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Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2019, Volume 29, Issue 3, Pages 351–362
DOI: https://doi.org/10.20537/vm190306
(Mi vuu688)
 

MATHEMATICS

Existence of Majorana bounded states in a simple Josephson transition model

T. S. Tinyukovaa, Yu. P. Chuburinb

a Udmurt State University, ul. Universitetskaya, 1, Izhevsk, 426034, Russia
b Udmurt Federal Research Center of the Ural Branch of the Russian Academy of Sciences, ul. T. Baramzinoi, 34, Izhevsk, 426067, Russia
References:
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.
Funding agency
The research was funded by Udmurt State University in the framework of the grant support program for young researchers “Scientific potential-2018”, project number 2018-03-02.
Received: 12.06.2019
Bibliographic databases:
Document Type: Article
UDC: 517.958, 530.145.6
MSC: 81Q10, 81Q15
Language: Russian
Citation: T. S. Tinyukova, Yu. P. Chuburin, “Existence of Majorana bounded states in a simple Josephson transition model”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 29:3 (2019), 351–362
Citation in format AMSBIB
\Bibitem{TinChu19}
\by T.~S.~Tinyukova, Yu.~P.~Chuburin
\paper Existence of Majorana bounded states in a simple Josephson transition model
\jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki
\yr 2019
\vol 29
\issue 3
\pages 351--362
\mathnet{http://mi.mathnet.ru/vuu688}
\crossref{https://doi.org/10.20537/vm190306}
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    Вестник Удмуртского университета. Математика. Механика. Компьютерные науки
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