Kondo Model Preview

The Kondo model couples a localized quantum spin to the spin density of a conduction-electron bath. In its minimal isotropic, single-channel form,

$$H_{\mathrm K} = \sum_{k,\sigma} \epsilon_k c_{k\sigma}^\dagger c_{k\sigma} + J_K \mathbf S\cdot\mathbf s_0.$$

With this plus-sign convention, $J_K>0$ is antiferromagnetic. Weak antiferromagnetic exchange grows as the energy scale is lowered, producing a nonperturbative crossover at the Kondo temperature $T_K$. For a spin-$1/2$ impurity and one ordinary metallic screening channel, the low-temperature state is a screened singlet described by a local Fermi-liquid fixed point.

This short statement hides the central lesson of the model: a coupling that looks weak microscopically can become strong in the infrared because repeated spin-flip scattering generates logarithms. The Kondo model is therefore a compact meeting point of many-body entanglement, impurity thermodynamics, transport, and the renormalization group.

Helpful background. Fermionic operators supplies mode algebra; spin one-half supplies singlet and triplet states.

A local spin in an electron bath

The model contains two distinct kinds of degrees of freedom:

• a localized impurity spin $\mathbf S$, usually taken to have fixed magnitude $S$;
• itinerant conduction electrons with modes $c_{k\sigma}$ and a band Hamiltonian $H_c$.

Spin operators are dimensionless, so $[S^a,S^b]=i\epsilon_{abc}S^c$ and $J_K$ has energy units for a normalized bath orbital. The impurity has a fixed location but no dynamical position or charge coordinate inside the minimal Kondo Hilbert space. Its charge fluctuations have already been removed. What remains is a spin that can exchange angular momentum with conduction electrons at one spatial location.

The full Hilbert space is

$$\mathcal H = \mathcal H_S \otimes \mathcal F_c,$$

where

$$\dim\mathcal H_S = 2S+1$$

and $\mathcal F_c$ is the fermionic Fock space of the bath. Even for $S=1/2$, this is not a two-spin problem: the bath contains a macroscopic number of occupied and empty modes near a Fermi surface.

Conduction Bath and Local Channel

Take the noninteracting bath to be

$$H_c = \sum_{k,\sigma} \epsilon_k c_{k\sigma}^\dagger c_{k\sigma}.$$

Only one linear combination of bath modes couples to a pointlike single-channel impurity. With uniform mode amplitudes, define

$$f_{0\sigma} = \frac1{\sqrt{\mathcal N}} \sum_k c_{k\sigma},$$

where $\mathcal N$ is the number of bath orbitals. Then

$$\left\{ f_{0\sigma}, f_{0\sigma'}^\dagger \right\} = \delta_{\sigma\sigma'}.$$

The corresponding local conduction-electron spin is

$$\mathbf s_0 = \frac12 \sum_{\alpha,\beta} f_{0\alpha}^\dagger \boldsymbol\sigma_{\alpha\beta} f_{0\beta}.$$

For a continuum field, one instead writes $\mathbf s_c(\mathbf 0)$ using $\psi_\sigma(\mathbf 0)$. The field normalization then changes the units assigned to $J_K$. A Hamiltonian is not fully specified until its local-field normalization, density of states, and ultraviolet cutoff are declared.

Local density of states

For a normalized local orbital with amplitudes $u_k$, define

$$\rho_0(\epsilon) = \sum_k \lvert u_k\rvert^2 \delta(\epsilon-\epsilon_k).$$

Unless stated otherwise, $\rho_0$ on this page means the local density of states per spin at the Fermi energy:

$$\rho_0 \equiv \rho_0(0).$$

The dimensionless bare coupling is

$$g_0 = \rho_0J_K.$$

For a metal with a smooth nonzero $\rho_0$, the simplest scaling analysis treats the density of states as constant within a band cutoff $D_0$.

