# Spectral Properties and Quantum Phase Transitions in Superconducting Junctions with a Ferromagnetic Link

## TL;DR

This paper theoretically investigates the spectral and transport properties of superconducting junctions with magnetic links, revealing quantum phase transitions characterized by changes in total spin and supercurrent sign.

## Contribution

It derives a general equation for the subgap spectrum in magnetic Josephson junctions and demonstrates the occurrence of quantum phase transitions in finite-length systems.

## Key findings

- Quantum phase transition occurs when bound states cross the Fermi level.
- Total spin changes discontinuously by integer jumps during the transition.
- Phase transitions are reflected in the sign change of the supercurrent.

## Abstract

We study theoretically the spectral and transport properties of a superconducting wire with a magnetic defect. We start by modelling the system as a one dimensional magnetic Josephson junction and derive the equation determining the full subgap spectrum in terms of the normal-state transfer matrix for arbitrary length and exchange field of the magnetic region. We demonstrate that the quantum phase transition predicted for a short-range magnetic impurity, and associated with a change of the total spin of the system, also occurs in junctions of finite length. Specifically, we find that the total spin changes discontinuously by integer jumps when bounds states cross the Fermi level. The spin can be calculated by using a generalization of Friedel sum rule for the superconducting state, which we also derive. With these tools, we analyze the subgap spectrum of a junction with the length of the magnetic region smaller than the superconducting coherence length and demonstrate how phase transitions also manifest as change of the sign of the supercurrent.