Striped phase in a quantum XY-model 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. 
A. W. Sandvik, S. Daul, R. R. P. Singh, D. J. Scalapino