Isotropic Landau levels of Dirac fermions in high dimensions

We generalize the Landau levels of two-dimensional Dirac fermions to three dimensions and above with the full rotational symmetry. Similarly to the two-dimensional case, there exists a branch of zero energy Landau levels of fractional fermion modes for the massless Dirac fermions. The spectra of other Landau levels distribute symmetrically with respect to the zero energy scaling with the square root of the Landau-level indices. This mechanism is a nonminimal coupling of Dirac fermions to the background fields. This high dimensional relativistic Landau-level problem is a square-root problem of its previous studied nonrelativistic version investigated in Li and Wu [arXiv:1103.5422 (2011)].