Scattering Dynamics of a Non-Hermitian Dirac Equation

Non-Hermitian systems can exhibit many interesting properties that differ qualitatively from their Hermitian counterparts. Here, we explore the properties of a non-Hermitian variant of the (2+1)-dimensional Dirac wave equation. This non-Hermitian Dirac equation (NHDE) can arise in the long-wavelength limit of a non-Hermitian lattice obeying a set of symmetries we call "semi-Hermiticity". Going beyond the more well-known parity/time-reversal (PT) symmetry, semi-Hermiticity allows a non-Hermitian system to exhibit not only a real energy spectrum but also pairwise-orthogonal eigenstates, which are required for the emergence of a Dirac cone. We derive the phase diagram of the NHDE, showing that it is governed by two Hermitian Dirac sub-systems with independently-variable Dirac masses. The NHDE exhibits an anomalous form of Klein tunneling, whereby flux conservation can be violated at the interface between two domains with real bulk spectra. Such an interface can even function as a simultaneous laser and coherent perfect absorber of infinitesimal thinness. We show that the key properties of the NHDE can be probed with experimentally-realizable non-Hermitian metamaterials based on acoustics or photonics.