Zero modes and index theorems in non-Hermitian Dirac materials

Dirac Hamiltonain by virtue of possessing a particle-hole symmetry features topological robust bound states at zero-energy under conducive environment. The number of zero energy modes is determined by an appropriate index theorem. Some celebrated examples are (a) Aharonov-Casher index theorem, relating the number of zero modes for planar noninteracting Dirac fermions with the total magnetic flux enclosed by the system, (b) Jackiw-Rebbi index theorem, counting the number of zero modes bound to a one-dimensional Dirac mass domain wall, and (c) Jackiw-Rossi index theorem, relating the number of zero modes bound to the core of the vortex of a U(1) Dirac mass in two dimensions. In this talk, first I will promote a Lorentz invariant formulation of non-Hermitian Dirac operator possessing either purely real or purely imaginary eigenvalues, depending on the strength of the anti-Hermitian parameter. In such systems, I will demonstrate that non-Hermitian Dirac materials continue to honor these index theorems at least when the eigenvalue spectrum is guaranteed to be purely real. Possible braoder implications of these findings will be discussed.