Master's Defense: Emil Jermiin Pedersen Frost

Interacting multiterminal Josephson junctions -- a study of the superconducting impurity Anderson model

Through the last couple of decades, quantum dots (QDs) embedded in Josephson junctions (JJs) has become a fruitful platform for the study of the competition between superconductivity – which favors the formation of Cooper pairs – and local Coulomb repulsion – which prevents electron pairing. Experimentally, the energy scales of these effects may be similar, and both the proximity effect and the Kondo effect has been observed. Therefore, the theoretical treatment of the problem is challenging and, often, one relies on Numerical Renormalization Group (NRG) techniques which are accurate but slow. In this thesis, we use the superconducting impurity Anderson model to describe interacting multiterminal JJs in the context of a single-level QD coupled to multiple s-wave superconductors. We derive the effective action for electrons on the QD and show -- in the limit of large superconducting gap (proximitized limit) -- that this leads to an effective low-energy Hamiltonian which describes proximity-induced superconductivity of the QD. In this limit, we obtain the eigenenergies, eigenstates, and the phase diagram, showing the transition from a BCS-like singlet to a magnetic doublet. Next, we consider the leading order corrections to describe finite-gap systems and find the energy level shifts through a self-consistent renormalization. We compare the phase diagram with NRG and find good agreement for single-lead systems in the proximitized limit and even for moderate interactions $U \sim \Delta$. We also develop a zero-bandwidth (ZBW) model which is shown to capture some qualitative features of the Anderson model, namely the transition from a BCS-like singlet to a YSR-screened singlet as $\Delta / U$ is decreased as well as similar bound state energy spectra, supercurrents, and phase diagrams. In the proximitized limit, we relate the ZBW model to the Anderson model on a quantitative level and compare with NRG. We find that the ZBW model accurately describes the proximitized QD but fails to capture Kondo physics due to the lack of a continuum of states in the leads.