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. We establish that the complex Berry phase is also quantized in the systems described by a family of non-Hermitian Hamiltonians in the presence of certain generalized symmetries. Specifically, we discuss two classes of non-Hermitian Hamiltonians with such generalized symmetries described by unitary and Hermitian operators. In one class, the complex Berry phase γ is quantized to 0 or π. In the other class, γ is not necessarily real, but its real part is shown to be quantized. Our results are quite general and apply to both interacting and noninteracting systems. We also argue that the quantized complex Berry phase can classify non-Hermitian topological phases and demonstrate this in one-dimensional many-body systems with and without interactions.