  The study of topological band structures has recently been extended from
electrons to many other systems with band-like excitations. One promising
avenue is the triplet excitations of quantum paramagnets, which can realize a
variety of topologically-nontrivial band structures. Recently, triplet bands
characterized by a Z2 topological invariant, analogous to that introduced by
Kane and Mele, have been discussed in a quantum paramagnet realized on a
bilayer honeycomb lattice [D. G. Joshi and A. P. Schnyder, Phys. Rev. B 100,
020407 (2019)]. In this case, triplet bands from time-reversal (TR) partners,
which play the role of the Kramers pairs of electrons in the Kane-Mele model.
However, Bosonic triplet excitations do not enjoy the same symmetry-protection
as Kramers pairs of electrons and can be mixed by interactions. The
consequences of such mixing for the Z2 topological phases of a quantum
paramagnet remains an open question. In this paper, we explore this problem on
the bilayer kagome lattice. We show that, as long as a TRxU(1) symmetry is
preserved, the triplet excitations provide a faithful analog of the Kane-Mele
model. 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 construct the most general TR invariant triplet Hamiltonian
allowed by the symmetry. We find that exchange interactions with quadrupolar
character, typical in realistic models, lift the required U(1) symmetry and
destroy the Z2 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 the formation of a triplet Chern insulator, which is stable against the
breaking of U(1) symmetry. We calculate the triplet-mediated spin Nernst and
thermal Hall signals, making predictions, which could be compared with
experiment.
