Two-cutoff renormalization and quantum versus classical aspects for the one-dimensional electron-phonon system

We study the $T=0\mathrm{}$ K properties of the electron-phonon system with Coulomb interaction using a two-cutoff renormalization procedure. We find that there exist two regimes. The "classical" regime is characterized by a classical amplitude order parameter and exists when the mean-field gap (pseudogap) $\Delta$ is larger than the phonon frequencies ${\omega }_D$. The order parameter for spinless fermions is increased in amplitude by a repulsive nearest-neighbor electron interaction. A repulsive Hubbard interaction in an incommensurate spin-$\frac{1}{2}$ fermion band will increase it at small ${\omega }_D$ and decrease it at finite $\frac{{\omega }_D}{E_F}$. For half-filled spin-$\frac{1}{2}$ fermion band, however, the molecular crystal (MC) and Su-Schrieffer-Heeger (SSH) models behave in opposite ways, the former having its order parameter decrease with increasing Coulomb interaction. We predict a maximum in the SSH amplitude order parameter as a function of the Hubbard interaction when $\Delta$ is equal to the Hubbard gap. This agrees quite well with the result of the simulation of Hirsch. For $\Delta <{\omega }_D$, we predict a change to a quantum behavior. In this regime long-range molecular order can only exist for a half-filled spin-$\frac{1}{2}$ fermion band whenever the effective electronic interaction is attractive and the umklapp processes are relevant. This quantum order is weakened and can be destroyed by a repulsive Coulomb interaction or a negative forward scattering. We predict the quantum to classical-amplitude crossover to occur when $\Delta \approx {\omega }_D$. In the case of the MC model with spinless noninteracting fermions, it corresponds to the disappearance of long-range order at values in agreement with the calculation of Hirsch and Fradkin. We analyze the implications of these two regimes on the properties of quasione-dimensional solids, more specifically on the effect of interchain potential or hopping ($t_{\perp }$) couplings. The existence of classical or quantum gaps favors interchain particle-pair tunneling whereas the single-particle interchain hopping is quite pertinent whenever $t_{\perp }$ is larger than these gaps.