General ab initio framework for electronic-order-induced lattice-dynamics symmetry breaking

Conventional \textit{ab initio} approaches are unable to describe phonon time-reversal symmetry ($\mathcal{T}$) breaking. Here, we develop an \textit{ab initio} framework, grounded in molecular Berry curvature (MBC) theory, that captures electronic-order-driven symmetry breaking in lattice dynamics. Using Co$_3$Sn$_2$S$_2$ as a model system, our \textit{ab initio} framework yields phonon spectra that break both $\mathcal{T}$ and mirror symmetries, quantitatively reproduce the observed phonon splittings observed in experiments, and reveal distinct microscopic origins for the $E_g$ and $E_u$ modes: $E_g$ splitting is governed by MBC and is accurately captured by our algorithm, whereas $E_u$ splitting is enhanced by the Fano resonance and matches the experimental data once the Fano-factor correction is included. Leveraging this algorithm, we predict several candidate materials with nonzero electronic-order-driven symmetry breaking in lattice dynamics, establishing a first-principles route to understand electron-phonon coupling, phonon magnetism, and related Hall-type lattice responses.