Symmetry-protected quantization of complex Berry phases in non-Hermitian 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 $H(theta)$ be a non-Hermitian Hamiltonian parameterized by $theta$. Suppose that there exists a unitary and Hermitian operator $P$ such that $PH(theta)P = H(-theta)$ or $PH(theta)P = H^dagger(-theta)$. We prove that in the former case, the complex Berry phase $gamma$ is $mathbb{Z}_2$-quantized, while in the latter, only the real part of $gamma$ is $mathbb{Z}_2$-quantized. The operator $P$ can be viewed as a generalized symmetry for $H(theta)$, 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.