Fragility of Z2 topological invariant characterizing triplet excitations in a bilayer kagome magnet

The discovery by Kane and Mele of a model of spinful electrons characterized by a Z(2) topological invariant had a lasting effect on the study of electronic band structures. Given this, it is natural to ask whether similar topology can be found in the bandlike excitations of magnetic insulators, and recently models supporting Z(2) topological invariants have been proposed for both magnon [H. Kondo et al., Phys. Rev. B 99, 04110(R) (2019)] and triplet [D. G. Joshi and A. P. Schnyder, Phv S. Rev. 13 100. 020407(R) (2019)] excitations. In both cases, magnetic excitations form time-reversal (TR) partners, which mimic the Kramers pairs of electrons in the KaneMele model but do not enjoy the same type of symmetry protection. In this paper, we revisit this problem in the context of the triplet excitations of a spin model on the bilayer kagome lattice. Here the triplet excitations provide a faithful analog of the Kane-Mele model as long as the Hamiltonian preserves the TR x U(1) symmetry. We find that exchange anisotropies, allowed by the point group and typical in realistic models, break the required TR x U(1) symmetry and instantly destroy the Z(2) band topology. We further consider the effects of TR breaking by an applied magnetic field. In this case, the lifting of spin degeneracy leads to a triplet Chern insulator, which is stable against the breaking of TR x U(1) symmetry. Kagome bands realize both a quadratic and a linear band touching, and we provide a thorough characterization of the Berry curvature associated with both cases. We also calculate the triplet-mediated spin Nernst and thermal Hall signals which could be measured in experiments. These results suggest that the Z(2) topology of bandlike excitations in magnets may be intrinsically fragile compared to their electronic counterparts.