Complex Berry curvature pair and quantum Hall admittance in non-Hermitian systems

We propose complex Berry curvatures associated with the non-Hermitian Hamiltonian and its Hermitian adjoint and use these to reveal new physics in non-Hermitian systems. We give the complex Berry curvature and Berry phase for the two-dimensional non-Hermitian Dirac model. The imaginary part of the complex Berry phase induces susceptance so that the quantum Hall conductance is generalized to admittance for non-Hermitian systems. This implies that the non-Hermiticity of physical systems can induce intrinsic capacitive or inductive properties, depending on the non-Hermitian parameters. We analyze the complex energy band structures of the two-dimensional non-Hermitian Dirac model, determine the point and line gaps, and identify the conditions for their closure. We find that closure is associated with the exceptional degeneracy of the energy bands in the parameter space, which, in turn, is associated with topological phase transitions. In the continuum limit, we obtain the complex Berry phase in the parameter space.