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 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 diagonal Born-Oppenheimer correction 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 Born-Oppenheimer approximation, but survive in renormalised form in the exact theory when the electronic levels remain sufficiently well separated.