Striped Phase in a QuantumXYModel with Ring Exchange

We present quantum Monte Carlo results for a square-lattice $S=1/2$ $XY$ model with a standard nearest-neighbor coupling $J$ and a four-spin ring exchange term $K$. Increasing $K/J$, we find that the ground state spin stiffness vanishes at a critical point at which a spin gap opens and a striped bond-plaquette order emerges. At still higher $K/J$, this phase becomes unstable and the system develops a staggered magnetization. We discuss the quantum phase transitions between these phases.