Making s -wave superconductors topological with magnetic field

We show that a two-dimensional $s$-wave superconductor may become topological in the presence of a magnetic field that leads to the formation of an Abrikosov vortex lattice. Below the upper critical field, a superconducting state with a nontrivial even topological number emerges, which we call the Abrikosov-Chern superconducting state. Deeper in the superconducting domain, the topological number changes in steps, always remaining even and thus not supporting Majorana states, and eventually reaches zero. Our theory uncovers the nature of evolution from an integer quantum Hall state having a cyclotron gap above the upper critical field to the topologically trivial $s$-wave superconductor carrying finite-energy Caroli-de Gennes-Matricon levels at low field. Topological transitions manifest as changes in the number of edge modes, detectable through tunneling spectroscopy and thermal or spin transport measurements.