Born-Oppenheimer Treatment of the Quadratic SSH Model at Half Filling

In order to tackle stronger electron-phonon coupling and address derivation inconsistencies, we extend the Su-Schrieffer-Heeger (SSH) model to second order in atomic displacements, and study it semi-analytically using the Born-Oppenheimer approximation. Including only linear terms in the SSH expansion can lead to un-physical "phase separation'' behavior at strong coupling, as we show using mean-field and DMRG approaches. Moreover, while the canonical SSH model successfully predicts Peierls distortion, the lattice is often treated semiclassically, as there is no known analytical solution for quantum phonons -- whereas to physically stabilize a dimerized chain, the phonon spectrum must be modified in response to electron influence. Using the quadratic SSH model, we find the undimerized phonon spectrum to be highly unstable due to a large Kohn anomaly at $2k_F$. This is lifted by Peierls dimerization, and a gap appears between the optical and acoustic modes. We proceed to study the multi-parameter phase space of the SSH chain, varying the coupling and phonon frequency, to analyze the impact on the phonon spectrum gap, optical phonon softening, and the zero-point energy.