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Interplay of Charge and Magnetic Orders in SmNiC$_2$ Mediated by Electron-Phonon Interaction

We investigate the interplay between charge density wave (CDW) instabilities and ferromagnetism in SmNiC$_2$ using diffuse and inelastic x-ray scattering together with Raman spectroscopy. We identify a soft acoustic phonon driving the incommensurate CDW (I-CDW) and uncover a second Kohn anomaly at the wave vector of the commensurate CDW (C-CDW) stabilized in other $R$NiC$_2$ members ($R=$ rare earth). The marked softening of both phonons and their contrasting evolution with temperature reveal a competition between the two ordering tendencies. Alongside pronounced anomalies in the temperature dependence of the zone center and soft phonons, we observe the collective amplitude mode of the CDW, which collapses abruptly as ferromagnetism sets in. Surprisingly, the Kohn anomalies persist in the ferromagnetic state despite the degradation of Fermi-surface nesting conditions. Our experimental findings, supported by ab initio calculations, highlight the central role of the electron-phonon interaction in driving the CDW formation and tuning the balance between competing charge and magnetic orders.