(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 α = 0.095. In each figure, from bottom to top in the small frequency regime, ε = −0.4, −0.3, −0.16, −0.12, −0.08, and , corresponding to the stars along the vertical dashed line in Fig. 1. The dashed lines are power functions of ω with the prescribed exponents. For panel (b), the zero frequency peak is not shown. NRG parameters are the same as Fig. 1 and the broadening parameter B = 1.0.