. 104 056402 (2010)
  167. Zazunov A, Yeyati A L, Egger R Phys. Rev. B 84 165440 (2011)
  168. Hützen R et al Phys. Rev. Lett. 109 166403 (2012)
  169. Khaymovich I M, Pekola J P, Melnikov A S New J. Phys. 19 123026 (2017)
  170. Chiu C-K, Sau J D, Das Sarma S Phys. Rev. B 96 054504 (2017)
  171. Aharonov Y, Bohm D Phys. Rev. 115 485 (1959)
  172. Yacoby A et al Phys. Rev. Lett. 73 3149 (1994)
  173. Kobayashi K et al Phys. Rev. B 68 235304 (2003)
  174. Cabosart D et al Phys. Rev. B 90 205433 (2014)
  175. Akhmerov A R et al Phys. Rev. Lett. 106 057001 (2011)
  176. Whiticar A M et al Nat. Commun. 11 3212 (2020)
  177. Ioselevich P A, Feigel’man M V Phys. Rev. Lett. 106 077003 (2011)
  178. Snelder M et al Phys. Rev. B 87 104507 (2013)
  179. Jacquod Ph, Büttiker M Phys. Rev. B 88 241409 (2013)
  180. Tripathi K M, Das S, Rao S Phys. Rev. Lett. 116 166401 (2016)
  181. Sau J D, Swingle B, Tewari S Phys. Rev. B 92 020511 (2015)
  182. Hell M, Flensberg K, Leijnse M Phys. Rev. B 97 161401 (2018)
  183. Flensberg K Phys. Rev. Lett. 106 090503 (2011)
  184. Liu D E, Baranger H U Phys. Rev. B 84 201308 (2011)
  185. Chiu C-K, Sau J D, Das Sarma S Phys. Rev. B 97 035310 (2018)
  186. Fano U Phys. Rev. 124 1866 (1961)
  187. Dessotti F A et al J. Appl. Phys. 116 173701 (2014)
  188. Shang E-M et al Chinese Phys. B 23 057201 (2014)
  189. Ueda A, Yokoyama T Phys. Rev. B 90 081405 (2014)
  190. Jiang C, Zheng Y-S Solid State Commun. 212 14 (2015)
  191. Schuray A, Weithofer L, Recher P Phys. Rev. B 96 085417 (2017)
  192. Zhang Y-J et al Superlattices Microstruct. 113 25 (2018)
  193. Rainis D et al Phys. Rev. Lett. 112 196803 (2014)
  194. Keldysh L V Zh. Eksp. Teor. Fiz. 47 1515 (1964); Keldysh L V Sov. Phys. JETP 20 1018 (1965)
  195. Arseev P I Usp. Fiz. Nauk 185 1271 (2015); Arseev P I Phys. Usp. 58 1159 (2015)
  196. Arseev P I i dr Usp. Fiz. Nauk 187 1147 (2017); Arseev P I et al Phys. Usp. 60 1067 (2017)
  197. Rogovin D, Scalapino D J Ann. Physics 86 1 (1974)
  198. Zeng Z Y, Li B, Claro F Phys. Rev. B 68 115319 (2003)
  199. Vonsovskii S V, Izyumov Yu A, Kurmaev E Z Sverkhprovodimost’ Perekhodnykh Metallov, Ikh Splavov i Soedinenii (M.: Nauka, 1977); Per. na angl. yaz., Vonsovsky S V, Izyumov Y A, Kurmaev E Z Superconductivity Of Transition Metals: Their Alloys And Compounds (Berlin: Springer-Verlag, 1982)
  200. von Neumann J, Wigner E Phys. Z. 30 465 (1929)
  201. Volya A, Zelevinsky V Phys. Rev. C 67 054322 (2003)
  202. Sadreev A F, Bulgakov E N, Rotter I Phys. Rev. B 73 235342 (2006)
  203. Lee H-W Phys. Rev. Lett. 82 2358 (1999)
  204. Sadreev A F, Rotter I J. Phys. A 36 11413 (2003)
  205. Nowak M P, Szafran B, Peeters F M Phys. Rev. B 84 235319 (2011)
  206. Lu H, Lü R, Zhu B Phys. Rev. B 71 235320 (2005)
  207. Landrón de Guevara M L, Orellana P A Phys. Rev. B 73 205303 (2006)
