Kondo screening in unconventional superconductors: The role of anomalous propagators

The Kondo effect in superconductors is frequently investigated using the local quasiparticle density of states as sole bath characteristics, i.e., the presence of anomalous propagators is ignored. Here we point out that this treatment is exact for a number of situations, including pointlike impurities in $d$-wave superconductors. We comment on recent investigations [M. Matsumoto and M. Koga, J. Phys. Soc. Jpn. 70, 2860 (2001); Phys. Rev. B 65, 024508 (2002)] which reached different conclusions: while their numerical results are likely correct, their interpretation in terms of two-channel Kondo physics and an “orbital effect of Cooper pairs” is incorrect.

Figure 1

Spatially extended impurities for which the anomalous propagator in $G_0$ drops out, i.e., ${\sum }_k{h}_k^2{\Delta }_k∕D_k=0$, for a square-lattice $d$-wave superconductor. The dot represents the impurity orbital, the dashed lines show the nonzero hybridization paths (in the notation of an Anderson model). The situations (b) and (c) correspond to the $s$-wave and $d$-wave linear combinations, i.e., screening channels, of the four host sites, which occur in a Kondo model for a spatially extended impurity, as discussed in Ref. 5.

Figure 2

Numerical results for the local susceptibility, obtained from a single-band NRG for a (semiconducting) bath with the hard-gap DOS of a $d+id$ (or $p+ip$) superconductor. The host bandwidth is unity, the NRG parameters are $\Lambda =3$ and $N_s=1000$. (a) $T{\chi }_{\mathrm{loc}}$ for different gap values $\Delta ={10}^{-2},{10}^{-3},{10}^{-4},{10}^{-5}$ (from top to bottom) where the Kondo temperature $T_K$ of the model at $\Delta =0$ is $3\times {10}^{-3}$. (b) The value of the local Curie constant $C$, normalized to its free-spin value $1∕4$, as a function of the ratio $T_K∕\Delta$. Different symbols correspond to different values of $\Delta$. (Deviations from a single universal curve are primarily due to NRG discretization effects.) The results can be compared with Fig. 4 of Ref. 8.