Frustrated magnetism and bicollinear antiferromagnetic order in FeTe

Iron chalcogenides display a rich variety of electronic orders in their phase diagram. A particularly enigmatic case is FeTe, a metal which possesses co-existing hole and electron Fermi surfaces as in the iron pnictides but has a distinct ($\pi$/2,$\pi$/2) bicollinear antiferromagnetic order in the Fe square lattice. While local-moment physics has been recognized as essential for the electronic order in FeTe, it has been a long-standing challenge to understand how the bicollinear antiferromagnetic ground state emerges in a proper quantum spin model. We demonstrate here that a bilinear-biquadratic spin-$1$ model on a square lattice with nonzero ring-exchange interactions stabilizes the bicollinear antiferromagnetic order over an extended parameter space in its phase diagram. Our results show that frustrated magnetism in the quantum spin model provides a unified description of the electronic orders in the iron chalcogenides and iron pnictides.