Zero modes and index theorems for non-Hermitian Dirac fermions

Dirac fermions, subject to external magnetic fields and in the presence of mass orders that assume topologically nontrivial spatial textures such as domain wall and vortices, for example, bind robust midgap states at zero energy, the number of which is governed by the Aharonov-Casher and Jackiw-Rebbi or Jackiw-Rossi index theorems, respectively. Here I extend the jurisdiction of these prominent index theorems to Lorentz invariant non-Hermitian (NH) Dirac operators, constructed by augmenting the celebrated Dirac Hamiltonian by a masslike anti-Hermitian operator that also scales linearly with momentum. The resulting NH Dirac operator manifests all-real eigenvalues over an extended NH parameter regime, characterized by a real effective Fermi velocity for NH Dirac fermions (${v}_{\mathrm{F}}$). From the explicit solutions of the zero-energy bound states, I show that in the presence of external magnetic fields of arbitrary shape such modes always exist when the system encloses a finite number of magnetic flux quanta, while in the presence of spatially nontrivial textures of the mass orders localized zero-energy modes can only be found in the spectrum when ${v}_{\mathrm{F}}$ is real. These findings pave a concrete route to realize nucleation of competing orders from the topologically robust zero-energy manifold in NH or open Dirac systems. Possible extensions of these outcomes to other index theorems and tabletop experimental setups to test these predictions are discussed.