Exotic quantum criticality in triangular lattice anti-ferromagnets

We introduce and study a generalized sign-problem free quantum anti-ferromagnet on the triangular lattice. Our Hamiltonian is shown to be a natural generalization of the popular bipartite SU($N$) anti-ferromagnet to non-bipartite lattices. At $N = 2$ our model is unitarily equivalent to a model of an XY superfluid (SF). Consistent with a large-N mapping to a certain quantum dimer model, we find evidence for valence bond solid (VBS) order with a large $\sqrt{12} \times\sqrt{12}$ unit cell. We show that there is a direct transition between these two phases that takes place between $N = 11$ and $N = 12$. For $N = 10, 11$ we use a four spin coupling parameter to tune through a new exotic ``deconfined'' continuous transition between SF and VBS.