Abstract
Odd-frequency superconductivity represents a truly unconventional ordered state which, in contrast to conventional superconductivity, exhibits pair correlations which are odd in relative time and, hence, inherently dynamical. In this review article we provide an overview of recent advances in the study of odd-frequency superconducting correlations in one-dimensional systems. In particular, we focus on recent developments in the study of nanowires with Rashba spin-orbit coupling and metallic edges of two-dimensional topological insulators in proximity to conventional superconductors. These systems have recently elicited a great deal of interest due to their potential for realizing one-dimensional topological superconductivity whose edges can host Majorana zero modes. We also provide a detailed discussion of the intimate relationship between Majorana zero modes and odd-frequency pairing. Throughout this review, we highlight the ways in which odd-frequency pairing provides a deeper understanding of the unconventional superconducting correlations present in each of these intriguing systems and how the study and control of these states holds the potential for future applications.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
H. Kamerlingh Onnes, Commun. Phys. Lab. Univ. Leiden, 120b (1911)
J. Bardeen, L.N. Cooper, J.R. Schrieffer, Phys. Rev. 108, 1175 (1957)
C.C. Tsuei, J.R. Kirtley, Rev. Mod. Phys. 72, 969 (2000)
A.P. Mackenzie, Y. Maeno, Rev. Mod. Phys. 75, 657 (2003)
G. Eliashberg, Sov. Phys. JETP 11, 696 (1960)
A.A. Abrikosov, L.P. Gorkov, I.E. Dzyaloshinski,Methods of quantum field theory in statistical physics (Courier Corporation, MA, 2012)
G.D. Mahan,Many-particle physics (Springer Science & Business Media, Berlin, 2013)
V.L. Berezinskii, JETP Lett. 20, 287 (1974)
T.R. Kirkpatrick, D. Belitz, Phys. Rev. Lett. 66, 1533 (1991)
D. Belitz, T.R. Kirkpatrick, Phys. Rev. B 46, 8393 (1992)
A. Balatsky, E. Abrahams, Phys. Rev. B 45, 13125 (1992)
E. Abrahams, A. Balatsky, J.R. Schrieffer, P.B. Allen, Phys. Rev. B 47, 513 (1993)
V.J. Emery, S. Kivelson, Phys. Rev. B 46, 10812 (1992)
A.V. Balatsky, J. Bonča, Phys. Rev. B 48, 7445 (1993)
N. Bulut, D.J. Scalapino, S.R. White, Phys. Rev. B 47, 6157 (1993)
P. Coleman, E. Miranda, A. Tsvelik, Phys. Rev. Lett. 70, 2960 (1993)
A. Balatsky, E. Abrahams, D. Scalapino, J. Schrieffer, Phys. B 199-200, 363 (1994)
P. Coleman, E. Miranda, A. Tsvelik, Phys. Rev. B 49, 8955 (1994)
E. Abrahams, A. Balatsky, D.J. Scalapino, J.R. Schrieffer, Phys. Rev. B 52, 1271 (1995)
E. Abrahams, A. Balatsky, J.R. Schrieffer, P.B. Allen, Phys. Rev. B 52, 15649 (1995)
A.V. Balatsky, E. Abrahams, Phys. Rev. Lett. 74, 1004 (1995)
