the Fe square lattice, and the bilinear and biquadratic interactions are chosen up to the third neighbors. The $P_{123}$ and $P_{1234}$ stand for the $3$-site and $4$-site ring exchanges with $\mathfrak{R}_3, \mathfrak{R}_4 >0$ [@Lai_SU(3)_gapless], which rotate the states such as $P_{123} | \alpha \beta \gamma\ra = |\gamma \alpha \beta\ra,~P_{1234} | \alpha \beta \gamma \xi \ra = | \xi \alpha \beta \gamma \ra$. The ring exchange operators can be re-expressed via physical spin operators [@Itoi97] as detailed in Supplemental Material [@supplemental]. The biquadratic terms also introduce the quadrupolar operator ${\bf Q}_i$, which has five components: $Q^{x^2 - y^2}_i = (S^x_i)^2 - (S^y_i)^2$, $Q^{3z^2 - r^2}_i = [ 2 (S^z_i)^2 - (S^x_i)^2 - (S^y_i)^2]/\sqrt{3}$, $Q^{xy} = S^x_i S^y_i + S^y_i S^x_i$, $Q^{yz} =  S^y_i,S^z_i +S^z_i S^y_i$, and, $Q^{zx} = S^z_i S^x_i + S^x_i S^z_i$. The biquadratic term can be re-expressed as $({\bf S}_i \cdot {\bf S}_j )^2 = ({\bf Q}_i \cdot {\bf Q}_j) /2 - ({\bf S}_i \cdot {\bf S}_j) /2 + ({\bf S}^2_i {\bf S}^2_j)/3$.

*Semi-classical phase diagram for spin-$1$ model.*— We first use the site-factorized wavefunction analysis to study the semi-classical phase diagram for $S = 1$. Based on the time-reversal invariant basis of the SU(3) fundamental representation, $
|x\ra 