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 which 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. In some cases, possible generalization of these index theorems for non-Hermitian chiral multi-fold fermions will also be highlighted. Possible material pertinence of these findings will be discussed as well.