Dynamical correlation functions of the quadratic coupling spin-Boson model

The spin-boson model with quadratic coupling is studied using the bosonic numerical renormalization group method. We focus on the dynamical auto-correlation functions C_O(ω), with the operator Ô taken as σ_x, σ_z, and X, respectively. In the weak-coupling regime α < α_c, these functions show power law ω-dependence in the small frequency limit, with the powers 1+2s, 1+2s, and s, respectively. At the critical point α=α_c of the boson-unstable quantum phase transition, the critical exponents y_O of these correlation functions are obtained as y_σx=y_σz=1-2s and y_X=-s, respectively. Here s is the bath index and X is the boson displacement operator. Close to the spin flip point, the high frequency peak of C_σx(ω) is broadened significantly and the line shape changes qualitatively, showing enhanced dephasing at the spin flip point.