Signatures of surface Majorana modes in the magnetic response of topological superconductors

We study the magnetic susceptibility of a two-dimensional cone of Majorana modes localized at the surface of a three-dimensional time-reversal invariant topological superconductor belonging to class DIII. A field parallel to the surface tilts the surface Majorana cone along the supercurrent direction. For fields larger than a critical threshold field H*, H > H*, a transition from a type I to a type II Dirac cone occurs and a finite current carried by the Majorana modes starts to flow, leading to an additional diamagnetic contribution to the surface magnetization. On a curved surface, interband transitions are promoted by the Majorana spin connection that couples to the external field, giving rise to a finite-frequency magnetic susceptibility.