  We study the Majorana bound states arising in quasi-one-dimensional systems
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 formal
calculation of the $\mathbb{Z}_2$ 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.

Mesoscale and Nanoscale Physics;Superconductivity