Magnetism in parent iron chalcogenides: Quantum fluctuations select plaquette order

We analyze magnetic order in Fe chalcogenide Fe _1+yTe, the parent compound of the high-temperature superconductor Fe _1+yTe _1-xSe _x. Experiments show that magnetic order in this material contains components with momentum Q ₁=(π/2,π/2) and Q ₂=(π/2,-π/2) in the Fe only Brillouin zone. The actual spin order depends on the interplay between these two components. Previous works assumed that the ordered state has a single Q (either Q ₁ or Q ₂). In such a state, spins form double stripes along one of the diagonals breaking the rotational C ₄ symmetry. We show that quantum fluctuations actually select another order-a double Q plaquette state with equal weight of Q ₁ and Q ₂ components, which preserves C ₄ symmetry. We argue that the order in Fe _1+yTe is determined by the competition between quantum fluctuations and magnetoelastic coupling.