Enhanced electron-phonon coupling near an electronic quantum phase transition: beyond the adiabatic approximation

We revisit the enhancement of electron-phonon coupling predicted near an electronic quantum phase transition in a two-state, one-phonon model. In an earlier analysis, one of us predicted that the diagonal Born-Oppenheimer correction (DBOC) generated a sharp barrier in the potential-energy surface, localised low-lying phonon states on either side of the transition point, and produced a pronounced hardening of the phonon modes. Here we ask whether this enhanced coupling survives in the exact coupled electron-phonon problem. We solve the full model by direct diagonalisation, obtaining the exact coupled electron-phonon excitation spectrum, and analyse its ground state using the parametric representation of the electron-nuclear wavefunction. We find that the exact solution recovers the adiabatic picture obtained by including the DBOC when the electronic levels are well separated, but that this picture is progressively smoothed and weakened as the electronic and phononic energy scales become comparable. In the near-degenerate regime, the exact solution crosses over to a diabatic description. Thus, the associated localisation and phonon-hardening signatures are not artefacts of the BO approximation, but survive in renormalised form in the exact theory when the electronic levels remain sufficiently well separated.