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Topological superconductors support Majorana modes, which are quasiparticlesthat are their own antiparticles and which obey non-Abelian statistics in whichsuccessive exchanges of particles do not always commute. Here we investigatewhether a two-dimensional superconductor with ordinary s-wave pairing can berendered topological by the application of a strong magnetic field. To addressthis, we obtain the self-consistent solutions to the mean field Bogoliubov-deGennes equations, which are a large set of nonlinearly coupled equations, forelectrons moving on a lattice. We find that the topological "quantum Hallsuperconductivity" 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 tothe Landau levels and also distorts the Abrikosov lattice. We find that, for arange of parameters, the Landau levels broadened by the external periodicpotential support topological superconductivity, which is typically accompaniedby 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 itnecessary 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 aperiodicity commensurate with the periodic potential. However, despite thisform of the pairing potential, the current in the ground state is zero. In theregion of ordinary superconductivity, we typically find a lattice of dimers of$h/2e$ vortices. Our work suggests a realistic proposal for achievingtopological superconductivity, as well as a helical order parameter and unusualAbrikosov lattices.

Topological superconductivity induced by spin-orbit coupling, perpendicular magnetic field and superlattice potential