On the spin--boson model with a sub--Ohmic bath

We study the spin--boson model with a sub--Ohmic bath using infinitesimal unitary transformations. Contrary to some results reported in the literature we find a zero temperature transition from an untrapped state for small coupling to a trapped state for strong coupling. We obtain an explicit expression for the renormalized level spacing as a function of the bare papameters of the system. Furthermore we show that typical dynamical equilibrium correlation functions exhibit an algebaric decay at zero temperature.