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

TL;DR

This paper demonstrates that complex Berry phases in certain non-Hermitian quantum systems are quantized due to generalized symmetries, providing a new topological classification method for non-Hermitian phases.

Contribution

It establishes the quantization of complex Berry phases in non-Hermitian systems with specific symmetries, extending topological classification beyond Hermitian cases.

Findings

Complex Berry phase is $ extbf{Z}_2$-quantized under certain symmetries.

The quantized phase classifies non-Hermitian topological phases.

Application demonstrated in one-dimensional strongly correlated 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 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.