Dynamical properties of the Anderson impurity model within a diagrammatic pseudoparticle approach

The Anderson model of a twofold spin degenerate impurity level in the limit of infinite Coulomb repulsion, $U\to \infty$, coupled to one and two degenerate conduction bands or channels, is considered in pseudo-particle representation. We extend the conserving T-matrix approximation (CTMA), a general diagrammatic approximation scheme based on a fully renormalized computation of two-particle vertex functions in the spin and in the charge channel, to the calculation of thermodynamic and spectral properties. In the single-channel case, the CTMA yields in the Kondo regime a temperature independent Pauli spin susceptibility for temperatures below the Kondo temperature $T_K$ and down to the lowest temperatures considered, reproducing the exact spin screening in the Fermi liquid state. The impurity spectral density appears to remain non-singular down to the lowest temperatures, in agreement with Fermi liquid behavior. However, the unitarity sum rule, which is crucial for an impurity solver like the CTMA to be applicable within dynamical mean field theories for strongly correlated lattice models, is overestimated at the lowest temperatures. We argue that this shortcoming may be due to numerical imprecision and discuss an appropriate scheme for its correction. In the two-channel case, the spectral density calculated within CTMA exhibits qualitatively the correct non-Fermi liquid behavior at low temperatures, i.e., a power law singularity.

Figure 1

Diagrammatic representation of the Bethe-Salpeter equations for the vertex functions $T^{\left(cf\right)}$ and $T^{\left(cb\right)}$ defining the CTMA. Dashed, wiggly, and solid lines represent here and in the following the renormalized pseudofermion, slave boson, and the local conduction electron propagators, respectively. The external lines are for clarity only and are not part of the vertices.

Figure 2

Generating functional of the CTMA. The first diagram corresponds to the NCA generating functional. Each diagram contains exactly one closed auxiliary particle loop; diagrams with more than one such loop vanish after projection onto $Q=1$. The class of CTMA diagrams beyond NCA is uniquely defined by the condition that each conduction electron line spans exactly two bare three-point hybridization vertices, as shown. The diagrams of the first and of the second columns generate, by second functional derivative, ${\delta }^2\Phi ∕\delta G_c\delta G_f$ and ${\delta }^2\Phi ∕\delta G_c\delta G_b$, the spin and the charge fluctuation T-matrices, $T^{\left(cf\right)}$, $T^{\left(cb\right)}$, respectively (Fig. 1).

Figure 3

Diagrams containing the leading logarithmic contributions to the $c\text{−}f$ vertex function. In contrast to Figs. 1, 2, the lines represent the bare Green’s functions here. a) Contribution from the NCA generating functional, b) additional contribution from the CTMA. Spurious logarithmic potential scattering terms cancel only when both terms are added. The $c\text{−}b$ vertex has no logarithmic terms and, hence, does not flow under perturbative RG.

Figure 4

The static magnetic susceptibility as a function of temperature. The temperature and the susceptibility scale are plotted in units of the conventional Kondo temperature $T_K^{\star }$ Eq. (13), that leads to the value $W=0.4128$ for the Wilson number; see the text. The CTMA curve agrees well with the exact BA result in both the high temperature and in the strong coupling regimes, with only a small spurious intermediate maximum in the crossover region.

Figure 5

Diagrams defining the $d$-electron Green’s function within CTMA. For details of their evaluation see Appendix A.

Figure 6

CTMA and NCA results for the local spectral density $A_{d\sigma }\left(\omega \right)$ for ${ϵ}_d∕D=-0.81$, $\Gamma ∕D=0.2$ at $T=T_K={6.1610}^{-4}D$. The decomposition of the CTMA result into $f\text{−}b$ bubble and vertex corrections arising from $T^{\left(cf\right)}$ (fc vertex) and $T^{\left(cb\right)}$ (bc vertex) is also shown. The NCA shows an incorrectly large shift of the Kondo peak of $O\left(T_K\right)$ toward positive frequencies, which is corrected by CTMA; see the text.

Figure 7

CTMA impurity spectral function and decomposition into $f\text{−}b$ bubble and vertex contributions for the same parameters as in Fig. 6, but at $T=0.01T_K$. The exact values for $A_{d\sigma }\left(0\right)$, $\tilde{\Gamma }$, ${\widetilde{ϵ}}_d$, Eqs. (16, 17, 18), are displayed for comparison.

Figure 8

Correlation part of the physical $d$-electron self-energy $\mathrm{Im}{\Sigma }_{d\sigma }\left(\omega \right)$ calculated in CTMA for various temperatures and ${ϵ}_d∕D=-0.81$, $\Gamma ∕D=0.2$. For the lowest temperature shown, $T=0.16T_K$, the NCA result (dot-dashed line) is shown for comparison, showing the cusp-like infrared singularity typical for NCA. The bold solid line centered at the origin represents the $T=0$ behavior expected for our parameter set based on Fermi liquid theory, Eqs. (19, 20).

