Proximity-induced superconductivity in Landau-quantized graphene monolayers

We consider massless Dirac fermions in a graphene monolayer subject to both a perpendicular magnetic field $B$ and a proximity-induced pairing gap $\mathrm{\Delta }$. When the chemical potential is at the Dirac point, our exact solution of the Bogoliubov-de Gennes equation yields $\mathrm{\Delta }$-independent relativistic Landau levels. Since eigenstates depend on $\mathrm{\Delta }$, many observables nevertheless are sensitive to pairing, e.g., the local density of states or the edge state spectrum. By solving the problem with an additional in-plane electric field, we also discuss how snake states are influenced by a pairing gap.