Non-Hermitian magnetic fields, zero modes, and index theorems in two-dimensional Dirac materials

Planar massless Dirac fermions residing on a flat Euclidean plane, subject to external perpendicular magnetic fields support topologically robust zero-energy modes, the number of which is exactly equal to the number of magnetic flux quanta enclosed by the system; a prominent result known as the Aharonov-Casher index theorem. In this talk, I will introduce the notion of non-Hermitian (NH) magnetic fields that minimally couple with massless Dirac fermions in a Lorentz invariant fashion. I will show that such a coupled system fosters robust zero-energy right or left eigenmodes in the femrionic spectrum, the number of which is determined by the quanta of NH flux enclosed by the system. Concrete lattice models for such NH magnetic fields will be presented to anchor these outcomes. If time permits, I will show a generalization of this scenario to hyperbolic Dirac systems, embedded on a curved space of constant negative curvature.