Z₂ topological invariant for magnon spin Hall systems

The classification and characterization of different phases of matter based on the topology of band structures has recently attracted much attention. It is known that some of the topological phenomena in electron systems are carried over to bosonic systems [1]. However, the topological invariant of bosonic systems with time-reversal symmetry has not yet been identified, as Kramers' theorem cannot be applied to them. To address this issue, we first introduce the pseudo-time-reversal operator which ensures the existence of ``Kramers pairs'' of bosons. Then, we define the Z₂ topological invariant for magnon spin Hall systems [2] using the Berry connection and curvature for bosons, which are different from those of electrons [3]. Furthermore, we propose two magnetic models with magnon bands carrying the nontrivial Z₂ topological invariant. We also demonstrate that the presence (absence) of helical edge states corresponds to the nontrivial (trivial) value of the Z₂ topological invariant.

[1] H. Katsura, N. Nagaosa, and P. A. Lee, Phys. Rev. Lett. 104, 066403 (2010).
[2] H. Kondo, Y. Akagi, and H. Katsura, arXiv:1808.09149.
[3] R. Matsumoto, R. Shindou, and S. Murakami, Phys. Rev. B 89,054420 (2014).