Thermodynamics of superconducting transition in a metal with a purely repulsive interaction

Superconductivity may arise in a system with a purely repulsive frequency-dependent electron interaction as the self-consistency equation can be solved by a superconducting gap function that changes sign as a function of frequency. Recently, it was shown that this solution disappears as the constant repulsive part of the interaction is increased, which establishes a novel type of the superconducting quantum phase transition in the absence of disorder. In our work, we study thermodynamics of this system at small temperatures right above the critical point. We identify a logarithmically singular part of the interaction, which appears in the Cooper channel and study its impact on the specific heat of the system. The latter has a weak logarithmic singularity as a function of temperature above the quantum phase transition.