Practical Guide to Quantum Phase Transitions in Quantum-Dot-Based Tunable Josephson Junctions

TL;DR

This paper provides analytical formulas and numerical methods for accurately determining the quantum phase transition boundary in quantum-dot-based Josephson junctions, aiding their experimental characterization and design.

Contribution

It introduces new analytical expressions for the phase boundary and compares numerical methods, enhancing understanding of quantum phase transitions in superconducting quantum dots.

Findings

Analytical formulas accurately predict phase boundary positions.

Two-level approximation effectively describes low-temperature physics.

Finite temperature numerical methods show good agreement and reliability.

Abstract

Quantum dots attached to BCS superconducting leads exhibit a $0-\pi$ impurity quantum phase transition, which can be experimentally controlled either by the gate voltage or by the superconducting phase difference. For the pertinent superconducting single-impurity Anderson model, we newly present two simple analytical formulae describing the position of the phase boundary in parameter space for the weakly correlated and Kondo regime, respectively. Furthermore, we show that the two-level approximation provides an excellent description of the low temperature physics of superconducting quantum dots near the phase transition. We discuss reliability and mutual agreement of available finite temperature numerical methods (Numerical Renormalization Group and Quantum Monte Carlo) and suggest a novel approach for efficient determination of the quantum phase boundary from measured finite temperature data. Our results enable fast and efficient, yet reliable characterization and design of such nanoscopic tunable Josephson junction devices.