Combining ferroelectricity with other properties such as visible light absorption or long-range magnetic order requires the discovery of new families of ferroelectric materials. Here, through the analysis of a high-throughput database of phonon band structures, we identify a new structural family of anti-Ruddlesden-Popper phases A$_4$X$_2$O (A=Ca, Sr, Ba, Eu, X=Sb, P, As, Bi) showing ferroelectric and anti-ferroelectric behaviors. The discovered ferroelectrics belong to the new class of hyperferroelectrics which polarize even under open-circuit boundary conditions. The polar distortion involves the movement of O anions against apical A cations and is driven by geometric effects resulting from internal chemical strains. Within this new structural family, we show that Eu$_4$Sb$_2$O combines coupled ferromagnetic and ferroelectric order at the same atomic site, a very rare occurrence in materials physics.
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