to the correct checkerboard AF state and $d$-wave pairing, showing that these problems can be attacked from a variety of view points. In addition, the newly discovered $A_y$Fe$_{2-x}$Se$_2$ ($A=$ K, Rb, Cs, Tl) iron-chalcogenide superconductors [@jgguo; @mhfang] do [*not*]{} exhibit hole pockets [@Wang_122Se; @Zhang_122Se; @Mou_122Se], but have strong AF ordered insulating phases with extremely high Néel transition temperatures [@wbao1; @fye]. Such a strong magnetism and high superconducting transition temperature ($T_c\approx 33$ K) cannot be explained by FS nesting since this is based on the enhancement of the particle-hole susceptibility due to an extended momentum space with nearly parallel Fermi surfaces, i.e. it applies only to particle and hole FS’s and not to purely electronic Fermi pockets.

Since iron-based superconductors have six electrons occupying the nearly degenerate $3d$ Fe orbitals, the system is intrinsically multi-orbital and therefore it is technically difficult to define and study a simple microscopic Hamiltonian to describe the electronic properties of these materials and characterize the strength of the electronic correlations. From optical conductivity measurements [@qazilbash09], it has been argued that electronic correlations in Fe pnictides are weaker than in underdoped copper-oxides but are stronger than those of Fermi liquid metals, contrary to the conclusion based on 