| We studied the structural instabilities of one-dimensional (1D) and quasi-one-dimensional (Q1D) electron-phonon systems at low temperature through two models, SuSchrieffer-Heeger (SSH) and molecular crystal (CM) with and without spin. The phase diagrams are obtained using a Kadanoff-Wilson renormalization group approach (GR).;We have extended this study to the quarter-filled Q1D Peierls systems at finite temperature. Within the SSH model, an unconventional superconducting phase with spin singlet symmetry SS-s emerges at low temperature when the deviation to the perfect nesting of the Fermi surface is strong enough. Peierls-SS transition is characterized by the presence of a quantum critical point at low frequency and by a power law behavior of the transition temperature as a function of frequency with an exponent identical to one of 1D system. This exponent which universality has been verified contrasts with the BCS result. Coulomb interactions have been introduced through the study of the extended SSH-Hubbard model. The extension of this work to half-filled SSH and CM cases was also performed. .;Keywords: one-dimensional and quasi-one-dimensional electron-phonon systems, Kadanoff-Wilson renormalization group, Su-Schrieffer-Heeger, molecular crystal, Holstein-Hubbard, stiffness factor, Peierls stat, Luttinger liquid, unconventional superconductivity, quantum-classical crossover, Kosterlitz-Thouless transition, quantum critical point.;For the 1D half-filled system the study of the frequency dependence of the electronic gap allowed us to connect continuously the two limits, adiabatic and non-adiabatic. The Peierls and Cooper channels interference and the quantum fluctuations reduce the gap. A regime change occurs when the frequency becomes of the order of mean field gap, marking a quantum-classical crossover that is the Kosterlitz-Thouless type. At this level, the effective coupling behaves in power law function on frequency. For the case with spin, a gapped Peierls state is maintained in the non-adiabatic limit, while for the case without spin, the system transits to ungapped disordered state, namely the Luttinger liquid stat (LL). For the SSH model without spin, the GR confirms the existence of a threshold phonon coupling beyond which the gap is restored. The study of the rigidities of the two models without spin allowed us to trace the main features of the LL state predicted by the bosonization method. The study of the Holstein-Hubbard model has allowed us not only to reproduce the phase diagrams already obtained by the Monte Carlo method, but to highlight two additional phases, namely, free fermions phase and the bond charge-density-wave phase. |
