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Detwinning of magnetic (nematic) domains in Fe-based superconductors has sofar only been obtained through mechanical straining, which considerablyperturbs the ground state of these materials. The recently discoverednon-mechanical detwinning in EuFe$_{2}$As$_{2}$ by ultra-low magnetic fieldsoffers an entirely different, non-perturbing way to achieve the same goal.However, this way seemed risky due to the lack of a microscopic understandingof the magnetically driven detwinning. Specifically, the following issuesremained 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$1$\,$T, andrestoration of the low-field orientation above 10$-$15$\,$T. In this paper, wepresent, using published as well as newly measured data, a full theory thatquantitatively explains all the observations. The key ingredient of this theoryis a biquadratic coupling between Fe and Eu spins, analogous to the Fe-Febiquadratic coupling that drives the nematic transition in this family ofmaterials.

Microscopic theory of magnetic detwinning in iron-based superconductors with large-spin rare earths