Kondo Model

The Kondo model couples a fixed localized quantum spin to the spin density of one or more conduction-electron channels, turning weak antiferromagnetic exchange into a nonperturbative low-energy screening problem.

The baseline is one spin-$1/2$ impurity, one metallic screening channel, isotropic exchange, a smooth density of states at the Fermi energy, zero field, and no potential scattering. A finite two-site example below checks the spin algebra; the continuum crossover requires a bath with many levels below its emergent energy scale.

Helpful background. Kondo screening explains why repeated spin flips generate a low-energy scale. Fermionic operators supplies the anticommutation rules used to normalize the screening channel.

Exchange between an Impurity and a Metallic Bath

The isotropic single-channel Hamiltonian is

$$H_{\mathrm K} = H_c + J_K\mathbf S\cdot\mathbf s_0.$$

The bath is noninteracting,

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

and the impurity spin satisfies

$$\mathbf S^2 = S(S+1)I.$$

With the displayed plus-sign convention,

$$J_K\gt0$$

is antiferromagnetic and

$$J_K\lt0$$

is ferromagnetic.

Historical and modern sources sometimes write the interaction with an overall minus sign. The physical sign must be inferred from the full Hamiltonian, not from the symbol $J$ alone.

Degrees of Freedom and Hilbert Space

The 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 conduction bath.

The impurity is a fixed spin in this model. It has no empty, singly occupied, or doubly occupied charge states. Those charge states are included in the Anderson impurity model.

For $N_b$ spinful bath orbitals, the complete finite-bath dimension is

$$\dim\mathcal H = (2S+1)4^{N_b}.$$

At fixed conduction-electron number $N_c$,

$$\dim\mathcal H_{N_c} = (2S+1) \binom{2N_b}{N_c}.$$

Total spin and spin projection can reduce this further when the bath and exchange are isotropic.

The Local Screening Channel

A pointlike impurity couples only to a particular normalized linear combination of bath orbitals. Define

$$f_{0\sigma} = \sum_k u_kc_{k\sigma}, \qquad \sum_k|u_k|^2 = 1.$$

Then

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

The local spin density is

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

The corresponding local density of states per spin is

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

For the baseline metallic model,

$$\rho_0 \equiv \rho_0(0) \gt0$$

is smooth near the Fermi energy. The dimensionless bare exchange is

$$g_0 = \rho_0J_K.$$

Here $\rho_0$ is per spin. If $\rho_{\mathrm{tot}}=2\rho_0$ is used instead, every displayed coefficient involving $\rho J_K$ must be converted consistently.

Flat-band baseline

For a normalized local orbital and a symmetric flat band of half-width $D_0$,

$$\rho_0(\epsilon) = \frac{1}{2D_0} \Theta(D_0-|\epsilon|)$$

per spin. A different band shape changes nonuniversal prefactors and may change the physics if the Fermi-level density of states vanishes or diverges.

Optional Terms

Potential scattering is

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

It is spin independent and, in the ordinary metallic one-channel problem, is marginal. It changes the elastic phase shift and particle–hole asymmetry but does not replace exchange-driven spin-flip physics.

A magnetic field can couple to impurity and bath moments,

$$H_B = -g_{\mathrm{imp}}\mu_{\mathrm B}B S^z - g_c\mu_{\mathrm B}B S_c^z.$$

The two $g$-factors need not be equal. Susceptibility conventions must state which moment the field probes and which bath background is subtracted.

For axial exchange,

$$H_{\mathrm{int}} = J_zS^zs_0^z + \frac{J_\perp}{2} \left( S^+s_0^- + S^-s_0^+ \right).$$

The isotropic baseline has $J_z=J_\perp=J_K$.

Exchange Sign and Spin Transfer

For one local conduction electron and a spin-$1/2$ impurity,

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

gives

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

The local exchange energies are therefore

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

Thus

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

Antiferromagnetic exchange favors a local singlet. Ferromagnetic exchange favors a triplet. This two-spin calculation fixes the sign convention, but the weak-coupling Kondo state is not a bound state of the impurity with one identifiable bath electron.

The spin-flip part is

$$\frac{J_K}{2} \left( S^+s_0^- + S^-s_0^+ \right).$$

It transfers one unit of angular momentum between impurity and bath. Replacing the exchange by $J_zS^zs_0^z$ removes that dynamical channel and produces a different impurity model.

