Variational Treatment Beyond the Mean-Field Approximation for the Half-Filled SSH model

The Su-Schrieffer-Heeger (SSH) model is a common approach to quantifying the effect of electron-phonon coupling on many-body behavior. Developed initially for polyacetylene chains, it assumes that the leading order effect of lattice motion is the modulation of the electronic tight-binding hopping amplitudes. While the SSH model successfully predicts Peierls distortion, the lattice is often treated semiclassically, as there is no known analytical solution for quantum phonons. Motivated by the successes of the momentum average approximation applied to a single Holstein polaron, and extensions to boson-modulated hopping and multiple phonon branches, we generalize the technique to a half-filled (one carrier per unit cell) one-dimensional SSH model. We begin by studying the SSH model in the Born-Oppenheimer approximation, and find that an expansion of the hopping amplitudes to second order in the atomic positions is needed for internal consistency. Using the Peierls distortion ansatz, we find that the acoustic and optical phonon branches emerge even at this mean-field level. We then systematically expand the variational space around this improved mean-field solution to study the nature of the ground state in the non-adiabatic limit.