First-order superfluid to valence bond solid phase transitions in easy-plane SU($N$) magnets for small-$N$

We consider the easy-plane limit of bipartite SU($N$) Heisenberg Hamiltonians which have a fundamental representation on one sublattice and the conjugate to fundamental on the other sublattice. For $N=2$ the easy plane limit of the SU(2) Heisenberg model is the well known quantum XY model of a lattice superfluid. We introduce a logical method to generalize the quantum XY model to arbitrary $N$, which keeps the Hamiltonian sign-free. We show that these quantum Hamiltonians have a world-line representation as the statistical mechanics of certain tightly packed loop models of $N$-colors in which neighboring loops are disallowed from having the same color. In this loop representation we design an efficient Monte Carlo cluster algorithm for our model. We present extensive numerical results for these models on the two dimensional square lattice, where we find the nearest neighbor model has superfluid order for $N\leq 5$ and valence-bond order for $N> 5$. By introducing SU($N$) easy-plane symmetric four-spin couplings we are able to tune across the superfluid-VBS phase boundary for all $N\leq 5$. We present clear evidence that this quantum phase transition is first order for $N=2$ and $N=5$, suggesting that easy-plane deconfined criticality runs away generically to a first order transition for small-$N$.