Exchange Hamiltonian

The isotropic interaction is

$$H_{\mathrm{int}} = J_K \mathbf S\cdot\mathbf s_0.$$

Writing it in longitudinal and spin-flip parts gives

$$\mathbf S\cdot\mathbf s_0 = S^zs_0^z + \frac12 \left( S^+s_0^- + S^-s_0^+ \right).$$

The transverse terms exchange one unit of spin between the impurity and bath. They are essential: replacing the interaction by $J_KS^zs_0^z$ produces an Ising impurity without Kondo spin-flip screening.

Potential scattering is often included as

$$H_W = W \sum_\sigma f_{0\sigma}^\dagger f_{0\sigma}.$$

It preserves spin but breaks particle–hole symmetry when $W\ne0$. In the ordinary single-channel metallic problem, potential scattering changes the elastic phase shift but does not by itself generate the Kondo logarithm.

Exchange-Sign Benchmark

Temporarily isolate the impurity and one singly occupied local conduction orbital, both with spin $1/2$. Their total spin is

$$\mathbf J = \mathbf S+\mathbf s_0.$$

Using

$$\mathbf S\cdot\mathbf s_0 = \frac12 \left( \mathbf J^2 -\mathbf S^2 -\mathbf s_0^2 \right),$$

the interaction energies are

$$E_{\mathrm s} = -\frac34J_K, \qquad E_{\mathrm t} = +\frac14J_K.$$

Thus

$$E_{\mathrm t}-E_{\mathrm s} = J_K.$$

Antiferromagnetic $J_K>0$ favors a singlet; ferromagnetic $J_K<0$ favors a triplet. This local calculation fixes the sign convention, but it does not explain the many-electron crossover or its exponentially small scale.

The one-loop solution $g(\ell)=g_0/(1-g_0\ell)$ grows for $g_0>0$ and approaches zero from below for $g_0<0$. The positive curve is drawn only to $g=0.5$: increasing coupling eventually invalidates the weak-coupling approximation, so extending the curve to its formal pole would not predict the exact crossover.

Symmetries and Conserved Quantities

For a spin-independent bath and isotropic exchange:

• conduction-electron number is conserved;
• total spin has global $SU(2)$ symmetry;
• time-reversal symmetry is present if the bath also respects it and there is no magnetic field or magnetic flux; spin independence alone does not guarantee it;
• charge $U(1)$ symmetry is present;
• spatial translation symmetry is broken by the impurity location.

The conserved electron number is

$$N_c = \sum_{k,\sigma} c_{k\sigma}^\dagger c_{k\sigma}.$$

The conserved total spin is

$$\mathbf J_{\mathrm{tot}} = \mathbf S + \frac12 \sum_{k,\alpha,\beta} c_{k\alpha}^\dagger \boldsymbol\sigma_{\alpha\beta} c_{k\beta},$$

so

$$\left[ H_{\mathrm K}, \mathbf J_{\mathrm{tot}} \right] =0.$$

The impurity spin alone is not conserved. Screening is a redistribution of spin correlations between impurity and bath that respects conservation of the total.

Microscopic Origin from a Local Orbital

The Kondo model is often obtained from an interacting localized orbital hybridized with a band. A representative parent Hamiltonian is

$$H_{\mathrm A} ={} H_c + \epsilon_d \sum_\sigma n_{d\sigma} + U n_{d\uparrow}n_{d\downarrow} + \sum_{k,\sigma} \left( V_kc_{k\sigma}^\dagger d_\sigma + \mathrm{h.c.} \right).$$

In the local-moment regime,

$$\epsilon_d<0, \qquad \epsilon_d+U>0,$$

the singly occupied impurity states lie below the empty and doubly occupied states. When hybridization is weak relative to both charge-excitation energies, a Schrieffer–Wolff transformation removes virtual charge fluctuations and generates antiferromagnetic exchange.

