Investigation of eigenvalues and scattering problem for the Bogoliubov–de Gennes Hamiltonian near the superconducting gap edge

T. S. Tinyukova   ; Yu. P. Chuburin

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 30 (2020) no. 2, pp. 259-269

Résumé

We consider the Bogolyubov–de Gennes Hamiltonian perturbed by a small potential, which describes quasiparticles of electron-hole type, in particular, Andreev bound states (ABSs) in a one-dimensional superconducting structure in the presence of an impurity. In the last 15–20 years, interest in such quasiparticles has increased sharply due to the discovery of Majorana bound states (MBSs) in topological superconductors. MBSs are neutral zero-energy quasiparticles resistant to external influences, which are very promising for future use in quantum computing. The study of the appearance and behavior, depending on the system parameters and the topological phase, of ABSs described by the eigenfunctions of the Bogolyubov–de Gennes Hamiltonian, is interesting both from a mathematical point of view, in comparison with the usual Schrödinger operator, and from a physical point of view, since it can clarify prerequisites for the occurrence of MBSs in the topologically nontrivial phase and marjoram-like states (often playing the role of MBSs) in the topologically trivial phase. The study of scattering is interesting due to the fact that the probability of a quasiparticle transmission through a potential barrier is proportional to the conductance, that can be measured experimentally, which in principle makes it possible to relate the conductance to the presence of ABS. In the paper, the conditions for the appearance of eigenvalues (energies of quasiparticles) in the superconducting gap in the continuous spectrum of the Hamiltonian, as well as their dependence on the parameters in both the topological nontrivial and topologically trivial phases, are found. In addition, the scattering problem for energies near the edge of the gap has been investigated, in particular, the probability of a quasiparticle transmission through a potential barrier as a function of system parameters has been found.

Détail

Keywords: Bogolyubov–de Gennes Hamiltonian, Green's function, spectrum, eigenvalue, Andreev bound states, scattering problem, transmission probability.

Bibliographie

[1] Elliot S. R., Franz M., “Colloquium: Majorana fermions in nuclear, particle, and solid-state physics”, Reviews of Modern Physics, 87:1 (2015), 137–163 | DOI | MR

[2] Alicea J., “New directions in the pursuit of Majorana fermions in solid state systems”, Reports on Progress in Physics, 75 (2012), 076501 | DOI

[3] Sato M., Fujimoto S., “Majorana fermions and topology in superconductors”, Journal of the Physical Society of Japan, 85 (2016), 072001 | DOI

[4] Lutchyn R. M., Bakkers E., Kouwenhoven L. P., Krogstrup P., Marcus C. M., Oreg Y., “Majorana zero modes in superconductor–semiconductor heterostructures”, Nature Reviews Materials, 3 (2018), 52–68 | DOI

[5] Oppen F., Peng Y., Pientka F., “Topological superconducting phases in one dimension”, Topological Aspects of Condensed Matter Physics, Lecture Notes of the Les Houches Summer School, 103, Oxford University Press, Oxford, 2017 | DOI

[6] Tinyukova T. S., Chuburin Yu. P., “Andreev reflection in the p-wave superconductor–normal metal contact”, Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, 54 (2019), 55–62 (in Russian) | DOI | MR | Zbl

[7] Chuburin Yu. P., “On small perturbations of the Schrödinger operator with a periodic potential”, Theoretical and Mathematical Physics, 110:3 (1997), 351–359 | DOI | MR | Zbl

[8] Chuburin Yu. P., “On levels of a weakly perturbed periodic Schrödinger operator”, Communications in Mathematical Physics, 249:3 (2004), 497–510 | DOI | MR | Zbl

[9] 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, 94 (2016), 035143 | DOI

[10] Karzig T., Refael G., “Boosting Majorana zero modes”, Physical Review X, 3 (2013), 041017 | DOI

[11] Sau J. D., Demler E., “Bound states at impurities as a probe of topological superconductivity in nanowires”, Physical Review B, 88 (2013), 205402 | DOI

[12] Kitaev A. Yu., “Unpaired Majorana fermions in quantum wires”, Physics-Uspekhi, 44 (2001), 131–136 | DOI

[13] Tinyukova T. S., “Majorana states in a $p$-wave superconducting nanowire”, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 28:2 (2018), 222–230 (in Russian) | DOI | MR | Zbl

[14] Reed M., Simon B., Methods of modern mathematical physics, v. I, Functional analysis, Mir, M., 1977

[15] Reed M., Simon B., Methods of modern mathematical physics, v. III, Scattering theory, Mir, M., 1982