  208. Shahbazyan T V, Raikh M E Phys. Rev. B 49 17123 (1994)
  209. Vorrath T, Brandes T Phys. Rev. B 68 035309 (2003)
  210. Landrón de Guevara M L, Claro F, Orellana P A Phys. Rev. B 67 195335 (2003)
  211. Orellana P A, Landrón de Guevara M L, Claro F Phys. Rev. B 70 233315 (2004)
  212. Dicke R H Phys. Rev. 89 472 (1953)
  213. Myoung N et al Phys. Rev. B 100 045427 (2019)
  214. Stoudenmire E M et al Phys. Rev. B 84 014503 (2011)
  215. Kim Ch S, Satanin A M Zh. Eksp. Teor. Fiz. 115 211 (1999); Kim C S, Satanin A M J. Exp. Theor. Phys. 88 118 (1999)
  216. Kim Ch S i dr Zh. Eksp. Teor. Fiz. 116 263 (1999); Kim C S et al J. Exp. Theor. Phys. 89 144 (1999)
  217. Aksenov S V, Kagan M Yu Pis’ma ZhETF 111 321 (2020); Aksenov S V, Kagan M Yu JETP Lett. 111 286 (2020)
  218. Miroshnichenko A E, Flach S, Kivshar Yu S Rev. Mod. Phys. 82 2257 (2010)
  219. Zubarev D N Usp. Fiz. Nauk 71 71 (1960); Zubarev D N Sov. Phys. Usp. 3 320 (1960)
  220. Miroshnichenko A E, Kivshar Yu S Phys. Rev. E 72 056611 (2005)
  221. Braunecker B et al Phys. Rev. B 82 045127 (2010)
  222. Kjaergaard M, Wölms K, Flensberg K Phys. Rev. B 85 020503 (2012)
  223. Egger R, Flensberg K Phys. Rev. B 85 235462 (2012)
  224. Kornich V et al Phys. Rev. B 101 125414 (2020)
  225. Braunecker B, Simon P Phys. Rev. Lett. 111 147202 (2013)
  226. Klinovaja J et al Phys. Rev. Lett. 111 186805 (2013)
  227. Vazifeh M M, Franz M Phys. Rev. Lett. 111 206802 (2013)
  228. Yu L Acta Phys. Sin. 4 113 (1965)
  229. Shiba H Prog. Theor. Phys. 40 435 (1968)
  230. Rusinov A I Zh. Eksp. Teor. Fiz. 56 2047 (1969); Rusinov A I Sov. Phys. JETP 29 1101 (1969)
  231. Choi D-J et al Rev. Mod. Phys. 91 041001 (2019)
  232. Nadj-Perge S et al Phys. Rev. B 88 020407 (2013)
  233. Nadj-Perge S et al Science 346 602 (2014)
  234. Ruby M et al Phys. Rev. Lett. 115 197204 (2015)
  235. Pawlak R et al Npj Quantum Inf. 2 16035 (2016)
  236. Ruby M et al Nano Lett. 17 4473 (2017)
  237. Feldman B E et al Nat. Phys. 13 286 (2017)
  238. Kim H et al Sci. Adv. 4 eaar5251 (2018)
  239. Spethmann J et al Phys. Rev. Lett. 124 227203 (2020)
  240. Bedow J et al Phys. Rev. B 102 180504 (2020)
  241. Crawford D et al Phys. Rev. B 101 174510 (2020)
  242. Rex S, Gornyi I V, Mirlin A D Phys. Rev. B 102 224501 (2020)
  243. Heimes A, Kotetes P, Schön G Phys. Rev. B 90 060507 (2014)
  244. Pawlak R et al Prog. Part. Nucl. Phys. 107 1 (2019)
  245. Buzdin A I i dr Usp. Fiz. Nauk 144 597 (1984); Buzdin A I et al Sov. Phys. Usp. 27 927 (1984)
  246. Iida K et al Phys. Rev. B 100 014506 (2019)
  247. Devizorova Zh, Buzdin A Phys. Rev. B 100 104523 (2019)
  248. Kim T K et al Phys. Rev. B 103 174517 (2021)
  249. Takada K et al Nature 422 53 (2003)
  250. Pyon S, Kudo K, Nohara M J. Phys. Soc. Jpn. 81 053701 (2012)
  251. McKenzie R H Comments Condens. Matter Phys. 18 309 (1998)
