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We discuss an even-odd effect for an impurity with an $N$-fold degenerateinternal states immersed in a two-dimensional superfluid--Mott-insulatorquantum critical bath, which is described by an spin-$S$ XY Bose-Kondo impuritymodel with $N=2S+1$. Using a dimensional- and momentum-cut off regularizedrenormalization group and unbiased large-scale Monte Carlo numericalsimulations, we establish the phase diagram for the $S=1$ impurity with all therelevant terms included. We show that the $S=1$ impurity with 3-fold degeneracyis fully screened by the critical bath, in qualitative contrast to the spin-1/2case where the impurity is only partially screened. We then argue that allimpurities with odd 2$S$ share the same universal physics as the spin-1/2 case,and all impurities with even 2$S$ are as the spin-1 case. We validate ourconjecture with unbiased Monte Carlo simulations up to $S=2$. A physicalconsequence of this even-odd effect is that two $N=2$ degenerate impurities inthe critical bath form a bound state at a sufficiently low temperature, whichcan be realized using ultracold atoms in an optical lattice.

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