Equilibrium Dynamics of the Sub-Ohmic Spin-boson Model At Finite Temperature

We use the full-density matrix (FDM) numerical renormalization group (NRG) method to calculate the equilibrium dynamical correlation function $C(\omega)$ of the spin operator $\sigma_z$ at finite temperature for the sub-Ohmic spin-boson model. A peak is observed at the frequency $\omega_{T}\sim T$ in the curve of $C(\omega)$. The curve merges with the zero temperature $C(\omega)$ in $\omega \gg \omega_{T}$ and deviate significantly from the power-law form $\omega^{\pm s}$ of the zero temperature curve in $\omega \ll\omega_{T}$.