Perturbation theory of a superconducting $0-\pi$ impurity quantum phase transition

TL;DR

This paper develops a perturbation theory approach for analyzing the quantum phase transition in a superconducting quantum dot system, achieving high accuracy in predicting phase boundaries and physical quantities.

Contribution

It introduces a second-order perturbation expansion in the spin-symmetric case that closely matches exact numerical results for the 0-$ ext{pi}$ phase transition in superconducting quantum dots.

Findings

Accurate phase boundary predictions for the 0-$ ext{pi}$ transition.

Precise calculations of level occupation and Josephson current.

Superior agreement with numerical methods compared to previous analytical approaches.

Abstract

A single-level quantum dot with Coulomb repulsion attached to two superconducting leads is studied via the perturbation expansion in the interaction strength. We use the Nambu formalism and the standard many-body diagrammatic representation of the impurity Green functions to formulate the Matsubara self-consistent perturbation expansion. We show that at zero temperature second order of the expansion in its $\it{spin-symmetric}$ version yields a nearly perfect agreement with the numerically exact calculations for the position of the $0-\pi$ phase boundary at which the Andreev bound states reach the Fermi energy as well as for the values of single-particle quantities in the 0-phase. We present results for phase diagrams, level occupation, induced local superconducting gap, Josephson current, and energy of the Andreev bound states with the precision surpassing any (semi)analytical approaches employed thus far.