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

Abstract

We study the Majorana bound states 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 (see Refs. [1-4]), 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 Majorana bound states pairs, and it is thus fully consistent with the phase diagrams of the disordered systems.

Formation of Majorana fermions in finite-size graphene strips

“…[39]. We would like to emphasize that unlike the case of a perpendicular field [39], the band structure is gapless (same as for a square lattice with an in-plane field [28]), and chiral Majorana edge modes can form, as shown in Fig. 2a.…”

“…[28,[48][49][50], applied to the lowenergy eigenstates of our systems. The way in which the MP criterion works is as follows: a Majorana state is an eigenstate of the particle-hole operator, therefore a Majorana state localized inside a spatial region must satisfy C = 1 where…”

“…[28] for an in-plane field. In the previous analyses it was found that for a purely one-dimensional system the Majorana states were forming mainly for B (µ − 2) 2 + ∆ 2 , while for a two-dimensional system (infinite ribbon) for B (µ − 4) 2 + ∆ 2 .…”

“…For finite-size strips the corresponding topological phases were described in Refs. [50] and [28], and were composed of multiple discrete topological regions in the (µ, B) parameter space, the details depending strongly on the direction of the magnetic field (in-plane or perpendicular to the lattice).…”

“…2 we present the topological phase diagram in a ribbon with zigzag edges (a similar situation is recovered for armchair edges), as well as the band structure for a set of parameters corresponding to the formation of Majoranas. The topological phase diagram was obtained by using the MatQ code [47] to diagonalize the Hamiltonian of the nanoribbon, and subsequently calculating the corresponding Majorana polarization of the lowest energy states [28,[48][49][50]. The pink regions correspond to lowest-energy edge states with a MP of 1, while the white region to a zero MP.…”