Ferroelectric $p$-wave magnets

Couplings between ferroelectric and magnetic orders offer promising routes toward low-dissipation electronics. However, such couplings are notably rare, largely due to the poor compatibility between insulating band structures and ferromagnetism. Here, we study a different strategy: we identify previously overlooked time-reversal-symmetric $p$- and $f$-wave spin-polarized insulating electronic states in ferroelectrics with noncollinear magnetic sublattices. We show that combining spin and magnetic group theory enables a systematic classification of the origin of polar symmetry breaking. We distinguish crystallographic, exchange-, or spin-orbit-driven mechanisms. Furthermore, we identify more than 50 candidate materials. Using first-principles calculations, we demonstrate a pristine, time-reversal-symmetric $p$-wave spin-polarized electronic structure in the well-known multiferroic $\mathrm{GdMn_2O_5}$. We further show that its $p$-wave order can be switched electrically, opening alternative paths toward spintronic and multiferroic functionalities in this class of materials.