Quantum Phase Transition and Universality Class for a Boson-Impurity Model: Implications for the Spin-Boson Model

We introduce a boson-impurity model, and prove that it is in the same universality class as the one-dimensional Ising model with algebraic decaying interaction. This boson-impurity model can be derived from the spin-boson model. The spin-boson model itself is therefore in the same universality class as well. Because the boson-impurity model contains only virtual spin-fluctuations, these are already enough to drive the quantum phase transition of the sub-ohmic spin-boson model.