Magnetic phase diagram of an extended J₁-J₂ model on a modulated square lattice and its implications for the antiferromagnetic phase of K_yFe_xSe₂

Motivated by the experimentally observed √5×√5 iron vacancy order and a block-spin antiferromagnetic phase with large magnetic moment in K_0.8Fe_1.6Se₂, we study the magnetic phase diagram of an extended J₁-J₂ model on a 15-depleted square lattice with √5×√5 vacancy order, using a classical Monte Carlo analysis. The magnetic phase diagram involves various antiferromagnetically ordered phases, and most of them have higher-order commensuration. We find that the experimentally relevant block-spin state occupies a significant portion of the phase diagram, and we discuss the spin dynamics of this phase using a linear spin-wave analysis. By comparing the calculated magnetization with the experimental values of magnetic moment, we determine the physical parameter regimes corresponding to the block-spin antiferromagnetic phase. Based on our spin-wave calculations in different parameter regimes, we show how spin-wave degeneracy along the high-symmetry directions of the magnetic Brillouin zone can provide information regarding the underlying exchange couplings. We have also analyzed the magnetic phase diagram of a J₁-J₂ model on two different modulated square lattices relevant to K_yFe_1.5Se₂, which respectively exhibit 14-depleted 2×2 and 4×2 vacancy ordering.