Low Ordered Magnetic Moment in Fe-As High-T$_{c}$ Superconductors by Violation of Hund's Rule

We study by exact diagonalization the J$_{0}$-J$_{1}$-J$_{2}$ model over the square lattice that Si and Abrahams introduced recently to describe magnetism in the newly discovered iron-arsenic class of high-T$_{c}$ superconductors. The case of maximum frustration between the nearest-neighbor and the next-nearest-neighbor Heisenberg exchange terms, J$_{2}=\vert $J$_{1}\vert $/2, over a 4 by 4 square lattice with periodic boundary conditions is focused on. Each site hosts two Fe orbitals. Hidden long-range antiferromagnetic order can appear in the absence of Hund's rule coupling, J$_{0}$ = 0. It shows no net ordered magnetic moment. The ordered collinear/SDW moment steadily increases from zero as Hund's rule coupling turns on: J$_{0} \quad <$ 0. This result compares well with recent determinations of a low ordered magnetic moment for the insulating parent compounds of Fe-As based high-T$_{c}$ superconductors by elastic neutron scattering. Our numerical results are also consistent with a quantum phase transition at intermediate Hund's rule coupling, J$_{0}$ = - 2$\vert $ J$_{1}\vert $, that separates the latter hidden-order state from a more familiar frustrated magnetic state that obeys Hund's rule.