bilinear $J_1-J_2-J_3$ interactions yields incommensurate spiral magnetic order, and the ordering wavevector can reach $(\pi/2, \pi/2)$ only for an infinitesimal $J_1$ [@Moreo1990; @Ferrer1993]. The multi-band nature of these systems makes it natural to consider the role of the biquadratic interactions, which have recently been discussed [@YuSi_AFQ; @FaWang2015; @Glasbrenner2015; @Wangzhentao2016; @Lai2016; @Wenjun_nematic2016; @Ergueta2016] as providing a route towards understanding the intriguing phenomenologies of the iron chalcogenides. Indeed, it is also known in classical spin models that the presence of the biquadratic $K$ interactions can make the ordering wavevector to be $(\pi/2,\pi/2)$ [@Hu2012]. The problem is that the BC state is degenerate with the plaquette AFM order (PL) shown in Fig. \[Fig:plaquette\]. How the BC order emerges as the true ground state in quantum spin models remains a long-standing puzzle.

In this *Letter*, we propose that ring exchange ($\mathfrak{R}$) interactions provide a robust mechanism to stabilize the BC order. Strongly correlated bad metals such as FeTe possess significant charge fluctuations [@Si2016]. We therefore expect that the ring exchange interactions involving cyclic permuting spin degrees of freedoms on more-than-two lattice sites can be significant, in addition to the biquadratic terms that only capture charge fluctuations between two lattice sites.

More specifically, we study 