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The magnetic and nematic properties of the iron chalcogenides have recentlybeen the subject of intense interest. Motivated by the proposedantiferroquadrupolar and Ising-nematic orders for the bulk FeSe, we study thephase diagram of an $S=1$ generalized bilinear-biquadratic model withmulti-neighbor interactions. We find a large parameter regime for a ($\pi$,0)antiferroquadrupolar phase, showing how quantum fluctuations stabilize it bylifting an infinite degeneracy of certain semiclassical states. Evidence forthis C$_4$-symmetry-breaking quadrupolar phase is also provided by an unbiaseddensity matrix renormalization group analysis. We discuss the implications ofour results for FeSe and related iron-based superconductors.

Antiferroquadrupolar order and rotational symmetry breaking in a generalized bilinear-biquadratic model on a square lattice