The spin-boson model with quadratic coupling is studied using the bosonic numerical renormalization group method. We focus on the dynamical auto-correlation functions CO(ω), with the operator Ô taken as σx, σz, and X, respectively. In the weak-coupling regime α < αc, 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=αc of the boson-unstable quantum phase transition, the critical exponents yO of these correlation functions are obtained as yσx=yσz=1-2s and yX=-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 Cσx(ω) is broadened significantly and the line shape changes qualitatively, showing enhanced dephasing at the spin flip point.
Project supported by the National Key Basic Research Program of China (Grant No. 2012CB921704), the National Natural Science Foundation of China (Grant No. 11374362), the Fundamental Research Funds for the Central Universities, China, and the Research Funds of Renmin University of China (Grant No. 15XNLQ03).
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