P. Coleman, E. Miranda, A. Tsvelik, Phys. Rev. Lett. 74, 1653 (1995)
P. Coleman, A. Georges, A.M. Tsvelik, J. Phys. Condens. Matter 9, 345 (1997)
M. Vojta, E. Dagotto, Phys. Rev. B 59, R713 (1999)
D. Belitz, T.R. Kirkpatrick, Phys. Rev. B 60, 3485 (1999)
K. Hashimoto, J. Phys. Soc. Jpn. 69, 2229 (2000)
K. Hashimoto, Phys. Rev. B 64, 132507 (2001)
Y. Fuseya, H. Kohno, K. Miyake, J. Phys. Soc. Jpn. 72, 2914 (2003)
H. Aizawa, K. Kuroki, Y. Tanaka, Phys. Rev. B 77, 144513 (2008)
D. Solenov, I. Martin, D. Mozyrsky, Phys. Rev. B 79, 132502 (2009)
K. Shigeta, S. Onari, K. Yada, Y. Tanaka, Phys. Rev. B 79, 174507 (2009)
T. Hotta, J. Phys. Soc. Jpn. 78, 123710 (2009)
K. Shigeta, Y. Tanaka, K. Kuroki, S. Onari, H. Aizawa, Phys. Rev. B 83, 140509 (2011)
H. Kusunose, Y. Fuseya, K. Miyake, J. Phys. Soc. Jpn. 80, 054702 (2011)
H. Kusunose, Y. Fuseya, K. Miyake, J. Phys. Soc. Jpn. 80, 044711 (2011)
M. Matsumoto, M. Koga, H. Kusunose, J. Phys. Soc. Jpn. 81, 033702 (2012)
Y. Yanagi, Y. Yamashita, K. Ueda, J. Phys. Soc. Jpn. 81, 123701 (2012)
H. Kusunose, M. Matsumoto, M. Koga, Phys. Rev. B 85, 174528 (2012)
K. Shigeta, S. Onari, Y. Tanaka, Phys. Rev. B 85, 224509 (2012)
T. Harada, Y. Fuseya, K. Miyake, J. Phys. Soc. Jpn. 81, SB015 (2012)
F.L. Pratt, T. Lancaster, S.J. Blundell, C. Baines, Phys. Rev. Lett. 110, 107005 (2013)
K. Shigeta, S. Onari, Y. Tanaka, J. Phys. Soc. Jpn. 82, 104702 (2013)
S. Hoshino, Y. Kuramoto, Phys. Rev. Lett. 112, 167204 (2014)
S. Hoshino, Phys. Rev. B 90, 115154 (2014)
H. Funaki, H. Shimahara, J. Phys. Soc. Jpn. 83, 123704 (2014)
J. Otsuki, Phys. Rev. Lett. 115, 036404 (2015)
M. Kashiwagi, M. Kato, Phys. Procedia Phys. 81, 29 (2016)
S. Hoshino, K. Yada, Y. Tanaka, Phys. Rev. B 93, 224511 (2016)
Y. Tanaka, S. Kashiwaya, T. Yokoyama, Phys. Rev. B 71, 094513 (2005)
Y. Tanaka, Y. Asano, A.A. Golubov, S. Kashiwaya, Phys. Rev. B 72, 140503 (2005)
Y. Tanaka, A.A. Golubov, Phys. Rev. Lett. 98, 037003 (2007)
Y. Tanaka, A.A. Golubov, S. Kashiwaya, M. Ueda, Phys. Rev. Lett. 99, 037005 (2007)
M. Eschrig, T. Löfwander, T. Champel, J.C. Cuevas, J. Kopu, G. Schön, J. Low Temp. Phys. 147, 457 (2007)
Y. Tanaka, Y. Tanuma, A.A. Golubov, Phys. Rev. B 76, 054522 (2007)
F.S. Bergeret, A.F. Volkov, K.B. Efetov, Phys. Rev. Lett. 86, 4096 (2001)
A. Kadigrobov, R.I. Shekhter, M. Jonson, Europhys. Lett. 54, 394 (2001)
V.T. Petrashov, V.N. Antonov, S.V. Maksimov, R.S. Shaľkhaľdarov, Pis’ma Zh. Eksp. Teor. Fiz. 59, 523 (1994)
M. Giroud, H. Courtois, K. Hasselbach, D. Mailly, B. Pannetier, Phys. Rev. B 58, R11872 (1998)
R.S. Keizer, S.T.B. Goennenwein, T.M. Klapwijk, G. Miao, G. Xiao, A. Gupta, Nature 439, 825 (2006)
