Three-body Fermi-liquid corrections for Andreev transport through quantum dots

We study crossed Andreev reflection occurring in quantum dots connected to one superconducting lead and two normal leads at low temperatures $T$. Specifically, we derive an exact formula for the conductance up to order $T^2$ in the large superconducting gap limit, which is expressed in terms of the transmission probabilities of Cooper pairs and interacting Bogoliubov quasiparticles. Our formulation is based on the latest version of Fermi-liquid theory for the Anderson impurity model, which has clarified the quasiparticle energy shifts of order ${\omega }^2$ and $T^2$ – that is, corrections of the same order as those arising from the finite lifetime of quasiparticles – can be exactly taken into account through three-body correlations of impurity electrons. We also demonstrate how the three-body contributions evolve and affect the Cooper-pair tunneling as the Andreev level moves away from the Fermi level, using the numerical renormalization group approach. The results demonstrate that the Cooper-pair contribution to the $T^2$ term of the nonlocal conductance becomes comparable to the Bogoliubov-quasiparticle contribution in the parameter region where superconducting proximity effects dominate over the Kondo effect.