  The $S_4$ symmetric microscopic model with two iso-spin components has been
studied via constrained-path quantum Monte Carlo simulation. Our results
demonstrate a stable $(\pi,0)$ or $(0,\pi)$ magnetic order which is
significantly enhanced on increasing both the Coulomb repulsion $U$ and Hund's
coupling strength $J$. Also, our simulation indicates that the magnetic order
tends to be in an orthomagnetic one, in which the nearest-neighbour magnetic
moment are orthogonal to each other, rather than in a collinear
antiferromagnetic state. Interestingly, when the system is doped away from half
filling, the magnetic order is obviously elevated in the low doping density,
and then significantly suppressed when more electrons are introduced.
Meanwhile, we find that an $A_{1g}$ $s_{\pm}$-wave pairing dominates all the
singlet nearest-neighbour pairings, and is significantly enhanced via electron
doping.

Strongly Correlated Electrons