FLOW EQUATIONS FOR THE SPIN-BOSON PROBLEM

Using continuous unitary transformations recently introduced by Wegner [1], we obtain flow equations for the parameters of the spin-boson Ha miltonian. Interactions not contained in the original Hamiltonian are generated by this unitary transformation. Within an approximation that neglects additional interactions quadratic in the bath operators, we can close the flow equations. Applying this formalism to the case of O hmic dissipation at zero temperature, we calculate the renormalized tu nneling frequency. We find a transition from an untrapped to trapped s tate at the critical coupling constant alpha(c) = 1. We also obtain th e static susceptibility via the equilibrium spin correlation function. Our results are both consistent with results known from the Kondo pro blem and those obtained from mode-coupling theories. Using this formal ism at finite temperature, we find a transition from coherent to incoh erent tunneling at T-2 approximate to 2T(1)*, where T-1* is the cross over temperature of the dynamics known from the NIBA.