the energies of BC/PL and AFM$^*_{a/b}$ by different amounts and make those of BC/PL lower. By contrast, the $4$-site ring exchange, $\mathfrak{R}_4$, *does* lift the degeneracy between BC/PL (as well as that between AFM$^*_{a/b}$).

To illustrate such effects of the ring exchanges, we take relatively small values $\mathfrak{R}_3 = \mathfrak{R}_4 = 0.1 \ll J_n, | K_n |$. The phase diagram of the $J$-$K$-$\mathfrak{R}$ model for this case is shown in Fig. \[Fig:PD\_wring\]. The red line is the phase boundary with $K - 2K_2 = -0.4$. We note that, in Fig. \[Fig:PD\_wring\], the BC order will remain stable over an extended regime in the presence of quantum fluctuations, based on what we have established in Fig. \[Fig:PD\_noring\]. Moreover, the energy splitting between BC and PL due to the presence of ring exchanges can already be observed at semi-classical level and we expect the splitting will be further enhanced if additional quantum fluctuation effects are incorporated by, *e.g.*, the DMRG method.

![(color online) The site-factorized wavefunction phase diagram with small ring exchange terms. Here we fix $(J_1, J_2, J_3, \mathfrak{R}_3,\mathfrak{R}_4) = (1, 0.8,1, 0.1, 0.1)$ and vary $K_1 = K_3=-K$ and $K_2$. The $4$-site ring exchange interaction is responsible for lifting the 