The dissipative phase transition in a pair of coupled noisy two-level systems

We study the renormalization group (RG) equations of a pair of spin-boson systems coupled in the z-direction with each other. Each spin is coupled to a different bath of harmonic oscillators. We introduce a systematic adiabatic RG, which generalizes the first-order adiabatic renormalization previously used for the single spin-boson model, and we obtain the flow equations for the tunneling constant, the dissipation strength and the inter-spin coupling up to third order in the tunneling. If one of the two spins is treated as a constant magnetization the other spin is described by a biased spin-boson Hamiltonian. In this case the RG equations we find coincide with the ones obtained via a mapping to a long-range Ising chain. If the whole Ohmic two-spin system is considered the Kosterlitz-Thouless phase transition is replaced by a second-order phase transition. In the case of a sub-Ohmic bath our approach predicts that the two-spin system is always localized.