Symmetries and Conserved Quantities

For an isotropic spin-independent bath at zero field:

• conduction-electron number has $U(1)$ symmetry;
• impurity plus bath spin has global $SU(2)$ symmetry;
• time reversal is present;
• the impurity breaks spatial translation invariance;
• particle–hole symmetry may be present for a symmetric bath with $W=0$.

The conserved conduction number is

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

The total spin is

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

and

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

The impurity spin alone is not conserved:

$$[H_{\mathrm K},\mathbf S] \ne 0.$$

Screening redistributes spin correlations while preserving the total spin.

Microscopic Origin and Model Boundary

In the local-moment regime of a spinful Anderson impurity,

$$\epsilon_d\lt0\lt\epsilon_d+U,$$

empty and doubly occupied impurity states are virtual at low energy. A Schrieffer–Wolff transformation generates exchange

$$J_K \simeq 2|V|^2 \left( \frac{1}{|\epsilon_d|} + \frac{1}{\epsilon_d+U} \right) \gt0$$

when $V$ couples the impurity to the normalized local bath orbital. With $H_W=W\sum_\sigma f_{0\sigma}^\dagger f_{0\sigma}$, the accompanying coefficient is

$$W=\frac{|V|^2}{2} \left(\frac{1}{|\epsilon_d|}-\frac{1}{\epsilon_d+U}\right).$$

The factor $1/2$ belongs to this potential-scattering definition; changing the exchange vertex or writing $2Wn_0$ changes quoted coefficients. The two virtual charge paths derive both terms using the Schrieffer–Wolff transformation.

This is a controlled low-energy reduction when hybridization is small compared with both charge-excitation energies. It is not an identity between the Anderson and Kondo Hamiltonians. The Kondo model cannot recover:

• impurity occupancy fluctuations;
• lower and upper charge-transfer peaks;
• mixed-valence behavior;
• gate dependence that crosses a charge degeneracy;
• an impurity electron spectral function at arbitrary energy.

At particle–hole symmetry, $\epsilon_d=-U/2$, the leading exchange becomes

$$J_K \simeq \frac{8|V|^2}{U},$$

while the leading potential-scattering contributions cancel.

Scale Generation

The exchange is classically marginal. Quantum spin-flip processes generate logarithms. For a smooth metallic bath and per-spin density of states $\rho_0$, define

$$\ell = \ln\frac{D_0}{D}, \qquad g(\ell) = \rho_0J_K(\ell).$$

One-loop scaling gives

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

Its solution is

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

For $g_0\gt0$, the running coupling grows and the weak-coupling coordinate reaches a pole. For $g_0\lt0$, it approaches zero from below.

The one-loop antiferromagnetic breakdown scale is

$$k_{\mathrm B}T_K^{(1)} = D_0 \exp\!\left( -\frac{1}{\rho_0J_K} \right).$$

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

$$k_{\mathrm B}T_K^{(1)} = D_0 \exp\!\left( -\frac{2}{\rho_{\mathrm{tot}}J_K} \right).$$

The coefficient follows from the declared $\mathbf s_0=c^\dagger\boldsymbol\sigma c/2$ vertex and per-spin density, as in the metallic limit of Cheng et al. (2017). In the alternative convention $H_{\mathrm{int}}=2J'\mathbf S\cdot\mathbf s_0$, used by Nevidomskyy (2015), $J_K=2J'$ and the beta function for $\rho_0J'$ has coefficient two. The teaching treatment checks the two shell contributions explicitly.

The one-loop pole is not a divergence of an exact observable. It marks the scale where weak-coupling perturbation theory stops being controlled.

No unique numerical Kondo temperature

Beyond the leading exponential, $T_K$ depends on:

• the bandwidth and cutoff convention;
• the density-of-states normalization;
• higher-order definition of the running coupling;
• whether it is defined by susceptibility, entropy, conductance, or a spectral width.

Different standard definitions are related by fixed factors only after the full model and observable conventions are aligned. A quoted value of $T_K$ without those conventions is incomplete.

Fixed Points and Controlled Limits

Zero exchange

At $J_K=0$, impurity and bath factorize. A spin-$S$ impurity contributes

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

and a Curie susceptibility at temperatures below any eliminated charge scale.

