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Russia Izhevsk
Section Mathematics
Title Existence of Majorana bounded states in a simple Josephson transition model
Author(-s) Tinyukova T.S.a, Chuburin Yu.P.b
Affiliations Udmurt State Universitya, Udmurt Federal Research Center, Ural Branch of the Russian Academy of Sciencesb
Abstract For the last 15 years, Majorana bounded states (MBSs) and associated phenomena, such as variation of conductance and the Josephson effect, have been actively studied in the physical literature. Research in this direction is motivated by a highly probable use of MBSs in quantum computing. The article studies the eigenfunctions of the one-dimensional Bogolyubov-de Gennes operator with a delta-shaped potential at zero, describing localized states with energy in the spectral gap (superconducting gap). The transmission probabilities are found in the scattering problem for this operator, when the energies are close to the boundary of the superconducting gap. These problems are studied both for a superconducting order that is the only one on the whole straight line and is defined by the real constant $\Delta,$ and for a superconducting order defined by the function $\Delta\theta(-x)+\Delta e^{i\varphi}\theta(x)$ for $\varphi=0,\pi$ (i.e., for zero superconducting current and for current close to critical). The Hamiltonian used can be considered as the simplest model of the Josephson junction. It is proved that in both cases there are two MBSs, but with certain values of the parameters, i.e., MBSs are unstable. Moreover, the probability of passage is zero in both cases.
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/vm190306
Received 12 June 2019
Language Russian
Citation Tinyukova T.S., Chuburin Yu.P. Existence of Majorana bounded states in a simple Josephson transition model, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2019, vol. 29, issue 3, pp. 351-362.
  1. Elliot S.R., Franz M. Colloquium: Majorana fermions in nuclear, particle, and solid-state physics, Reviews of Modern Physics, 2015, vol. 87, issue 1, pp. 137-163.
  2. Alicea J. New directions in the pursuit of Majorana fermions in solid state systems, Reports on Progress in Physics, 2012, vol. 75, no. 7, 076501.
  3. Sato M., Fujimoto S. Majorana fermions and topology in superconductors, Journal of the Physical Society of Japan, 2016, vol. 85, no. 7, 072001.
  4. 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, Physical Review B, 2016, vol. 94, issue 3, 035143.
  5. 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.
  6. Kitaev A.Yu. Unpaired Majorana fermions in quantum wires, Physics-Uspekhi, 2001, vol. 44, no. 10S, pp. 131-136.
  7. Karzig T., Refael G., von Oppen F. Boosting Majorana zero modes, Physical Review X, 2013, vol. 3, issue 4, 041017.
  8. 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, Physical Review B, 2012, vol. 86, issue 22, 220506.
  9. 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 (in Russian).
  10. Cayao J., San-Jose P., Black-Schaffer A.M., Aguado R., Prada E. Majorana splitting from critical currents in Josephson junctions, Physical Review B, 2017, vol. 96, issue 20, 205425.
  11. Cayao J., Black-Schaffer A.M., Prada E., Aguado R. Andreev spectrum and supercurrents in nanowire-based SNS junctions containing Majorana bound states, 2018, arXiv: 1712.08127v2 [cond-mat.mes-hall].
  12. Olund C.T., Zhao E. Current-phase relation for Josephson effect through helical metal, Physical Review B, 2012, vol. 86, issue 21, 214515.
  13. Peng Y., Pientka F., Berg E., Oreg Y., von Oppen F. Signatures of topological Josephson junctions, Physical Review B, 2016, vol. 94, issue 8, 085409.
  14. Schmidt V.V. Vvedenie v fiziku sverkhprovodnikov (Introduction to the physics of superconductors), Moscow, 2000, 402 p.
  15. Golubov A.A., Kupriyanov M.Yu., Il’ichev E. The current-phase relation in Josephson junctions, Reviews of Modern Physics, 2004, vol. 76, issue 2, pp. 411-469.
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