Even-Odd effect of an Spin-S impurity coupled to a quantum critical system

We discuss an even-odd effect for an impurity with $N$ degenerate internal states immersed in a two-dimensional superfluid--Mott-insulator quantum critical bath, which is described by a spin-$S$ XY Bose Kondo impurity model with $N=2S+1$. Using a dimensionally and momentum-cut-off regularized renrmalization group and an unbiased large-scale Monte Carlo numerical simulations, we establish the phase diagram of an $S=1$ impurity with relevant terms included. We show that the impurity with $N=3$-fold degeneracy is fully screened by the critical bath, which is qualitatively different from the $N=2$ case where the impurity is only partially screened. We then argue that all even-$N$ impurities share the same universal physics as the $N=2$ case, and all odd-$N$ impurities are as the $N=3$ case. We validate our conjuecture with unbiased Monte Carlo simulations up to $N=5$.