We study fixed points of the easy-plane $mathbb{CP}^{N-1}$ field theory by combining quantum Monte Carlo simulations of lattice models of easy-plane SU($N$) superfluids with field theoretic renormalization group calculations, by using ideas of deconfined criticality. From our simulations, we present evidence that at small $N$ our lattice model has a first order phase transition which progressively weakens as $N$ increases, eventually becoming continuous for large values of $N$. Renormalization group calculations in $4-epsilon$ dimensions provide an explanation of these results as arising due to the existence of an $N_{ep}$ that separates the fate of the flows with easy-plane anisotropy. When $N<N_{ep}$ the renormalization group flows to a discontinuity fixed point and hence a first order transition arises. On the other hand, for $N > N_{ep}$ the flows are to a new easy-plane $mathbb{CP}^{N-1}$ fixed point that describes the quantum criticality in the lattice model at large $N$. Our lattice model at its critical point thus gives efficient numerical access to a new strongly coupled gauge-matter field theory.
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