T.S. Khaire, M.A. Khasawneh, W.P. Pratt, N.O. Birge, Phys. Rev. Lett. 104, 137002 (2010)
J. Wang, M. Singh, M. Tian, N. Kumar, B. Liu, C. Shi, J.K. Jain, N. Samarth, T.E. Mallouk, M.H.W. Chan, Nat. Phys. 6, 389 (2010)
J.W.A. Robinson, J.D.S. Witt, M.G. Blamire, Science 329, 59 (2010)
M.S. Anwar, F. Czeschka, M. Hesselberth, M. Porcu, J. Aarts, Phys. Rev. B 82, 100501 (2010)
C. Cirillo, S. Voltan, E.A. Ilyina, J.M. Hernández, A. García-Santiago, J. Aarts, C. Attanasio, New J. Phys. 19, 023037 (2017)
Y. Proshin, M. Avdeev, J. Magn. Magn. Mater. 459, 359 (2018)
M.V. Avdeev, Y.N. Proshin, Phys. Rev. B 97, 100502 (2018)
W. Belzig, F.K. Wilhelm, C. Bruder, G. Schon, A.D. Zaikin, Superlattices Microstruct. 25, 1251 (1999)
A. Buzdin, Phys. Rev. B 62, 11377 (2000)
M.L. Kulić, M. Endres, Phys. Rev. B 62, 11846 (2000)
M. Zareyan, W. Belzig, Y.V. Nazarov, Phys. Rev. Lett. 86, 308 (2001)
F.S. Bergeret, A.F. Volkov, K.B. Efetov, Phys. Rev. B 68, 064513 (2003)
D. Huertas-Hernando, Y.V. Nazarov, W. Belzig, Phys. Rev. Lett. 88, 047003 (2002)
M. Eschrig, J. Kopu, J.C. Cuevas, G. Schön, Phys. Rev. Lett. 90, 137003 (2003)
Y.V. Fominov, A.A. Golubov, M.Y. Kupriyanov, JETP Lett. 77, 510 (2003)
A.I. Buzdin, Rev. Mod. Phys. 77, 935 (2005)
F.S. Bergeret, A.F. Volkov, K.B. Efetov, Rev. Mod. Phys. 77, 1321 (2005)
V. Braude, Y.V. Nazarov, Phys. Rev. Lett. 98, 077003 (2007)
Y. Asano, Y. Tanaka, A.A. Golubov, Phys. Rev. Lett. 98, 107002 (2007)
T. Yokoyama, Y. Tanaka, A.A. Golubov, Phys. Rev. B 75, 134510 (2007)
S.Y. Hwang, P. Burset, B. Sothmann, Phys. Rev. B 98, 161408 (2018)
F.S. Bergeret, A.F. Volkov, K.B. Efetov, Phys. Rev. B 64, 134506 (2001)
A. Di Bernardo et al., Phys. Rev. X 5, 041021 (2015)
M. Alidoust, K. Halterman, O.T. Valls, Phys. Rev. B 92, 014508 (2015)
A.D. Bernardo, S. Diesch, Y. Gu, J. Linder, G. Divitini, C. Ducati, E. Scheer, M. Blamire, J. Robinson, Nat. Commun. 6, 8053 (2015)
M. Sigrist, K. Ueda, Rev. Mod. Phys. 63, 239 (1991)
A.P. Higginbotham, S.M. Albrecht, G. Kirsanskas, W. Chang, F. Kuemmeth, P. Krogstrup, T.S.J.J. Nygård, K. Flensberg, C.M. Marcus, Nature 372, 532 (1994)
H. Tou, Y. Kitaoka, K. Ishida, K. Asayama, N. Kimura, Y. Ōnuki, E. Yamamoto, Y. Haga, K. Maezawa, Phys. Rev. Lett. 80, 3129 (1998)
S. Kashiwaya, H. Kashiwaya, H. Kambara, T. Furuta, H. Yaguchi, Y. Tanaka, Y. Maeno, Phys. Rev. Lett. 107, 077003 (2011)
J. Linder, J.W.A. Robinson, Nat. Phys. 11, 307 (2015)
M. Eschrig, Phys. Today 64, 43 (2011)
M. Eschrig, T. Löfwander, Nat. Phys. 4, 138 (2008)
M. Eschrig, Rep. Prog. Phys. 78, 104501 (2015)
L. Fu, C.L. Kane, Phys. Rev. Lett. 100, 096407 (2008)
L. Fu, C.L. Kane, Phys. Rev. B 79, 161408 (2009)
R.M. Lutchyn, J.D. Sau, S. Das Sarma, Phys. Rev. Lett. 105, 077001 (2010)
