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 ca
lculated. 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.
Quantum spin systems with strong geometric restrictions give rise to rich quantum phases such as valence bond solids and spin liquid states. However, the geometric restrictions often hamper the application of sophisticated numerical approaches. Based
on the stochastic series expansion method, we develop an efficient and exact quantum Monte Carlo sweeping cluster algorithm which automatically satisfies the geometrical restrictions. Here we use the quantum dimer model as a benchmark to demonstrate the reliability and power of this algorithm. Comparing to existing numerical methods, we can obtain higher accuracy results for a wider parameter region and much more substantial system sizes.
