Vesignieite: a $S=\frac{1}{2}$ kagome antiferromagnet with dominant third-neighbor exchange

The spin-$\frac{1}{2}$ kagome antiferromagnet is an archetypal frustrated system predicted to host a variety of exotic magnetic states. We show using neutron scattering measurements that deuterated vesignieite BaCu${}_3$V${}_2$O${}_8$(OD)${}_2$, a fully stoichiometric $S=1\mathrm{/}2$ kagome magnet with $<$1% lattice distortion, orders magnetically at $T_{\mathrm{N}}=9$K into a multi-k coplanar variant of the predicted triple-k octahedral structure. We find this structure is stabilized by a dominant antiferromagnetic 3${}^{\mathrm{r}\mathrm{d}}$-neighbor exchange $J_3$ with minor 1${}^{\mathrm{s}\mathrm{t}}$- or 2${}^{\mathrm{n}\mathrm{d}}$–neighbour exchange. The spin-wave spectrum is well described by a $J_3$-only model including a tiny symmetric exchange anisotropy.