The Andreev states of a superconducting quantum dot: mean field vs exact numerical results

TL;DR

This paper compares mean field and exact numerical methods to analyze Andreev states in a superconducting quantum dot, revealing the reliability of mean field in certain phases and identifying multiple bound states within the gap.

Contribution

It demonstrates the correspondence between mean field and exact results for Andreev states in a superconducting quantum dot, highlighting the conditions where mean field is reliable.

Findings

Up to four bound states within the gap in the spin doublet ( phase)

Mean field approximation is reliable within the  phase

Complete correspondence between exact and mean field quasiparticle spectra

Abstract

We analyze the spectral density of a single level quantum dot coupled to superconducting leads focusing on the Andreev states appearing within the superconducting gap. We use two complementary approaches: the numerical renormalization group and the Hartree-Fock approximation. Our results show the existence of up to four bound states within the gap when the ground state is a spin doublet (\pi\ phase). Furthermore the results demonstrate the reliability of the mean field description within this phase. This is understood from a complete correspondence that can be established between the exact and the mean field quasiparticle excitation spectrum