Weak antiferromagnetic exchange

For

$$T\gg T_K,$$

the impurity behaves approximately as a local moment with logarithmic corrections. Perturbative scaling is controlled while $|g(D)|\ll1$.

Fully screened one-channel limit

For $S=1/2$, one channel, and $J_K\gt0$, the infrared state is fully screened. The impurity entropy tends to zero and the low-energy theory is a local Fermi liquid.

At particle–hole symmetry without extra potential scattering, the zero-energy phase shift is

$$\delta_\sigma(0) = \frac\pi2.$$

The quasiparticle phase shift generally has a term linear in energy. At particle–hole symmetry its zero-energy value is $\pi/2$, so the elastic scattering probability $\sin^2\delta_\sigma$ has no linear correction: its leading departures are quadratic in $\hbar\omega/(k_{\mathrm B}T_K)$ and $T/T_K$. Inelastic Fermi-liquid rates are likewise quadratic. With potential scattering away from that symmetry point, an energy-odd term in the scattering probability is allowed; not every scattering observable is an even function of energy.

Strong antiferromagnetic exchange

If $J_K$ is much larger than the hopping out of the local bath orbital, the impurity and one local electron form a singlet with exchange energy $-3J_K/4$. The rest of the bath couples perturbatively to this strong-coupling reference state.

The strong-coupling singlet is a useful fixed-point picture. It should not be projected backward into a claim that weak-coupling screening uses one localized electron.

Ferromagnetic exchange

Weak $J_K\lt0$ is marginally irrelevant. The effective coupling approaches zero from below, and a residual local moment survives with logarithmic corrections. There is no antiferromagnetic Kondo scale of the preceding form.

Finite bath

A finite bath has a level spacing $\delta$. To resolve a continuum-like screening crossover,

$$\delta \ll k_{\mathrm B}T_K$$

is generally required. Bath parity, boundaries, and the local spectral weights $|u_k|^2$ also matter. A small exact-diagonalization cluster can validate operators and multiplets without reproducing universal Kondo thermodynamics.

Screening Classification

Let $K$ denote the number of equivalent antiferromagnetic screening channels and $S$ the impurity spin. Under the standard channel-symmetric metallic assumptions:

Relation	Screening class	Infrared character
$K=2S$	exactly screened	local Fermi liquid
$K\lt2S$	underscreened	residual moment with singular corrections
$K\gt2S$	overscreened	non-Fermi-liquid fixed point when channel symmetry is maintained

This table is a classification, not a substitute for an RG analysis. Channel anisotropy can destabilize overscreened fixed points, and a nonmetallic bath can change the phase diagram entirely.

For the baseline $S=1/2$, $K=1$ model,

$$K = 2S,$$

so screening is exact.

Exact Solution Status

The phrase “the Kondo model is exactly solvable” requires qualification.

Bethe-ansatz-integrable continuum model

After partial-wave reduction, a pointlike isotropic impurity couples to an effective one-dimensional radial channel. Special continuum formulations admit Bethe-ansatz solutions. They yield exact equilibrium thermodynamics, magnetization, susceptibility, and universal crossover information.

The exact solution relies on the integrable structure. Arbitrary band curvature, momentum-dependent exchange, several impurities, generic channel mixing, or nonequilibrium reservoirs need not preserve it.

Toulouse limit

At a special anisotropic coupling, bosonization and refermionization map the problem to a quadratic resonant-level form. This is an exact solvable point inside the anisotropic family, not the generic isotropic Hamiltonian.

Numerical renormalization group

Wilson’s numerical renormalization group is not a closed-form exact solution. It is a systematically improvable nonperturbative method tailored to exponentially separated impurity scales. Quantitative work must converge the discretization parameter, kept states, broadening, and interleaved discretizations.

What is not exact

The one-loop beta function, the leading exponential estimate of $T_K$, a slave-particle saddle, and a small finite bath are approximations or controlled representations in stated regimes. They should not be promoted to exact solutions of every Kondo variant.

Typical Observables

Impurity thermodynamics

An impurity contribution is defined by subtraction,

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

For the fully screened spin-$1/2$ model,

$$S_{\mathrm{imp}}(T) : k_{\mathrm B}\ln2 \longrightarrow 0$$

across the crossover.

