On the equivalence of the bond and optical SSH models in the adiabatic limit

The interactions between electrons and phonons play an important role in conventional superconductivity. In general, lattice vibrations can modulate the electron potential energy, as in the Holstein model, or the electron kinetic energy ,as in the Su-Schrieffer-Heeger (SSH) model. SSH-like models have received significant attention recently as they produce qualitatively different physics in comparison to Holstein-like models. Many QMC treatments of the SSH model focus on the "bond" SSH model, which places the phonons on the atomic bonds, for various technical reasons. This is in contrast to the "optical" SSH model, where the phonons live on the individual atomic sites. Here we study the equivalence of these models in one- and two-dimensions in the adiabatic limit using semi-classical Monte Carlo. Specifically, we will present results for a systematic comparison of the models as a function of e-ph coupling, density, and dimensionality.