  The discovery of superconductivity in LaFeAsO introduced the ferropnictides
as a major new class of superconducting compounds with critical temperatures
second only to cuprates. The presence of magnetic iron makes ferropnictides
radically different from cuprates. Antiferromagnetism of the parent compounds
strongly suggests that superconductivity and magnetism are closely related.
However, the character of magnetic interactions and spin fluctuations in
ferropnictides, in spite of vigorous efforts, has until now resisted
understanding within any conventional model of magnetism. Here we show that the
most puzzling features can be naturally reconciled within a rather simple
effective spin model with biquadratic interactions, which is consistent with
electronic structure calculations. By going beyond the Heisenberg model, this
description explains numerous experimentally observed properties, including the
peculiarities of the spin wave spectrum, thin domain walls, crossover from
first to second order phase transition under doping in some compounds, and
offers new insight in the occurrence of the nematic phase above the
antiferromagnetic phase transition.
