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 $C(\omega)$ of the spin operator $\sigma_z$ for the biased sub-Ohmic spin-boson model. The small-$\omega$ behavior $C(\omega) \propto \omega^s$ is found to be universal and independent of the bias $\epsilon$ and the coupling strength $\alpha$ (except at the quantum critical point $\alpha =\alpha_c$ and $\epsilon=0$). Our NRG data also show $C(\omega) \propto \chi^{2}\omega^{s}$ for a wide range of parameters, including the biased strong coupling regime ($\epsilon \neq 0$ and $\alpha > \alpha_c$), supporting the general validity of the Shiba relation. Close to the quantum critical point $\alpha_c$, the dependence of $C(\omega)$ on $\alpha$ and $\epsilon$ is understood in terms of the competition between $\epsilon$ and the crossover energy scale $\omega_{0}^{\ast}$ of the unbiased case. $C(\omega)$ is stable with respect to $\epsilon$ for $\epsilon \ll \epsilon^{\ast}$. For $\epsilon \gg \epsilon^{\ast}$, it is suppressed by $\epsilon$ in the low frequency regime. We establish that $\epsilon^{\ast} \propto (\omega_0^{\ast})^{1/\theta}$ holds for all sub-Ohmic regime $0 \leqslant s < 1$, with $\theta=2/(3s)$ for $0 < s \leqslant 1/2$ and $\theta = 2/(1+s)$ for $1/2 < s < 1$. The variation of $C(\omega)$ with $\alpha$ and $\epsilon$ is summarized into a crossover phase diagram on the $\alpha-\epsilon$ plane.