the regime of $(\pi/2,\pi/2)$ AFM$^*_{a/b}$ the nearest-neighbor spins be tilted on the plane from the spin patterns shown in Figs. 1(c)-(d) by a small angle $\theta$ in such a way that each second-neighbor spins remain orthogonal, while third-neighbor spins are always anti-parallel to each other. Despite of the angle $\theta$, the energy of AFM$_{a/b}^*$ is independent of $\theta$.

We note that in numerics, there are indeed some fluctuations in the quadrupolar degrees of freedom and the dipolar magnetization can be slightly smaller than one, $|\la{\bf S}\ra| \leq 1$. The energy difference between the phases obtained in the exact numerical result and the AFM$^*_{a/b}$ are within $O(10^{-2})$, which may be due to the numerical errors in searching for a global minimum in the $4\times L^2$ parameter space. On other other hand, this may suggests there are many competing local minima in the AFM$^*_{a/b}$ regime in the parameter space in the $S=1$ model. Enlarging the spin size $S >1$ sufficiently suppresses the fluctuations in the quadrupolar degrees of freedom and the spin pattern can be unbiasedly confirmed in the density matrix renormalization group analysis. [@FeTe-long]

In the presence of ring exchanges, the $3$-site ring exchanges within the site-factorized wavefunction analysis can not 