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Non-Hermiticity and topological invariants of magnon Bogoliubov-de Gennes systems

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.