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Abstract In this study, we investigate the quantum phase transition in a Josephson junction consisting of parallel-connected quantum dots (QDs) with a magnetic flux φ B threading the central region. Employing the surrogate model approach, we discretize the superconducting (SC) electrode into three discrete energy levels, which enables direct diagonalization of the Hamiltonian to determine the system’s ground state. Our calculations demonstrate that when Coulomb interaction is present in only one QD, the system exhibits a phase transition between singlet and doublet states. The magnetic flux exerts a minor influence on the singlet state while substantially affecting the doublet state. When both QDs possess Coulomb interactions, the system undergoes two successive phase transitions as the SC phase difference increases: the ground state evolves from a doublet to a singlet and subsequently to a triplet state at ϕ = π . Enhancement of the magnetic flux suppresses the doublet and triplet phases, ultimately stabilizing the singlet state. Within this regime, increasing the Coulomb interaction strength U does not trigger a singlet-triplet transition; rather, it induces a transition between upper and lower singlet states, resulting in a pronounced peak in the critical current as U increases. Finally, we explore the scenario where the tunneling self-energy Γ between the QDs and SCs exceeds the SC pairing potential Δ. In this case, doublet states prevail, and the system manifests a phase transition exclusively between doublet and triplet states when ϕ B = 0 . Upon application of a magnetic flux, all three states converge, giving rise to a triple point in the ( φ , φ B ) parameter space.
Wang et al. (Tue,) studied this question.