Padé approximant approach for obtaining finite-temperature spectral functions of quantum impurity models using the numerical renormalization group technique

We introduce an alternative approach for obtaining smooth finite-temperature spectral functions of quantum impurity models using the numerical renormalization group (NRG) technique. It is based on calculating first the Green's function on the imaginary-frequency axis, followed by an analytic continuation to the real-frequency axis using Padé approximants. The arbitrariness in choosing a suitable kernel in the conventional broadening approach is thereby removed and, furthermore, we find that the Padé method is able to resolve fine details in spectral functions with less artifacts on the scale of $\omega \sim T$. We discuss the convergence properties with respect to the NRG calculation parameters (discretization $\Lambda$, $z$-averaging, truncation cutoff) and the number of Matsubara points taken into account in the analytic continuation. We test the technique on the single-impurity Anderson model and the Hubbard model (within the dynamical mean-field theory). For the Anderson impurity model, we discuss the shape of the Kondo resonance and its temperature dependence. For the Hubbard model, we discuss the inner structure of the Hubbard bands in metallic and insulating solutions at half-filling, as well as in the doped Mott insulator. Based on these test cases, we conclude that the Padé approximant approach provides improved results for spectral functions at low-frequency scales of $\omega \lesssim T$ and that it is capable of resolving sharp spectral features also at high frequencies.

Figure 1

(a) Spectral function of the noninteracting resonant-level model obtained with broadening ($\alpha =0.2$) and with the Padé approximant. The Padé approximant overlaps nearly perfectly with the analytic result. (b) Spectral functions calculated at different temperatures $T$ using the Padé approximant [i.e., using the sets of Matsubara points $i{\omega }_n=i(2n+1)\pi T$ with the raw results of NRG calculations performed at chosen temperatures $T$]. The analytic result does not depend on the temperature. Deviations in the numerical results occur for $T\gtrsim \Gamma$.

Figure 2

(a) Imaginary part of the Green's function for the noninteracting resonant-level model (same parameters as in Fig. 1) evaluated at the Matsubara frequencies for different temperatures. The analytic solution is shown with solid black line. (b) Relative errors of raw Matsubara spectral functions (dots) and the corresponding Padé approximants (solid lines). The dots quantify the intrinsic NRG errors, while the lines measure the full systematic error (intrinsic+analytic continuation errors).

Figure 3

Spectral function of the noninteracting resonant-level model with large hybridization $\Gamma /D=0.3$, where the behavior at the band edges $\omega =-D$ and $D$ may be compared.

Figure 4

(a) Spectral function of the noninteracting resonant-level model with a peak centered far away from the Fermi level, $\epsilon \gg \Gamma$. Standard approach produces a severely overbroadened result since the kernel width for $\alpha =0.2$ is ${\Gamma }_{\mathrm{br}}\approx \alpha \epsilon ={10}^{-2}$. The Padé method is much more successful, although the peak is slightly shifted away from its correct position. (b) Comparison of the spectral functions computed with the Padé approximant for a range of temperatures. Similar degree of agreement is produced for all $T$ considered.

Figure 5

Spectral functions for the interacting single-impurity Anderson model (SIAM) obtained with broadening ($\alpha =0.2$) and by Padé approximation, with and without the $\Sigma$ trick. Two sets of parameters with the same $\Gamma /U$ ratio are used, thus the effective Kondo exchange coupling constant $J$ is the same. For $U/D=1$, the Hubbard peaks are located close to the band edges, and additional spectral features can be resolved at the edge. For $U/D=0.5$, the Padé results with and without the $\Sigma$-trick overlap nearly perfectly, while for $U/D=1$ there are some small differences.

Figure 6

Spectral function of the single-impurity Anderson model for different numbers of kept states, $N_{\mathrm{keep}}$, in the NRG procedure. The artifacts (for instance, the dip at $\omega /D=0.5$ for $N_{\mathrm{keep}}=500$) are truncation-cutoff dependent, thus they stem from the raw NRG results.

Figure 7

(a) Spectral functions of the single-impurity Anderson model obtained with Padé approximant using the first $N_m$ Matsubara frequencies. (b) Relative difference between the spectral functions computed with $N_m=1000$ Matsubara frequencies compared to those for lower $N_m$.

Figure 8

Spectral functions of the single-impurity Anderson model obtained with Padé approach using different shifts from the real axis.

Figure 9

Close-up view of the top of the Kondo resonance at a temperature greater than $T_K$. Note the deviation of the peak shape from parabolic behavior when broadening is used.

Figure 10

(a) Spectral functions of the SIAM at very low temperature obtained using Padé approximations, which use nonconsecutive Matsubara frequencies as input: $i{\omega }_n=i(2n+1)\pi sT$. The parameter $s$ quantifies the step length. The inset (b) shows a close-up on the Kondo resonance. (c) Green's function on the imaginary-frequency axis, indicating the Matsubara points taken into account in the Padé approximant construction. The black line is the asymptotic $1/z$ form.

Figure 11

Spectral function of the SIAM for different number $N_z$ of the $z$-averaging discretization meshes. For small $N_z$, the spectra exhibit a large number of spurious resonances, which are eliminated at larger $N_z$.

Figure 12

Spectral function of the SIAM for different values of the discretization parameters $\Lambda$.

Figure 13

Dynamical charge susceptibility of the noninteracting resonant-level model computed with broadening and with the Padé continuation. We also compare analytic results for the wide-band limit ($D\to \infty$, dashed line) and finite flat band (full line), $T/D={10}^{-4}$.

Figure 14

Dynamical spin (top) and charge (bottom) susceptibility of the single-impurity Anderson model obtained with Padé continuation for a range of temperatures. The Kondo temperature is $T_K=2.5{10}^{-3}D$.

Figure 15

Dynamical spin susceptibility obtained using the broadening procedure (to be compared with the upper panel of Fig. 14).

Figure 16

Cumulant function $C\left(x\right)$ for the raw dynamical spin susceptibility spectral data. The slope of the fit function for $T/D={10}^{-8}$ is $1.991$.

Figure 17

Spectral functions of the particle-hole symmetric Hubbard model at half-filling, $\langle n\rangle =1$, at temperatures (a) $T/D=0.01$ and (b) $T/D=0.1$ for a range of repulsion strengths $U/D$ computed using the dynamical mean-field theory (DMFT). We compare spectra computed using the Padé approach (solid lines) and with broadening (dots). (Padé continuation is used in all steps of the DMFT calculation, not just to obtain the final spectrum.)

Figure 18

Spectral functions of the particle-hole symmetric Hubbard model at half-filling for $U/D=2.5$: dependence on the NRG discretization parameter $\Lambda$. Dashed line show $A(-\omega )D$.

Figure 19

Spectral functions of the Hubbard model describing a doped Mott insulator for a range of temperatures. We show curves obtained by broadening ($\alpha =0.2$) and Padé approximation.

Figure 20

Shape of the Kondo resonance for $T\ll T_K$: close-up on the resonance peak and fits to various analytic expressions. ISR stands for the inverse-square-root (Frota) form.

Figure 21

(a) Kondo resonance height as a function of temperature, calculated using broadening and Padé approximation. (b) Spectra for a range of temperatures from $T/T_K={10}^{-2}$ to $T/T_K=10$ in 13 equally spaced temperatures in the logarithmic scale.

Figure 22

Conductance $G\left(T\right)/G_0$ of a quantum dot described by the single-impurity Anderson model and a fit to the phenomenological Goldhaber-Gordon et al. formula.