Energy resolution and discretization artifacts in the numerical renormalization group

We study the limits of the energy resolution that can be achieved in the calculations of spectral functions of quantum impurity models using the numerical renormalization group (NRG) technique with interleaving ($z$ averaging). We show that overbroadening errors can be largely eliminated, that higher-moment spectral sum rules are satisfied to a good accuracy, and that positions, heights and widths of spectral features are well reproduced; the NRG approximates very well the spectral-weight distribution. We find, however, that the discretization of the conduction-band continuum nevertheless introduces artifacts. We present a modified discretization scheme which removes the band-edge discretization artifacts of the conventional approach and significantly improves the convergence to the continuum $\left(\Lambda \to 1\right)$ limit. Sample calculations of spectral functions with high energy resolution are presented. We follow in detail the emergence of the Kondo resonance in the Anderson impurity model as the electron-electron repulsion is increased, and the emergence of the phononic side peaks and the crossover from the spin Kondo effect to the charge Kondo effect in the Anderson-Holstein impurity model as the electron-phonon coupling is increased. We also compute the spectral function of the Hubbard model within the dynamical mean-field theory, confirming the presence of fine structure in the Hubbard bands.

Figure 1

(Color online) Spectral function of the noninteracting model for a range of discretization parameters $\Lambda$ compared with the exact solution, Eq. (18).

Figure 2

(Color online) Spectral function of the symmetric Anderson impurity model for a range of the discretization parameter $\Lambda$.

Figure 3

(Color online) Spectral function of (a) symmetric and (b) asymmetric Anderson impurity model: comparison of raw spectral function with that obtained using the self-energy trick.

Figure 4

(Color online) (a) High-energy artifacts in the spectral function of the resonant-level model. Inset: the spectral function on the first site of the Wilson chain, $f_0$. (b) Spectral function on the first site of the free Wilson chain, $A_{f_0}$, for different values of the discretization parameter $\Lambda$. (c) Spectral function $A_{f_0}$ computed for different values of the spectral patching parameter $p$. (d) Spectral function $A_{f_0}$ in the case of semielliptic DOS, $\rho \left(ϵ\right)={\rho }_0\sqrt{1-{\left(ϵ/D\right)}^2}$. (e) Spectral function $A_{f_0}$ in the case of semielliptic DOS, $\rho \left(ϵ\right)={\rho }_0\sqrt{1-{\left(2ϵ/D\right)}^2}$ with support $\left[-0.5:0.5\right]D$. (f) Comparison of spectral function $A_{f_0}$ computed with the complete-Fock-space NRG approach and the conventional density-matrix NRG approach.

Figure 5

(Color online) Comparison of spectral functions $A_{f_0}$.

Figure 6

(Color online) Analytically computed spectral function $A_{f_0}$ for semielliptical DOS in the conventional and Campo-Oliveira discretization scheme compared with the exact DOS.

Figure 7

(Color online) Comparison between the Campo-Oliveira and the improved discretization (“this work”) approaches.

Figure 8

(Color online) Spectral function of the resonant-level model: comparison between the exact analytical result and two different discretization approaches.

Figure 9

(Color online) Spectral function $A_{f_0}$ in the case of semielliptic DOS computed using the modified discretization scheme.

Figure 10

(Color online) Spectral function of the symmetric Anderson model with semielliptic DOS computed using the modified discretization scheme.

Figure 11

(Color online) Convergence of the expectation value of the level occupancy in the resonant-level model with decreasing $\Lambda$ in different discretization schemes. Dashed lines are fits with quadratic functions which serve to perform a $\Lambda \to 1$ extrapolation.

Figure 12

(Color online) Convergence of the spectral function of the resonant-level model to the exact result with decreasing $\Lambda$ in the case of the proposed discretization scheme.

Figure 13

(Color online) Spectral function of the resonant-level model in the case where the resonance is outside the conduction band.

Figure 14

(Color online) Spectral functions of the Anderson model for increasing $U$.

Figure 15

(Color online) Close-up on the Kondo resonance of (a) symmetric and (b) asymmetric Anderson impurity model and a fit to a Lorentzian (red/gray curve) in the Fermi-liquid regime for $\omega ⪡T_K$.

Figure 16

(Color online) (a) Spectral functions for the Anderson-Holstein model in the particle-hole symmetric case, $\delta =ϵ+U/2=0$. (b) Close-up on the crossover range. (c) Phononic side-peak positions in the crossover range as a function of the electron-phonon coupling.

Figure 17

(Color online) Local spectral functions of the Hubbard model on the hypercubic lattice. Patching parameter $p=2.1$.

Figure 18

Local spectral functions of the Hubbard model on the Bethe lattice. NRG parameters are $\Lambda =1.8$, $p=2.0$, $\eta =0.01$, and $N_z=64$.

Figure 19

(Color online) Spectral function of the symmetric Anderson impurity model: comparison of results obtained using different broadening kernels. Reference results are calculated using Gaussian broadening with sufficiently narrow kernel so that very little change is obtained by further narrowing.