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The discovery by Kane and Mele of a model of spinful electrons characterizedby a $\mathcal{Z}_2$ topological invariant had a lasting effect on the study ofelectronic band structures. Given this, it is natural to ask whether similartopology can be found in the band-like excitations of magnetic insulators, andrecently models supporting $\mathcal{Z}_2$ topological invariants have beenproposed for both magnon [Kondo et al. Phys. Rev. B 99, 041110(R) (2019)] andtriplet [D. G. Joshi and A. P. Schnyder, Phys. Rev. B 100, 020407 (2019)]excitations. In both cases, magnetic excitations form time--reversal (TR)partners, which mimic the Kramers pairs of electrons in the Kane-Mele model butdo not enjoy the same type of symmetry protection. In this paper, we revisitthis problem in the context of the triplet excitations of a spin model on thebilayer kagome lattice. Here the triplet excitations provide a faithful analogof the Kane-Mele model as long as the Hamiltonian preserves the TR$\times$U(1)symmetry. We find that exchange anisotropies, allowed by the point group andtypical in realistic models, break the required TR$\times$U(1) symmetry andinstantly destroy the $\mathcal{Z}_2$ band topology. We further consider theeffects of TR breaking by an applied magnetic field. In this case, the liftingof spin-degeneracy leads to a triplet Chern insulator, which is stable againstthe breaking of TR$\times$U(1) symmetry. Kagome bands realize both a quadraticand a linear band touching, and we provide a thorough characterization of theBerry curvature associated with both cases. We also calculate thetriplet-mediated spin Nernst and thermal Hall signals which could be measuredin experiments. These results suggest that the $\mathcal{Z}_2$ topology ofband-like excitations in magnets may be intrinsically fragile compared to theirelectronic counterparts.

Fragility of $\mathcal{Z}_2$ topological invariant characterizing triplet excitations in a bilayer kagome magnet