Influence of finite superconducting gaps on the proximity effect in interacting quantum dots

The tunnel coupling of superconductors to an interacting quantum dot leads to the formation of Andreev-bound states (ABS), which are the excitation energies of the proximized dot. In the limit of large superconducting gaps in the leads, Δ→∞, the hybrid system of dot and superconductors can be described exactly by an effective Hamiltonian. Since in real systems the superconducting gap is usually not a large quantity it is an interesting question whether the predictions of the Δ→∞ calculations still have significance for real systems with finite gaps. In order to investigate this question we perform two different calculations: first, a systematic 1/Δ expansion where we find that the occurring currents show only small changes but we find indications that a finite Δ renormalizes the ABSs. To quantify the last statement we perform a second calculation where we resume the coupling to the superconductors at arbitrary values for Δ neglecting vertex corrections.