Effective low-energy models for superconducting impurity systems

TL;DR

This paper introduces two new methods for calculating Andreev bound state energies in superconducting quantum dots, enabling efficient and accurate analysis without extensive computational resources.

Contribution

It presents a low-energy mapping and an exactly solvable atomic limit approach, improving speed and reliability in modeling superconducting impurity systems.

Findings

Methods accurately predict Andreev bound states.

Predictions agree with numerical renormalization group results.

Approach enables quick parameter space exploration.

Abstract

We present two complementary methods to calculate the Andreev bound state energies of a single-level quantum dot connected to superconducting leads described by the superconducting impurity Anderson model. The first method, which is based on a mapping to a low-energy model, can be utilized to extract the Andreev bound state energies from finite-temperature, imaginary-time quantum Monte Carlo data without the necessity of any analytic continuation technique. The second method maps the full model on an exactly solvable superconducting atomic limit with renormalized parameters. As such, it represents a fast and reliable method for a quick scan of the parameter space. We demonstrate that after adding a simple band correction this method can provide predictions for measurable quantities, including the Josephson current, that are in a solid quantitative agreement with precise results obtained by the numerical renormalization group and quantum Monte Carlo.