Classical Ring-exchange Processes on the Triangular Lattice
  The effects of the ring-exchange Hamiltonian H_3 = J_3 sum_{<ijk>} (S_i * S_j) (S_i * S_k) on the triangular lattice are studied using classical Monte Carlo simulations. Each spin $S_i$ is treated as a classical XY spin taking on Q equally spaced angles (Q-states clock model). For Q=6, a first-order transition into a stripe-ordered phase preempts the macroscopic classical degeneracy. For Q > 6, a finite window of critical phase exists, intervening between the low-temperature stripe phase and the high-temperature paramagnetic phase. 
June Seo Kim, Jung Hoon Han