Magnetoelectric responses are related, in general, to magnetic multipoles and, in particular, the off-diagonal linear response is proportional to the toroidization order parameter. In multiferroics with three order parameters (polarization, magnetization, and toroidization), such response turns out to also be proportional to the trilinear coupling between the order parameters. Here we explore this scenario, discussing the effects of such trilinear coupling within a Landau theory framework, and using ab initio calculations to discuss a specific model system for this effect, namely, the recently predicted three-order-parameter multiferroic metal ${\mathrm{Bi}}_5{\mathrm{Mn}}_5{\mathrm{O}}_{17}$.

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• Ferroelectricity
• Landau theory
• Magnetoelectric effect
• Order parameters
• Ferromagnets
• Half-metals
• Multiferroics
• Density functional theory
• First-principles calculations