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.

我们提出了与非埃尔米特哈密顿量和其埃尔米特伴随子相关的复杂贝瑞曲率，并利用它们来揭示非埃尔米特系统中的新物理学。对于二维非Hermitian Dirac模型，我们给出了复杂的Berry曲率和Berry相。复杂的贝里相的虚部感应出电纳，因此量子霍尔电导被推广为非厄米系统的导纳。这意味着，物理系统的非赫米特性可以根据非赫米特参数来诱发固有的电容性或电感性。我们分析了二维非Hermitian Dirac模型的复杂能带结构，确定了点和线的间隙，并确定了它们闭合的条件。我们发现闭合与参数空间中能带的异常退化有关，而退化又与拓扑结构相变有关。在连续极限中，我们获得了参数空间中的复Berry相。