Complex Berry curvature and complex energy band structures in non-Hermitian graphene model

Non-Hermitian quantum systems exhibit many novel physical properties of quantum states. We consider a non-Hermtian graphene model based on the tight-binding approximation with the coupling of the graphene and the substrate. We analyze the complex energy structure of this model and its exceptional points as well as relevant topological invariants. We give the analytic complex Berry connection and Berry curvature in the Brillouin zone and investigate numerically the relationships between the complex Berry curvature and the complex energy band structures. We find that the behaviors of the complex Berry curvature depend on the complex energy band structures. The occurrence of the peaks of both real and imaginary parts of the complex Berry curvature corresponds to the exceptional (gapless) points in the Brillouin zone. In particular, the Dirac cone of the imaginary part of the Berry curvature occurs and corresponding to the occurrence of the flat real energy band for the non-Hermitian parameter \(\eta =3\). These results provide some novel insights to the relationship between the non-Hermitian graphene, geometry, and topological invariants.