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

Strongly Correlated Electrons;Superconductivity