Yu–Shiba–Rusinov States

A magnetic impurity in a superconductor breaks time-reversal symmetry and creates sub-gap bound states (YSR states) at energies ±ε = ±Δ(1 − α²)/(1 + α²), where α = πJρ₀ is the dimensionless Kondo coupling. As α → 1, the YSR energy crosses zero — a quantum phase transition.

Kondo coupling α = πJρ₀ 0.50

SC Gap Δ 1.00

Broadening Γ 0.05

YSR energy ε/Δ

—

QPT crossed?

Ground state

Singlet

ε_YSR = Δ · (1 − α²)/(1 + α²) ; QPT at α = 1 (singlet → doublet ground state)