Effective tight-binding model for the iron vacancy ordered A$_{y}$Fe$%_{1.6}$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 $\sqrt{5}$ vacancy ordered structure. Based on three orbitals of $$, 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.