Quantum effects on the phonon excitations of the Su-Schrieffer-Heeger model

The quantum effects on the phonon excitations of the one-dimensional Su-Schrieffer-Heeger model with a dimerized lattice are studied in a similar way by treating the σ and π modes in the Hubbard model. The phonon energies are obtained for all values of the ionic mass ranging from the M = 0 (Gross-Neveu model) to the M = ∞ (adiabatic case) limit. In the adiabatic (large M) limit, the phonon spectrum is the same as that of Nakahara and Maki. However, at the Gross-Neveu limit, the frequency of the optical phonon at zero momentum approaches zero: for weak electron-phonon couplings. This result is consistent with the fact that the quantum reduction of the dimerization magnitude is almost 100% at Gross-Neveu limit.