Quantum phonons and the Peierls transition temperature

The effect of quantum phonons on the Peierls transition temperature in a one-dimensional commensurate system is studied. The nonadiabatic effect due to finite phonon frequency ωₚ>0 are treated through an energy-dependent electron-phonon scattering function δ(k^',k). By using the Green's-function perturbation theory we have shown that our theory gives a good description of the effect of quantum phonons: (i) There exists a critical electron-phonon coupling gc². The Peierls transition temperature Tₚ=0 when the coupling g²<gc² and Tₚ increases with increasing g²>gc². (ii) The ratio 2Δ/Tₚ could be much larger than the BCS value 3.527.