Strong-Coupling Screening, Phase Shifts, and the Friedel Sum Rule

The one-channel spin-$1/2$ Kondo model does not remain a gas of weakly coupled moments as the exchange grows. Its impurity spin and one conduction-electron combination form a singlet, and the surviving low-energy particles scatter from a new boundary condition. At the spin-symmetric, particle–hole-symmetric fixed point that boundary condition is a phase shift $\delta_\sigma=\pi/2$ modulo $\pi$. The Friedel sum rule turns the phase shift into a count of all charge displaced by the impurity, while Nozières’ local Fermi liquid organizes the first corrections. Quadratic frequency dependence is special to the unitary point: away from particle–hole symmetry, a linear spectral term is generally allowed even though the ground state remains a Fermi liquid.

Required background. Kondo RG flow supplies the strong-coupling destination.

Helpful background. Landau Fermi-liquid theory supplies quasiparticle scattering and thermodynamic parameters.

From the screened singlet to a π/2 phase shift

Only the local, $s$-wave combination of a three-dimensional metal reaches a point impurity; after radial reduction it behaves like a one-dimensional channel ending at the impurity. In the formal $J\to+\infty$ limit, the impurity spin and the electron on the end orbital make

$$\lvert s\rangle=\frac{1}{\sqrt2} \left(\lvert\uparrow\rangle_d\lvert\downarrow\rangle_0 -\lvert\downarrow\rangle_d\lvert\uparrow\rangle_0\right).$$

Low-energy electrons cannot occupy that singlet orbital. Relative to the free chain, their standing-wave quantization is shifted by half a level. Equivalently, the elastic quasiparticle $S$-matrix at the Fermi energy is

$$S_\sigma(0)=e^{2i\delta_\sigma}.$$

The changed boundary condition gives $S_\sigma(0)=-1$, hence $\delta_\sigma=\pi/2$ modulo $\pi$. This strong-coupling picture is not a controlled expansion at finite $J$, but the one-channel antiferromagnetic model is adiabatically connected to it: no symmetry or level crossing intervenes between the Kondo limit and the infrared fixed point. Potential scattering adds a nonuniversal charge-sector phase without destroying the local Fermi liquid.

The phrase “modulo $\pi$” matters. Scattering determines $e^{2i\delta}$, so $\delta$ and $\delta+\pi$ are indistinguishable in $S$. A charge count is not modulo anything. Fix its branch by following the phase continuously from a stated reference, such as the empty-orbital limit of the Anderson model.

Friedel counting and spectral pinning

Put a large radial system in a box. If an allowed wave number obeys $k_nL+\delta_\sigma(k_n)=n\pi$, then turning on a one-body scatterer shifts the number of levels below the Fermi surface by $\delta_\sigma(0)/\pi$. For an interacting Anderson impurity, define the displaced charge by

$$n_{\mathrm{imp},\sigma} =\langle n_{d\sigma}\rangle +\sum_p\left( \langle c_{p\sigma}^\dagger c_{p\sigma}\rangle -\langle c_{p\sigma}^\dagger c_{p\sigma}\rangle_0 \right)$$

This is the impurity-induced change in the whole system, not just the charge on the local orbital. In Green-function language,

$$n_{\mathrm{imp},\sigma} =\langle n_{d\sigma}\rangle +\frac{1}{\pi}\operatorname{Im}\int_{-\infty}^{0} \mathrm d\omega\, G^R_{d\sigma}(\omega)\,\partial_\omega\Delta^R_\sigma(\omega).$$

With the Dyson convention $[G^R_{d\sigma}]^{-1}=\omega^+-\varepsilon_d-\Delta^R_\sigma-\Sigma^R_\sigma$ and the phase chosen continuously from the empty-orbital branch, the generalized Friedel–Luttinger identity is

$$\delta_\sigma(0) =\pi n_{\mathrm{imp},\sigma}+I_{L,\sigma}\pmod\pi, \qquad I_{L,\sigma}=\operatorname{Im}\int_{-\infty}^{0} \mathrm d\omega\, G^R_{d\sigma}(\omega)\,\partial_\omega\Sigma^R_\sigma(\omega).$$

