I will present a brief overview of relatively simple theoretical approaches developed in our Prague group in past several years and applied to the problem of description of correlated quantum dots attached to the BCS superconducting leads. As a thorough Quantum Monte Carlo analysis (1) of the experimental data (2) showed realistic experimental setups can be even quantitatively captured by the Single Impurity Anderson Model (SIAM) with superconducting leads. Pioneering semi-analytical approaches have not matched the so far employed heavy numerical tools such as Numerical Renormalization Group (NRG) and/or Quantum Monte Carlo (QMC) in the ability of quantitatively predicting the properties of this model. However, we have shown that self-consistent perturbation expansion up to the second order in the interaction strength (3) yields at zero temperature and for a wide range of other parameters excellent results for the position of the 0 − π impurity quantum phase transition boundary and the Josephson current as well as the energy of Andreev bound states in the 0-phase. Furthermore, we have discovered exact identities connecting symmetric and asymmetric coupling situations which significantly reduce computational requirements in experimentally generic asymmetric setups (4) and provided simple approximate analytical formulas for the fitting of the phase boundaries from finite-temperature experimental data (5). I will also briefly mention an exact mapping of a half-filled superconducting SIAM onto a normal SIAM with a structured semiconducting lead which simplifies some technical aspects of its NRG solution significantly (6) and a simple way of determination of the quantum critical point from finite-temperature QMC statistics (7). Finally, the most recent extensions of those methods to more quantum dots (8) or superconducting leads (9) will be mentioned.



(1) David J. Luitz, Fakher F. Assaad, Tomáš Novotný, Christoph Karrasch, and Volker Meden, Understanding the Josephson current through a Kondo-correlated quantum dot, Phys. Rev. Lett. 108, 227001 (2012).

(2) H. Ingerslev Jørgensen, T. Novotný, K. Grove-Rasmussen, K. Flensberg, and P. E. Lindelof, Critical Current 0-π Transition in Designed Josephson Quantum Dot Junctions, Nano Lett. 7 (8), 2441 (2007).

(3) M. Žonda, V. Pokorný, V. Janiš, and T. Novotný, Perturbation theory of a superconducting 0-π impurity quantum phase transition, Scientific Reports 5, 8821(2015); Perturbation theory for an Anderson quantum dot asymmetrically attached to two superconducting leads, Phys. Rev. B 93, 024523 (2016).

(4) Alžběta Kadlecová, Martin Žonda, and Tomáš Novotný, Quantum dot attached to superconducting leads: Relation between symmetric and asymmetric coupling, Phys. Rev. B 95, 195114 (2017).

(5) Alžběta Kadlecová, Martin Žonda, Vladislav Pokorný, and Tomáš Novotný, Practical Guide to Quantum Phase Transitions in Quantum-Dot-Based Tunable Josephson Junctions, Phys. Rev. Applied 11, 044094 (2019).

(6) Peter Zalom, Vladislav Pokorný, and Tomáš Novotný, Spectral and transport properties of a half-filled Anderson impurity coupled to phase-biased superconducting and metallic leads, Phys. Rev. B 103, 035419 (2021); Peter Zalom and Tomáš Novotný, Tunable reentrant Kondo effect in quantum dots coupled to metal-superconducting hybrid reservoirs, Phys. Rev. B 104, 035437 (2021).

(7) V. Pokorný and T. Novotný, Footprints of impurity quantum phase transitions in quantum Monte Carlo statistics, Phys. Rev. Research 3, 023013 (2021).

(8) M. Žonda, P. Zalom, T. Novotný, G. Loukeris, J. Bätge, and V. Pokorný, Generalized atomic limit of a double quantum dot coupled to superconducting leads, Phys. Rev. B 107, 115407 (2023).

(9) Peter Zalom, Martin Žonda, and Tomáš Novotný, Hidden Symmetry in Interacting-Quantum-Dot-Based Multiterminal Josephson junctions, Phys. Rev. Lett. 132, 126505 (2024).