We investigate the quantization of the complex-valued Berry phases in non-Hermitian quantum systems with certain generalized symmetries. In Hermitian quantum systems, the real-valued Berry phase is known to be quantized in the presence of certain symmetries, and this quantized Berry phase can be regarded as a topological order parameter for gapped quantum systems. In this Letter, on the other hand, we establish that the complex Berry phase is also quantized in the systems described by a family of non-Hermitian Hamiltonians. Let $H\left(\theta \right)$ be a non-Hermitian Hamiltonian parametrized by $\theta$. Suppose that there exists a unitary and Hermitian operator $P$ such that $PH\left(\theta \right)P=H(-\theta )$ or $PH\left(\theta \right)P=H^{†}(-\theta )$. We prove that in the former case, the complex Berry phase $\gamma$ is ${\mathbb{Z}}_2$ quantized, whereas in the latter, only the real part of $\gamma$ is ${\mathbb{Z}}_2$ quantized. The operator $P$ can be viewed as a generalized symmetry operation for $H\left(\theta \right)$, and, in practice, $P$ can be, for example, a spatial inversion. Our results are quite general and apply to both interacting and noninteracting systems. We also argue that the quantized complex Berry phase is capable of classifying non-Hermitian topological phases and demonstrate this in one-dimensional many-body systems with and without interactions.

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• Open quantum systems & decoherence
• Topological phase transition
• Topological phases of matter
• 1-dimensional systems
• Non-Hermitian systems
• Exact diagonalization
• Inversion symmetry