Kondo physics of the Anderson impurity model by distributional exact diagonalization

The distributional exact diagonalization (DED) scheme is applied to the description of Kondo physics in the Anderson impurity model. DED maps Anderson's problem of an interacting impurity level coupled to an infinite bath onto an ensemble of finite Anderson models, each of which can be solved by exact diagonalization. An approximation to the self-energy of the original infinite model is then obtained from the ensemble-averaged self-energy. Using Friedel's sum rule, we show that the particle number constraint, a central ingredient of the DED scheme, ultimately imposes Fermi liquid behavior on the ensemble-averaged self-energy, and thus is essential for the description of Kondo physics within DED. Using the numerical renormalization group (NRG) method as a benchmark, we show that DED yields excellent spectra, both inside and outside the Kondo regime for a moderate number of bath sites. Only for very strong correlations ($U/\mathrm{\Gamma }\gg 10$) does the number of bath sites needed to achieve good quantitative agreement become too large to be computationally feasible.

Figure 1

Schematic representation of the DED method. The noninteracting density of the impurity level ${\rho }_0\left(\omega \right)$ is interpreted as a probability distribution for the poles of the noninteracting Green's function $G_0\left(\omega \right)$. A finite number of poles $b_i^{\nu }$ is then generated randomly according to the distribution ${\rho }_0\left(\omega \right)$. The selected $n$ poles uniquely define a finite Anderson model $H^{\nu }$ with $n-1$ bath sites. Diagonalization of $H^{\nu }$ yields the self-energy ${\mathrm{\Sigma }}^{\nu }\left(\omega \right)$ corresponding to the finite Anderson model. This process is repeated many times ($N$). An approximation to the self-energy of the original infinite Anderson model (1) is obtained from the ensemble average of the self-energies of the finite Anderson model samples (13).

Figure 2

Effect of constraint: (a) Comparison of spectra calculated with (blue line) and without (red dashed line) imposing the particle number constraint for $n=4$ sites, $\mathrm{\Gamma }=0.3, U=3, {\epsilon }_d=-U/2$. (b) Comparison of sampled noninteracting DOS ${\overline{\rho }}_0\left(\omega \right)=-\mathrm{Im}{\overline{G}}_0\left(\omega \right)/\pi$ for different number of sites with original Lorentzian noninteracting DOS ${\rho }_0\left(\omega \right)$.

Figure 3

Comparison between DED and NRG spectra for the symmetric Anderson model ($U=3, {\epsilon }_d=-1.5$). (a) DED ($n=8$) and NRG spectra for $\mathrm{\Gamma }=0.3$. (b) NRG and DED spectra with different number of sites $n$ on a half-log scale for $\mathrm{\Gamma }=0.3$. (c) DED ($n=6$) and NRG spectra for $\mathrm{\Gamma }=0.9$. (d) DED ($n=6$) and NRG spectra for $\mathrm{\Gamma }=0.5$. (e) DED ($n=8$) and NRG spectra for $\mathrm{\Gamma }=0.2$. (f) NRG and DED spectra for different number of sites $n$ on a half-log scale for $\mathrm{\Gamma }=0.2$. (g) Half-width of Kondo peak estimated by fitting with Frota line shapes [45] versus $\mathrm{\Gamma }$ calculated by DED ($n=8$ for $\mathrm{\Gamma }\le 0.3$ and $n=6$ for $\mathrm{\Gamma }\ge 0.5$) compared to NRG and the exact expression [46] on half-log scale. (h) Half-width of Kondo peak versus number of sites $n$ compared to NRG for $\mathrm{\Gamma }=0.3$.

Figure 4

Comparison between DED and NRG spectra for the asymmetric Anderson model ($U=3, {\epsilon }_d<-1.5, \mathrm{\Gamma }=0.3$). (a) DED (for $n=8$ sites) and NRG spectra for ${\epsilon }_d=-2$. (b) Half-width of Kondo peak versus number of sites $n$ compared to NRG (${\epsilon }_d=-2$). (c) NRG and DED spectra with different number of sites $n$ for ${\epsilon }_d=-2$ on a half-log scale for negative energies. (d) NRG and DED spectra with different number of sites $n$ for ${\epsilon }_d=-2$ on a half-log scale for positive energies. (e) DED (for $n=8$ sites) and NRG spectra for ${\epsilon }_d=-2.5$. (e) DED (for $n=8$ sites) and NRG spectra for ${\epsilon }_d=-3$.

Figure 5

Comparison of interacting (blue) and noninteracting (red) finite Anderson model spectra ($n=2$ sites) for the case that (a) the particle constraint is not fulfilled, and (b) when it is fulfilled. Only in the latter case a 1:1 correspondence between the interacting and noninteracting system can be established.

Figure 6

Spectral function close to the Fermi level for different values of ${\mathrm{\Sigma }}_0$ calculated by DED ($n=4$ sites) compared to NRG for $U=3, {\epsilon }_d=-3$, and $\mathrm{\Gamma }=0.3$ on a large energy scale (left) and at low energies (right). Note that on the left panel the curves for ${\mathrm{\Sigma }}_0=-{\epsilon }_d$ (green) and for ${\mathrm{\Sigma }}_0=V_{\mathrm{HF}}$ (cyan) have been offset by 0.2 and 0.4, respectively, in order to increase the visibility.