# Lecture: Odd-frequency pairing in topological superconductors

- Date:
- Location: Ångströmlaboratoriet, Lägerhyddsvägen 1 The villa
- Lecturer: Prof. Yukio Tanaka, Nagoya University
- Contact person: Annica Black-Schaffer
- Föreläsning

It is known that odd-frequency pairing ubiquitously presents in superconductor junctions [1]. In the presence of zero energy surface Andreev bound state (ZESABS) realized in topological superconductors, the odd-frequency pairing is amplified near the surface or interface. One of the remarkable property generated by odd-frequency pairing is anomalous proximity effect in diffusive normal metal (DN) / superconductor junction where quasiparticle density of states in DN has a zero energy peak (ZEP) of LDOS due to the penetration of odd-frequency spin-triplet s-wave pairing [2,3]. The anomalous proximity effect can trigger exotic response against applied magnetic effect and so called paramagnetic Meissner effect appears [4-6]. Since spin-triplet p-wave superconductor is rare, it is desired to generate anomalous proximity effect from spin-singlet superconductor. It has been shown that proximity coupled nano-wire junction [7] is an idealistic system to study anomalous proximity effect due to odd-frequency triplet-s wave pairing [8]. Also, high Tc cuprate on the substrate with large Rashba spin-orbit coupling generated anomalous proximity effect [9].

Finally, we mention about the new type of bulk edge correspondence in the presence of chiral symmetry of the Hamiltonian. We have clarified the relation between induced odd-frequency pairing and the bulk quantity defined by Green’s function [10]. Odd-frequency Cooper pairs with chiral symmetry emerging at the edges are a useful physical quantity. We have shown that the odd-frequency Cooper pair amplitudes can be expressed by a winding number extended to a nonzero frequency and can be evaluated from the spectral features of the bulk. We have found that the odd-frequency Cooper pair amplitudes are classified into two categories: the amplitudes in the first category have the singular functional form proportional to 1/z (where z is a complex frequency) that reflects the presence of ZESABS, whereas the amplitudes in the second category have the regular form proportional to z. It has been shown that the topological phase transition is characterized by using the coefficient in the latter category [10].

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[3] Y. Tanaka and S. Kashiwaya, Phys. Rev. B 70 012507 (2004).

[4] Y. Asano and Y. Tanaka, Phys. Rev. B 87 104513 (2013).

[5] Y. Tanaka, et al, Phys. Rev. B 72, 140503(R) (2005).

[6] S-I. Suzuki and Y. Asano, Phys. Rev. B 89, 184508 (2014).

[7] Shu-Ping Lee, R. M. Lutchyn, and J. Maciejko, Phys. Rev. B 95, 184506 (2017).

[8] R. M. Lutchyn, J. D. Sau, and S. Das Sarma, Phys. Rev. Lett. 105, 077001 (2010),

Y. Oreg, G. Refael, and F. von Oppen, Phys. Rev. Lett. 105, 177002 (2010).

[9] S. Tamura and Y. Tanaka, Phys. Rev. B 99 184501 (2019).

[10] S. Tamura, S. Hoshino and Y. Tanaka, Phys. Rev. B 99, 184512 (2019).