  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.

Statistical Mechanics