A Kadanoff-Wilson renormalization group analysis of half-filled one-dimensional quantum electron-phonon models

We study the zero temperature phase diagrams of the half-filled one-dimensional Su- Schrieffer-Heeger (SSH) and molecular crystal (CM) models using the Kadanoff-Wilson renormalization group approach. At the one-loop level, the full frequency dependence of the phonon induced electron-electron coupling constants is taken into account in the vertex corrections and the quantum interference between the Cooper and Peierls diffusion channels. This enters as a key ingredient for the description of the quantum to classical transition for the Peierls instability. Our results confirm that finite phonon frequency introduces quantum fluctuations that depress the Peierls gap $\Delta$ compared to the classical - mean field - limit $\Delta_0$. It is found that in the spinless fermion case, the Peierls gap vanishes at the threshold $\omega_D\sim \pi \Delta_0$, whereas for fermions with spins, the gap remains in the quantum spin- charge separated regime. We extend our study to the XY spin-Peierls chain and confirm the DMRG result about the existence of a power law relation between the critical spin-phonon coupling $\alpha_c$ and frequency at the quantum-classical boundary, namely $\alpha_c\sim \omega_D^{0.7}$.