Y. Oreg, G. Refael, F. von Oppen, Phys. Rev. Lett. 105, 177002 (2010)
A.Y. Kitaev, Phys. Usp. 44, 131 (2001)
C. Nayak, S.H. Simon, A. Stern, M. Freedman, S. Das Sarma, Rev. Mod. Phys. 80, 1083 (2008)
S.D. Sarma, M. Freedman, C. Nayak, Npj Quantum Information 1, 15001 (2015)
R. Aguado, La Rivista del Nuovo Cimento 40, 523 (2017)
R.M. Lutchyn, E.P.A.M. Bakkers, L.P. Kouwenhoven, P. Krogstrup, C.M. Marcus, Y. Oreg, Nat. Rev. Mater. 3, 52 (2018)
H. Zhang, D.E. Liu, M. Wimmer, L.P. Kouwenhoven, https://arXiv:1905.07882 (2019)
D. Culcer, A.C. Keser, Y. Li, G. Tkachov, https://arXiv:1907.10058 (2019)
T. Yokoyama, Phys. Rev. B 86, 075410 (2012)
A.M. Black-Schaffer, A.V. Balatsky, Phys. Rev. B 86, 144506 (2012)
A.M. Black-Schaffer, A.V. Balatsky, Phys. Rev. B 87, 220506 (2013)
P. Burset, B. Lu, G. Tkachov, Y. Tanaka, E.M. Hankiewicz, B. Trauzettel, Phys. Rev. B 92, 205424 (2015)
B. Lu, P. Burset, K. Yada, Y. Tanaka, Supercond. Sci. Technol. 28, 105001 (2015)
F. Crépin, P. Burset, B. Trauzettel, Phys. Rev. B 92, 100507 (2015)
J. Cayao, A.M. Black-Schaffer, Phys. Rev. B 96, 155426 (2017)
D. Kuzmanovski, A.M. Black-Schaffer, Phys. Rev. B 96, 174509 (2017)
B. Lu, Y. Tanaka, Phil. Trans. R. Soc. A 376, 20150246 (2018)
F. Keidel, P. Burset, B. Trauzettel, Phys. Rev. B 97, 075408 (2018)
D. Breunig, P. Burset, B. Trauzettel, Phys. Rev. Lett. 120, 037701 (2018)
C. Fleckenstein, N.T. Ziani, B. Trauzettel, Phys. Rev. B 97, 134523 (2018)
C.R. Reeg, D.L. Maslov, Phys. Rev. B 92, 134512 (2015)
H. Ebisu, B. Lu, J. Klinovaja, Y. Tanaka, Prog. Theor. Exp. Phys. 2016, 083I01 (2016)
I.V. Bobkova, A.M. Bobkov, Phys. Rev. B 95, 184518 (2017)
J. Cayao, A.M. Black-Schaffer, Phys. Rev. B 98, 075425 (2018)
S. Tamura, Y. Tanaka, Phys. Rev. B 99, 184501 (2019)
B. Sothmann, S. Weiss, M. Governale, J. König, Phys. Rev. B 90, 220501 (2014)
P. Burset, B. Lu, H. Ebisu, Y. Asano, Y. Tanaka, Phys. Rev. B 93, 201402 (2016)
C. Triola, A.V. Balatsky, Phys. Rev. B 94, 094518 (2016)
C. Triola, A.V. Balatsky, Phys. Rev. B 95, 224518 (2017)
A.M. Black-Schaffer, A.V. Balatsky, Phys. Rev. B 88, 104514 (2013)
L. Komendová, A.V. Balatsky, A.M. Black-Schaffer, Phys. Rev. B 92, 094517 (2015)
Y. Asano, A. Sasaki, Phys. Rev. B 92, 224508 (2015)
A.V. Samokhvalov, A.S. Melnikov, A.I. Buzdin, Physics-Uspekhi 59, 571 (2016)
A.A. Golubov, Y. Tanaka, Y. Asano, Y. Tanuma,Odd-Frequency Pairing in Superconducting Heterostructures (Springer, Berlin, 2011)
Y. Tanaka, S. Tamura, J. Low Temp. Phys. 191, 61 (2018)
Y. Tanaka, M. Sato, N. Nagaosa, J. Phys. Soc. Jpn 81, 011013 (2012)
J. Linder, A.V. Balatsky, https://arXiv:1709.03986 (2017)
