We propose an adiabatic approach for Hubbard models in the Fermi-liquid regime coupled to phonons. The Hubbard parameters are associated with the bandwidth $W$ via an interpolation formula between the trivial for $W$ strong- and weak-coupling limits of Hubbard models. Phonons are introduced via an adiabatic random-phase approximation scheme. We obtain simple conditions for phonon-driven instabilities in a Fermi liquid with short-ranged interactions. We report phonon-driven instabilities without nesting and describe the elimination of the Peierls instability in a nested system by Coulomb correlations. We also report the possibility for a phonon-driven phase separation (PS) instability as well as the strong enhancement of forward processes in the effective electron-phonon scattering near the phase separation instability. We show that the proximity to PS induces momentum decoupling (MD) in superconductivity which implies a tendency for decorrelation between the physics in the different regions of the Fermi surface. MD could induce anisotropic superconductivity with unconventional gap symmetry such as $d$ wave. Whether anisotropy in the high-$T_c$ oxides is driven by MD or by anisotropic scattering (for example with spin fluctuations) becomes a crucial question. We discuss some qualitative implications of MD that explain puzzling qualitative aspects of superconductivity in the oxides and could advocate that MD is at the origin of anisotropies. Such effects are the marginality of the superconducting gap symmetry for the condensation free energy and the resulting possibility of gap symmetry transitions with the doping, the temperature dependence of the shape of the anisotropy, and the behavior of the anomalous dip above the gap in the density of states. We also show that in the MD regime the orthorhombic distortion of the ${\mathrm{CuO}}_2$ planes in ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_7$ could be sufficient to explain the mixing of $s$-gap components in the dominantly $d$-wave gap. On the other hand, if spin fluctuations mediate the pairing in ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_7,$ at least 25% of the condensate must be located in the chains. Our analysis could rehabilitate phonons as potential mediators of the pairing in all “unconventional” superconductors including heavy-fermion and organic compounds.

• Received 26 June 1997

DOI:https://doi.org/10.1103/PhysRevB.57.13743