Non-Hermitian Weyl fermions of types III and IV: Hamiltonian, topological protection, and Landau levels

We adopt the non-Hermitian Hamiltonian formalism to describe Weyl fermions of types III and IV. The spectrum of Hamiltonian has an unusual type of anisotropy. Namely, the hermiticity of Hamiltonian strongly depends on the direction in momentum space: for some directions, the spectrum is real, in contrast for other directions it becomes complex. It is shown that the type III and IV Weyl points are topologically stable and the Chern number is equal to +/- 1 despite to the fact that the Hamiltonian is not Hermitian. Furthermore, we calculate the Landau levels and demonstrate that zero Landau level is real, which means that there is a real spectral flow between electronlike and holelike states. Due to the formal analogy with the index theorem, the presence of such a flow as well indicates a nonzero Chern number. In addition, we illustrate that the non-Hermitian Hamiltonian can be regarded as a one-particle problem in the context of topological band theory.