  Majorana quasi-particles may arise as zero-energy bound states in vortices on
the surface of a topological insulator that is proximitized by a conventional
superconductor. Such a system finds its natural realization in the iron-based
superconductor FeTe$_{0.55}$Se$_{0.45}$ that combines bulk $s$-wave pairing
with spin helical Dirac surface states, and which thus comprises the
ingredients for Majorana modes in absence of an additional proximitizing
superconductor. In this work, we investigate the emergence of Majorana vortex
modes and lattices in such materials depending on parameters like the magnetic
field strength and vortex lattice disorder. A simple 2D square lattice model
here allows us to capture the basic physics of the underlying materials system.
To address the problem of disordered vortex lattice, which occurs in real
systems, we adopt the technique of the singular gauge transformation which we
modify such that it can be used in a system with periodic boundary conditions.
This approach allows us to go to larger vortex lattices than otherwise
accessible, and is successful in replicating several experimental observations
of Majorana vortex bound states in the FeTe$_{0.55}$Se$_{0.45}$ platform.
Finally it can be related to a simple disordered Majorana lattice model that
should be useful for further investigations on the role of interactions, and
towards topological quantum computation.
