Zero modes of the generalized fermion-vortex system in magnetic field

We show that Dirac fermions moving in two spatial dimensions with a generalized dispersion $E\sim p^N$, subject to an external magnetic field and coupled to a complex scalar field carrying a vortex defect with winding number $Q$ acquire $NQ$ zero modes. This is the same as in the absence of the magnetic field. Our proof is based on selection rules in the Landau level basis that dictate the existence and the number of the zero modes. We show that the result is insensitive to the choice of geometry and is naturally extended to general field profiles, where we also derive a generalization of the Aharonov-Casher theorem. Experimental consequences of our results are briefly discussed.