We discuss the stability of ferromagnetic long-range order in three-dimensional classical $XY$ ferromagnets upon substitution of a small subset of equally oriented bonds by impurity bonds, on which the ferromagnetic exchange $J_{\perp }>0$ is replaced by a strong antiferromagnetic coupling $J_{\mathrm{imp}}<0$. In the impurity-free limit, the effective low-energy Hamiltonian is that of spin waves. In the presence of a single, sufficiently strongly frustrating impurity bond, the ground state is twofold degenerate, corresponding to either clockwise or counterclockwise canting of the spins in the vicinity of the impurity bond. For a small but finite concentration of impurity bonds, the effective low-energy Hamiltonian is that of Ising variables encoding the sense of rotation of the local canting around the impurities. Those degrees of freedom interact pairwise through a dipolar interaction mediated by spin waves. A spatially random distribution of impurities leads to a ferromagnetic Ising ground state, which indicates the instability of the $XY$ ferromagnet towards a spiral state, with wave vector and transition temperature both proportional to the concentration of impurity bonds. This mechanism of spiral order by disorder is relevant for magnetic materials such as ${\mathrm{YBaCuFeO}}_5$, for which our theory predicts a ratio between the spiral ordering temperature and the modulus of the spiral wave vector close to the measured ones.

• Received 18 December 2019
• Accepted 4 February 2020

DOI:https://doi.org/10.1103/PhysRevResearch.2.013273

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Published by the American Physical Society