Temperature-dependent spectral function of a Kondo impurity in an $s$-wave superconductor

Using the numerical renormalization group method, the effect due to a Kondo impurity in an $s$-wave superconductor is examined at finite temperature ($T$). The $T$-behaviors of the spectral function and the magnetic moment at the impurity site are calculated. At $T$=0, the spin due to the impurity is in singlet state when the ratio between the Kondo temperature $T_k$ and the superconducting gap $Delta$ is larger than 0.26. Otherwise, the spin of the impurity is in a doublet state. We show that the separation of the double Yu-Shiba-Rusinov peaks in the spectral function shrinks as $T$ increases if $T_k/Delta<0.26$ while it is expanding if $T_k/Delta>0.26$ and $Delta$ remains to be a constant. These features could be measured by experiments and thus provide a unique way to determine whether the spin of the single Kondo impurity is in singlet or doublet state at zero temperature.