A. Zagoskin,Quantum Theory of Many-Body Systems: Techniques and Applications (Springer, Berlin, 2014)
W.L. McMillan, Phys. Rev. 175, 559 (1968)
A. Larkin, Y.N. Ovchinnikov, Pis’ma Zh. Eksp. Teor. Fiz. 55, 2262 (1968)
A. Andreev, Zh. Eksp. Teor. Fiz. 46, 1823 (1964)
G.E. Blonder, M. Tinkham, T.M. Klapwijk, Phys. Rev. B 25, 4515 (1982)
B. Pannetier, H. Courtois, J. Low Temp. Phys. 118, 599 (2000)
T.M. Klapwijk, J. Supercond. 17, 593 (2004)
L. Covaci, F. Marsiglio, Phys. Rev. B 73, 014503 (2006)
A.M. Black-Schaffer, S. Doniach, Phys. Rev. B 81, 014517 (2010)
C. Triola, A.M. Black-Schaffer, Phys. Rev. B 100, 024512 (2019)
C. Triola, A.M. Black-Schaffer, https://arXiv:1905.00955 (2019)
J.M. Rowell, W.L. McMillan, Phys. Rev. Lett. 16, 453 (1966)
A. Tsintzis, A.M. Black-Schaffer, J. Cayao, https://arXiv:1905.01171 (2019)
I.O. Kulik, Sov. Phys. JETP 30, 944 (1970)
C. Beenakker,Transport phenomena in mesoscopic systems: Proceedings of the 14th Taniguchi symposium, Shima, Japan, Nov. 10–14, 1991 (Springer-Verlag, Berlin, 1992), p. 235
A. Furusaki, H. Takayanagi, M. Tsukada, Phys. Rev. Lett. 67, 132 (1991)
A. Furusaki, M. Tsukada, Solid State Commun. 78, 299 (1991)
A.V. Balatsky, S.S. Pershoguba, C. Triola, https://arXiv:1804.07244 (2018)
G. Dresselhaus, Phys. Rev. 100, 580 (1955)
E. Rashba, Sov. Phys. Solid. State 2, 1109 (1960)
Y.A. Bychkov, E.I. Rashba, Sov. Phys. JETP 39, 78 (1984)
L.P. Gor’kov, E.I. Rashba, Phys. Rev. Lett. 87, 037004 (2001)
E. Bauer, G. Hilscher, H. Michor, C. Paul, E.W. Scheidt, A. Gribanov, Y. Seropegin, H. Noël, M. Sigrist, P. Rogl, Phys. Rev. Lett. 92, 027003 (2004)
P.A. Frigeri, D.F. Agterberg, A. Koga, M. Sigrist, Phys. Rev. Lett. 92, 097001 (2004)
N. Reyren et al., Science 317, 1196 (2007)
S. Fujimoto, J. Phys. Soc. Jpn 76, 051008 (2007)
M. Sato, S. Fujimoto, Phys. Rev. B 79, 094504 (2009)
X. Liu, J.K. Jain, C.X. Liu, Phys. Rev. Lett. 113, 227002 (2014)
M. Smidman, M.B. Salamon, H.Q. Yuan, D.F. Agterberg, Rep. Prog. Phys. 80, 036501 (2017)
M. Schultz, F. Heinrichs, U. Merkt, T. Colin, T. Skauli, S. Løvold, Semicond. Sci. Technol. 11, 1168 (1996)
J. Nitta, T. Akazaki, H. Takayanagi, T. Enoki, Phys. Rev. Lett. 78, 1335 (1997)
D. Liang, X.P. Gao, Nano Lett. 12, 3263 (2012)
K. Takase, Y. Ashikawa, G. Zhang, K. Tateno, S. Sasaki, Sci. Rep. 7, 930 (2017)
W. Chang, S.M. Albrecht, T.S. Jespersen, F. Kuemmeth, P. Krogstrup, J. Nygård, C.M. Marcus, Nat. Nanotech. 10, 232 (2015)
A.P. Higginbotham, S.M. Albrecht, G. Kirsanskas, W. Chang, F. Kuemmeth, P. Krogstrup, T.S.J.J. Nygård, K. Flensberg, C.M. Marcus, Nat. Phys. 11, 1017 (2015)
P. Krogstrup, N.L.B. Ziino, W. Chang, S.M. Albrecht, M.H. Madsen, E. Johnson, J. Nygård, C.M. Marcus, T.S. Jespersen, Nat. Mater. 14, 400 (2015)
