Since the theoretical prediction and experimental observation of the thermal Hall effect of magnons, a variety of novel phenomena that may occur in magnonic systems have been proposed. In this paper, we review the recent advances in the study of topological phases of magnon Bogoliubov-de Gennes (BdG) systems. After giving an overview of the previous works on electronic topological insulators and the thermal Hall effect of magnons, we provide the necessary background for bosonic BdG systems, with a particular emphasis on their non-Hermiticity arising from the diagonalization of the BdG Hamiltonian. After that, we introduce the definitions of $ \mathbb{Z}_2 $ topological invariants for bosonic systems with pseudo-time-reversal symmetry, which ensures the existence of bosonic counterparts of "Kramers pairs". Because of the intrinsic non-Hermiticity of the bosonic BdG systems, these topological invariants have to be defined in terms of the bosonic Berry connection and curvature. We then introduce theoretical models that can be thought of as magnonic analogues of two- and three-dimensional topological insulators in class AII. We demonstrate analytically and numerically that the $ \mathbb{Z}_2 $ topological invariants precisely characterize the presence of gapless edge/surface states. We also predict that bilayer CrI$_3$ with a particular stacking would be an ideal candidate for realization of a two-dimensional magnon system characterized by a nontrivial $ \mathbb{Z}_2 $ topological invariant. For three-dimensional topological magnon systems, the thermal Hall effect of magnons is expected to occur when a magnetic field is applied to the surface.

中文翻译：

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magnon Bogoliubov-de Gennes 系统的非厄密性和拓扑不变量

由于磁振子的热霍尔效应的理论预测和实验观察，人们提出了磁振子系统中可能发生的各种新现象。在本文中，我们回顾了磁振子 Bogoliubov-de Gennes (BdG) 系统拓扑相研究的最新进展。在概述了以前关于电子拓扑绝缘体和磁振子的热霍尔效应的工作之后，我们为玻色 BdG 系统提供了必要的背景，特别强调了它们由 BdG 哈密顿量的对角化引起的非厄米性。之后，我们介绍了具有伪时间反转对称性的玻色系统的 $\mathbb{Z}_2 $ 拓扑不变量的定义，这确保了“Kramers 对”的玻色对应物的存在。由于玻色 BdG 系统固有的非厄米性，必须根据玻色 Berry 连接和曲率来定义这些拓扑不变量。然后，我们介绍了可以被认为是 AII 类中二维和三维拓扑绝缘体的磁子类似物的理论模型。我们通过分析和数值证明 $\mathbb{Z}_2 $ 拓扑不变量精确地表征了无间隙边缘/表面状态的存在。我们还预测具有特定堆叠的双层 CrI$_3$ 将是实现以非平凡 $\mathbb{Z}_2 $ 拓扑不变量为特征的二维磁子系统的理想候选者。对于三维拓扑磁振子系统，当磁场施加到表面时，磁振子的热霍尔效应预计会发生。