Renormalization effects in interacting quantum dots coupled to superconducting leads

We study subgap transport through an interacting quantum dot tunnel coupled to one normal and two superconducting leads. To check the reliability of an approximation of an infinitely large gap $\Delta$ in the superconducting leads and weak tunnel coupling to the normal lead, we perform a $1/\Delta$ expansion, and we analyze next-to-leading-order corrections in the tunnel coupling to the normal lead. Furthermore, we propose a resummation approach to calculate the Andreev bound states for finite $\Delta$. The results are substantially more accurate than those obtained by mean-field treatments and favorably compare with numerical exact results.

Figure 1

Setup: A quantum dot is tunnel coupled to one normal and two superconducting leads.

Figure 2

Example diagrams contributing to the rate $W_{\uparrow -}^{\uparrow +}$.

Figure 3

Density plot of the Josephson current in the limit of $\Delta \to \infty$ as a function of the level position $\epsilon$ and the chemical potential of the normal lead ${\mu }_N$. The other parameters are ${\Gamma }_S=0.2U$, ${\Gamma }_N=0.001U$, $\Phi =\pi /2$, and $k_{B}T=0.01U$.

Figure 4

Density plot of the Andreev current in the limit of $\Delta \to \infty$ as a function of the level position $\epsilon$ and the chemical potential of the normal lead ${\mu }_N$. The other parameters are ${\Gamma }_S=0.2U$, ${\Gamma }_N=0.001U$, $\Phi =\pi /2$, and $k_{B}T=0.01U$.

Figure 5

Plot of (a) the Josephson current and (b) the Andreev current as a function the chemical potential of the normal lead ${\mu }_N$. The solid (blue) curve shows the $\Delta \to \infty$ limit; the dashed (red) curve includes the first $1/\Delta$ correction. The other parameters are $\epsilon =-0.4U$, ${\Gamma }_S=0.2U$, ${\Gamma }_N=0.001U$, $\Delta =5U$, $\Phi =\pi /2$, and $k_{B}T=0.01U$.

Figure 6

Plot of the $U=0$ Andreev bound states for a tunnel-coupling strength to the superconducting lead of ${\Gamma }_S=0.25\Delta$ as a function of the level position $\epsilon$. All energies are normalized to $\Delta$. The solid (blue) line shows the exact result, the dashed (black) line the bound states computed by means of the $\Delta \to \infty$ expression, which in the noninteracting case is simply $\pm {\epsilon }_A$.

Figure 7

Examples of diagrams that contribute to the self-energy.

Figure 8

Plot of the Andreev bound states from the resummation approach, NRG (Ref. 70), and the Hartree-Fock approximation as a function of the level position $\epsilon$ for (a) $U=\Delta$ and ${\Gamma }_S=0.5\Delta$, (b) $U=\Delta$ and ${\Gamma }_S=0.2\Delta$, and (c) $U=4\Delta$ and ${\Gamma }_S=0.2\Delta$.

Figure 9

Plot of (a) the position of the 0-$\pi$ transition of the Josephson current and (b) the extreme values of the Andreev current. The chosen parameters are $k_{B}T/U=0.05$, ${\Gamma }_N/U=0.05$, ${\Gamma }_S/U=0.2$, and $\Phi =\pi /2$. In (a) the $\pi$ junctions are indicated by the symbol $\pi$. The thin black lines in both plots show the position of the ABSs and $CB$ marks the Coulomb-blockaded area between the two inner ABSs.

Figure 10

Plot of the average dot charge. The chosen parameters are $k_{B}T/U=0.05$, ${\Gamma }_N/U=0.05$, ${\Gamma }_S/U=0.2$, and $\Phi =\pi /2$. The thin black lines show the position of the ABSs and $CB$ marks the Coulomb-blockaded area between the two inner ABSs.