Even-odd effect of a spin- S impurity coupled to a quantum critical system

We discuss an even-odd effect for an impurity with $N$-fold-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 dimensional- and momentum-cutoff regularized renormalization group and unbiased large-scale Monte Carlo numerical simulations, we establish the phase diagram for the $S=1$ impurity with all relevant terms included. We show that a threefold-degenerate $S=1$ impurity coupled to a critical bath has a nondegenerate ground state with a quantized charge, in qualitative contrast to the spin-1/2 case where the impurity is partially screened. We then argue that all impurities with odd-$2S$ degeneracy share the same universal physics as the spin-1/2 case and all impurities with even-$2S$ degeneracy are the same as the spin-1 case. We validate our conjecture with unbiased Monte Carlo simulations up to $S=2$. A physical consequence of this even-odd effect is that two spin-$1/2$ impurities in the critical bath form a long-range entangled state at a low temperature, which can be realized in ultracold atoms in an optical lattice.