The Anderson Impurity Model and Hybridization

The single-impurity Anderson model describes a correlated local orbital whose empty, singly occupied, and doubly occupied states hybridize with a fermionic bath. The competition among level energy $\varepsilon_d$, repulsion $U$, and width $\Gamma$ distinguishes empty-orbital, mixed-valence, and local-moment regimes. Hybridization produces both lifetime broadening and virtual charge fluctuations; only in the local-moment window can those fluctuations be eliminated in favor of a Kondo exchange.

Required background. Impurity models and local moments supplies the bath, channel, and scale contract.

Helpful background. Dyson equations supplies the hybridization self-energy.

Model and exact noninteracting checkpoint

For one spinful orbital,

$$H=\sum_{k\sigma}\varepsilon_kc_{k\sigma}^\dagger c_{k\sigma} +\varepsilon_d\sum_\sigma n_{d\sigma} +Un_{d\uparrow}n_{d\downarrow} +\sum_{k\sigma}(V_kc_{k\sigma}^\dagger d_\sigma+\mathrm{h.c.}).$$

The atomic energies are $E_0=0$, $E_\sigma=\varepsilon_d$, and $E_2=2\varepsilon_d+U$. With the Fermi level at zero, single occupancy is lowest when

$$\varepsilon_d<0<\varepsilon_d+U.$$

For $U=0$, the retarded propagator is exactly

$$G_d^R(\omega)= \frac{1}{\omega-\varepsilon_d-\Delta^R(\omega)}, \qquad \mathcal A_d(\omega)=-\frac1\pi\operatorname{Im}G_d^R(\omega).$$

In a wide flat band, $\Delta^R\simeq-i\Gamma$ with $\Gamma=\pi\rho\lvert V\rvert^2$, so $\mathcal A_d$ is a Lorentzian of unit integrated weight. It is related to the volume’s correlator convention by $A_d=2\pi\mathcal A_d$. This is the first solver test: the sign of $\operatorname{Im}\Delta^R$ must be negative and $\int \mathcal A_d\,\mathrm d\omega=1$ per spin.

Charge and moment regimes

Define the charge-removal and addition scales from the singly occupied sector,

$$E_-=-\varepsilon_d, \qquad E_+=\varepsilon_d+U.$$

The local-moment limit requires $E_\pm\gg\Gamma,T$. Then $\langle n_d\rangle\simeq1$, the charge susceptibility is small, and an intermediate-temperature Curie response appears. At particle–hole symmetry $\varepsilon_d=-U/2$, potential scattering vanishes after matching, but the Kondo resonance at low temperature still represents a many-body scale rather than a bare orbital.

If either $E_-$ or $E_+$ is of order $\Gamma$, charge sectors overlap: this is mixed valence. A Schrieffer–Wolff spin-only model then omits active states and can give misleading Kondo scales. Empty-orbital and doubly occupied regimes occur when the corresponding atomic sector is well below the others.

The interacting spectrum contains broad charge-transfer features near $\varepsilon_d$ and $\varepsilon_d+U$ and, in the screened metallic regime, a low-energy resonance. Spectral peak positions and weights depend on the bath, temperature, and self-energy; the three-peak cartoon is not a theorem for every parameter set.

Anderson 1961, §§2–4 introduced the model and moment criterion; Hewson 1993, chs. 2–4 develops the hybridization, charge regimes, and low-energy reduction. A quantum dot maps to the model only after charging energies, level spacing, lead modes, and voltage coupling are calibrated.

Wide-band limits and failures

The wide-band approximation assumes $\Gamma(\omega)$ and the real hybridization shift vary slowly over the impurity scales. It fails near a band edge, superconducting gap, pseudogap, van Hove singularity, or structured mesoscopic lead. Finite bandwidth also sets the upper cutoff for Kondo matching.

The structure diagram marks the Anderson model as the charge-fluctuating parent of the Kondo model.

The ratios $E_-/\Gamma$ and $E_+/\Gamma$ decide whether charge fluctuations are virtual or active. A spin-only Kondo reduction is controlled only on the local-moment branch. Original schematic, not to scale.

The impurity claim test matrix compares the regimes and diagnostics.

Exercise

Classify three points. For $U=10\Gamma$, classify (a) $\varepsilon_d=-5\Gamma$, (b) $\varepsilon_d=-0.5\Gamma$, and (c) $\varepsilon_d=2\Gamma$ at $T\ll\Gamma$.

Solution

(a) has $E_-=E_+=5\Gamma$ and is a reasonably developed, particle–hole-symmetric local-moment regime, though asymptotic control improves at still larger ratios. (b) has $E_-=0.5\Gamma$ and active empty/singly occupied fluctuations, so it is mixed valent. (c) places the empty state below the singly occupied state and is empty-orbital. A label based only on $U/\Gamma$ would miss (b) and (c).

References

• Anderson, P. W. (1961). “Localized magnetic states in metals.” Physical Review 124, 41–53. doi:10.1103/PhysRev.124.41.
• Hewson, A. C. (1993). The Kondo Problem to Heavy Fermions. Cambridge University Press. doi:10.1017/CBO9780511470752.