From the Anderson Model to Kondo Exchange

In the local-moment regime, a Schrieffer–Wolff transformation removes virtual empty and doubly occupied Anderson states order by order in the hybridization. To second order it produces antiferromagnetic Kondo exchange and potential scattering. The signs and denominators are fixed by the two charge gaps; the reduction fails in mixed valence, where those charge states are not high-energy degrees of freedom.

Required background. The Anderson impurity model supplies the atomic sectors and hybridization.

Helpful background. Integrating out heavy fields supplies the general matching logic.

Canonical transformation

Split $H=H_0+H_V$, where $H_V$ changes the impurity charge. Choose an anti-Hermitian generator $S=O(V)$ satisfying

$$[S,H_0]=-H_V.$$

Then

$$e^SHe^{-S}=H_0+\frac12[S,H_V]+O(V^3).$$

Project the commutator onto the singly occupied impurity subspace. For momenta near the Fermi surface and slowly varying $V_k\simeq V$, the result is

$$H_{\mathrm{eff}}=\sum_{k\sigma}\varepsilon_kc_{k\sigma}^\dagger c_{k\sigma} +J\,\mathbf S_d\mathbin{\cdot}\mathbf s(0) +K\sum_\sigma c_{0\sigma}^\dagger c_{0\sigma},$$

with $\mathbf s(0)=\frac12c_0^\dagger\boldsymbol\sigma c_0$ and

$$J=2\lvert V\rvert^2 \left(\frac1{-\varepsilon_d}+\frac1{\varepsilon_d+U}\right)>0,$$ $$K=\frac{\lvert V\rvert^2}{2} \left(\frac1{-\varepsilon_d}-\frac1{\varepsilon_d+U}\right).$$

These prefactors follow the stated spin-density convention. Sources defining $c^\dagger\boldsymbol\sigma c$ without $1/2$ print an exchange smaller by two. At particle–hole symmetry $\varepsilon_d=-U/2$, $K=0$, while $J=8\lvert V\rvert^2/U$.

Origin of the antiferromagnetic sign

An incoming conduction electron can exchange spin with the impurity through either an empty intermediate state of cost $E_-=-\varepsilon_d$ or a doubly occupied state of cost $E_+=\varepsilon_d+U$. Both paths lower the singlet relative to the triplet, so their positive inverse denominators add in $J$. They subtract in the spin-independent amplitude $K$ because particle and hole paths carry opposite potential-scattering contributions.

This is matching, not RG. $J$ is defined at an upper cutoff

$$D_0\lesssim\min(D_{\mathrm{band}},E_-,E_+),$$

after charge fluctuations above that scale have been removed. Running $J$ to the Kondo scale is the job of the next page. Using the full bandwidth when a charge gap is smaller double-counts degrees of freedom.

Schrieffer and Wolff 1966, pp. 491–492 gives the transformation and exchange. Observables must also be transformed: $O_{\mathrm eff}=P(e^SOe^{-S})P$. A local charge or tunnelling operator generally acquires conduction-electron pieces, so matching the Hamiltonian alone is insufficient for spectroscopy.

Validity and extensions

Require $\Gamma/E_\pm\ll1$ and external energies below both charge gaps. Degenerate orbitals can yield Coqblin–Schrieffer exchange, orbital Kondo effects, Hund-coupled moments, or several anisotropic channels; their generators and projectors must be constructed in the full multiplet space. Superconducting or pseudogap baths change the subsequent infrared flow but not the need for controlled charge elimination.

The structure map makes the matching step and its cutoff explicit.

Schrieffer–Wolff matching converts virtual charge fluctuations into low-energy exchange. Its denominators, potential scattering, transformed observables, and matching cutoff are part of the result. Original schematic, not to scale.

See the impurity claim test matrix for the mixed-valence failure test.

Exercise

Evaluate the symmetric point. Put $\varepsilon_d=-U/2$ into the amplitudes and explain the result.

Solution

Both charge gaps equal $U/2$. Thus $J=2\lvert V\rvert^2(2/U+2/U)=8\lvert V\rvert^2/U$, while the two terms in $K$ cancel. Exchange remains antiferromagnetic because the singlet-lowering virtual processes add. The cancellation of $K$ relies on particle–hole symmetry and is lost when the level is detuned.

References

• Schrieffer, J. R., and Wolff, P. A. (1966). “Relation between the Anderson and Kondo Hamiltonians.” Physical Review 149, 491–492. doi:10.1103/PhysRev.149.491.