For the normalized local-orbital convention

$$V_k = \frac{V}{\sqrt{\mathcal N}},$$

the leading exchange is

$$J_K \simeq 2\lvert V\rvert^2 \left( \frac1{\lvert\epsilon_d\rvert} + \frac1{\epsilon_d+U} \right) >0.$$

It also generates potential scattering, which vanishes at particle–hole symmetry in the simplest model. Exact numerical factors depend on the normalization of the local conduction orbital and hybridization.

This mapping explains why the Kondo Hilbert space contains a spin but no impurity charge state. It is controlled only below the eliminated charge-excitation scales. The charge-path derivation is developed in Effective Hamiltonians in Many-Body Systems; the Schrieffer–Wolff Transformation explains the general method. See the Anderson impurity treatment for the charge spectrum and its relation to screening.

Why Weak Coupling Becomes Strong

At first order, exchange scatters a conduction electron and may flip its spin together with the impurity. At higher orders, intermediate particle and hole states can lie anywhere between the observation scale $E$ and the band cutoff $D_0$.

For a smooth metallic density of states, the phase-space integral contains

$$\int_E^{D_0} \frac{d\epsilon}{\epsilon} = \ln\frac{D_0}{E}.$$

Consequently, the effective antiferromagnetic coupling has the perturbative form

$$J_{\mathrm{eff}}(E) = J_K + \rho_0J_K^2 \ln\frac{D_0}{E} + O(J_K^3).$$

Here $\rho_0$ is the local density of states per spin, and the vertex is $J_K\mathbf S\cdot(c^\dagger\boldsymbol\sigma c/2)$. For $\rho_{\mathrm{tot}}=2\rho_0$, the logarithmic coefficient is $\rho_{\mathrm{tot}}J_K^2/2$. The exchange normalization matters independently of the density-of-states normalization.

To check the shell counting, write the spin-$1/2$ scattering matrix as $M=\sum_a S_a\otimes(\sigma_a/2)$. Lower the positive half-bandwidth $D$ by $dD>0$, with $d\ell=dD/D$, eliminating one empty particle shell and one occupied hole shell. Each shell has weight $\rho_0dD$ for each fixed spin. Their second-order spin matrices are

$$\begin{aligned} \delta V_{\mathrm p} &=\rho_0J_K^2d\ell \left(-\frac{3}{16}I+\frac12M\right),\\ \delta V_{\mathrm h} &=\rho_0J_K^2d\ell \left(+\frac{3}{16}I+\frac12M\right). \end{aligned}$$

The first follows from $-M^2$. The hole term has the fermionic exchange sign and reversed impurity-matrix ordering, $\sum_{ab}S_bS_a\otimes(\sigma_a\sigma_b/4)$. The spin-independent parts cancel in this symmetric-shell benchmark and the exchange parts add to $\rho_0J_K^2d\ell\,M$. Both band edges and the intermediate spin sum have already been counted; there is no further factor of two.

This normalization agrees with the metallic limit of Cheng et al. (2017), Eqs. (9), (66) and (67). Nevidomskyy (2015) instead writes $H_{\mathrm{int}}=2J'\mathbf S\cdot\mathbf s_0$. Since $J_K=2J'$, its equation $dJ'/d\ell=2\rho_0J'^2$ gives exactly the same running Hamiltonian.

The logarithm grows as $E$ decreases. Perturbation theory fails when

$$\rho_0J_K \ln\frac{D_0}{E} \sim1.$$

The failure does not mean that an observable physically diverges. It says that the weak-coupling expansion is being used beyond its regime.

