Semi-analytical theories for a correlated quantum dot attached to superconducting leads

I will present an overview of relatively simple theoretical approaches developed in our group in past several years and applied to the problem of description of correlated quantum dots attached to the BCS supercon-ducting 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 with super-conducting leads. Pioneering semi-analytical approaches have not matched the so far employed heavy nu-merical tools such as Numerical Renormalization Group and/or Quantum Monte Carlo in the ability of quan-titatively predicting the properties of this model. However, we have shown recently that self-consistent per-turbation expansion up to the second order in the interaction strength [3,4] yields at zero temperature and for a wide range of other parameters excellent results for the position of the 0 − π impurity quantum phase tran-sition boundary and the Josephson current as well as the energy of Andreev bound states in the 0-phase. This method can be also extended to the three-terminal situation with an extra normal lead corresponding to the experimentally interesting STM setup [5], where it allows to study phase-dependent Kondo physics [6,7]. Furthermore, we have discovered exact identities connecting symmetric and asymmetric coupling situations which significantly reduce computational requirements in experimentally generic asymmetric setups [8] and provided simple approximate analytical formulas for the fitting of the phase boundaries from finite-temperature experimental data [9].

[1] David J. Luitz, Fakher F. Assaad, Tomáš Novotný, Christoph Karrasch, and Volker Meden, Understand-ing 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 Cur-rent 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; DOI:10.1038/srep08821 (2015)
[4] M. Žonda, V. Pokorný, V. Janiš, and T. Novotný, Perturbation theory for an Anderson quantum dot asymmetrically attached to two superconducting leads, Phys. Rev. B 93, 024523 (2016)
[5] T. Domański, M. Žonda, V. Pokorný, G. Górski, V. Janiš, and T. Novotný, Josephson-phase-controlled interplay between correlation effects and electron pairing in a three-terminal nanostructure, Phys. Rev. B 95, 045104 (2017)
[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)
[7] Peter Zalom and Tomáš Novotný, Tunable reentrant Kondo effect in quantum dots coupled to metal-superconducting hybrid reservoirs, Phys. Rev. B 104, 035437 (2021)
[8] 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)
[9] 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)