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

Abstract

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 $H(\ensuremath{\theta})$ be a non-Hermitian Hamiltonian parametrized by $\ensuremath{\theta}$. Suppose that there exists a unitary and Hermitian operator $P$ such that $PH(\ensuremath{\theta})P=H(\ensuremath{-}\ensuremath{\theta})$ or $PH(\ensuremath{\theta})P={H}^{\ifmmode\dagger\else\textdagger\fi{}}(\ensuremath{-}\ensuremath{\theta})$. We prove that in the former case, the complex Berry phase $\ensuremath{\gamma}$ is ${\mathbb{Z}}_{2}$ quantized, whereas in the latter, only the real part of $\ensuremath{\gamma}$ is ${\mathbb{Z}}_{2}$ quantized. The operator $P$ can be viewed as a generalized symmetry operation for $H(\ensuremath{\theta})$, 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.