Section

Mathematics

Title

Twodimensional difference Dirac operator in the strip

Author(s)

Tinyukova T.S.^{a}

Affiliations

Udmurt State University^{a}

Abstract

In the last decade, a new class of materials  topological insulators  is extensively studied in the physics literature. Topological insulators have remarkable physical properties, in particular, nearzero resistance, and are expected to be applied in microelectronics. Unlike conventional metals and semiconductors, an electron in topological insulators is described not by the Schrodinger operator (Hamiltonian), but by the massless Dirac operator. Such operators in quasionedimensional structures (for example, strips with different boundary conditions) are very interesting from a mathematical point of view, but they are not well studied by mathematicians yet. This article discusses the Dirac Hamiltonian of a topological insulator of somewhat more general form, namely in the presence of a ferromagnetic layer. The spectrum of such an operator is described; its Green's function (the kernel of the resolvent) and (generalized) eigenfunctions are established.

Keywords

discrete difference Dirac operator, resolution, spectrum

UDC

517.958, 530.145.6

MSC

81Q10, 81Q15

DOI

10.20537/vm150110

Received

1 February 2015

Language

Russian

Citation

Tinyukova T.S. Twodimensional difference Dirac operator in the strip, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2015, vol. 25, issue 1, pp. 93100.

References

 Hasan M.Z., Kane C.L. Colloquium: topological insulators, Rev. Mod. Phys., 2010, vol. 82, pp. 30453067.
 Bardarson J.H., Moore J.E. Quantum interference and AharonovBohm oscillations in topological insulators, Rep. Prog. Phys., 2012, vol. 76, 056501.
 Yokoyama T., Tanaka Y., Nagaosa N. Anomalous magnetoresistance of a twodimensional ferromagnet / ferromagnet junction on the surface of a topological insulator, Phys. Rev. B., 2010, vol. 81, 121401 (R).
 Chuburin Y.P. Electron scattering on the surface of a topological insulator, Phys. A.: Math. Theor., 2014, vol. 47, 255203 (13 p).
 Morozova L.E., Chuburin Yu.P. On levels of the onedimensional discrete Schrödinger operator with a decreasing small potential, Izv. Inst. Mat. Inform. Udmurt. Gos. Univ., 2004, no. 1 (29), pp. 8594.

Full text

