Exact dynamics of the spin-boson model at the Toulouse limit

Starting from the stochastic Liouville equation, we obtain an integral functional equation for describing the finite-temperature dynamics of the asymmetric spin-boson model. Like the zero-temperature case, this equation displays a hierarchical structure characterizing the multiple-timescale nature of quantum dissipative dynamics. We are thereby able to establish a perturbation series and illustrate the contributions with the conventional concepts of blips and sojourns. We clarify the speciality of the Toulouse limit by presenting the analytical properties of intrablip and blip-sojourn interactions. This speciality leads to a closed-form equation for the blip, from which the exact result of the dissipative dynamics can be obtained. We also demonstrate the exact solution by summing the perturbative series. Moreover, we calculate the exact equilibrium distribution at the Toulouse limit and compare it with the Boltzmann distribution valid for weak dissipation.