More recently, the iron chalcogenides have occupied the center stage, in part because they have provided a new record of the superconducting transition temperature ($T_c$) [@QYWang2012; @JJLee_Nature; @SLHe2013; @YWang2015] and a renewed hope of reaching even higher $T_c$. Because superconductivity in these materials occurs at the border of correlation-induced electronic orders [@Johnston2010; @Si2016], it is vitally important to understand the origin and nature of the ordered states.

One of the outstanding puzzles in the field arises in the structurally simplest iron chalcogenide FeTe. Compared to the iron pnictides, it has a similar Fermi surface with hole and electron pockets [@Subedi2008; @Xia2009]. Yet, instead of having a ($\pi$, $0$) collinear antiferromagnetic (AFM) order as in the latter case, FeTe has a ($\pi/2$,$\pi/2$) bicollinear AFM order (BC) illustrated in Fig. \[Fig:bicollinear\] [@Bao2009; @ShiliangLi2009; @JWen_FeTe]. This order is entirely unexpected in the weak-coupling Fermi-surface nesting picture. Thus, it is widely believed that the origin lies in the frustrated magnetism of correlation-induced local moments [@SiAbrahams08; @Ma2009; @Fang2009]. A natural starting point would be to consider bilinear exchange interactions between the local moments with nearest-neighbor ($J_1$) and further neighbor (second-neighbor $J_2$ and third-neighbor $J_3$) interactions on the square lattice. However, a classical spin model with 