Eur. Phys. J. B (2017) 90: 211
https://doi.org/10.1140/epjb/e2017-80103-y

Regular Article

Majorana fermions in finite-size strips with in-plane magnetic fields

¹ Institut de Physique Théorique, CEA/Saclay, Orme des Merisiers, 91190 Gif-sur-Yvette Cedex, France
² Laboratoire de Physique des Solides, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay Cedex, France
³ Laboratoire Charles Fabry, Institut d’Optique, CNRS, Univ. Paris-Saclay, 2 avenue Augustin Fresnel, 91127 Palaiseau Cedex, France
⁴ Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland

^a e-mail: vardan.kaladzhyan@cea.fr

Received: 20 February 2017
Received in final form: 19 June 2017
Published online: 7 November 2017

Abstract

We study the Majorana bound states (MBS) arising in finite-size strips with Rashba spin-orbit coupling in the presence of an in-plane Zeeman magnetic field. Using two different methods, first, the numerical diagonalization of the tight-binding Hamiltonian, and second, finding the singular points of the Hamiltonian [I. Mandal, Europhys. Lett. 110, 67005 (2015); I. Mandal, S. Tewari, Physica E 79, 180 (2016); I. Mandal, Condens. Matter Phys. 19, 33703 (2016); P. San-Jose, E. Prada, R. Aguado, Sci. Rep. 6, 21427 (2016)], we obtain the topological phase diagram for these systems as a function of the chemical potential and the magnetic field, and we demonstrate the consistency of these two methods. By introducing disorder into these systems we confirm that the states with even number of Majorana pairs are not topologically protected. Finally, we show that a calculation of the ℤ₂ topological invariants recovers correctly the parity of the number of MBS pairs, and it is thus fully consistent with the phase diagrams of the disordered systems.