  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.
