David J. Fernández C.
Darboux transformations applied to graphene in external magnetic fields
The Dirac electron in graphene with magnetic fields which are orthogonal to the graphene surface is studied. The eigenvalue problem is reduced to two one-dimensional Schrödinger Hamiltonians which are Darboux-transformed to each other. The magnetic field is initially chosen such that the associated Schrödinger potentials turn out to be shape invariant. Then, more general first-order Darboux transformations are used to generate magnetic fields leading to new analytic solutions for the graphene problem. The iterations of the method are discussed in the real, singular, and complex cases, looking for new graphene Hermitian solvable Hamiltonians. The case when the magnetic fields are complex, leading to non-Hermitian graphene Hamiltonians with complex eigenvalues, is addressed. Finally, the magnetic periodic superlattices are explored.
