Quantitative determination of the discretization and truncation errors in numerical renormalization-group calculations of spectral functions

In the numerical renormalization-group (NRG) calculations of spectral functions of quantum impurity models, the results are always affected by discretization and truncation errors. The discretization errors can be alleviated by averaging over different discretization meshes (“$z$-averaging”), but since each partial calculation is performed for a finite discrete system, there are always some residual discretization and finite-size errors. The truncation errors affect the energies of the states and result in the displacement of the delta-peak spectral contributions from their correct positions. The two types of errors are interrelated: for coarser discretization, the discretization errors increase, but the truncation errors decrease since the separation of energy scales is enhanced. In this work, it is shown that by calculating a series of spectral functions for a range of the total number of states kept in the NRG truncation, it is possible to estimate the errors and determine the error bars for spectral functions, which is important when making accurate comparison to the results obtained by other methods and for determining the errors in the extracted quantities (such as peak positions, heights, and widths). The closely related problem of spectral broadening is also discussed: it is shown that the overbroadening contorts the results without, surprisingly, reducing the variance of the curves. It is thus important to determine the results in the limit of zero broadening. The method is applied to determine the error bounds for the Kondo peak splitting in an external magnetic field. For moderately strong fields, the results are consistent with the Bethe ansatz study by Moore and Wen [Phys. Rev. Lett. 85, 1722 (2000)]. We also discuss the regime of large $U/\Gamma$ ratio. It is shown that in the strong-field limit, a spectral step is observed in the spectrum precisely at the Zeeman frequency until the field becomes so strong that the step merges with the atomic spectral peak.

Figure 1

Spectral function $A_{\uparrow }\left(\omega \right)$ of the Anderson impurity model calculated for a range of $N$, the number of states kept in the truncation after each NRG iteration step. $N$ ranges from 1800 to 3600 in steps of 100. The Kondo temperature (defined as in Refs.9 and10) is $T_K=6.9\times {10}^{-5}$, and thus $B/T_K=4.7$.

Figure 2

Ground-state energy $E_{\mathit{GS}}$ as a function of the number of states kept, $N$. $E_{\mathit{GS}}$ is computed as described in Ref.33. $E_{\mathit{GS}}^{\left(0\right)}$ is the ground-state energy of the conduction band without the impurity; thus what is plotted is the impurity binding energy. The variation of the binding energy (in the large-$N$ limit) with $\Lambda$ is a direct consequence of the discretization errors. These results indicate the degree of convergence of the NRG calculation as a function of $N$ and $\Lambda$.

Figure 3

Upper panel: Spectral function calculated by averaging the CFS spectra over all values of $N$ (thick black line) with a confidence interval determined by the standard deviation of the data at each $\omega$ (thinner lines; red online). For comparison, the spectral function calculated using the patching approach is also shown (dashed line; green online). The gray lines in the background are the individual CFS spectra from Fig. 1. Lower panel: Standard deviation as a function of frequency.

Figure 4

Estimation of the true position of the peak maximum; see the text for details. The data are taken from the left inset in Fig. 3.

Figure 5

Raw binned spectral data in the energy range of the Kondo peak for a fine-grained range of $z$ values ($N_z=256$). These are raw spectral weights of the delta peaks forming the spectral functions which have been binned (using a logarithmic mesh, with 600 bins per decade, the intervals of the binning grid being much narrower than the broadening kernels used to postprocesses these raw results). Left and middle panels show results for $N=2000$ and $N=3000$, while the right panel shows the difference between the two. The grayscale ranges are approximately equal in the three plots. In the plots $z$ ranges from 0 to 1 (bottom to top) and $\omega$ ranges from $0.0002$ to $0.0003$ (across the Kondo peak). The model parameters are as in Fig. 1.

Figure 6

The Kondo peak position (with “pessimistic” error bars) as a function of the broadening parameter. The (average) peak position is extracted by determining the maximum of the average of the spectral functions for different $N$; the maximum is located using the conjugate gradient method. The smallest $\alpha$, for which the Kondo peak maximum can still be extracted, is determined by the number $N_z$ of different values of the twist parameter (here $N_z=12$ and ${\alpha }_{\min }\approx 0.05$). For completeness, the results for larger broadening parameters $\Lambda =3$ and $\Lambda =4$ are also shown; smaller error bars result from the fact that for coarser broadening the truncation errors decrease (while the discretization errors increase). Thus larger values of $\Lambda$ actually lead to less scatter in the calculated spectral functions as $N$ is varied.

Figure 7

The Kondo peak position (with “optimistic” error bars) as a function of the broadening parameter. The peak position is extracted as an average of the maxima of the spectral functions for different $N$; each maximum is located using the conjugate gradient method. The error bars are determined by calculating the standard deviation of the extracted values for ${\omega }_K$. The error bars increase significantly for $\alpha <0.1$ because at given number of the twist parameters, $N_z=12$, the oscillations in the spectral functions become severe for $\alpha \lesssim 1/\sqrt{\Lambda }{N}_z$.

Figure 8

The Kondo resonance splitting as a function of the external magnetic field. The error bars correspond to the “pessimistic” and “optimistic” error estimates. Model parameters are $U=0.1$, ${\epsilon }_d=-U/2$, and $\Gamma =0.008$ in units of half-bandwidth of the (flat) conduction band. The broadening parameter is $\alpha =0.075$. The data labeled as “Moore Wen” are taken from Fig. 2 in Ref.37.

Figure 9

Spectral function of the Anderson impurity model in a strong magnetic field, $B\gg T_K$. The black line corresponds to the NRG results, the two lighter (brown online) lines indicate the confidence region, while the light (red online) curve corresponds to the analytical perturbative RG function from Ref.39.

Figure 10

Spectral function of the Anderson impurity model in very strong magnetic field, $B\gg T_K$ and $B\sim U/2$. The dark lines are the averaged NRG results, while the lighter (brown online) lines indicate the confidence region. The discretization errors are more pronounced on the high-frequency side of the spectral step at $\omega =b\equiv g{\mu }_{B}B$.