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 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(??) be a non-Hermitian Hamiltonian parametrized by ??. Suppose that there exists a unitary and Hermitian operator P such that PH(??)P = H (?????) or PH(??)P = H???(?????). We prove that in the former case, the complex Berry phase ?? is Z2 quantized, whereas in the latter, only the real part of ?? is Z2 quantized. The operator P can be viewed as a generalized symmetry operation for H (??), 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.