One-Loop Scaling Flow

Let the running half-bandwidth be

$$D = D_0e^{-\ell}, \qquad \ell = \ln\frac{D_0}{D}.$$

Eliminating a thin shell of high-energy conduction states and preserving low-energy scattering gives, for the isotropic one-channel model,

$$\frac{dg}{d\ell} = g^2 + O(g^3),$$

where

$$g(\ell) = \rho_0J_K(\ell).$$

At one loop,

$$g(\ell) = \frac{g_0}{ 1-g_0\ell }.$$

The flow distinguishes the two exchange signs:

Bare coupling	One-loop flow	Low-energy interpretation
$g_0>0$	grows toward strong coupling	marginally relevant antiferromagnetic exchange
$g_0<0$	approaches $0$ from below	marginally irrelevant ferromagnetic exchange
$g_0=0$	remains zero	decoupled local moment

For $g_0>0$, the one-loop denominator vanishes at a finite $\ell$. That pole marks the breakdown of weak-coupling scaling, not a literal singularity of the exact finite-temperature system.

Kondo Temperature

The one-loop breakdown scale defines the exponential structure of the Kondo energy:

$$E_K \equiv k_{\mathrm B}T_K \sim D_0 \exp \left( -\frac1{\rho_0J_K} \right), \qquad J_K>0.$$

If $\rho_{\mathrm{tot}}=2\rho_0$ denotes the spin-summed density of states, the same exponent is

$$E_K \sim D_0 \exp \left( -\frac2{\rho_{\mathrm{tot}}J_K} \right).$$

These are the same convention, not competing physical predictions.

The exponential dependence means that a modest change in $J_K$, density of states, or cutoff can change $T_K$ by orders of magnitude. It also means that exact diagonalization with a uniformly spaced finite bath may miss screening unless its level spacing is much smaller than $E_K$.

There is no unique numerical definition

Beyond the leading weak-coupling exponent, the prefactor depends on:

• the cutoff scheme and band shape;
• whether $\rho$ is local, per spin, or spin summed;
• higher-loop conventions;
• the operational definition of $T_K$.

Common definitions use the zero-temperature impurity susceptibility, the half-width of a spectral resonance, an entropy crossover, a conductance scale, or Wilson’s numerical normalization. In the universal Kondo scaling regime, well-specified definitions can be related by fixed conversion factors. Outside that regime, charge fluctuations, band structure or an asymmetric line shape can prevent such a simple conversion.

Screening Crossover

For

$$k_{\mathrm B}T \gg E_K,$$

the impurity behaves approximately as a free local moment with logarithmic corrections. As the temperature or probe energy approaches $E_K$, repeated spin-flip scattering entangles the impurity with increasingly low-energy bath states.

For the ordinary spin-$1/2$, single-channel, antiferromagnetic model,

$$k_{\mathrm B}T \ll E_K$$

is a fully screened regime with no residual impurity moment. In a finite system with the compatible electron-number parity, the ground state is a global singlet. The crossover is not spontaneous symmetry breaking and has no local order parameter.

The phrase “Kondo singlet” is useful but must be interpreted carefully. At strong bare coupling, the impurity and the local conduction orbital provide an intuitive singlet benchmark. At weak bare coupling, screening involves a coherent many-electron state spread over a large range of energies and distances.

Impurity Thermodynamics

An impurity contribution is defined by subtracting the bath without the impurity:

$$X_{\mathrm{imp}} = X_{\mathrm{full}} - X_{\mathrm{bath}}.$$

This subtraction is necessary because the bath contribution is extensive while the impurity contribution is order unity.

Entropy

For a decoupled spin $S$,

$$S_{\mathrm{imp}} \longrightarrow k_{\mathrm B}\ln(2S+1)$$

in the window $E_K\ll k_{\mathrm B}T\ll D_0$ and below eliminated charge excitations. The high-temperature plateau is approximate at nonzero exchange; an exactly decoupled spin has this entropy at every temperature. For the fully screened spin-$1/2$ model,

$$S_{\mathrm{imp}} \longrightarrow0$$

as $T\to0$.

The entropy is released continuously across a crossover. A finite residual impurity entropy can occur in overscreened or other nonstandard impurity fixed points, but it is not a property of the minimal one-channel spin-$1/2$ model.

