Magnetic flux controlled current phase relationship in double Quantum Dot Josephson junction

In this work, we study a Josephson junction with parallel-connected quantum dots (QDs) threaded by a magnetic flux in the central region. We discretize the superconducting (SC) electrode into three discrete energy levels and modify the tunneling coefficients to construct a finite-dimensional surrogate Hamiltonian. By directly diagonalizing this Hamiltonian, we compute the physical quantities of the system. Additionally, we employ a low-energy effective model to gain deeper physical insight. Our findings reveal that when only one QD exhibits Coulomb interaction, the system undergoes a phase transition between singlet and doublet states. The magnetic flux has a minor influence on the singlet state but significantly affects the doublet state. When both QDs have interactions, the system undergoes two phase transitions as the SC phase difference increases: the ground state evolves from a doublet to a singlet and finally into a triplet state at $\phi = \pi$. Increasing the magnetic flux suppresses the doublet and triplet phases, eventually stabilizing the singlet state. In this regime, enhancing the interaction strength does not induce a singlet-doublet transition but instead drives a transition between upper and lower singlet states, leading to a critical current peak as $U$ increases. Finally, we examine the case where the tunneling coefficient $\Gamma$ exceeds the SC pairing potential $\Delta$. Here, doublet states dominate, and the system only exhibits a phase transition between doublet and triplet states when $\phi_B = 0$. In the presence of a magnetic flux, the three states converge, resulting in a triple point in the ($\phi$, $\phi_B$) parameter space.