Adaptive broadening to improve spectral resolution in the numerical renormalization group

We propose an adaptive scheme of broadening the discrete spectral data from numerical renormalization group (NRG) calculations to improve the resolution of dynamical properties at finite energies. While the conventional scheme overbroadens narrow features at large frequency by broadening discrete weights with constant width in log-frequency, our scheme broadens each discrete contribution individually based on its sensitivity to a $z$-shift in the logarithmic discretization intervals. We demonstrate that the adaptive broadening better resolves various features in noninteracting and interacting models at comparable computational cost. The resolution enhancement is more significant for coarser discretization as typically required in multiband calculations. At low frequency below the energy scale of temperature, the discrete NRG data necessarily needs to be broadened on a linear scale. Here we provide a method that minimizes transition artifacts in between these broadening kernels.

Figure 1

(a) The hopping amplitudes $t_n^{\left(z\right)}$ vs Wilson shell index $n$ and (b) their derivatives $d{t}_n/dz$ (obtained numerically using $\delta z=0.01/n_z$), for the hybridization $\mathrm{\Gamma }\left(\epsilon \right)=\mathrm{\Gamma }\mathrm{\Theta }(D-|\epsilon \left|\right)$ in units of half-bandwidth, i.e., $D=1$. Since $t_{n=0}$ is independent of $z$, only data for $n>0$ is shown, which itself is independent of the value of $\mathrm{\Gamma }$. For large $n\gtrsim 8, t_n$ and $d{t}_n/dz$ follow an exponential scaling, i.e., $t_n\simeq {\mathrm{\Lambda }}^{-z}{\mathrm{\Lambda }}^{(1-n)/2}(\mathrm{\Lambda }-1)/ln\mathrm{\Lambda }$ and $(d{t}_n/dz)/t_n=dln\left(t_n^{\left(z\right)}\right)/dz\simeq -ln\mathrm{\Lambda }$. For small $n$, the values for $t_n$ and $d{t}_n/dz$ deviate from the exponential scaling. The deviation is larger for smaller $z$, since the discretization interval at the band edge becomes narrower, i.e., is more weakly coupled to the impurity.

Figure 2

Discrete spectral data $A_z\left({\omega }_k\right)\equiv {\sum }_{{\sigma }_k}{A}_z({\omega }_k,{\sigma }_k)$ for (a) the noninteracting resonant level model [see Eq. (22)] and (b) the SIAM at $T=0$. Different colors denote different $z$-shifts. Since the bin $k$ at frequency ${\omega }_k$ has width proportional to $|{\omega }_k|$ in the frequency domain, $A_z\left({\omega }_k\right)/\left|{\omega }_k\right|$ is related with the height of a subsequently broadened curve. The discrete spectral weights are bunched, and most of the bunches are uniformly spaced in log-frequency space with the spacing $\mathrm{\Delta }(ln|\omega \left|\right)\simeq (ln\mathrm{\Lambda })/n_z$ after $z$-averaging. But in certain regimes $[{\omega }_k\sim {\epsilon }_d=-0.05$ for (a) and ${\omega }_k\sim U/2=0.25, {\omega }_k\sim T_{\mathrm{K}}\simeq 0.01$ for (b)], bunches are widely spread or irregularly distributed: the adaptive broadening scheme systematically applies the narrower broadening width to these bunches, leading to the improved spectral resolution in such regimes (see Figs. 3 and 5). See Secs. 4a and 4b for details.

Figure 3

Adaptive broadening for the $T$-matrix in the spinless noninteracting resonant level model. Panel (a) shows the exact $T\left(\omega \right)$ [Eq. (23), black dashed line] and the NRG results with the conventional and adaptive broadening schemes for different ${\epsilon }_d$. For small $n_z$, the conventional scheme (strongly) overbroadens the resonance peaks (red dashed line), while the adaptive scheme (blue solid line) nicely reproduces the exact $T\left(\omega \right)$. In contrast, the conventional broadening scheme can capture the sharp peak shapes only for much larger $n_z$ (purple dash-dotted line), but also acquires noise (cf. Sec. 2d). Panels (b) and (c) show the binned discrete data $\overline{A}({\omega }_k,{\sigma }_k)/\left|{\omega }_k\right|$ ($\overline{A}/\omega$ for short; see text) for ${\epsilon }_d=-0.45$ and ${\epsilon }_d=-0.05$, respectively. The weights near ${\omega }_k\simeq {\epsilon }_d$ as well as near the band edge $-D$ have clearly reduced adaptive broadening width ${\sigma }_k<ln\mathrm{\Lambda }$. Thus the adaptive broadening can capture the sharp features near ${\omega }_k\simeq {\epsilon }_d$ and band edges $|{\omega }_k|=D$ [not shown in (a)] much better than the conventional broadening scheme for fixed $n_z$.

