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 CO（ω）, with the operator 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.

Equilibrium dynamics of the sub-Ohmic spin-boson model under bias

Using the bosonic numerical renormalization group method, we studied the equilibrium dynamical correlation function C（ω） of the spin operator σz for the biased sub-Ohmic spin-boson model. The small-ω behavior C（ω） ∝ ω～s is found to be universal and independent of the bias ε and the coupling strength α（except at the quantum critical point α = αc and ε = 0）. Our NRG data also show C（ω） ∝ χ～2ω～s for a wide range of parameters, including the biased strong coupling regime（ε = 0 and α 〉 αc）, supporting the general validity of the Shiba relation. Close to the quantum critical point αc,the dependence of C（ω） on α and ε is understood in terms of the competition between ε and the crossover energy scale ω0^＊of the unbiased case. C（ω） is stable with respect to ε for ε《ε^＊. For ε 》ε^＊, it is suppressed by ε in the low frequency regime. We establish that ε^＊∝（ω0^＊）^1/θ holds for all sub-Ohmic regime 0≤s 〈 1, with θ = 2/（3s） for 0 〈 s≤1/2 and θ = 2/（1 ＋ s） for 1/2 〈 s 〈 1. The variation of C（ω） with α and ε is summarized into a crossover phase diagram on the α–ε plane.

Equilibrium dynamics of the sub-ohmic spin-boson model at finite temperature

We use the full-density matrix numerical renormalization group method to calculate the equilibrium dynamical correlation function C(ω) of the spin operator σ_(z) at finite temperature for the sub-ohmic spin-boson model.A peak is observed at the frequency ω_(T)~T in the curve of C(ω).The curve merges with the zero-temperature C(ω) in ω>>ω_(T) and deviates significantly from the zero-temperature curve in ω<<ω_(T).