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.

Figure 1

Plaquette (left) and bond (right) strengths at the center of a $64\times 128$ open-boundary lattice at $K/J=10$ and $T=J/8$. The plaquette strengths are represented by shades of gray with the weakest $⟨P_{ijkl}⟩=0.222$ (white squares) and strongest $0.468$ (black squares). The bond strengths are indicated by the width of the line segments, with the weakest $⟨B_{ij}⟩=0.181$ and the strongest $0.505$.

Figure 2

Spin stiffness and plaquette-stripe order parameter vs ring-exchange coupling for a $64\times 64$ system with periodic boundary conditions at $T=J/8$.

Figure 3

Finite-size scaling of the ground state staggered magnetization (open circles) and the plaquette-stripe order parameter (solid circles) at $K/J=8.5$ and $64$. The dotted straight lines show extrapolations of the infinite-size order parameters. The dashed curves show the form $\sim 1/L^2$ expected asymptotically when there is no long-range order.

Figure 4

Temperature dependence of the uniform magnetic susceptibility for $L=80$ systems close to the superfluid-striped transition. Statistical errors are of the order of the size of the symbols. The dashed line shows the linear behavior expected for a quantum phase transition with $z=1$.