On the Truncation Error of Numerical Renormalization Group

Using the recently developed exact numerical renormalization group (NRG) method, we analyse the NRG truncation errors $\delta \chi$ of the local magnetic susceptibility and $\delta F$ of the free energy for the spin-boson model (SBM). We find that for temperatures higher than a crossover temperature $T_{cr}$, as the number of kept states $M$ increases, both errors have oscillations with quasi period $\ln M\mathrm{/}\ln N_b$ and the envelopes decrease as ${ϵ}_{tr}={\mathrm{\Lambda }}^{-\ln M\mathrm{/}\ln N_b}$ ($N_b$ is the number of boson states used for each bath site). For $T\ll T_{cr}$, they decrease slower than the power law. We extract that $T_{cr}=T^{\ast }{ϵ}_{tr}$, with $T^{\ast }$ being the crossover energy scale between the declocalized and the critical fixed points of SBM. The same rule applies to $\delta \chi$ and $\delta F$ calculated from the full density matrix NRG method and is expected to hold for general impurity models, allowing accurate removal of NRG truncation errors in static quantities at high temperatures.