Effective tight-binding model for the iron vacancy ordered A${}_y$Fe$$Se${}_2$

We investigate the electronic structure of the ternary iron selenide K${}_y$% Fe${}_{1.6}$Se${}_2$ by considering the spatial symmetry of the $\sqrt{5}% \times \sqrt{5}$vacancyorderedstructure\mathrm{.}Basedonthreeorbitalsof$% t_{2g}$, which are believed to play major physics in iron-based superconductors, an effective two-dimensional tight binding Hamiltonian is constructed with the vacancy ordered structure being explicitly included. It is shown that the constructed band model, when combined with generalized Hubbard interactions, yields a spin susceptibility which exhibits both the block-checkerboard antiferromagnetism instability and the stripe antiferromagnetism instability. In particular, for large Hund’s rule couplings, the block-checkerboard antiferromagnetism wins over the stripe antiferromagnetism, in agreement with the observation in experiments. We argue that such a model with correct symmetry and Fermi surface structures should be the starting point to model K${}_y$Fe${}_{1.6}$Se${}_2$. The spin fluctuations at $\mathbf{q}$=($\pi ,\pi$) suggest that interblock fluctuations of spins might play an important role in the mechanism of superconductivity occurring in this system.