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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 $C(ω)$ of the spin operator $σ_z$ at finite temperature for the sub-Ohmic spin-boson model. A peak is observed at the frequency $ω_{T}\sim T$ in the curve of $C(ω)$. The curve merges with the zero temperature $C(ω)$ in $ω\gg ω_{T}$ and deviate significantly from the power-law form $ω^{\pm s}$ of the zero temperature curve in $ω\llω_{T}$.