Superconductivity in a Fermi liquid from repulsive interactions: The role of electron–phonon interaction

• The sign reversal of pair interaction in momentum space is proved.
• It is also shown that electron-phonon interaction in fact leads to the pairing-break effect.
• Transition temperature into superconductivity depends on competition between electron-phonon and Coulomb interactions.
• Calculated exponent α of the isotope effect shows the possibility equal to, greater or less than 0.5, and even negative.

Based on our previously proven theorem that the interaction between a pair of quasiparticles in the normal Fermi liquid has an opposite sign to the interaction between particles, we consider pair correlation between a pair of quasiparticles when the interaction between particles is repulsive. For the convenience of statements, we have presented in this article once again the proof of the theorem in terms of an exact equation for the thermodynamic potential due to interaction between particles and based on the Green’s function method. Further, we have derived the Landau expansion of the thermodynamic potentials in terms of the variation of the quasiparticle distribution function. We have also derived the expansion of the thermodynamic potential in terms of the variation of an exact single particle (not quasiparticles), these derivations lead to the relationship between the interaction function for two quasiparticles and the interaction energy between two particles as shown.According to the proven theorem the interaction between a pair of quasiparticles is attractive in this case, the pairing – Cooper’s pairing between a pair of quasiparticles is possible. We solve the Bethe–Salpeter type equation for paring of two quasiparticles when both interactions – the Coulomb repulsive and electron–phonon interaction are present. We show that the electron–phonon interaction, in fact, leads to the pair breaking effect, in contrast to the common belief that electron–phonon interaction is the main mechanism for Cooper’s pair formation. We have calculated the transition temperature and the isotope effect on the transition temperature in terms of the Bethe–Salpeter equation.