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Shift of the critical temperature in superconductors: a self-consistent approach

Within the Ginzburg-Landau functional framework for the superconducting transition, we analyze the fluctuation-driven shift of the critical temperature. In addition to the order parameter fluctuations, we also take into account the fluctuations of the vector potential above its vacuum. We detail the approximation scheme to include the fluctuating fields contribution, based on the Hartree-Fock-Bogoliubov-Popov framework. We give explicit results for $d=2$ and $d=3$ spatial dimensions, in terms of easily accessible experimental parameters such as the Ginzburg-Levanyuk number $\text{Gi}_{(d)}$, which is related to the width of the critical region where fluctuations cannot be neglected, and the Ginzburg-Landau parameter $κ$, defined as the ratio between the magnetic penetration length and the coherence one.