  It has been shown previously that a finite-length topological insulator
nanowire, proximity-coupled to an ordinary bulk s-wave superconductor and
subject to a longitudinal applied magnetic field, realizes a one-dimensional
topological superconductor with an unpaired Majorana fermion (MF) localized at
each end of the nanowire. Here, we study the stability of these MFs with
respect to various perturbations that are likely to occur in a physical
realization of the proposed device. We show that the unpaired Majorana fermions
persist in this system for any value of the chemical potential inside the bulk
band gap of order 300 meV in Bi$_2$Se$_3$ by computing the Majorana number.
From this calculation, we also show that the unpaired Majorana fermions persist
when the magnetic flux through the nanowire cross-section deviates
significantly from half flux quantum. Lastly, we demonstrate that the unpaired
Majorana fermions persist in strongly disordered wires with fluctuations in the
on-site potential ranging in magnitude up to several times the size of the bulk
band gap. These results suggest this solid-state system should exhibit unpaired
Majorana fermions under accessible conditions likely important for experimental
study or future applications.