For the ordinary adiabatically connected Fermi liquid, $I_{L,\sigma}=0$. A wide, flat hybridization also has $\partial_\omega\Delta^R=0$, so $n_{\mathrm{imp},\sigma}=\langle n_{d\sigma}\rangle$ and the familiar shortcut $\delta_\sigma=\pi\langle n_{d\sigma}\rangle$ follows on the continuously chosen branch. With energy-dependent hybridization, a pseudogap, a degenerate non-Fermi-liquid ground state, or a nonzero Luttinger integral, one or both steps can fail. Langreth 1966, pp. 516–518 proves the interacting Anderson-model relation under its stated analyticity assumptions; Mitchell, Logan, and Krishnamurthy 2011, §III E, Eqs. (58)–(62) explicitly keeps the displaced bath charge and Luttinger integral in the generalized identity.

For a spin-symmetric screened Kondo impurity, one electron is displaced in total. Equal spin sectors therefore give $n_{\mathrm{imp},\sigma}=1/2$ and recover $\delta_\sigma=\pi/2$. For a wide-band Anderson orbital with retarded hybridization $\Delta^R(0)=-i\Gamma$, the same phase fixes the zero-energy local spectrum,

$$\mathcal A_{d\sigma}(0) =-\frac{1}{\pi}\operatorname{Im}G^R_{d\sigma}(0) =\frac{\sin^2\delta_\sigma}{\pi\Gamma}.$$

Thus particle–hole symmetry pins $\pi\Gamma\mathcal A_{d\sigma}(0)=1$. This is a zero-energy statement, not a claim that the entire Kondo resonance is a noninteracting Lorentzian.

Local Fermi-liquid expansion

Nozières describes the fixed point using quasiparticles that already include the constant phase shift. Their residual scattering is encoded in a phase-shift functional. To first order in energy and in the change of the opposite-spin distribution,

$$\delta_\sigma(\varepsilon,\delta n) =\delta_0+\alpha_1\varepsilon -\phi_1\int\mathrm d\varepsilon'\,\delta n_{\bar\sigma}(\varepsilon')+\cdots.$$

Here $\alpha_1$ measures the elastic energy dependence and $\phi_1$ the residual interaction with an added opposite-spin quasiparticle; both have dimensions of inverse energy. If the reference Fermi level is shifted by $\delta\mu$, then $\varepsilon$ and the filled quasiparticle sea both shift. The physical phase cannot depend on this bookkeeping choice. At the symmetric Kondo fixed point the two first-order changes cancel, giving $\alpha_1=\phi_1\sim T_K^{-1}$. Away from particle–hole symmetry, two additional second-order parameters are needed. Nozières 1974, pp. 31–42 constructs the local Fermi liquid; Mora et al. 2015, §§II–III gives the asymmetric Anderson-model expansion in terms of susceptibilities.

To see why the unitary spectrum begins quadratically, first isolate its elastic origin. At $\omega=T=0$, scattering at a local-Fermi-liquid fixed point is elastic, so define

$$S_\sigma=1-2\pi i\rho\,T_\sigma=e^{2i\delta_\sigma}, \qquad -\pi\rho\operatorname{Im}T_\sigma=\sin^2\delta_\sigma,$$

with $\rho$ the bath density of states per spin in the active channel. At nonzero energy or temperature, residual interactions also open inelastic channels, and the full one-particle $S$ matrix is no longer a pure phase. Expanding only the elastic phase-shift contribution, write $\delta_\sigma=\delta_0+\Delta\delta_\sigma$. Then

$$\sin^2(\delta_0+\Delta\delta) =\sin^2\delta_0 +\sin(2\delta_0)\Delta\delta +\cos(2\delta_0)(\Delta\delta)^2+\cdots.$$

At $B=0$ and the particle–hole-symmetric unitary point, $\delta_0=\pi/2$: the linear elastic term vanishes, while elastic curvature and inelastic quasiparticle collisions together give the full low-energy $T$ matrix

$$-\pi\rho\operatorname{Im}T(\omega,T) =1-c_\omega\frac{\omega^2}{T_K^2} -c_T\frac{(\pi k_{\mathrm B}T)^2}{T_K^2}+\cdots.$$

The positive constants depend on the definition of $T_K$. A small bias produces another even correction proportional to $(eV/T_K)^2$. If potential scattering or Anderson-level asymmetry makes $\delta_0\ne\pi/2$, the $\sin(2\delta_0)\alpha_1\omega$ term is generally present. “Fermi liquid” therefore means analytic low-energy scattering; it does not by itself mean an even spectrum.