The high-temperature Curie form is

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

while $\chi_{\mathrm{imp}}(0)$ is finite in the screened regime.

The low-temperature impurity heat capacity is

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

In the wide-band scaling limit $k_{\mathrm B}T_K\ll D_0$, the spin-$1/2$ one-channel Wilson ratio is

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

Here impurity and bath have the same $g$-factor, a uniform field probes both, and $\chi_{\mathrm{imp}}$ is the full susceptibility minus that of the uncoupled bath. A field applied only to the impurity defines a different response and should not be inserted into this formula without relating the conventions.

Spin correlations and screening cloud

The real-space screening diagnostic is

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

Its crossover length is

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

The cloud is an extended, oscillatory many-body correlation profile. It is not a one-electron orbital.

Scattering and transport

The conduction-electron $T$-matrix controls impurity scattering. In a dilute metal it produces the logarithmic high-temperature Kondo correction and a finite unitarity-limited low-temperature response. In a quantum-dot realization, conductance additionally depends on lead geometry, hybridization asymmetry, potential scattering, bias, and the parent Anderson model.

Entanglement and finite-size spectra

Impurity–bath entanglement, total-spin multiplets, finite-size level flows, and boundary-condition changes diagnose the crossover in numerical calculations. Entanglement is partition dependent and should be reported together with the bath mapping and subsystem definition.

Physical Phenomena

Resistance minimum

Spin-flip scattering from dilute magnetic impurities generates logarithmic corrections as temperature is lowered. Combined with a decreasing phonon resistivity, this can produce a resistance minimum. The perturbative logarithm cannot be extrapolated through $T_K$ to zero temperature.

Local-moment screening

The impurity crosses from a Curie-like moment to an entangled screened state. This is a crossover, not spontaneous symmetry breaking, and it has no local order parameter in the baseline single-impurity problem.

Quantum dots

A Coulomb-blockaded quantum dot with an odd occupancy can enter a local-moment regime and display Kondo-enhanced conductance. The Kondo Hamiltonian describes its low-energy spin sector; charge addition spectra remain properties of the Anderson-type parent.

Heavy-fermion bridge

An array of moments introduces interimpurity exchange, lattice coherence, and possible magnetic or heavy-Fermi-liquid phases. Those effects are absent from a single-impurity Hamiltonian. The single-impurity scale remains an input to, not a solution of, the Kondo-lattice problem.

Minimal Worked Example and Benchmark: Two-Site Kondo Box

Consider one spin-$1/2$ impurity and one conduction electron on a two-site open bath:

$$H_{\mathrm{box}} ={} -t \sum_\sigma \left( c_{1\sigma}^\dagger c_{2\sigma} + c_{2\sigma}^\dagger c_{1\sigma} \right) + J_K\mathbf S\cdot\mathbf s_1, \qquad t\gt0.$$

The fixed-$N_c=1$ Hilbert space has dimension

$$2\binom41 = 8.$$

Use basis states

$$\lvert r,\sigma;\sigma_S\rangle, \qquad r\in\{1,2\},$$

where $r$ is the electron site, $\sigma$ its spin, and $\sigma_S$ the impurity spin.

Because the Hamiltonian is $SU(2)$ invariant, total spin separates the problem into one singlet orbital block and three identical triplet orbital blocks.

Singlet block

In the basis where the electron and impurity form a singlet and the electron occupies site $1$ or $2$,

$$H_{S_{\mathrm{tot}}=0} = \begin{pmatrix} -\frac34J_K & -t\\ -t & 0 \end{pmatrix}.$$

Its energies are

$$E_{\mathrm s,\pm} = -\frac{3J_K}{8} \pm \sqrt{ t^2+ \frac{9J_K^2}{64} }.$$

Triplet blocks

For each $m=-1,0,1$,

$$H_{S_{\mathrm{tot}}=1} = \begin{pmatrix} \frac14J_K & -t\\ -t & 0 \end{pmatrix},$$

with energies

$$E_{\mathrm t,\pm} = \frac{J_K}{8} \pm \sqrt{ t^2+ \frac{J_K^2}{64} }.$$

Each triplet energy is threefold degenerate.

