$N_s$ is the total number of lattice site). This nematic order parameter is nonzero for the BC phase, but vanishes for the PL phase. The two orders, AFM$^*_{a/b}$, do not possess the spin dipolar nematicity. Still, they have broken $C_4$ symmetry due to the nonzero “spin quadrupolar nematicity", defined in terms of $\sigma^Q_1 = 1/N_s \sum_j {\bf Q}_j \cdot \left[ {\bf Q}_{j + \hat{x}} - {\bf Q}_{j + \hat{y}}\right]$. The spin quadrupolar nematicity along off-diagonal direction are also nonzero but are much smaller.

Since our main focus is to examine the mechanism for stabilizing the BC order relevant to FeTe, below we focus on the BC/PL regime.

*DMRG calculations for the $J$-$K$ model.*— To analyze the robustness of the degeneracy in the BC and PL orders, we next turn to analyzing the quantum fluctuations in an unbiased way using the DMRG [@White1992] method with spin rotational SU(2) symmetry [@Mcchlloch2002; @gong2014square]. We study cylinder system on two different geometries–the rectangular cylinder (RC) and the $\pi/4$-rotated tilt cylinder (TC) [@DMRG]. We show the spin correlations for spin-$1$ and spin-$2$ models at the parameter $(J_1,J_2,J_3,K_1,K_2,K_3)=(1,0.8,1,-0.5,0.1,-0.5)$ in Fig. \[Fig:DMRG\_main\] (see Supplemental Material for spin-$3/2$ [@supplemental]).

For the models with different spin-$S$, we find the 