Spontaneous cycloidal order mediating a spin-reorientation transition in a polar metal

We show how, in a polar crystal with uniform Dzyaloshinskii-Moriya interaction, modulated order can spontaneously emerge when magnetic anisotropies compete at a spin-reorientation transition. Combining neutron and resonant x-ray scattering, we reveal that the spin-reorientation in Ca$_3$Ru$_2$O$_7$ is mediated by an incommensurate cycloid whose eccentricity evolves smoothly with temperature. Our results suggest that the application of a magnetic field continuously generates higher harmonic modulations, establishing a rich phase diagram and providing a new, unified interpretation of incommensurate structures previously seen under doping and magnetic field.