Renormalized phonon spectrum in the Su–Schrieffer–Heeger model

Motivated to understand phonon spectrum renormalization in the ground state of the half-filled Su–Schrieffer–Heeger model, we use the Born–Oppenheimer approximation together with the harmonic approximation to evaluate semi-analytically the all-to-all real-space ionic force constants generated through both linear and quadratic electron-phonon coupling. We then compute the renormalized phonon spectrum and the corresponding lattice zero-point energy (ZPE) as a function of the lattice dimerization. Crucially, the latter is included in the system’s total energy, and thus has a direct effect on the equilibrium dimerization. We find that inclusion of a small quadratic coupling leads to very significant changes in the predicted equilibrium dimerization, calling into question the use of the linear approximation for this model. We also argue that inclusion of the ZPE is key for systems with comparable lattice and electronic energies, and/or for finite size chains. Our method can be straightforwardly generalized to study similar problems in higher dimensions.