include an orbital order [@Turner2009; @Singh2009] or spin-lattice interactions [@Fang2009; @EDagotto_FeTe].

*Conclusion.*— We have studied the quantum bilinear-biquadratic model in the presence of ring exchange interactions, and identified two mechanisms that stabilize the $(\pi/2,\pi/2)$ bicollinear antiferromagnetic order experimentally observed in FeTe. In the absence of ring exchanges, we demonstrate a relatively narrow parameter range where quantum fluctuations select the bicollinear order, by destabilizing the classically-degenerate $(\pi/2,\pi/2)$ plaquette antiferomagnetic order. We also identify a larger parameter range where the bicollinear and plquettes orders are degenerate and both are stable. In this regime, the presence of a $4$-site ring exchange interaction breaks the degeneracy and selects the bicollinear order. Because the second mechanism operates over a considerably more extended parameter regime, it represents a more robust mechanism to understand the magnetic and nematic orders of FeTe. Our work unifies the electronic order of FeTe with those of the other iron chalcogenides and iron pnictides within a single Hamiltonian, and highlights the importance of spin frustration to the magnetism and superconductivity of the iron-based systems.

*Acknowledgement.*— The work was supported in part by the NSF Grant No. DMR-1611392 and the Robert A. Welch Foundation Grant No. C-1411 (W.-J.H., H.-H.L. and Q.S.), the NSF 