Figure 4

$T$-matrix of the Kondo model for $B=T=0$. Due to total spin and particle-hole symmetries, $T_{\nu }(\pm \omega )=T\left(\omega \right)$. (a) Broadened curves with different broadening schemes (adaptive or conventional) and number of $z$-shifts ($n_z=4$ and 8). Inset: Discrete spectral data $\overline{A}/\omega$ for the data in the main panel. (b) Close-up near the band edge $D=1$. The adaptive broadening better resolves the sharp edge at $\left|\omega \right|=D$, which stems from the reduced broadening ${\sigma }_k<ln\mathrm{\Lambda }$ at $|{\omega }_k|\simeq D$ in the inset of (a).

Figure 5

$T$-matrix of the SIAM for $B=T=0$. Due to total spin and particle-hole symmetries, $T_{\nu }(\pm \omega )=T\left(\omega \right)$. (a) Broadened curves with different broadening schemes (adaptive or conventional), number of $z$-shifts ($n_z=4$ and 8), and partly improved by self-energy ($\mathrm{\Sigma }$). (b) Close-up of the Kondo peak at $\omega \ll T_{\mathrm{K}}$, showing convergence to within 1%. Here the blue solid (dashed) line coincides with the red solid (dashed) line. (c) Close-up of the Hubbard side peak centered at $\omega =U+{\epsilon }_d=U/2$. The self-energy removes discretization-related noise at $\omega \ll T_{\mathrm{K}}$ and sharpens the Hubbard side peaks. The inset to panel (a) presents the discrete spectral data $\overline{A}/\omega$ for the data in the main panel. It shows that the adaptive broadening reduces the broadening ${\sigma }_k$ for frequencies ${\omega }_k\simeq U/2$, and hence enhances the resolution of the side peak. This is more significant for smaller $n_z$ or without self-energy.

Figure 6

$T$-matrix of the Kondo model for large magnetic field $B\gg T_{\mathrm{K}}$ at $T=0$. Due to particle-hole symmetry, $T_{\uparrow }\left(\omega \right)=T_{\downarrow }(-\omega )$. (a), (b) Broadened curves at $B/T_{\mathrm{K}}={10}^2$ and ${10}^3$, plotted versus (a) $\omega /B$ and (b) $\omega$, with different broadening schemes (adaptive or conventional) and number of $z$-shifts ($n_z=4$ and 8). The blue and red cross hairs in (a) indicate the value and uncertainty for the peak positions and heights extrapolated to the limit ${\alpha }_z\to 0$ for adaptive and conventional schemes, respectively, as derived from Fig. 8. The extrapolated peak position at $B={10}^2{T}_{\mathrm{K}}$ is consistent with the DMRG result (dotted line, taken from Ref. [22]). (c) Discrete spectral data $\overline{A}/\omega$ for the data in panel (a) for the parameters as indicated.

Figure 7

$T$-matrix of the Anderson model for large magnetic field $B\gg T_{\mathrm{K}}$ at $T=0$. Due to particle-hole symmetry, $T_{\uparrow }\left(\omega \right)=T_{\downarrow }(-\omega )$. (a), (b) Broadened curves at $B/T_{\mathrm{K}}={10}^2$ and ${10}^3$, versus (a) $\omega /B$ and (b) $\omega$, with different broadening schemes (adaptive or conventional) and number of $z$-shifts ($n_z=4$ and 8). All curves are improved by utilizing the self-energy $\mathrm{\Sigma }$. The blue and red cross hairs in (a) indicate value and uncertainty of the peak positions and heights extrapolated to the limit ${\alpha }_z\to 0$ for adaptive and conventional schemes, respectively, as extracted from Fig. 9. (c) Discrete spectral data $\overline{A}/\omega$ for the data in panel (a) for the parameters as indicated.

Figure 8

Extrapolation of the Kondo peak in $T_{\uparrow }\left(\omega \right)$ for the Kondo model towards the “continuum limit” ${\alpha }_z\equiv \alpha /n_z\to 0$, in terms of (a) heights $T_{max}$ and (b) positions ${\omega }_{max}$ vs $B/T_{\mathrm{K}}$. For ${\omega }_{max}$, we compare with the analytic prediction (dashed line). At $B=100T_{\mathrm{K}}$, we show (c) $T_{max}$ and (d) ${\omega }_{max}$ vs ${\alpha }_z$ (see Fig. 6 for remaining parameters). For each magnetic field, we extrapolate $T_{max}$ and ${\omega }_{max}$ to ${\alpha }_z\to 0$ by fitting the data points from $n_z=4,8$ within the fitting range $[0.25,0.95]$ as indicated by the vertical dotted lines with a Padé approximant of order [2/1] (dashed and dash-dotted lines). The inset in (c) shows the adaptively (blue) and the conventionally (red) broadened curves of $T_{\uparrow }\left(\omega \right)$ at $B=100T_{\mathrm{K}}, \alpha =2$, and $n_z=8$, where ${\alpha }_z=0.25$ is on the lower edge of the fitting range. The extrapolated peak positions and heights are depicted as cross hairs. The inset in (d) shows a close-up of the data points and the fitted curves of ${\omega }_{max}/B$ vs ${\alpha }_z$ at small ${\alpha }_z$. The upturn of ${\omega }_{max}$ for the smallest ${\alpha }_z$ appears with the onset of underbroadening.

