Competition between Kondo and Josephson effects in a triangular triple quantum dot connected to normal and superconducting leads

We study low-energy properties of a triangle triple quantum dot (TTQD) connected to one normal and two superconducting (SC) leads, using the Wilson numerical renormalization group approach. This system has various types of quantum phases in the normal state such as the SU(4) and the $S=1$ Nagaoka-high-spin Kondo effects, depending on the electron filling [1]. The ground state evolves as additional SC leads are connected, and interesting competition between the Kondo and Cooper-pairing singlets occurs [2]. It also causes the Andreev scattering which takes place at the interface between the TTQD and normal lead. We examine how the Andreev scattering affects the quantum phase transition, in a wide range of the gate voltage $\epsilon_d$. Near half-filling, the Josephson phase $\phi$ between the two SC leads lifts an orbital degeneracy of the TTQD in a different way at $\phi \simeq 0$ and $\pi$. We also discuss the conduction-electron under-screening of the local Nagaoka high-spin state. \noindent [1] T.\ Numata, Y.\ Nisikawa, A.\ Oguri, and A.\ C.\ Hewson, PRB {\bf 80}, 155330 (2009). [2] A.\ Oguri, I. Sato, M. Shimamoto, and Yoichi Tanaka, J.\ Phys.: Conf.\ Ser.\ {\bf 592}, 012143 (2015).