The magnetic phase diagram of an extended J1-J2 model on a modulated square lattice and its implications for the antiferromagnetic phase of KyFexSe2

Motivated by the experimentally observed $\sqrt{5}\times \sqrt{5}$ iron vacancy order and a block spin antiferromagnetic phase with large magnetic moment in ${\mathrm{K}}_{0.8}{\mathrm{F}\mathrm{e}}_{1.6}{\mathrm{S}\mathrm{e}}_2$, we study the magnetic phase diagram of an extended $J_1-J_2$ model on a 1/5-depleted square lattice with $\sqrt{5}\times \sqrt{5}$ 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. Based on our spin wave calculations in different parameter regimes corresponding to the block spin antiferromagnetic phase, 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 $J_1-J_2$ model on two different modulated square lattices relevant for ${\mathrm{K}}_y{\mathrm{F}\mathrm{e}}_{1.5}{\mathrm{S}\mathrm{e}}_2$, which respectively exhibit 1/4-depleted $2\times 2$ and $4\times 2$ vacancy ordering.