First order phase transitions in the square lattice "easy-plane" J-Q model

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.