in Fig. \[Fig:FLWT\_lattice\]. In addition, we associate $3$ Schwinger-bosons at each site $i$, $b_{i\alpha=x,y,z}$, to the states under SU(3) time-reversal invariant basis, $|x\ra,~|y\ra,~|z\ra$, where $b^\dagger_{i\alpha} |vac\ra = |\alpha\ra$ with $|vac\ra$ being the vacuum state of the Schwinger bosons. The bosons satisfy a local constraint, $\sum_{\alpha} b^\dagger_{i \alpha}b_{i \alpha} = 1.$ The model Hamiltonian in terms of spin operators can be re-expressed in terms of the bosons, $$\begin{aligned}
H \simeq \sum_{i, \delta_n, \alpha, \beta} \left[ J_nb^\dagger_{i \alpha} b_{i \alpha} b^\dagger_{j\beta} b_{i\beta} + \left(K_n - J_n\right) b_{i\alpha}^\dagger b^\dagger_{j \alpha} b_{i\beta} b_{i\beta}\right],~~~\end{aligned}$$ where $j = i + \delta_n,$ and $\delta_n$ (with $n = 1,~2,~3$) connects site $i$ to its $n$th nearest neighbor sites, and we have ignored the constant terms $K_n - J_n$ in the above equation. We also introduce different local rotations for different site $j$. For site $j \in | S^z = \pm1\ra$, we introduce $$\begin{aligned}
\begin{pmatrix}
b_{jx} \\
b_{jy} \\
b_{jz}
\end{pmatrix} =
\begin{pmatrix}
\mp \frac{i}{\sqrt{2}} & \frac{1}{\sqrt{2} }& 0\\
\frac{1}{\sqrt{2}} & \mp \frac{i}{\sqrt{2}} & 0 \\
0 & 0 & 1
\end{pmatrix}
\begin{pmatrix}
d_{jx}\\
d_{jy}\\
d_{jz}
\end{pmatrix},\end{aligned}$$ where for each site there is only one flavor of bosons $d_{ix}$ condensing, and we replace $d^\dagger_{ix}$ and $d_{ix}$ by $(M - 