Quantum phase transitions in a resonant-level model with dissipation: Renormalization-group studies

We study a spinless level that hybridizes with a fermionic band and is also coupled via its charge to a dissipative bosonic bath. We consider the general case of a power-law hybridization function $\mathrm{\Gamma }(\text{w})\propto \mathrm{\mid }\text{w}{\mathrm{\mid }}^r$ with $r\ge 0$, and a bosonic bath spectral function $B(\text{w})\propto {\text{w}}^s$ with $s\ge -1$\mathrm{.}For$r<1$and$$\mathrm{m}\mathrm{a}\mathrm{x}$(0,2r-1)<s<1$, this Bose-Fermi quantum impurity model features a continuous zero-temperature transition between a delocalized phase, with tunneling between the impurity level and the band, and a localized phase, in which dissipation suppresses tunneling in the low-energy limit. The phase diagram and the critical behavior of the model are elucidated using perturbative and numerical renormalization-group techniques, between which there is excellent agreement in the appropriate regimes. For $r=0$ this model’s critical properties coincide with those of the spin-boson and Ising Bose-Fermi Kondo models, as expected from bosonization.