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Since the theoretical prediction and experimental observation of the thermalHall effect of magnons, a variety of novel phenomena that may occur in magnonicsystems have been proposed. In this paper, we review the recent advances in thestudy of topological phases of magnon Bogoliubov-de Gennes (BdG) systems. Aftergiving an overview of the previous works on electronic topological insulatorsand the thermal Hall effect of magnons, we provide the necessary background forbosonic BdG systems, with a particular emphasis on their non-Hermiticityarising from the diagonalization of the BdG Hamiltonian. After that, weintroduce the definitions of $ \mathbb{Z}_2 $ topological invariants forbosonic systems with pseudo-time-reversal symmetry, which ensures the existenceof bosonic counterparts of "Kramers pairs". Because of the intrinsicnon-Hermiticity of the bosonic BdG systems, these topological invariants haveto be defined in terms of the bosonic Berry connection and curvature. We thenintroduce theoretical models that can be thought of as magnonic analogues oftwo- and three-dimensional topological insulators in class AII. We demonstrateanalytically and numerically that the $ \mathbb{Z}_2 $ topological invariantsprecisely characterize the presence of gapless edge/surface states. We alsopredict that bilayer CrI$_3$ with a particular stacking would be an idealcandidate for realization of a two-dimensional magnon system characterized by anontrivial $ \mathbb{Z}_2 $ topological invariant. For three-dimensionaltopological magnon systems, the thermal Hall effect of magnons is expected tooccur when a magnetic field is applied to the surface.

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