Ferroelectric polarization can be harnessed to record binary data in a nonvolatile manner, where the electric field switching of polarization offers a possibility for an energy-efficient data writing process. The detection of ferroelectric polarization—associated with the data readout process—is achievable in ferroelectric or multiferroic tunnel junctions. In single-phase multiferroics, it is anticipated that ferroelectric polarization may be detected by measuring anomalous Hall conductivity (AHC) or nonlinear Drude conductivity (NDC) that is intrinsically hosted by the material and is ferroelectrically switchable. Yet, selection rules for the identification of such multiferroics are lacking. Here, we use a group-theory-based approach to established a theory on ferroelectrically switchable AHC and NDC in multiferroics. We further work with polar magnetic point groups (MPGs) and provide a symmetry classification of these MPGs with respect to ferroelectrically switchable AHC or NDC. We identify various multiferroic materials (including multiferroic altermagnets) that exhibit the aforementioned intriguing AHC or NDC, and confirm by first-principles simulations that ${\mathrm{YMnO}}_3$ and ${\mathrm{LuFeO}}_3$ showcase ferroelectrically switchable NDC and AHC, respectively. Our Letter opens a door for developing multiferroic memory devices based on AHC or NDC transport phenomena.

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• Anomalous Hall effect
• Magnetism
• Transport phenomena
• Multiferroics
• First-principles calculations