Magnetic susceptibility

Specify how the field couples before taking a derivative. The local susceptibility responds to a field on the impurity alone. The impurity contribution used below subtracts the bath response to a uniform field from the full uniform-field response, with the same bath and field convention. They are different quantities at finite bandwidth. The simple Wilson-ratio expression assumes equal impurity and conduction-electron magnetic factors and the universal wide-band Kondo limit.

At high temperature, the impurity approximately obeys the Curie law

$$\chi_{\mathrm{imp}}(T) \sim \frac{ (g_{\mathrm{imp}}\mu_{\mathrm B})^2 S(S+1) }{ 3k_{\mathrm B}T }.$$

For $S=1/2$,

$$\chi_{\mathrm{imp}}(T) \sim \frac{ (g_{\mathrm{imp}}\mu_{\mathrm B})^2 }{ 4k_{\mathrm B}T }.$$

Screening cuts off the Curie divergence. At low temperature, $\chi_{\mathrm{imp}}(0)$ is finite and scales inversely with a conventionally defined $T_K$.

Heat capacity and Wilson ratio

The screened fixed point has an impurity heat capacity

$$C_{\mathrm{imp}} = \gamma_{\mathrm{imp}}T + O(T^3).$$

In the universal spin-$1/2$, single-channel Kondo limit, the ratio of susceptibility enhancement to heat-capacity enhancement approaches the universal Wilson ratio

$$R_W = \frac{ 4\pi^2k_{\mathrm B}^2 }{ 3(g_{\mathrm{imp}}\mu_{\mathrm B})^2 } \frac{ \chi_{\mathrm{imp}}(0) }{ \gamma_{\mathrm{imp}} } =2.$$

Comparing quoted values requires checking whether the bath background, $g$-factor, and susceptibility units have been subtracted and normalized consistently.

The Screening Cloud

A spatial diagnostic is the impurity–bath spin correlation

$$C(\mathbf r) = \left\langle \mathbf S\cdot\mathbf s_c(\mathbf r) \right\rangle.$$

The crossover defines a length scale

$$\xi_K \sim \frac{ \hbar v_F }{ k_{\mathrm B}T_K },$$

where $v_F$ is an appropriate Fermi velocity. Because $T_K$ may be exponentially small, $\xi_K$ can be much larger than a microscopic lattice spacing.

The screening cloud is not one identifiable electron bound in a hydrogen-like orbital. It is a many-body correlation pattern with oscillatory, dimension-dependent, and observable-dependent structure. Finite temperature, sample size, channel geometry, and boundaries can cut it off.

In a finite spin-singlet system, the integrated bath spin compensates the impurity. For $S=1/2$, this implies the global sum rule

$$\left\langle \mathbf S\cdot \mathbf S_c^{\mathrm{tot}} \right\rangle = -\frac34,$$

provided $\mathbf S_c^{\mathrm{tot}}$ includes the full screening channel and the total state is a singlet.

Low-Energy Local Fermi Liquid

The fully screened single-channel model flows to a strong-coupling fixed point with well-defined low-energy quasiparticle scattering. The impurity is absorbed into a change of boundary condition plus irrelevant local interactions.

At particle–hole symmetry and without additional potential scattering, the zero-energy phase shift per spin is

$$\delta_\sigma(0) = \frac{\pi}{2}.$$

Potential scattering adds a nonuniversal phase shift, and asymmetric Anderson-model realizations require the corresponding Friedel-sum-rule accounting.

At particle–hole symmetry without potential scattering, the spin-averaged spectral $T$-matrix weight has the structure

$$\mathcal A(\omega,T) = \mathcal A(0,0) \left[ 1 - c_\omega \left( \frac{\hbar\omega}{E_K} \right)^2 - c_T \left( \frac{k_{\mathrm B}T}{E_K} \right)^2 + \cdots \right],$$

with positive coefficients whose numerical values depend on the observable and $T_K$ convention. Analytic low-energy corrections characterize a local Fermi liquid. Away from particle–hole symmetry a spectral quantity can also have a term linear in energy, and quadratic coefficients need not describe a maximum. Thus the displayed symmetric peak is not a formula for every observable.

