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We elucidate the existing controversies in the newly discovered K-doped iron selenide (K(x)Fe(2-y)Se(2-z)) superconductors. The stoichiometric KFe(2)Se(2) with √2 × √2 charge ordering was identified as the parent compound of K(x)Fe(2-y)Se(2-z) superconductor using scanning tunneling microscopy and spectroscopy. The superconductivity is induced in KFe(2)Se(2) by either Se vacancies or interacting with the antiferromagnetic K(2)Fe(4)Se(5) compound. In total, four phases were found to exist in K(x)Fe(2-y)Se(2-z): parent compound KFe(2)Se(2), superconducting KFe(2)Se(2) with √2 × √5 charge ordering, superconducting KFe(2)Se(2-z) with Se vacancies, and insulating K(2)Fe(4)Se(5) with √5 × √5 Fe vacancy order. The phase separation takes place at the mesoscopic scale under standard molecular beam epitaxy conditions.

The spin-3/2 fermion models with contact interactions have a generic SO5 symmetry without any fine-tuning of parameters. Its physical consequences are discussed in both the continuum and lattice models. A Monte Carlo algorithm free of the sign problem at any doping and lattice topology is designed when the singlet and quintet interactions satisfy U0< or = U2< or = -3/5 U0 (U0< or = 0), thus making it possible to study different competing orders with high numerical accuracy. This model can be accurately realized in ultracold atomic systems.