Numerical renormalization group approach to Green’s functions for quantum impurity models

We present a technique for the calculation of dynamical correlation functions of quantum impurity systems in equilibrium with Wilson’s numerical renormalization group. Our formulation is based on a complete basis set of the Wilson chain. In contrast to all previous methods, it does not suffer from overcounting of excitations. By construction, it always fulfills sum rules for spectral functions. Furthermore, it accurately reproduces local thermodynamic expectation values, such as occupancy and magnetization, obtained directly from the numerical renormalization group calculations.

Figure 1

(Color online) The full Wilson chain of length $N$ is divided into a subchain of length $m$ and the “environment” $R_{m,N}$. The Hamiltonian ${\mathcal{H}}_m$ can be viewed either as acting only on the subchain of length $m$, or as acting on the full chain of length $N$, but with the hopping matrix elements $t_m,\dots ,t_{N-1}$ all set to zero. The former picture is the traditional one. In this paper, we adopt the latter point of view.

Figure 2

(Color online) Comparison of the spectral function for the three different methods for the symmetric case ${ϵ}_f∕D=-U∕D2=0.5$. The inset on the left side emphasizes the broad charge excitation at ${ϵ}_f$, the inset on the right side zooms in at the Abrikosov-Suhl resonance (ASR). NRG parameters: $\Gamma ∕D=\pi V^2{\rho }_0∕D=0.1$, $\Lambda =2.5$, $N_S=500$, $b=0.8$, and $T=0$.

Figure 3

(Color online) Comparison of the spectral function for the three different methods for the same model and NRG parameters as in Fig. 2 but with only $N_S=50$ states kept in each NRG iteration. The inset zooms in onto the ASR.

Figure 4

(Color online) Comparison of the spectral function obtained from CFS and HA approaches in a finite magnetic field (a) $H=0.005$ and (b) $H=0.1$. The inset in (a) is a zoom into the region around the Fermi level showing the splitting and suppression of the Kondo resonance. Model and NRG parameters as in Fig. 3. The norm and occupancies obtained are compared in Table .

Figure 5

(Color online) Comparison of the spectral function for the three different methods for the asymmetric case (a) $U∕D=1$ and ${ϵ}_f∕D=-0.9$, and (b) $U∕D=1000$ and ${ϵ}_f∕D=-0.4$. NRG parameters: $\Gamma ∕D=\pi V^2{\rho }_0∕D=0.1$, $\Lambda =2.5$, and $N_S=500$. The results for norm and occupancies obtained are collected in Table .

Figure 6

(Color online) Spectral function for temperatures $T=0$, $0.00453\approx T_K$, 0.01133 and $0.112⪢T_K$ calculated with CFS. The solid curves result from calculations $N_S=500$, the dashed ones from calculations with $N_S=50$. The upper panel shows the full frequency spectrum, the lower panel the magnification of the vicinity around $\omega =0$. The arrow in the lower panel indicates the increasing temperature. Model and NRG parameters are otherwise the same as in Fig. 2.

Figure 7

(Color online) Spectral function for temperatures $T=0$, $0.00453\approx T_K$, 0.011 33 and $0.112⪢T_K$, and finite magnetic field $H=0.005$ calculated with CFS. The right panel shows an enlarged view of the region around the Fermi energy $\omega =0$. Model and NRG parameters are otherwise the same as in Fig. 3. The values for norm and occupancies are listed in Table .