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Dynamical renormalization group approach to the spin-boson model

We develop a semi-analytical approach beyond the Born-Markov approximation to study the quench dynamics of the spin-boson model in the strong-coupling regime ($α\leq1/2$) for the Ohmic bath. The basic idea in our approach is to write an effective time-dependent model for the dynamics of the system coupled to the bosonic bath after integrating out high-frequency bath modes. By applying this procedure to the Heisenberg equations of motion, we derive a set of flow equations for the system parameters as a function of time. The final flow equations look similar to those of the equilibrium renormalization-group theory; however, in our derivation the scaling parameter is set by the real time. We solve the equations of motion with time-dependent renormalized parameters and show that the resulting dynamics is in decent agreement with the exact NRG calculations as well as the non-interacting blip approximation that is a well-known good solution in this limit.