Perturbation theory of a superconducting 0 − π impurity quantum phase transition

Nanostructures attached to leads with specific properties display interesting and important quantum effects at low temperatures. Much attention, both from experimentalists and theorists has been paid in recent years to a quantum dot with well separated energy levels attached to BCS superconductors. In particular, the behavior of the supercurrent (Josephson current) that can flow through the impurity in equilibrium without any external voltage bias between two superconducting leads was in the center of interest. The Josephson current through quantum dots with tangible on-dot Coulomb repulsion can induce a transition signalled by the sign reversal of the supercurrent observed experimentally.

We use the Nambu formalism and the standard many-body diagrammatic representation of the impurity Green functions to formulate the Matsubara self-consistent perturbation expansion. We show that at zero temperature second order of the expansion in its spin-symmetric version yields a nearly perfect agreement with the numerically exact calculations for the position of the 0 − π phase boundary at which the Andreev bound states reach the Fermi energy as well as for the values of single-particle quantities in the 0-phase. We present results for phase diagrams, level occupation, induced local superconducting gap, Josephson current and energy of the Andreev bound states with the precision surpassing any (semi)analytical approaches employed thus far.

Description

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) double occupancy  and locally induced SC gap Δ_d ≡ −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).

Contact person: Václav Janiš