M.T. Deng, S. Vaitiekėnas, E.B. Hansen, J. Danon, M. Leijnse, K. Flensberg, J. Nygård, P. Krogstrup, C.M. Marcus, Science 354, 1557 (2016)
S.M. Albrecht, A.P. Higginbotham, M. Madsen, F. Kuemmeth, T.S. Jespersen, J. Nygård, P. Krogstrup, C.M. Marcus, Nature 531, 206 (2016)
S. Vaitiek ėnas, M.T. Deng, J. Nygård, P. Krogstrup, C.M. Marcus, Phys. Rev. Lett. 121, 037703 (2018)
M.T. Deng, S. Vaitiekėnas, E. Prada, P. San-Jose, J. Nygård, P. Krogstrup, R. Aguado, C.M. Marcus, Phys. Rev. B 98, 085125 (2018)
H. Zhang et al., Nat. Commun. 8, 16025 (2017)
Ö. Gül, D.J. van Woerkom, I. van Weperen, D. Car, S.R. Plissard, E.P.A.M. Bakkers, L.P. Kouwenhoven, Nanotechnology 26, 215202 (2015)
S. Gazibegovic et al., Nature 548, 434 (2017)
H. Zhang et al., Nature 556, 74 (2018)
J.E. Sestoft et al., Phys. Rev. Mater. 2, 044202 (2018)
S.T. Gill, J. Damasco, D. Car, E.P.A.M. Bakkers, N. Mason, Appl. Phys. Lett. 109, 233502 (2016)
O. Gül et al., Nano Lett. 17, 2690 (2017)
L. Santos, T. Neupert, C. Chamon, C. Mudry, Phys. Rev. B 81, 184502 (2010)
C.L. Kane, E.J. Mele, Phys. Rev. Lett. 95, 146802 (2005)
C.L. Kane, E.J. Mele, Phys. Rev. Lett. 95, 226801 (2005)
B.A. Bernevig, T.L. Hughes, S.C. Zhang, Science 314, 1757 (2006)
B.A. Bernevig, S.C. Zhang, Phys. Rev. Lett. 96, 106802 (2006)
M. König, S. Wiedmann, C. Brüne, A. Roth, H. Buhmann, L.W. Molenkamp, Science 318, 766 (2007)
A. Roth, C. Brüne, H. Buhmann, L.W. Molenkamp, J. Maciejko, X.L. Qi, S.C. Zhang, Science 325, 294 (2009)
C. Brüne, A. Roth, H. Buhmann, E.M. Hankiewicz, L.W. Molenkamp, J. Maciejko, X.L. Qi, S.C. Zhang, Nat. Phys. 8, 485 (2012)
K.C. Nowack et al., Nat. Mater. 12, 787 (2013)
C. Liu, T.L. Hughes, X.L. Qi, K. Wang, S.C. Zhang, Phys. Rev. Lett. 100, 236601 (2008)
I. Knez, R.R. Du, G. Sullivan, Phys. Rev. Lett. 107, 136603 (2011)
M.Z. Hasan, C.L. Kane, Rev. Mod. Phys. 82, 3045 (2010)
X.L. Qi, S.C. Zhang, Rev. Mod. Phys. 83, 1057 (2011)
G. Tkachov, E.M. Hankiewicz, Phys. Status Solidi B 250, 215 (2013)
Y. Ando, J. Phys. Soc. Jpn 82, 102001 (2013)
M. Sato, Y. Ando, Rep. Prog. Phys. 80, 076501 (2017)
S. Hart, H. Ren, T. Wagner, P. Leubner, M. Mühlbauer, C. Brüne, H. Buhmann, L.W. Molenkamp, A. Yacoby, Nat. Phys. 10, 638 (2014)
V.S. Pribiag, A.J.A. Beukman, F. Qu, M.C. Cassidy, C. Charpentier, W. Wegscheider, L.P. Kouwenhoven, Nat. Nanotech. 10, 593 (2015)
P. Adroguer, C. Grenier, D. Carpentier, J. Cayssol, P. Degiovanni, E. Orignac, Phys. Rev. B 82, 081303 (2010)
I. Knez, R.R. Du, G. Sullivan, Phys. Rev. Lett. 109, 186603 (2012)
T.D. Stanescu, J.D. Sau, R.M. Lutchyn, S. Das Sarma, Phys. Rev. B 81, 241310 (2010)
G. Tkachov, E.M. Hankiewicz, Phys. Rev. B 88, 075401 (2013)
