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We study ground-state properties of a triangular triple quantum dot connectedto two superconducting (SC) leads. In this system orbital motion along thetriangular configuration causes various types of quantum phases, such as aKondo effect with a four-fold degenerate state and the Nagaoka ferromagneticmechanism, depending on the electron filling. The ground state also evolves asthe Cooper pairs penetrate from the SC leads. We describe the phase diagram ina wide range of the parameter space, varying the gate voltage, the couplingsbetween the dots and leads, and also the Josephson phase between the SC gaps.The results are obtained in the limit of large SC gap, carrying out exactdiagonalization of an effective Hamiltonian. We also discuss a classificationof the quantum states according to the fixed point of the Wilson numericalrenormalization group (NRG). Furthermore, we show that the Bogoliubovzero-energy excitation determines the ground state of a $\pi$ Josephsonjunction at small electron fillings.

Ground-state properties of a triangular triple quantum dot connected to superconducting leads