In the following, we will drop the irrelevant constant terms.

To illustrate the energetic degeneracy of the BC and PL orders, we first set ring exchanges $\mathfrak{R}_3, \mathfrak{R}_4=0$ and without loss of generality we study the $J$-$K$ model with $(J_1,J_2, J_3) = (1, 0.8, 1)$. We vary $K_1 = K_3\equiv -K <0$ and $K_2 \in [-0.5,0.5]$ on an $L\times L$ square lattice with $L$ up to $8$ to obtain the site-factorized wavefunction phase diagram, Fig. \[Fig:PD\_noring\]. Besides the intriguing doubly degenerate BC/PL orders, we find another doubly degenerate orders dubbed $(\pi/2, \pi/2)$ AFM$^*_{a/b}$ and abbreviated as AFM$^*_{a/b}$ below, whose spin patterns are illustrated in Figs. \[Fig:pi/2pi/2\_phase\](c)-(d). [@AFMs_note] The semi-classical energies for these two orders in the absence of the ring exchange interactions are $ \mathcal{E}_{BC/PL}= K_1  + K_2 + 2(K_3 - J_3)$ and $ \mathcal{E}_{AFM_{a/b}^*} = \frac{3}{4}K_1 + \frac{1}{2} K_2 + 2(K_3 - J_3)$. This gives the phase boundary, $K_1 + 2K_2 =0$, consistent with the numerical result in Fig. \[Fig:PD\_noring\].

The BC/PL orders have different “spin dipolar nematicity" along the off-diagonal directions of the square lattice, defined as $\sigma^s_{2} \equiv (1/N_s) \sum_j {\bf S}_j \cdot \left[{\bf S}_{j + \hat{x} + \hat{y}} -  {\bf S}_{j - \hat{x} + \hat{y}}\right]$ (where 