${\mathbb Z}_2$ Topological Invariant for Magnon Spin Hall Systems

We propose a definition of a ${\mathbb Z}_2$ topological invariant for magnon spin Hall systems which are the bosonic analog of two-dimensional topological insulators in class AII. The existence of "Kramers pairs" in these systems is guaranteed by pseudo-time-reversal symmetry which is the same as time-reversal symmetry up to some unitary transformation. The ${\mathbb Z}_2$ index of each Kramers pair of bands is expressed in terms of the bosonic counterparts of the Berry connection and curvature. We construct explicit examples of magnon spin Hall systems and demonstrate that our ${\mathbb Z}_2$ index precisely characterizes the presence or absence of helical edge states. The proposed ${\mathbb Z}_2$ index and the formalism developed can be applied not only to magnonic systems but also to other non-interacting bosonic systems.