Coherent-incoherent transition in the sub-Ohmic spin-boson model

We study the spin-boson model with a sub-Ohmic bath using a variational method. The transition from coherent dynamics to incoherent tunneling is found to be abrupt as a function of the coupling strength α and to exist for any power $0<s<1$, where the bath coupling is described by J(ω)~αωˢ. We find nonmonotonic temperature dependence of the two-level gap ̃ \~K and a reentrance regime close to the transition due to nonadiabatic low-frequency bath modes. Differences between thermodynamic and dynamic conditions for the transition as well as the limitations of the simplified bath description are discussed.