  Topological superconductors support Majorana modes, which are quasiparticles
that are their own antiparticles and which obey non-Abelian statistics in which
successive exchanges of particles do not always commute. Here we investigate
whether a two-dimensional superconductor with ordinary s-wave pairing can be
rendered topological by the application of a strong magnetic field. To address
this, we obtain the self-consistent solutions to the mean field Bogoliubov-de
Gennes equations, which are a large set of nonlinearly coupled equations, for
electrons moving on a lattice. We find that the topological "quantum Hall
superconductivity" is facilitated by a combination of spin-orbit coupling,
which locks an electron's spin to its momentum as it moves through a material,
and a coupling to an external periodic potential which gives a dispersion to
the Landau levels and also distorts the Abrikosov lattice. We find that, for a
range of parameters, the Landau levels broadened by the external periodic
potential support topological superconductivity, which is typically accompanied
by a lattice of "giant" $h/e$ vortices as opposed to the familiar lattice of
$h/2e$ Abrikosov vortices. In the presence of a periodic potential, we find it
necessary to use an ansatz for the pairing potential of the form
$\Delta(\vec{r})e^{i2\vec{Q}\cdot\vec{r}}$ where $\Delta(\vec{r})$ has a
periodicity commensurate with the periodic potential. However, despite this
form of the pairing potential, the current in the ground state is zero. In the
region of ordinary superconductivity, we typically find a lattice of dimers of
$h/2e$ vortices. Our work suggests a realistic proposal for achieving
topological superconductivity, as well as a helical order parameter and unusual
Abrikosov lattices.
