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Motivated to understand the phonon spectrum renormalization in the groundstate of the half-filled SSH model, we use the Born-Oppenheimer approximationtogether with the harmonic approximation to evaluate the all-to-all real-spaceionic force constants generated through the electron-phonon interaction. Usingthese force constants, we compute the renormalized phonon spectrum and studyits behaviour as a function of the Peierls distortion. For the undimerizedchain we confirm the presence of a large Kohn anomaly at $2k_F$, signalling astrong lattice instability. For the dimerized chain, we find an optical branchseparated by a gap from the acoustic one, while the Kohn anomaly manifests asphonon softening. To find the equilibrium dimerization, we minimize the groundstate energy, crucially including the contribution of the renormalized phononzero-point energy (ZPE). Our results show strong agreement with prior\textit{ab initio} studies for trans-polyacetylene and linear acetylenic carbon(carbyne), validating our method which is much simpler, and moreover can beeasily generalized to study other problems in higher dimensions.

Renormalized phonon spectrum of polyacetylene and similar materials