Non-perturbative microscopic theory of superconducting fluctuations near a quantum critical point

We consider an inhomogeneous anisotropic gap superconductor in the vicinity of the quantum critical point, where the transition temperature is suppressed to zero by disorder. Starting with the BCS Hamiltonian, we derive the Ginzburg-Landau action for the superconducting order parameter. It is shown that the critical theory corresponds to the marginal case in two dimensions and is formally equivalent to the theory of an antiferromagnetic quantum critical point, which is a quantum critical theory with the dynamic critical exponent, z=2. This allows us to use a parquet method to calculate the non-perturbative effect of quantum superconducting fluctuations on thermodynamic properties. We derive a general expression for the fluctuation magnetic susceptibility, which exhibits a crossover from the logarithmic dependence, $\chi ~ ln(dn)$, valid beyond the Ginzburg region to $\chi ~ ln^{1/5}(dn)$ valid in the immediate vicinity of the transition (where $dn$ is the deviation from the critical disorder concentration). We suggest that the obtained non-perturbative results describe the low-temperature critical behavior of a variety of diverse superconducting systems, which include overdoped high-temperature cuprates, disordered p-wave superconductors, and conventional superconducting films with magnetic impurities.