Characteristic polynomial

The complete eight-dimensional characteristic polynomial is

$$p(E) ={} \left( E^2- \frac{J_K}{4}E -t^2 \right)^3 \times \left( E^2+ \frac{3J_K}{4}E -t^2 \right).$$

For $J_K\gt0$, the ground state is the lower singlet branch.

The two-site Kondo box separates into one singlet block and three identical triplet blocks. At $J_K=t>0$, the exact ground state is a singlet. The finite box is an operator and symmetry benchmark, not a discretization fine enough to exhibit an exponentially small continuum Kondo scale.

Numerical target at equal exchange and hopping

Set

$$J_K = t \gt0.$$

The two singlet energies are

$$\frac{E_{\mathrm s,\pm}}{t} = -\frac38 \pm \frac{\sqrt{73}}8,$$

or

$$\frac{E_{\mathrm s,-}}{t} = -1.443000468165, \qquad \frac{E_{\mathrm s,+}}{t} = 0.693000468165.$$

The triplet energies are

$$\frac{E_{\mathrm t,\pm}}{t} = \frac18 \pm \frac{\sqrt{65}}8,$$

or

$$\frac{E_{\mathrm t,-}}{t} = -0.882782218537, \qquad \frac{E_{\mathrm t,+}}{t} = 1.132782218537,$$

each with degeneracy three.

The singlet–triplet gap above the ground state is

$$\frac{\Delta_{\mathrm{st}}}{t} = 0.560218249627.$$

Useful complete-matrix invariants are

$$\operatorname{tr}H_{\mathrm{box}} = 0,$$ $$\operatorname{tr}H_{\mathrm{box}}^2 = 8t^2 + \frac34J_K^2,$$

and

$$\det H_{\mathrm{box}} = t^8.$$

At $J_K=t$, the ground-state probability that the conduction electron occupies site $1$ is

$$p_1 = \frac12 \left( 1+ \frac3{\sqrt{73}} \right) = 0.675561720794.$$

Only site $1$ participates in exchange, so

$$\langle\mathbf S\cdot\mathbf s_1\rangle = -\frac34p_1 = -0.506671290596.$$

Hellmann–Feynman gives the same result:

$$\langle\mathbf S\cdot\mathbf s_1\rangle = \frac{\partial E_{\mathrm s,-}}{\partial J_K}.$$

Checking the finite matrix

1. Build the eight-dimensional fixed-$N_c=1$ matrix from fermionic operators and Pauli matrices.
2. Verify Hermiticity, particle-number conservation, and $SU(2)$ multiplet degeneracies.
3. Reproduce the factorized characteristic polynomial.
4. Check the trace, trace-square, and determinant before diagonalization.
5. At $J_K=t$, reproduce all four distinct energies and their degeneracies.
6. Check the ground-state site occupation and exchange correlation independently.

This benchmark does not test the one-loop Kondo temperature, a screening cloud, or the Wilson ratio. Those require a bath with controlled scale separation.

Numerical and Analytical Methods

Poor-man’s scaling

Shell elimination controls the weak-coupling regime and derives logarithms and the leading exponential scale. It stops being perturbative near $T_K$.

Numerical renormalization group

NRG logarithmically discretizes the bath and maps it to a Wilson chain whose hoppings fall asymptotically as

$$t_n \propto D_0\Lambda^{-n/2}, \qquad \Lambda\gt1.$$

Iterative diagonalization then resolves exponentially separated scales. Spectra require additional broadening and discretization checks beyond thermodynamic convergence.

Bethe ansatz

Integrable formulations give exact equilibrium results and universal ratios. Their power does not automatically extend to arbitrary real-frequency observables, generic bands, or nonequilibrium driving.

Tensor networks

After star-to-chain or energy-space mappings, tensor networks can resolve screening correlations and finite-size dynamics. Chain length, bath discretization, entanglement growth, and the ratio $\delta/(k_{\mathrm B}T_K)$ control reliability.

Quantum Monte Carlo

Impurity Monte Carlo is often formulated for an Anderson parent and can be highly accurate on the imaginary-time axis. Analytic continuation remains ill conditioned, and sign or ergodicity behavior depends on the variant.

Exact diagonalization

Exact diagonalization is decisive for small-cluster operator checks such as the Kondo-box benchmark. A uniformly spaced bath grows exponentially in size and resolves an exponentially small scale inefficiently.

