Zero-energy states of massive Dirac equation in magnetic fields

TL;DR

This paper analyzes zero-energy solutions of the massive Dirac equation in magnetic fields, revealing conditions for their existence and implications for graphene and vortex charge.

Contribution

It demonstrates that the Dirac equation always has one zero-energy state in magnetic fields and explores how pseudo-magnetic fields affect zero-energy states.

Findings

One zero-energy state exists in arbitrary magnetic fields.

Pseudo-magnetic fields can alter the number of zero-energy states.

Explicit examples illustrate the dependence on field profiles.

Abstract

The Dirac equation with a U(1) vortex in the mass-term is solved in the presence of magnetic-like fields at zero energy. By drawing an analogy to classical mechanics, it is shown that the four-component Dirac equation in arbitrary magnetic field always yields one zero-energy state. In the time-reversal preserving, pseudo-magnetic field, however, the number of zero-energy states may depend on the field's profile and sign. Some explicit examples are worked out. Possible implications of these results for the charge of the vortex and for the behavior of graphene in magnetic field are discussed.