Making s-wave superconductors topological with magnetic field

We show that an s-wave superconductor may become topological in the presence of a magnetic field that induces the formation of an Abrikosov vortex lattice. Specifically, we demonstrate nontrivial topology of a two-dimensional superconductor near the upper critical field, or when a proximity-induced Δ is smaller than or comparable to the cyclotron energy ω_c. Deeper in the superconducting domain, the topological number changes in steps, eventually reaching zero at small fields. Our theory elucidates the evolution from an integer quantum Hall state, characterized by a cyclotron gap above the upper critical field, to a topologically trivial s-wave superconductor that exhibits finite-energy Caroli-de Gennes-Matricon levels at low fields. Topological transitions are marked by gap closures in the bulk and change in the number of edge modes, detectable through tunneling spectroscopy and thermal or spin transport measurements.