Equilibrium Dynamics of the Sub-Ohmic Spin-boson Model Under Bias

Abstract

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 ($\epsilon \ne 0$ 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 $0⩽s<1$, with $\theta =2/(3s)$ for $0<s⩽1/2$ 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.

8 pages, 8 figures