  Certain types of topological superconductors and superfluids are known to
host protected Majorana zero modes in cores of Abrikosov vortices. When such
vortices are arranged in a dense periodic lattice one expects zero modes from
neighboring vortices to hybridize and form dispersing bands. Understanding the
structure of these bands is essential for the schemes that aim to employ the
zero modes in quantum computation applications and in studies of their strongly
interacting phases. We investigate here the band formation phenomenon in two
concrete models, describing a two dimensional p + ip superconductor and a
superconducting surface of a three-dimensional strong topological insulator
(Fu-Kane model), using a combination of analytical and numerical techniques. We
find that the physics of the Majorana bands is well described by tight binding
models of Majorana fermions coupled to a static Z2 gauge field with a
non-trivial gauge flux through each plaquette, in accord with expectations
based on very general arguments. In the case of the Fu-Kane model we also find
that, irrespective of the lattice geometry, the Majorana band becomes
completely flat at the so called neutrality point (chemical potential
coincident with the Dirac point) where the model exhibits an extra chiral
symmetry. In this limit the low energy physics will be dominated by
four-fermion interaction terms which are permitted by symmetries and may arise
from the Coulomb interaction between the constituent electron degrees of
freedom.
