On the Truncation Error of Numerical Renormalization Group

TL;DR

This paper analyzes the truncation errors in the numerical renormalization group method for the spin-boson model, revealing oscillatory behavior and decay patterns of errors at different temperature regimes, and proposes a rule to estimate and correct these errors.

Contribution

It introduces a detailed analysis of NRG truncation errors in the spin-boson model and proposes a universal rule for estimating and correcting these errors in static quantities.

Findings

Errors oscillate with quasi period related to the number of states kept and boson states.

Error envelopes decay as a power law with respect to the number of states.

The error correction rule applies across different temperature regimes and impurity models.

Abstract

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 $\ln{M}/\ln{N_b}$ and the envelopes decrease as $\epsilon_{tr}=\Lambda^{-\ln{M}/\ln{N_b}}$ ($N_b$ 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 $T_{cr} = T^{\ast} \epsilon_{tr}$, 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.