Wilson ratio and quantum-dot conductance

Two thermodynamic coefficients probe the same local quasiparticles. With $\chi_{\mathrm{imp}}$ the impurity contribution to the uniform spin susceptibility and $\gamma_{\mathrm{imp}}=\lim_{T\to0}C_{\mathrm{imp}}/T$, define

$$R_W=\frac{4\pi^2k_{\mathrm B}^2}{3(g\mu_{\mathrm B})^2} \frac{\chi_{\mathrm{imp}}}{\gamma_{\mathrm{imp}}}.$$

In the normalization of the phase-shift functional above, the spin-symmetric flat-band Ward identities give $R_W=1+\phi_1/\alpha_1$. Thus $R_W=1$ for noninteracting spin-$1/2$ quasiparticles, where $\phi_1=0$, while suppressed charge fluctuations in the Kondo limit give $\phi_1=\alpha_1$ and $R_W\to2$. Mora et al. 2015, Supplemental §S-II derives these relations from the spin and charge susceptibilities. A quoted number is meaningful only when the bulk subtraction, $g$ factor, and definition of $\chi_{\mathrm{imp}}$ are the same in numerator and reference.

For a dot coupled proportionally to left and right leads, form even and odd combinations

$$c_{e}=\frac{\sqrt{\Gamma_L}c_L+\sqrt{\Gamma_R}c_R}{\sqrt{\Gamma_L+\Gamma_R}}, \qquad c_{o}=\frac{\sqrt{\Gamma_R}c_L-\sqrt{\Gamma_L}c_R}{\sqrt{\Gamma_L+\Gamma_R}}.$$

Only $c_e$ couples to the dot; $c_o$ is a spectator. Recombining an even wave with phase $e^{2i\delta}$ and an unchanged odd wave gives the transmission probability

$$\mathcal T(0)= \frac{4\Gamma_L\Gamma_R}{(\Gamma_L+\Gamma_R)^2}\sin^2\delta.$$

Landauer’s formula then yields

$$G=\frac{2e^2}{h} \frac{4\Gamma_L\Gamma_R}{(\Gamma_L+\Gamma_R)^2} \sin^2\delta.$$

At $\delta=\pi/2$ and $\Gamma_L=\Gamma_R$, $G=2e^2/h$. Lead asymmetry lowers the measured conductance even at perfect screening; nonproportional contacts, extra paths, finite bias, and finite temperature require more than this one-channel formula. Meir and Wingreen 1992, Eqs. (4)–(6), pp. 2512–2514 gives the interacting current formula from which this equilibrium limit follows.

The fixed-point map contrasts this analytic endpoint with underscreened and overscreened alternatives, and shows where a perturbed two-channel system ultimately returns to a Fermi liquid.

Screening count selects the candidate infrared endpoint. The exactly screened branch has zero residual entropy and an elastic phase shift; at its symmetric unitary point the first spectral corrections are quadratic. The overscreened branch instead has fractional boundary entropy and nonanalytic corrections, but a finite two-channel device shows that behavior only in the window $\max(T^\ast,\Delta_L)\ll E\ll T_K$. At exact symmetry, the non-Fermi-liquid endpoint persists to $E\to0$ only in the thermodynamic limit $\Delta_L\to0$. Original schematic for metallic SU(2) channels; not to scale.

When the conclusion changes

The local-Fermi-liquid argument needs a unique screened ground state and a bath with finite low-energy density of states. It changes in three important ways:

• $k<2S$ leaves a residual moment and singular corrections;
• $k>2S$ with exact channel symmetry can reach the overscreened non-Fermi-liquid fixed point developed next; and
• a pseudogapped or superconducting bath can prevent ordinary metallic screening or create a phase transition.

