Fermi-Liquid Approach for Superconducting Kondo Problems

We present a Fermi liquid approach to superconducting Kondo problems applicable when the Kondo temperature is large compared to the superconducting gap. To illustrate the theory, we study the current-phase relation and the Andreev level spectrum for an Anderson impurity between two $s$-wave superconductors. In the particle-hole symmetric Kondo limit, we find a $4\pi$ periodic Andreev spectrum. The $4\pi$ periodicity persists under a small voltage bias which however causes an asymmetric distortion of Andreev levels. The latter distinguishes the present $4\pi$ effect from the one in topological Majorana junctions.

Figure 1

Schematic setup. (a) Semi-infinite left and right ($j=L/R$, blue or red) superconducting leads at $x<0/x>0$, respectively, harbor one-dimensional (1D) right ($+$) and left ($-$) movers and are tunnel coupled (dashed lines) to an Anderson dot (shaded circle) at $x=0$. (b) Unfolded representation with 1D chiral fermions.

Figure 2

ABS spectrum vs phase $\varphi$. Main panel: Black dotted curves show the particle-hole symmetric limit with $T_K/\mathrm{\Delta }\to \infty$ [8]. Blue and red solid curves depict $4\pi$ periodic Andreev levels for $T_K/\mathrm{\Delta }=5$, $\lambda =0$, and ${\alpha }_1=\gamma =1/T_K$. Green dashed curves illustrate the gap $E_g$ formed away from particle-hole symmetry ($\lambda =0.2$), leading to $2\pi$ periodicity. Inset: Asymmetry of $4\pi$-periodic adiabatic Andreev levels near the crossing at $\varphi =\pi$ with voltage $V=0.33\mathrm{\Delta }$ and $\mathrm{Im}(\gamma )=3/T_K$.