|
Section
|
Mathematics
|
|
Title
|
Majorana states in a $p$-wave superconducting nanowire
|
|
Author(-s)
|
Tinyukova T.S.a
|
|
Affiliations
|
Udmurt State Universitya
|
|
Abstract
|
At the present time, the Majorana bounded states (MBSs) and associated phenomena, such as the variation of the conductance, are being actively studied in the physical literature because of the highly probable use of MBSs in quantum computations. In spite of the urgency, a rigorous mathematical study of the spectral properties and scattering for the one-particle Bogolyubov-de Gennes operator $H,$ commonly used for investigation of MBS's, has almost never been carried out. The methods proposed in the article allow one to obtain mathematically and physically interesting results. In this paper, we study the problem of the existence of MBSs (that is, the existence of a zero eigenvalue) for the Bogolyubov-de Gennes Hamiltonian in the case of an infinite one-dimensional superconducting structure in the presence of a potential. Conditions for the existence of MBSs are obtained. The scattering problem for the Bogolyubov-de Gennes operator with a potential is studied. The Green's function of the operator $H$ used in solving these problems is also found.
|
|
Keywords
|
Bogolyubov-de Gennes Hamiltonian, Green's function, spectrum, eigenvalue, scattering problem, transmission probability, Majorana bounded states
|
|
UDC
|
517.958, 530.145.6
|
|
MSC
|
81Q10, 81Q15
|
|
DOI
|
10.20537/vm180208
|
|
Received
|
10 May 2018
|
|
Language
|
Russian
|
|
Citation
|
Tinyukova T.S. Majorana states in a $p$-wave superconducting nanowire, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2018, vol. 28, issue 2, pp. 222-230.
|
|
References
|
- Elliot S.R., Franz M. Colloquium: Majorana fermions in nuclear, particle, and solid-state physics, Rev. Mod. Phys., 2015, vol. 87, issue 1, pp. 137-163. DOI: 10.1103/RevModPhys.87.137
- Alicea J. New directions in the pursuit of Majorana fermions in solid state systems, Rep. Prog. Phys., 2012, vol. 75, issue 7, 076501. DOI: 10.1088/0034-4885/75/7/076501
- Sato M., Fujimoto S. Majorana fermions and topology in superconductors, Journal of the Physical Society of Japan, 2016, vol. 85, no. 7, 072001. DOI: 10.7566/JPSJ.85.072001
- Sarma S.D., Nag A., Sau J.D. How to infer non-Abelian statistics and topological visibility from tunneling conductance properties of realistic Majorana nanowires, Phys. Rev. B., 2016, vol. 94, issue 3, 035143. DOI: 10.1103/PhysRevB.94.035143
- Chuburin Yu.P. Existence of Majorana bound states near impurities in the case of a small superconducting gap, Physica E: Low-dimensional Systems and Nanostructures, 2017, vol. 89, pp. 130-133. DOI: 10.1016/j.physe.2017.02.017
- Kitaev A.Yu. Unpaired Majorana fermions in quantum wires, Phys.-Usp., 2001, vol. 44, pp. 131-136. DOI: 10.1070/1063-7869/44/10S/S29
- Karzig T., Refael G., von Oppen F. Boosting Majorana zero modes, Phys. Rev. X., 2013, vol. 3, issue 4, 041017. DOI: 10.1103/PhysRevX.3.041017
- Sarma S.D., Sau J.D., Stanescu T.D. Splitting of the zero-bias conductance peak as smoking gun evidence for the existence of the Majorana mode in a superconductor-semiconductor nanowire, Phys. Rev. B., 2012, vol. 86, issue 22, 220506. DOI: 10.1103/PhysRevB.86.220506
|
|
Full text
|
|
+7 (3412) 91 60 92
Russian