(color online) Equilibrium auto-correlation functions of the operators σ_x (a), σ_z (b), and (c), obtained from NRG for s = 0.3, Δ = 0.1, and ε = 0.0. In each figure, from bottom to top, α = 0.075, 0.085, 0.09, and , corresponding to the stars along the horizontal dashed line in Fig. 1. The dashed lines are power functions of ω with the prescribed exponents. For (b), the zero frequency peak is not shown. NRG parameters are the same as Fig. 1 and the broadening parameter B = 1.0.