Principal Variants

Anisotropic Kondo model

Independent $J_z$ and $J_\perp$ produce coupled RG flows and include the Toulouse solvable point. Setting $J_\perp=0$ removes spin flips and changes the infrared problem.

Multichannel model

Several conserved bath channels can compete to screen the same moment. Channel symmetry is essential to the canonical overscreened non-Fermi-liquid fixed point.

Higher-spin impurity

Changing $S$ changes the channel count required for exact screening. A spin larger than $1/2$ in one channel is underscreened rather than simply a rescaled baseline model.

Two-impurity model

Two moments experience both Kondo screening and a conduction-mediated RKKY interaction. RKKY Interaction derives that weak-coupling pair term from the host susceptibility. Its competition with screening and special two-impurity critical behavior are absent from the single-impurity Hamiltonian.

Kondo lattice

A moment on every lattice site introduces coherence, magnetism, and collective Fermi-surface questions. A lattice is not obtained by multiplying a single-impurity thermodynamic function by the number of sites. Kondo Lattices develops that dense many-body problem.

Pseudogap and structured baths

If

$$\rho_0(\epsilon) \propto |\epsilon|^r,$$

the metallic logarithmic flow changes and impurity quantum phase transitions can occur. Superconducting, Luttinger-liquid, and topological baths likewise require separate analyses.

Coqblin–Schrieffer and orbital Kondo models

Larger impurity multiplets and orbital or flavor exchange generalize the spin-$1/2$ algebra. The relevant symmetry group, representation, and number of channels must be declared.

Exercises

1. Normalize the local channel

Starting from

$$f_{0\sigma} = \sum_k u_kc_{k\sigma},$$

derive the condition on $u_k$ required for canonical anticommutation.

Solution

Using the bath anticommutator,

$$\begin{aligned} \{f_{0\sigma},f_{0\sigma'}^\dagger\} ={}& \sum_{k,q} u_ku_q^* \{c_{k\sigma},c_{q\sigma'}^\dagger\} \\ ={}& \delta_{\sigma\sigma'} \sum_k|u_k|^2. \end{aligned}$$

Therefore $f_{0\sigma}$ is normalized when

$$\sum_k|u_k|^2 = 1.$$

The same weights enter the local density of states, so changing them changes both the local orbital and the coupling normalization.

2. Derive the Kondo-box blocks

Explain why the two-site, one-electron benchmark reduces to one singlet $2\times2$ block and three identical triplet $2\times2$ blocks.

Solution

The hopping is spin independent and the exchange is rotationally invariant, so total spin is conserved. Combining the impurity spin $1/2$ and electron spin $1/2$ gives

$$\frac12\otimes\frac12 = 0\oplus1.$$

For either total-spin value, the remaining orbital coordinate is whether the electron occupies site $1$ or site $2$. Hopping contributes $-t$ between those two orbital states. Exchange acts only on site $1$, with eigenvalue $-3J_K/4$ in the singlet and $J_K/4$ in the triplet. This gives the displayed $2\times2$ blocks. The singlet has one spin state, while the triplet has three $m$ values and therefore three identical blocks.

3. Check the Kondo-box invariants

Use the two block matrices to derive $\operatorname{tr}H$, $\operatorname{tr}H^2$, and $\det H$ for the complete eight-state benchmark.

Solution

The singlet block has trace $-3J_K/4$ and each triplet block has trace $J_K/4$. Including three triplets,

$$\operatorname{tr}H = -\frac{3J_K}{4} + 3\frac{J_K}{4} = 0.$$

For a real symmetric block

$$\begin{pmatrix} a&-t\\ -t&0 \end{pmatrix},$$

the trace of the square is $a^2+2t^2$. Hence

$$\begin{aligned} \operatorname{tr}H^2 ={}& \frac{9J_K^2}{16} +2t^2 +3 \left( \frac{J_K^2}{16} +2t^2 \right) \\ ={}& \frac34J_K^2 +8t^2. \end{aligned}$$

Every block determinant is $-t^2$. Four blocks therefore give

$$\det H = (-t^2)^4 = t^8.$$

4. Ferromagnetic one-loop flow

Let $g_0\lt0$. Use the one-loop solution to show that the running exchange approaches zero from below as $D\to0$ within the perturbative coordinate.

