The N\'{e}el-VBS transition in three-dimensional SU($N$) antiferromagnets

We present results for the phase diagram of an SU($N$) generalization of the Heisenberg antiferromagnet on a bipartite three-dimensional anisotropic cubic (tetragonal) lattice as a function of $N$ and the lattice anisotropy $\gamma$. In the “isotropic” $\gamma =1$ cubic limit, we find a transition from N'{e}el to valence bond solid (VBS) between N=9 and N=10. We follow the N'{e}el-VBS transition to the limiting cases of $\gamma \ll 1$ (weakly coupled layers) and $\gamma \gg 1$ (weakly coupled chains). Throughout the phase diagram we find a direct first-order transition from N'{e}el at small-$N$ to VBS at large-$N$. In the three-dimensional models studied here, we find no evidence for either an intervening spin-liquid “photon” phase or a continuous transition, even close to the limit $\gamma \ll 1$ where the isolated layers undergo continuous N'{e}el-VBS transitions.