Observable Signatures

Dilute-alloy resistivity

At temperatures above $T_K$ but below microscopic band scales, the Kondo contribution to electron scattering grows approximately logarithmically as the system is cooled. Combined with a phonon contribution that decreases on cooling, this can produce a resistance minimum.

The perturbative logarithm cannot be extrapolated to $T=0$. In the fully screened model, scattering approaches a finite unitarity-limited value with Fermi-liquid corrections.

Spectral and scattering response

The conduction-electron $T$-matrix describes scattering from the impurity. Its low-energy resonance has a characteristic width of order $E_K$, but the Kondo model itself contains no elementary impurity-charge spectral function. A localized-electron resonance is naturally discussed in the Anderson impurity model.

Magnetic response

Static susceptibility tracks the crossover from a Curie moment to a finite screened response. Dynamic spin susceptibility probes the redistribution of spectral weight over frequencies of order $E_K/\hbar$.

Quantum-dot conductance

A quantum dot in a local-moment regime can realize Kondo correlations, and its low-temperature conductance can approach a unitary limit under symmetric coupling. The measured conductance depends on lead geometry, hybridization asymmetry, potential scattering, nonequilibrium bias, and the parent Anderson model. It is not determined by $J_K$ alone.

Resolving an exponentially small scale

Perturbative renormalization

Poor-man’s scaling and field-theoretic RG organize the high-energy regime where the running coupling remains small. They determine the exponential scale and universal logarithms but do not justify using a truncated weak-coupling series deep below $T_K$.

Numerical renormalization group

Wilson’s numerical renormalization group logarithmically discretizes the bath and maps it to a chain with decreasing energy scales. A representative asymptotic hopping is

$$t_n \propto D_0 \Lambda^{-n/2}, \qquad \Lambda>1.$$

Iterative diagonalization then resolves exponentially separated impurity scales. Discretization, state truncation, broadening, and $z$-averaging must be converged for quantitative spectra.

Bethe ansatz

Bethe ansatz gives exact equilibrium results for integrable versions. It is especially powerful for thermodynamics and universal ratios, but extracting arbitrary real-frequency observables or nonintegrable perturbations can remain difficult.

Tensor networks and finite baths

Matrix-product-state and related methods can resolve real-space screening correlations after mapping the bath to a chain. Finite-size level spacing, chain length, bath discretization, and entanglement growth must all be compared with $E_K$ and $\xi_K$.

Quantum Monte Carlo

Impurity Monte Carlo methods can be highly effective for Anderson-type formulations. Analytic continuation from imaginary time is ill-conditioned, and sign or ergodicity difficulties depend on the chosen variant.

Principal Variants

Anisotropic exchange

The axial model is

$$H_{\mathrm{int}} = J_\perp \left( S^xs_0^x + S^ys_0^y \right) + J_zS^zs_0^z.$$

Its coupled scaling equations reveal flows toward isotropy in part of the antiferromagnetic regime and connect to boundary sine-Gordon and spin-boson descriptions. The general fixed-point and phase-portrait language belongs in Renormalization Group Preview.

Higher impurity spin and several channels

For $K$ equivalent screening channels and impurity spin $S$:

• $K=2S$ suggests exact screening;
• $K<2S$ gives underscreening and a residual moment;
• $K>2S$ gives overscreening and can produce a non-Fermi-liquid fixed point.

These rules assume channel symmetry and antiferromagnetic coupling. Channel anisotropy and magnetic fields can destabilize overscreened fixed points. Ordinary spin-independent potential scattering is not automatically such a relevant perturbation; in the metallic multichannel model it can instead change a charge-sector phase shift.