Figure 9

${\mathrm{Log}}_{10}{\mathrm{-log}}_{10}$ plot of $\mathrm{Im}{\Sigma }_{d\sigma }(\omega -i0)-\mathrm{Im}{\Sigma }_{d\sigma }({\omega }_0-i0)$ versus frequency, $(\omega -{\omega }_0)∕T_K$, for $T=0.16T_K$. ${\omega }_0$ is the position of the minimum of $\mathrm{Im}{\Sigma }_{d\sigma }(\omega -i0)$ in Fig. 8. The dashed line is the fit to the low-frequency quadratic behavior of $\mathrm{Im}{\Sigma }_{d\sigma }(\omega -i0)$ and represents the function $y=0.02440{(\omega -{\omega }_0)}^2∕T_K^2$; see the text.

Figure 10

${\mathrm{Log}}_{10}{\mathrm{-log}}_{10}$ plot of the minimum value of the imaginary part of the self-energy, $\mathrm{Im}{\Sigma }_{d\sigma }(\omega ={\omega }_{min}(T),T)-\mathrm{Im}{\Sigma }_{d\sigma }(\omega ={\omega }_0,T=0.01T_K)$ versus $T∕T_K$. ${\omega }_{min}\left(T\right)$ is the position of the minimum for a given temperature $T$; see Fig. 8. Quadratic in $T$ behavior is clearly visible for $T\lesssim T_K$. The dashed line is given by $y=0.013D{\left(\pi T∕T_K\right)}^2$.

Figure 11

CTMA (solid line) and NCA (dotted line) result for the local spectral density in the empty impurity regime, ${ϵ}_d∕D=+0.81$, $\Gamma ∕D=0.2$ for $T={1.010}^{-4}D$. The resulting occupation per spin is $n_{d\sigma }=0.037$. The insets show the CTMA (left panel) and the NCA (right panel) spectral functions over the complete band width.

Figure 12

CTMA spectral functions with a constant potential scattering term added to the real part of ${\Sigma }_{d\sigma }(\omega -i0)$ according to the effective potential scattering method; see the text. No low-energy singularity occurs, and the unitarity limit (arrow) is accurately fulfilled at the lowest temperatures.

Figure 13

Real and imaginary part of the total impurity self-energy ${\Sigma }_{d\sigma }^{tot}(\omega -i0)$, corrected by the effective potential scattering method. The curvature of $\mathrm{Im}{\Sigma }_{d\sigma }^{tot}(\omega -i0)$ at the Fermi energy does not change considerably from $T=0.16T_K$ down to $T=0.01T_K$. The small bump in the curve for $T=0.01T_K$ at positive $\omega$ is attributed to numerical inaccuracy. At $\omega =0,T\to 0$, $\mathrm{Im}{\Sigma }_{d\sigma }^{tot}(0-i0)$ assumes the effective hybridization ${\Gamma }_{eff}\approx 0.139D$, somewhat smaller than the exact value $\Gamma =0.2D$. ${\Gamma }_{eff}$ is to be subtracted from $\mathrm{Im}{\Sigma }_{d\sigma }^{tot}(\omega -i0)$ in order to obtain the interaction part of the local self-energy.

Figure 14

CTMA results for the impurity spectral density of the two-channel SIAM, with ${ϵ}_d=-0.81D,\Gamma =0.2D$ at various temperatures. The NCA result is shown for comparison. The $\mid \omega {\mid }^{1∕2}$ cusp develops at the Fermi level, with a weight asymmetry due to potential scattering. The inset shows the spectrum over a wider frequency range.

Figure 15

Diagrammatic representation of the integral equations comprising the T-matrices that enter in the expressions for the auxiliary particle self-energies, see the text for details.

Figure 16

Auxiliary particle self-energies ${\Sigma }_f$ and ${\Sigma }_b$: The first diagram is the NCA contribution, where however fully CTMA-renormalized Green functions have to be used. The second and third diagrams constitute the vertex corrections from $T^{\left(fc\right)}$ and $T^{\left(bc\right)}$.

Figure 17

Diagrammatic representation of the equation for the local $d$-electron Green function. The first diagram is the NCA contribution. As described in the text, because of the analytical structure of the Adsigma it is necessary to multiply with a rung at each end of the vertex functions $V_{(\pm )}^{\left(cf\right)}$, $V_{(\pm )}^{\left(cb\right)}$, as shown in the last two diagrams on the right-hand side. Hence the sum of these diagrams contains at least four rungs, i.e., the three-rung term must be added separately (second diagram on the right-hand side.)