Dynamic approach to finite-temperature magnetic phase transitions in the extended J1- J2 model with vacancy order

The recently discovered iron-based superconductors A$_{y}$Fe$_{2-x}$Se$_{2}$ ($A$=K, Rb, Cs, Tl) show a long-range antiferromagnetic order with an unexpected high transition temperature $T_N \sim 550$ K and a unique $\sqrt{5} \times \sqrt{5}$ vacancy order. Taking the extended $J_1$-$J_2$ model as a minimal model, we investigate the finite-temperature magnetic phase transitions in a square lattice with a $\sqrt{5} \times \sqrt{5}$ vacancy superstructure by using large-scale Monte Carlo simulations. By the parallel tempering technique, the block spin checkerboard and stripe antiferromagnetic states are detected to be the groundstates for three representative sets of model parameters. The short-time dynamic approach is applied to accurately determine the critical temperature as well as the static and dynamic exponents. Our results indicate that the dramatic enhancement of the critical temperature as observed in experiments should be mainly due to a combination effect of the vacancy order and the block lattice contraction.