Ab initio density-functional calculations show that orthorhombic $Pca{2}_1$ hafnia ${\mathrm{HfO}}_2$ mixed with vanadium at low concentration is a ferroelectric and ferromagnetic insulator. The multiorbital degeneracy of singly-occupied V states in the nominally $4+$ ionic state is broken by magnetism, reduced symmetry, and local distortion, causing a single one-electron majority state per V atom to be occupied. A gap of order 1 eV thus survives at all V concentrations, and intrinsic polarization is preserved at the level of 60–70% of the undoped-${\mathrm{HfO}}_2$ value. Magnetization is found to increase linearly with V content, with values of 30–40 $\mathrm{emu}/{\mathrm{cm}}^3$ at concentrations near the end of the stability range ($\sim 16\%$ V).

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• Electric polarization
• Order parameters
• Ferroelectrics
• Ferromagnets
• Multiferroics
• Density functional theory