Figure 3
From: Perturbation theory of a superconducting 0 − π impurity quantum phase transition
Comparing various methods of calculation of one-particle quantities.
Panels (a) and (b) show supercurrent at half-filling as a function of the phase difference Φ for U = 4Δ (a) and U = 8Δ (b) calculated by numerically exact NRG and analytically approximative fRG, spin-symmetric HF and, finally, the second-order PT/dynamical corrections (DC) showing a nearly perfect agreement with NRG (unlike the other two methods). Inset in panel (a) depicts the ABS energies ω0 as functions of Φ for the two values of the Coulomb interaction U. The green dashed line in panel (b) represents the HF tunneling current component. In panel (c) the occupation number 
and locally induced SC gap Δd ≡ −U 〈d↓d↑〉 (inset) are plotted as functions of the level energy for two values of the phase difference Φ = 0 (with no phase transition) and Φ = π (exhibiting phase transition). fRG data in panels (a), (b) were graphically taken from Fig. 4b of Ref. 27.
