Three-dimensional topological magnon systems

We propose a concept of three-dimensional topological magnon systems which are the magnonic analog of topological insulators in three dimensions. We define a set of Z(2) topological invariants that characterizes different topological phases and determines the presence or absence of surface Dirac cones. The validity of the classification scheme based on these Z(2) invariants is demonstrated by considering three concrete examples. One of them is a bosonic counterpart of the Fu-Kane-Mele model on a diamond lattice, which is found to exhibit three distinct phases analogous to strong topological, weak topological, and trivial insulator phases of the original fermionic model. We also discuss a possible realization of the thermal Hall effect of surface magnons in the presence of a magnetic field in proximity to a normal ferromagnet. The topological characterization in this paper can also be applied to other systems, such as spin liquids or paramagnets, described by Schwinger bosons.