  We study the quantum phase transition between the superfluid and valence bond
solid in "easy-plane" J-Q models on the square lattice. The Hamiltonian we
study is a linear combination of two model Hamiltonians: (1) an SU(2) symmetric
model, which is the well known J-Q model that does not show any direct signs of
a discontinuous transition on the largest lattices and is presumed continuous,
and (2) an easy plane version of the J-Q model, which shows clear evidence for
a first order transition even on rather small lattices of size $L\approx16$. A
parameter $0\leq\lambda\leq 1$ ($\lambda=0$ being the easy-plane model and
$\lambda=1$ being the SU(2) symmetric J-Q model) allows us to smoothly
interpolate between these two limiting models. We use stochastic series
expansion (SSE) quantum Monte Carlo (QMC) to investigate the nature of this
transition as $\lambda$ is varied - here we present studies for
$\lambda=0,0.5,0.75,0.85,0.95$ and $1$. While we find that the first order
transition weakens as $\lambda$ is increased from 0 to 1, we find no evidence
that the transition becomes continuous until the SU(2) symmetric point,
$\lambda=1$. We thus conclude that the square lattice superfluid-VBS transition
in the two-component easy-plane model is generically first order.
