Quantum phase transition of the sub-Ohmic rotor model

We investigate the behavior of an $N$-component quantum rotor coupled to a bosonic dissipative bath having a sub-Ohmic spectral density $J(\omega) \propto \omega^s$with$s<1$. With increasing dissipation strength, this system undergoes a quantum phase transition from a delocalized phase to a localized phase. We determine the exact critical behavior of this transition in the large-$N$ limit. For $1>s>1\mathrm{/}2$, we find nontrivial critical behavior corresponding to an interacting renormalization group fixed point while we find mean-field behavior for $s<1\mathrm{/}2$. The results agree with those of the corresponding long-range interacting classical model. The quantum-to-classical mapping is therefore valid for the sub-Ohmic rotor model.