  Detwinning of magnetic (nematic) domains in Fe-based superconductors has so
far only been obtained through mechanical straining, which considerably
perturbs the ground state of these materials. The recently discovered
non-mechanical detwinning in EuFe$_{2}$As$_{2}$ by ultra-low magnetic fields
offers an entirely different, non-perturbing way to achieve the same goal.
However, this way seemed risky due to the lack of a microscopic understanding
of the magnetically driven detwinning. Specifically, the following issues
remained unexplained: (i) ultra-low value of the first detwinning field of
$\sim$0.1$\,$T, two orders of magnitude below that of BaFe$_{2}$As$_{2}$, (ii)
reversal of the preferential domain orientation at $\sim$1$\,$T, and
restoration of the low-field orientation above 10$-$15$\,$T. In this paper, we
present, using published as well as newly measured data, a full theory that
quantitatively explains all the observations. The key ingredient of this theory
is a biquadratic coupling between Fe and Eu spins, analogous to the Fe-Fe
biquadratic coupling that drives the nematic transition in this family of
materials.
