(color online) Equilibrium auto-correlation functions of the operators σ_x (a), σ_z (b), and (c), obtained from NRG for a series of s values, Δ = 0.1, ε = 0.1, and . For each figure, from bottom to top in the small frequency regime, s = 0.1, 0.2, 0.3, 0.4, 0.5, and 0.7. The dashed lines are power functions of ω with the exponents in panels (a) and (b), and −s in panel (c). 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.