E. Bocquillon, R.S. Deacon, J. Wiedenmann, P. Leubner, T.M. Klapwijk, C. Brüne, K. Ishibashi, H. Buhmann, L.W. Molenkamp, Nat. Nanotech. 12, 137 (2017)
C.T. Olund, E. Zhao, Phys. Rev. B 86, 214515 (2012)
G. Tkachov, E.M. Hankiewicz, Phys. Status Solidi B 250, 215 (2013)
C. Benjamin, Phys. Rev. B 74, 180503 (2006)
B.H. Wu, W. Yi, J.C. Cao, G.C. Guo, Phys. Rev. B 90, 205435 (2014)
D. Beckmann, H.B. Weber, H. v. Löhneysen, Phys. Rev. Lett. 93, 197003 (2004)
S. Russo, M. Kroug, T.M. Klapwijk, A.F. Morpurgo, Phys. Rev. Lett. 95, 027002 (2005)
J. Strunz et al., https://arXiv:1905.08175 (2019)
E. Majorana, Nuovo Cim. 14, 171 (1937)
J. Klinovaja, D. Loss, Phys. Rev. B 86, 085408 (2012)
E. Prada, P. San-Jose, R. Aguado, Phys. Rev. B 86, 180503 (2012)
S. Das Sarma, J.D. Sau, T.D. Stanescu, Phys. Rev. B 86, 220506 (2012)
D. Rainis, L. Trifunovic, J. Klinovaja, D. Loss, Phys. Rev. B 87, 024515 (2013)
T.P. Choy, J.M. Edge, A.R. Akhmerov, C.W.J. Beenakker, Phys. Rev. B 84, 195442 (2011)
D.J. Choi, N. Lorente, J. Wiebe, K. von Bergmann, A.F. Otte, A.J. Heinrich, https://arXiv:1904.09941 (2019)
Z. Huang, P. Wölfle, A.V. Balatsky, Phys. Rev. B 92, 121404 (2015)
S.P. Lee, R.M. Lutchyn, J. Maciejko, Phys. Rev. B 95, 184506 (2017)
D. Takagi, S. Tamura, Y. Tanaka, https://arXiv:1809.09324 (2018)
S. Tamura, S. Hoshino, Y. Tanaka, Phys. Rev. B 99, 184512 (2019)
Y. Asano, Y. Tanaka, Phys. Rev. B 87, 104513 (2013)
Y. Tanaka, S. Kashiwaya, Phys. Rev. B 70, 012507 (2004)
X. Liu, J.D. Sau, S. Das Sarma, Phys. Rev. B 92, 014513 (2015)
Y. Tanaka, S. Kashiwaya, Phys. Rev. B 53, 9371 (1996)
H.J. Kwon, K. Sengupta, V.M. Yakovenko, Eur. Phys. J. B 37, 349 (2004)
J. Klinovaja, D. Loss, Phys. Rev. B 90, 045118 (2014)
S. Baba et al., New J. Phys. 20, 063021 (2018)
O. Kashuba, B. Sothmann, P. Burset, B. Trauzettel, Phys. Rev. B 95, 174516 (2017)
D. Kuzmanovski, A.M. Black-Schaffer, J. Cayao, in preparation.
N.V. Gnezdilov, Phys. Rev. B 99, 024506 (2019)
S. Sachdev, J. Ye, Phys. Rev. Lett. 70, 3339 (1993)
A. Kitaev, a simple model of quantum holography, Talks at KITP, April 7, 2015 and May 27, 2015, http://online.kitp.ucsb.edu/online/entangled15/
A. Chew, A. Essin, J. Alicea, Phys. Rev. B 96, 121119 (2017)
A. Abrikosov, L. Gor’kov, Sov. Phys. JETP 12, 1243 (1961)
F. Reif, M.A. Woolf, Phys. Rev. Lett. 9, 315 (1962)
J.C. Phillips, Phys. Rev. Lett. 10, 96 (1963)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Cayao, J., Triola, C. & Black-Schaffer, A.M. Odd-frequency superconducting pairing in one-dimensional systems. Eur. Phys. J. Spec. Top. 229, 545–575 (2020). https://doi.org/10.1140/epjst/e2019-900168-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1140/epjst/e2019-900168-0