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A nonlocal Landau-type free energy functional of the spin density isdeveloped to model the large variety of magnetic states which occur in themagnetic field-temperature phase diagram of magnetoelectric CuFeO$_2$.Competition among long-range quadratic exchange, biquadratic anti-symmetricexchange, and trigonal anisotropy terms, consistent with the high-temperaturerhombohedral R$\bar{3}$m crystal symmetry, are shown to all play importantroles in stabilizing the unusual combination of commensurate and incommensuratespin structures in this highly frustrated triangular antiferromagnet. It isargued that strong magnetoelastic coupling is largely responsible for thenonlocal nature of the free energy. A key feature of the analysis is that anelectric polarization is induced by a canting of the non-collinearincommensurate spin structure. Application of the model to ordered spin statesin the triangular antiferromagnets MnBr$_2$ and NaFeO$_2$ is also discussed.

Nonlocal Landau theory of the magnetic phase diagram of highly frustrated magnetoelectric CuFeO$_2$