Perfectly harmonic spin cycloid and multi-$Q$ textures in the Weyl semimetal GdAlSi

A fundamental question concerns how topological electronic states are influenced by many-body correlations, and magnetic Weyl semimetals represent an important material platform to address this problem. However, the magnetic structures realized in these materials are limited, and in particular, no clear example of an undistorted helimagnetic state has been definitively identified. Here, we report clear evidence of a harmonic helimagnetic cycloid with an incommensurate magnetic propagation vector in the Weyl semimetal GdAlSi via resonant elastic X-ray scattering, including rigorous polarization analysis. This cycloidal structure is consistent with the Dzyaloshinskii-Moriya (DM) interaction prescribed by the polar crystal structure of GdAlSi. Upon applying a magnetic field, the cycloid undergoes a transition to a novel multi-$Q$ state. This field-induced, noncoplanar texture is consistent with our numerical spin model, which incorporates the DM interaction and, crucially, anisotropic exchange. The perfectly harmonic Weyl helimagnet GdAlSi serves as a prototypical platform to study electronic correlation effects in periodically modulated Weyl semimetals.