Quantum critical point of spin-boson model and infrared catastrophe in bosonic bath

An analytic ground state is proposed for the unbiased spin-boson Hamiltonian, which is non-Gaussian and beyond the Silbey-Harris ground state with lower ground state energy. The infrared catastrophe in Ohmic and sub-Ohmic bosonic bath plays an important role in determining the degeneracy of the ground state. We show that the infrared divergence associated with the displacement of the nonadiabatic modes in bath may be removed from the proposed ground state for the coupling $\alpha<\alpha_c$. Then $\alpha_c$ is the quantum critical point of a transition from non-degenerate to degenerate ground state and our calculated $\alpha_c$ agrees with previous numerical results.