Nonlocal Andreev transport through a quantum dot in a magnetic field: Interplay between Kondo, Zeeman, and Cooper-pair correlations
We study the nonlocal magnetotransport through a strongly correlated quantum dot, connected to multiple terminals consisting of two normal and one superconducting (SC) leads. Specifically, we present a comprehensive view on the interplay between the crossed Andreev reflection (CAR), the Kondo effect, and the Zeeman splitting at zero temperature in the large SC gap limit. The ground state of this network shows an interesting variety, which varies continuously with the system parameters, such as the coupling strength $\Gamma_S^{}$ between the SC lead and the quantum dot, the Coulomb repulsion $U$, the impurity level $\varepsilon_d^{}$, and the magnetic field $b$. We show, using the many-body optical theorem which is derived from the Fermi-liquid theory, that the nonlocal conductance is determined by the transmission rate of the Cooper pairs $\mathcal{T}_{\mathrm{CP}}^{} = \frac{1}{4} \sin^2 \Theta\, \sin^2 \bigl(\delta_{\uparrow}+ \delta_{\downarrow})$ and that of the Bogoliubov particles $\mathcal{T}_{\mathrm{BG}}^{}= \frac{1}{2}\sum_{\sigma} \sin^2 \delta_{\sigma}^{}$. Here, $\delta_\sigma^{}$ is the phase shift of the renormalized Bogoliubov particles, and $\Theta \equiv \cot^{-1} (\xi_d^{}/ \Gamma_S^{})$ is the Bogoliubov-rotation angle in the Nambu pseudo spin space, with $\xi_d^{} =\varepsilon_d^{}+U/2$. It is also demonstrated, using Wilson's numerical renormalization group approach, that the CAR is enhanced in the crossover region between the Kondo regime and the SC-proximity-dominated regime at zero magnetic field. The magnetic fields induce another crossover between the Zeeman-dominated regime and the SC-dominated regime. We find that the CAR is enhanced and becomes less sensitive to magnetic fields in the SC-dominated regime close to the crossover region spreading over the angular range of $\pi/4 \lesssim \Theta \lesssim 3\pi/4$.