(for a point deep inside this regime, the energy splitting is $\leq 0.1\%$; near the boundary, the energy splitting is $\sim 1\%$). Because this common regime, where both orders are stable, covers a large parameter space, we need to explore additional inputs for another mechanism that breaks the degeneracy and stabilizes the BC. This leads us to discuss the effect of ring exchanges in the next subsection.

We close this subsection with a remark on the limit of vanishing $K,~K_2$ (and in the absence of the ring exchange interactions); further details are given in the Supplemental Material [@supplemental]. For $(J_1, J_2, J_3) = (1, 0.8, 1)$, which we have so far focused on, Fig. \[Fig:PD\_noring\] shows that the BC/PL phases are unstable at $K, K_2 \rightarrow 0$. In this limit, the flavor-wave theory suggests that neither of the BC/PL phases can be stable except for infinitesimally small ratios of $J_1/J_3$ and $J_1/J_2$ in the $J_1$-$J_2$-$J_3$-only model. Consider, for example, $(J_2, J_3) =( 0.8,1.0)$, $K,K_2 = 0$: we find that the BC and PL can only be stable for a very small range of $J_1 \leq 0.03$. (Within this regime, the energies between the two orders are degenerate, with their difference 