Even within a Fermi liquid, particle–hole asymmetry shifts $\delta_0$, energy-dependent hybridization separates displaced charge from local occupancy, and an extra transmission path invalidates the single-phase conductance formula.

The broader chapter diagram locates this endpoint after charge matching and weak-coupling RG.

The exactly screened branch is the destination analyzed on this page. The one-orbital Anderson route supplies one active hybridization eigenchannel; genuinely multichannel branches require a channel-preserving microscopic realization. Original workflow schematic, not to scale.

The impurity claim test matrix lists the exceptions.

Common pitfalls

Counting only the dot charge. Friedel’s count concerns the impurity-induced charge of the full system. Replacing it by $n_d$ is justified in the conventional wide-flat-band Anderson Fermi liquid, not by the name “impurity model” alone.

Calling every Fermi-liquid spectrum quadratic. Analyticity permits a linear term when $\delta_0\ne\pi/2$. The linear term disappears at the symmetric unitary point because $\sin(2\delta_0)=0$.

Equating perfect screening with perfect transmission. The impurity can have $\delta=\pi/2$ while lead asymmetry or a parallel path makes the measured conductance nonunitary.

Exercises

Use the Friedel branch. A spin-symmetric wide-band Anderson impurity has $n_d=0.84$ and is continuously connected to the empty-orbital limit. Find $\delta_\sigma$ and $\pi\Gamma\mathcal A_{d\sigma}(0)$.

Solution

Spin symmetry gives $n_{d\sigma}=0.42$. On the branch that starts from $\delta_\sigma=0$ at zero occupancy,

$$\delta_\sigma=\pi n_{d\sigma}=0.42\pi.$$

Therefore $\pi\Gamma\mathcal A_{d\sigma}(0)=\sin^2(0.42\pi)\simeq0.94$. Choosing $1.42\pi$ would reproduce the same $S$-matrix but the wrong continuously tracked charge.

Locate the linear spectral term. Let $\delta(\omega)=\delta_0+a\omega/T_K+O(\omega^2)$. Expand $-\pi\rho\operatorname{Im}T(\omega)$ through first order. Compare $\delta_0=\pi/2$ with $\delta_0=0.4\pi$.

Solution

Using $-\pi\rho\operatorname{Im}T=\sin^2\delta$ gives

$$-\pi\rho\operatorname{Im}T(\omega) =\sin^2\delta_0 +a\sin(2\delta_0)\frac{\omega}{T_K}+O(\omega^2).$$

For $\delta_0=\pi/2$, $\sin(2\delta_0)=0$, so the first correction is quadratic. For $\delta_0=0.4\pi$, the coefficient is $a\sin(0.8\pi)$ and a linear term is allowed.

Separate screening from lead asymmetry. Evaluate the zero-temperature conductance for $\delta=\pi/2$ and $\Gamma_L=3\Gamma_R$.

Solution

The asymmetry factor is $4(3\Gamma_R)(\Gamma_R)/(4\Gamma_R)^2=3/4$. Thus $G=(3/4)(2e^2/h)$. The impurity can be fully screened with a $\pi/2$ phase shift even though the measured two-terminal conductance is below the unitary symmetric-lead value.

References

• Langreth, D. C. (1966). “Friedel sum rule for Anderson’s model of localized impurity states.” Physical Review 150, 516–518. doi:10.1103/PhysRev.150.516.
• Meir, Y., and Wingreen, N. S. (1992). “Landauer formula for the current through an interacting electron region.” Physical Review Letters 68, 2512–2515. doi:10.1103/PhysRevLett.68.2512.
• Mitchell, A. K., Logan, D. E., and Krishnamurthy, H. R. (2011). “Two-channel Kondo physics in odd impurity chains.” Physical Review B 84, 035119. doi:10.1103/PhysRevB.84.035119.
• Mora, C., Moca, C. P., von Delft, J., and Zaránd, G. (2015). “Fermi-liquid theory for the single-impurity Anderson model.” Physical Review B 92, 075120. doi:10.1103/PhysRevB.92.075120.
• Nozières, P. (1974). “A ‘Fermi-liquid’ description of the Kondo problem at low temperatures.” Journal of Low Temperature Physics 17, 31–42. doi:10.1007/BF00654541.