  We attain a renormalized and iterative expression of the Andreev level in a
quantum-dot Josephson junction, which is bound to have significant implications
due to several significant advantages. The renormalized form of the Andreev
level not only allows us to extend beyond the limitations of small tunnel
coupling, quantum dot energy, magnetic field, and mean-field Coulomb
interaction but also enables the capturing of subgap levels that leak out of
the superconducting gap into the continuous spectrum. Furthermore, the
iterative form of the Andreev level provides an intuitive understanding of the
spin-split and superconducting proximity effects of the superconducting leads.
We find a singlet-doublet quantum phase transition (QPT) in the ground state
due to the intricate competition between the superconducting and spin-split
proximity effects, that differs from the typical QPT arising from the
competition between the superconducting proximity effect (favoring singlet
phase) and the quantum dot Coulomb interaction (favoring doublet phase). This
QPT has a diverse phase diagram owing to the spin-split proximity effects which
favors the doublet phase akin to the quantum-dot Coulomb interaction but can be
also enhanced by the tunneling coupling like the superconducting proximity
effect. Unlike the typical QPT, where tunnel coupling prefers singlet ground
state, this novel QPT enables strong tunnel coupling to suppress the singlet
ground state via the spin-split proximity effect, allowing a
singlet-doublet-singlet transition with increasing tunnel coupling. Our
renormalized and iterative formalism of the Andreev level is crucial for the
electrostatic gate, external flux, and magnetic field modulations of the
Andreev qubits.
