Equilibrium dynamics of the sub-Ohmic spin-boson model under bias

Using the bosonic numerical renormalization group method, we studied the equilibrium dynamical correlation function C(ω) of the spin operator σ_z for the biased sub-Ohmic spin-boson model. The small-ω behavior C(ω)∝ω^s is found to be universal and independent of the bias ε and the coupling strength α (except at the quantum critical point α=α_c and ε=0). Our NRG data also show C(ω)∝χ²ω^s for a wide range of parameters, including the biased strong coupling regime (ε≠0 and α > α_c), supporting the general validity of the Shiba relation. Close to the quantum critical point α_c, the dependence of C(ω) on α and ε is understood in terms of the competition between ε and the crossover energy scale ω₀^* of the unbiased case. C(ω) is stable with respect to ε for ε≪ε^*. For ε≫ε^*, it is suppressed by ε in the low frequency regime. We establish that ε^*∝(ω₀^*)^1/θ holds for all sub-Ohmic regime 0≤s < 1, with θ=2/(3s) for 0 < s≤1/2 and θ=2/(1+s) for 1/2 < s < 1. The variation of C(ω) with α and ε is summarized into a crossover phase diagram on the α-ε plane.