Magnetic state of K$_{0.8}$Fe$_{1.6}$Se$_2$ from a five-orbital Hubbard model in the Hartree-Fock approximation

The five-orbital Hubbard model (without lattice distortions) is investigated to study theoretically the recently discovered Fe-based superconductors K$_{0.8}$Fe$_{1.6}$Se$_2$, by using the real-space Hartree-Fock approximation and employing a 10$\times$10 Fe cluster with Fe vacancies in a $\sqrt{5}\times\sqrt{5}$ pattern[1]. The phase diagram contains an insulating state with the same spin pattern as observed experimentally, involving 2$\times$2 ferromagnetic plaquettes coupled with one another antiferromagnetically. The magnetic moment $\sim$3$\mu_B$/Fe is in good agreement with experiments. Several other competing phases are also stabilized in the phase diagram, in agreement with recent calculations using phenomenological models. [1] Qinlong Luo {\it et al.}, Phys. Rev. B {\bf 84}, 140506(R) (2011), and references therein.