  252. Golosov D I, Chubukov A V Pis’ma ZhETF 50 416 (1989); Golosov D I, Chubukov A V JETP Lett. 50 451 (1989)
  253. Ritchey I, Coleman P J. Phys. Condens. Matter 2 9227 (1990)
  254. Chubukov A V, Golosov D I J. Phys. Condens. Matter 3 69 (1991)
  255. Barabanov A F, Mikheyenkov A V Pis’ma ZhETF 56 470 (1992); Barabanov A F, Mikheyenkov A V JETP Lett. 56 454 (1992)
  256. Chubukov A V, Senthil T, Sachdev S Phys. Rev. Lett. 72 2089 (1994)
  257. Chubukov A V, Sachdev S, Senthil T J. Phys. Condens. Matter 6 8891 (1994)
  258. Mikheenkov A V i dr Zh. Eksp. Teor. Fiz. 153 483 (2018); Mikheyenkov A V et al J. Exp. Theor. Phys. 126 404 (2018)
  259. Lu Y-M, Wang Z Phys. Rev. Lett. 110 096403 (2013)
  260. Yao M-Y et al Phys. Rev. B 91 161411 (2015)
  261. Buzdin A I, Bulaevskii L N Usp. Fiz. Nauk 149 45 (1986); Buzdin A I, Bulaevskii L N Sov. Phys. Usp. 29 412 (1986)
  262. Pfleiderer C Rev. Mod. Phys. 81 1551 (2009)
  263. Val’kov V V, Zlotnikov A O Pis’ma ZhETF 95 390 (2012); Val’kov V V, Zlotnikov A O JETP Lett. 95 350 (2012)
  264. Lu Y-M, Xiang T, Lee D-H Nat. Phys. 10 634 (2014)
  265. Lau A, Timm C Phys. Rev. B 88 165402 (2013)
  266. Lau A, Timm C Phys. Rev. B 90 024517 (2014)
  267. Ghaemi P, Wang F, Vishwanath A Phys. Rev. Lett. 102 157002 (2009)
  268. Youmans C et al Phys. Rev. B 98 144517 (2018)
  269. Balatsky A V, Abrahams E Phys. Rev. B 45 13125 (1992)
  270. Coleman P, Miranda E, Tsvelik A Phys. Rev. Lett. 70 2960 (1993)
  271. Asano Y, Tanaka Y Phys. Rev. B 87 104513 (2013)
  272. Val’kov V V, Val’kova T A, Mitskan V A Pis’ma ZhETF 102 399 (2015); Val’kov V V, Val’kova T A, Mitskan V A JETP Lett. 102 361 (2015)
  273. Ovchinnikov S G, Val’kov V V Hubbard Operators In The Theory Of Strongly Correlated Electrons (London: Imperial College Press, 2004)
  274. Ishikawa K, Matsuyama T Nucl. Phys. B 280 523 (1987)
  275. Val’kov V V, Zlotnikov A O Pis’ma ZhETF 104 512 (2016); Val’kov V V, Zlotnikov A O JETP Lett. 104 483 (2016)
  276. Val’kov V V, Zlotnikov A O Fiz. Tverd. Tela 59 2100 (2017); Val’kov V V, Zlotnikov A O Phys. Solid State 59 2120 (2017)
  277. Val’kov V V, Zlotnikov A O Pis’ma ZhETF 109 769 (2019); Val’kov V V, Zlotnikov A O JETP Lett. 109 736 (2019)
  278. Reeg C, Loss D, Klinovaja J Phys. Rev. B 96 125426 (2017)
  279. Kopasov A A, Khaymovich I M, Mel’nikov A S Beilstein J. Nanotechnol. 9 1184 (2018)
  280. Theiler A, Björnson K, Black-Schaffer A M Phys. Rev. B 100 214504 (2019)
  281. Jiang K, Zhou S, Wang Z Phys. Rev. B 90 165135 (2014)
  282. Pasrija K, Kumar S Phys. Rev. B 93 195110 (2016)
  283. Weber C et al Phys. Rev. B 73 014519 (2006)
  284. Val’kov V V et al J. Magn. Magn. Mater. 440 37 (2017)
  285. Val’kov V V, Zlotnikov A O, Shustin M S J. Magn. Magn. Mater. 459 112 (2018)
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