Kondo effect in a side-coupled double quantum dot system embedded in a mesoscopic ring

We study finite-size effect of the Kondo screening cloud in a double quantum dot setup via a large-$N$ slave-boson mean-field theory. In this setup, one of the dots is embedded in a close metallic ring with a finite size $L$ and the other dot is side coupled to the embedded dot via an antiferromagnetic spin-spin-exchange coupling with a strength $K$. The antiferromagnetic coupling favors local spin-singlet state and suppresses the Kondo screening. The mean-field phase diagram as a function of $1/L$ and $K$ shows a crossover to the local spin-singlet ground state from the Kondo phase to occur at $K<{K}_{c}$ $(L=4n,4n+1,4n+3)$ and at $K>{K}_{c}$ $(L=4n+2)$. The effective Kondo temperature ${T}_{k}$ (proptotional to inverse of the Kondo screening cloud size) shows the Kosterlitz-Thouless (KT) scaling at finite sizes for $L=4n,4n+2$, indicating quantum transition of the KT type between the Kondo screened phase for $K\ensuremath{\le}{K}_{c}$ and a local spin-singlet phase for $K\ensuremath{\ge}{K}_{c}$ in the thermodynamic limit with ${K}_{c}$ being the critical value. The local density of states on the embedded quantum dot and the persistent current at finite sizes with different values of $K$ are calculated in the crossover region.