Figure 9

Same as Fig. 8, but for the SIAM. The fitting range in (c), (d) is ${\alpha }_z\in [0.2,0.9]$.

Figure 10

Dependence of $T$-matrices on $\mathrm{\Lambda }\in [2,16]$, for the models discussed in Secs. 4b–4c and Figs. 5–7. Here we use $\alpha =2$, and for comparable accuracy across the wide range of $\mathrm{\Lambda }$, we employ an energy-based truncation with $E_{\mathrm{trunc}}=10$, throughout. (a) Zoom into the Hubbard side peaks of the SIAM at $B=0$ (see Fig. 5 for system parameters). Here we choose different pairs of $(\mathrm{\Lambda },n_z)$ with a constant ratio $(ln\mathrm{\Lambda })/n_z$. This implies the same spectral resolution within the conventional broadening scheme in that the brown and purple lines (lower two rows in the legend) (nearly) lie on top of each other. Panels (b) and (c) show the $T$-matrix for (b) the Kondo and (c) the SIAM at large magnetic field ($B={10}^2{T}_{\mathrm{K}}$; see Figs. 6 and 7 for system parameters, respectively). Here again $ln\left(\mathrm{\Lambda }\right)/n_z=ln\left(10\right)/26\simeq ln\left(2\right)/8$ was chosen comparable. Note that by truncating with respect to energy ($E_{\mathrm{trunc}}=10$) rather than fixed number of states, the curves of $\mathrm{\Lambda }=2$ here slightly differ from the ones in Figs. 5, 6, 7. For comparison, we again also plot the extrapolated peak position and height (blue and red cross hairs) and the DMRG data (black dotted line), adapted from Figs. 6 and 7.

Figure 11

$T$-matrix of the SIAM at finite $T\ll T_{\mathrm{K}}$. (a) Broadened curves for different $\gamma$ and (b) binned discrete data $\overline{A}/\omega$ for $T\left(\omega \right)$. Contrary to the confined distribution of $\overline{A}/\omega$ at $T\lesssim {\omega }_k\lesssim T_{\mathrm{K}}$ and ${\sigma }_k\simeq ln\mathrm{\Lambda }$, the weights at $\left|\omega \right|\ll T$ spread widely along ${\sigma }_k$. This leads to distinct irregular behavior in $T\left(\omega \right)$ at $\left|\omega \right|\ll T$ after the logarithmic Gaussian broadening [blue solid line in (a)]. The secondary convolution with width $\gamma \lesssim T$ smears out sharp fluctuations without the blip artifact, keeping features at higher $\omega$ untouched [red and brown solid lines in (a)]. (c) Spectral function values $T_i\equiv T\left({\omega }_i\right)$ at frequencies ${\omega }_i=\gamma {\mathrm{\Lambda }}^2\times \{-1,-0.8,-0.6,...,1\}$ vs $\gamma /T$ (see text). (d) Error ${\varepsilon }_{\mathrm{rms}}$ of quadratic polynomial fit vs $\gamma /T$ [see Eq. (24)]. In (c), (d), solid lines are obtained by applying first the log-Gaussian and then the linear broadening kernels, as in Eq. (16). In comparison, dashed lines are obtained by the opposite order of broadening; the order of broadening kernels is essential to obtain smooth curves at $\left|\omega \right|\lesssim T$ (cf. Sec. 3b).

Figure 12

$T$-matrix of the SIAM at $T=D/30\simeq 2T_{\mathrm{K}}$. (a) Comparison of NRG and quantum Monte Carlo results (data from Ref. [25]) (b) Low-frequency region of $T\left(\omega \right)$ for different $\gamma$, where $\times$ symbols are the discrete points $T_i$ at ${\omega }_i=\gamma {\mathrm{\Lambda }}^2\times \{-1,-0.8,-0.6,...,1\}$ vs $\gamma /T$ to estimate the smoothness of the curve at $\left|\omega \right|\sim \gamma$. (c) Error ${\varepsilon }_{\mathrm{rms}}$ of quadratic polynomial fit for discrete points vs $\gamma /T$ [cf. Eq. (24)].

Figure 13

$T$-matrix of the SIAM at large temperature ($T\simeq 100T_{\mathrm{K}}$). (a) Broadened curves for various $\gamma$. Due to high temperature, the Kondo peak is completely suppressed while the Hubbard side peaks persist at $\omega =\pm U/2$. Using the self-energy ($\mathrm{\Sigma }$) improves the spectral resolution of the Hubbard peaks. The $\mathrm{\Sigma }$-improved curve for $n_z=2$ at $\gamma /T=0.1$ (purple line) shows a height of the Hubbard peaks that is comparable with a non-$\mathrm{\Sigma }$-improved curve for $n_z=8$ and the same $\gamma /T=0.1$ (not shown). (b) Binned discrete data $\overline{A}/\omega$ for $T\left(\omega \right)$. While the weights at $|{\omega }_k|<T$ spread widely along ${\sigma }_k$ for $T\ll T_{\mathrm{K}}$ [see Fig. 11], here a certain spread also appears at $|{\omega }_k|\gtrsim T$. (c) Same as Fig. 12. (d) Same as Fig. 12.