Universal phase diagram of topological superconductors subjected to magnetic flux

Abstract: We perform a theoretical study of the orbital effect of a magnetic field on a proximity-coupled islands array of $p_{x}+ip_{y}$ topological superconductors. To describe the system, we generalize the tight-binding model of the Hofstadter butterfly to include the effect of the superconducting islands. The quantum Hall topological phases, appearing in the absence of superconductivity, are characterized by integer fermionic Chern numbers corresponding to the number of occupied bulk Landau levels. As the strength of the superconducting pairing increases a series of transitions occurs, with one less chiral Majorana edge mode at each consecutive phase, leading to a reduction of the fermionic Chern number by a half. When the pairing potential exceeds the tight-binding model bandwidth, Cooper pairs are localized in the islands, the Chern number is zero, and there are no low-energy edge modes. We identify domains in the model's parameter space for which the system is topological and supports an odd number of chiral Majorana edge modes. While the precise shape of the domains depends on the details of the model, the general structure of the phase diagram is robust, and it is obtained numerically and in several simplified traceable analytical models. We discuss the relevance of this study to recent experimental research of two-dimensional superconductor arrays on semiconductor systems.