Solution

The solution is

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

For $g_0\lt0$ and $\ell\gt0$, the denominator is

$$1+|g_0|\ell,$$

which grows without crossing zero. Thus

$$g(\ell) \longrightarrow 0^-$$

as $\ell\to\infty$. The exchange is marginally irrelevant and the impurity is not driven to the fully screened antiferromagnetic fixed point.

5. Convert Kondo-scale conventions

Show that the per-spin expression

$$D_0e^{-1/(\rho_0J_K)}$$

equals the spin-summed expression

$$D_0e^{-2/(\rho_{\mathrm{tot}}J_K)}$$

when $\rho_{\mathrm{tot}}=2\rho_0$.

Solution

Substitution gives

$$\rho_{\mathrm{tot}}J_K = 2\rho_0J_K,$$

so

$$\frac{2}{\rho_{\mathrm{tot}}J_K} = \frac{1}{\rho_0J_K}.$$

The exponents are identical. Redefining the density of states or the exchange vertex requires a corresponding formula change.

6. Finite-bath resolution

A calculation has $k_{\mathrm B}T_K/D_0=10^{-4}$ and a uniformly spaced bath with level spacing $\delta/D_0=2\times10^{-3}$. Can it resolve a continuum-like Kondo crossover?

Solution

The ratio is

$$\frac{\delta}{k_{\mathrm B}T_K} = 20.$$

There are too few levels within an energy window $k_{\mathrm B}T_K$ to approximate the continuum crossover. Even a level placed exactly at the Fermi energy does not supply the missing continuum. It may still provide a valid finite-box spectrum and operator test. A logarithmic discretization, a much larger bath, or another impurity solver is needed to resolve the exponentially small scale.

References

• P. W. Anderson, “A Poor Man’s Derivation of Scaling Laws for the Kondo Problem”, Journal of Physics C 3, 2436–2441 (1970) — shell-elimination scaling argument.
• N. Andrei, “Diagonalization of the Kondo Hamiltonian”, Physical Review Letters 45, 379–382 (1980) — exact Bethe-ansatz solution.
• N. Andrei, K. Furuya, and J. H. Lowenstein, “Solution of the Kondo Problem”, Reviews of Modern Physics 55, 331–402 (1983) — exact-solution framework and thermodynamics.
• M. Cheng, T. Chowdhury, A. Mohammed, and K. Ingersent, “Phase boundaries of power-law Anderson and Kondo models: A poor man’s scaling study,” Physical Review B 96, 045103 (2017), doi:10.1103/PhysRevB.96.045103, author manuscript — per-spin density, half-Pauli exchange vertex and scaling coefficients in Eqs. (9), (66) and (67).
• A. C. Hewson, The Kondo Problem to Heavy Fermions, Cambridge University Press (1993) — standard monograph on impurity models, scaling, exact methods, and heavy-fermion connections.
• J. Kondo, “Resistance Minimum in Dilute Magnetic Alloys”, Progress of Theoretical Physics 32, 37–49 (1964) — perturbative spin-flip logarithm and the resistance minimum in the paper’s sign convention.
• A. H. Nevidomskyy, “The Kondo Model and Poor Man’s Scaling,” in Many-Body Physics: From Kondo to Hubbard, edited by E. Pavarini, E. Koch, and P. Coleman, Modeling and Simulation 5, Forschungszentrum Jülich (2015), author lecture — explicitly uses the alternative exchange $2J\mathbf S\cdot\mathbf s$.
• P. Nozières, “A Fermi-Liquid Description of the Kondo Problem at Low Temperatures”, Journal of Low Temperature Physics 17, 31–42 (1974) — local Fermi-liquid fixed-point description.
• J. R. Schrieffer and P. A. Wolff, “Relation between the Anderson and Kondo Hamiltonians”, Physical Review 149, 491–492 (1966) — canonical low-energy transformation from a charge-fluctuating impurity.
• P. B. Wiegmann, “Exact Solution of the s–d Exchange Model (Kondo Problem)”, Journal of Physics C 14, 1463–1478 (1981) — integrability and exact equilibrium properties.
• K. G. Wilson, “The Renormalization Group: Critical Phenomena and the Kondo Problem”, Reviews of Modern Physics 47, 773–840 (1975) — nonperturbative numerical renormalization-group solution.