Two impurities and Kondo lattices

Two or more moments can interact indirectly through the conduction bath. The induced RKKY scale is schematically

$$E_{\mathrm{RKKY}} \sim J_K^2 \chi_c(R),$$

where $\chi_c(R)$ is a static conduction-spin response to an energy-valued source, with units of inverse energy for normalized lattice spins. Its sign and tensor structure determine the induced exchange; this expression only states its second-order scale.

Competition between intermoment correlations and Kondo screening is absent from the one-impurity model. In a Kondo lattice, coherence, magnetism, heavy quasiparticles, and possible Kondo-breakdown phenomena require a distinct many-body analysis.

RKKY interaction derives the sign, spatial oscillations and normalization of this induced exchange.

Nonmetallic baths

If the local density of states vanishes or diverges at the Fermi energy, the constant-$\rho_0$ logarithmic analysis changes. Pseudogap, superconducting, Luttinger-liquid, and topological baths can support impurity quantum phase transitions or altered screening.

The metallic formula for $T_K$ must not be transplanted unchanged into these cases.

Worked Example: One-Loop Scale

Suppose the per-spin dimensionless coupling is

$$g_0 = \rho_0J_K = 0.08.$$

The one-loop scale is

$$\frac{E_K}{D_0} \sim \exp \left( -\frac1{g_0} \right) = e^{-12.5} \approx 3.73\times10^{-6}.$$

A bare coupling of order $10^{-1}$ therefore produces a scale of order $10^{-6}D_0$. This separation explains both the power of logarithmic methods and the numerical difficulty of resolving $T_K$ with a uniformly spaced bath.

Exercises

Solve the one-loop flow

Integrate $dg/d\ell=g^2$ with $g(0)=g_0$. Describe the flows for positive and negative $g_0$.

Solution

Separate variables:

$$\frac{dg}{g^2} = d\ell.$$

Integrating gives

$$-\frac1{g(\ell)} + \frac1{g_0} = \ell,$$

so

$$g(\ell) = \frac{g_0}{ 1-g_0\ell }.$$

For $g_0>0$, the running coupling grows and the one-loop expression breaks down at $\ell=1/g_0$. For $g_0<0$, the denominator grows without vanishing and $g(\ell)\to0^-$.

Screening length estimate

Take $v_F=1.0\times10^6\,\mathrm{m\,s^{-1}}$ and $T_K=10\,\mathrm K$. Estimate $\xi_K=\hbar v_F/(k_{\mathrm B}T_K)$.

Solution

Using

$$\hbar \approx 1.055\times10^{-34}\,\mathrm{J\,s}$$

and

$$k_{\mathrm B} \approx 1.381\times10^{-23}\,\mathrm{J\,K^{-1}},$$

one obtains

$$\xi_K \approx \frac{ 1.055\times10^{-34} \times 1.0\times10^6 }{ 1.381\times10^{-23} \times 10 } \mathrm m.$$

Thus

$$\xi_K \approx 7.6\times10^{-7}\,\mathrm m = 0.76\,\mu\mathrm m.$$

The result is much larger than an atomic spacing, illustrating why direct spatial observation of the screening cloud is subtle.

Integrated screening in a singlet

Let the full finite system be a spin singlet with impurity spin $S$. Derive the integrated impurity–bath correlation without assuming that one particular bath electron forms a pair with the impurity.

Solution

The singlet satisfies $(\mathbf S+\mathbf S_c^{\mathrm{tot}})\lvert\Psi\rangle=0$. Taking its scalar product with $\langle\Psi|\mathbf S$ gives

$$\langle\mathbf S\cdot\mathbf S_c^{\mathrm{tot}}\rangle =-\langle\mathbf S^2\rangle=-S(S+1).$$

For $S=1/2$ this is $-3/4$. It constrains the integral over the full bath, not its spatial distribution or the identity of a screening electron. A finite system in a nonsinglet sector need not obey it.

References

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