Quantum phase transition in a clean superconductor with repulsive dynamical interaction

We consider a model of electrons at zero temperature, with a repulsive interaction which is a function of the energy transfer. Such an interaction can arise from the combination of electron–electron repulsion at high energies and the weaker electron–phonon attraction at low energies. As shown in previous works, superconductivity can develop despite the overall repulsion due to the energy dependence of the interaction, but the gap Δ(ω) must change sign at some (imaginary) frequency ω₀ to counteract the repulsion. However, when the constant repulsive part of the interaction is increased, a quantum phase transition towards the normal state occurs. We show that, as the phase transition is approached, Δ and ω₀ must vanish in a correlated way such that 1/∣log[Δ(0)]∣~ω02. We discuss the behavior of phase fluctuations near this transition and show that the correlation between Δ(0) and ω₀ locks the phase stiffness to a non-zero value.