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Symmetry-protected quantization of complex Berry phases in non-Hermitian many-body systems

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 paper, 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 $H(θ)$ be a non-Hermitian Hamiltonian parameterized by $θ$. Suppose that there exists a unitary and Hermitian operator $P$ such that $PH(θ)P = H(-θ)$ or $PH(θ)P = H^\dagger(-θ)$. We prove that in the former case, the complex Berry phase $γ$ is $\mathbb{Z}_2$-quantized, while in the latter, only the real part of $γ$ is $\mathbb{Z}_2$-quantized. The operator $P$ can be viewed as a generalized symmetry for $H(θ)$, and in practice, $P$ can be, for example, a spatial inversion. We also argue that this quantized complex Berry phase is capable of classifying non-Hermitian topological phases, and we demonstrate this in some one-dimensional strongly correlated systems.