Open Wilson chain numerical renormalization group approach to Green's functions

By combining Wilson's numerical renormalization group (NRG) with a modified Bloch-Redfield approach (BRA), we are able to eliminate the artificial broadening of the Lehmann representation of quantum impurity spectral functions required by the standard NRG algorithm. Our approach is based on the exact reproduction of the continuous coupling function in the original quantum impurity model. It augments each chain site of the Wilson chain by a coupling to an additional reservoir. This open Wilson chain is constructed by a continuous fraction expansion, and the coupling function is treated in second order in the context of the BRA. The eigenvalues of the resulting Bloch-Redfield tensor generate a finite lifetime of the NRG excitations that leads to a natural broadening of the spectral functions. We combine this approach with z-averaging and an analytical exact expression for the correlation part of the self-energy to obtain an accurate representation of the spectral function of the original continuum model in the absence and presence of an external magnetic field.

Figure 1

Spectral function $A\left(\omega \right)$ in dimensionless units $\omega /U$ for a particle-hole symmetric SIAM with $U/\mathrm{\Gamma }=20$, $D/\mathrm{\Gamma }=100$ for $T/T_{\mathrm{K}}=5.75\times {10}^{-3}$ and a Kondo temperature of $T_{\mathrm{K}}/\mathrm{\Gamma }=5.50\times {10}^{-4}$. The NRG parameters are $\mathrm{\Lambda }=2$, $N_s=1000$, and $N=50$. A $z$-averaging with $N_z=12$ has been performed. The solid line represents our hybrid open-chain (OC) approach without artificial broadening, while the dashed line shows the conventional NRG result with a Gaussian broadening parameter $b=0.5$—see the literature [29, 34, 35] for details. The inset shows the Kondo resonance.

Figure 2

Spectral function $A\left(\omega \right)$ in dimensionless units $\omega /U$. Parameters and color coding are identical to Fig. 1. The solid lines represent our hybrid OC approach without artificial broadening, while the dashed lines show the conventional NRG result with a Gaussian broadening parameter $b=0.5$. The black dashed line is a fit for the $U/\mathrm{\Gamma }\to \infty$ Hubbard peak at $\omega =U/2$.

Figure 3

Spectral function $A\left(\omega \right)$ in dimensionless units $\omega /T_{\text{K}}$ with a logarithmic horizontal axis. Parameters are identical to Fig. 1. The solid lines represent our hybrid approach without artificial broadening, while the dashed lines show the conventional NRG result with a Gaussian broadening parameter $b=0.5$. The effect of $z$-averaging is shown by variation of the number $N_z$ of configurations.

Figure 4

Spectral function $A\left(\omega \right)$ of the minority spin in dimensionless units vs $\omega /U$ in a magnetic field of (a) $H/T_{\text{K}}=5$ and (b) $H/T_{\text{K}}=100$, respectively. Parameters and color coding as in Fig. 1.

Figure 5

Data of Fig. 4 plotted vs $\omega /T_{\text{K}}$, (a) $H/T_{\text{K}}=5$ and (b) $H/T_{\text{K}}=100$. Parameters and color coding as in Fig. 1.