# (p.241) Appendix E Scaling

# (p.241) Appendix E Scaling

# E.1 Elimination of high-energy conduction states

In the Kondo and Coqblin–Schrieffer models, the transitions between the low- and high-energy conduction states induced by the exchange term in the Hamiltonian give rise to logarithmic corrections to the correlation functions, which diverge when the high-energy cutoff is taken to infinity. To remove these divergences, we eliminate the conduction states close to the band edges but impose the condition that the low-energy excitations of the effective Hamiltonian remain unchanged. This procedure leads to scaling equations for the effective coupling constants, which we first derive for the Kondo model with an anisotropic exchange scattering and a constant density of conduction states.

The scaling is carried out by dividing the conduction band into a “low-energy” sector, where $0<|{\u03f5}_{\mathbf{\text{k}}}|<D-|\mathrm{\delta}D|$, and a “high-energy” sector of width $|\mathrm{\delta}D|$. This is sketched in Fig. E.1, where the chemical potential $\mathrm{\mu}$ separates the occupied from the unoccupied states.

To derive the effective Hamiltonian, we write the Hilbert space of the full model, $\mathcal{H}$, as a direct sum of three subspaces ${\mathcal{H}}_{0}$, ${\mathcal{H}}_{1}$, and ${\mathcal{H}}_{2}$ and represent the exact many-body wavefunction of the system by the three components $\mathrm{\psi}=\{{\mathrm{\psi}}_{0},{\mathrm{\psi}}_{1},{\mathrm{\psi}}_{2}\}$. The wavefunction ${\mathrm{\psi}}_{1}\in {\mathcal{H}}_{1}$ describes a low-energy state, such that there are no conduction (p.242) electrons excited close to the upper band edge, $D-|\mathrm{\delta}D|<{\u03f5}_{\mathbf{\text{k}}}<D$, or holes excited close to the lower band edge, $-D<{\u03f5}_{\mathbf{\text{k}}}<-D+|\mathrm{\delta}D|$; ${\mathrm{\psi}}_{0}\in {\mathcal{H}}_{0}$ describes a state with at least one hole in the lower band edge; and ${\mathrm{\psi}}_{2}\in {\mathcal{H}}_{2}$ describes a state with at least one conduction electron in the upper band edge. The full Hilbert space is spanned by ${\mathcal{H}}_{0},{\mathcal{H}}_{1}$, and ${\mathcal{H}}_{2}$, which, however, are not invariant subspaces of ${H}_{K}={H}_{c}+{H}_{sd}$, because ${H}_{sd}$ can transfer a state from ${\mathcal{H}}_{1}$ to ${\mathcal{H}}_{0}$ or ${\mathcal{H}}_{2}$, or from ${\mathcal{H}}_{0}$ to ${\mathcal{H}}_{2}$, and vice versa. The Schrödinger equation ${H}_{K}\mathrm{\psi}=E\mathrm{\psi}$ can be written in matrix form

and for the ${\mathrm{\psi}}_{1}$ component we get

where ${H}_{11}$ contains ${H}_{c}$ and the part of ${H}_{sd}$ that does not transfer ${\mathrm{\psi}}_{1}$ outside ${\mathcal{H}}_{1}$. The two terms ${H}_{12}{\mathrm{\psi}}_{2}+{H}_{10}{\mathrm{\psi}}_{0}$ provide the correction to the low-energy dynamics due to the presence of the high-energy states and are treated as a perturbation. The effective Schrödinger equation for ${\mathrm{\psi}}_{1}$ is obtained by eliminating ${\mathrm{\psi}}_{0}$ and ${\mathrm{\psi}}_{2}$ from Eq. (E.2) using the exact expressions

and

This yields

Iteration of this equation produces a series expansion in which each successive term has one additional propagator in the upper or lower cutoff region:

Keeping only the lowest-order corrections in *J*, we approximate
$(E-{H}_{00}{)}^{-1}{H}_{01}{\mathrm{\psi}}_{1}\simeq (E-{H}_{c}{)}^{-1}{H}_{01}{\mathrm{\psi}}_{1}\simeq {H}_{01}/|D|{\mathrm{\psi}}_{1}$ and
$(E-{H}_{22}{)}^{-1}{H}_{21}{\mathrm{\psi}}_{1}\simeq (E-{H}_{c}{)}^{-1}{H}_{21}{\mathrm{\psi}}_{1}\simeq {H}_{21}/|D|{\mathrm{\psi}}_{1}$, so that each successive term in the series (E.6) has one more power of
$1/D$. Repeating the same procedure for
${\mathrm{\psi}}_{2}$, substituting the ensuing series together
(p.243)
with
${\mathrm{\psi}}_{0}$ into (E.2 ), and keeping only the first-order term in
$1/|D|$ yields an effective Schrödinger equation for
${\mathrm{\psi}}_{1}$:

where

A detailed picture of the action of $\mathrm{\Delta}H$ is obtained if we introduce the explicit band energies ${\u03f5}_{\mathbf{\text{k}}}$. The first term in Eq. (E.8) generates an intermediate state with a high-energy hole close to the bottom of the band and a low-energy electron at, say, $|{\u03f5}_{{\mathbf{\text{k}}}^{\mathrm{\prime}}}|\le D-|\mathrm{\delta}D|$. This intermediate state is propagated by $(E-{H}_{00}{)}^{-1}$, after which ${H}_{10}$ lets the hole relax to a state with, for example, wavevector $\mathbf{\text{k}}$ and (low) energy $|{\u03f5}_{\mathbf{\text{k}}}|\le D-|\mathrm{\delta}D|$. The net effect is a scattering event with a scattering vector $\mathbf{\text{q}}={\mathbf{\text{k}}}^{\mathrm{\prime}}-\mathbf{\text{k}}$ in the subspace ${\mathcal{H}}_{1}$. The second term in Eq. (E.8) involves an intermediate state with an excited electron at energy $D-|\mathrm{\delta}D|\le {\u03f5}_{\mathbf{\text{q}}}\le D$ and a hole at energy $|{\u03f5}_{\mathbf{\text{k}}}|<D-|\mathrm{\delta}D|$. The propagation of this configuration is described by $(E-{H}_{22}{)}^{-1}$. Finally, ${H}_{12}$ removes the electron with wavevector $\mathbf{\text{q}}$ from the upper band edge and creates an electron with energy $|{\u03f5}_{{\mathbf{\text{k}}}^{\mathrm{\prime}}}|<D-|\mathrm{\delta}D|$, so that the total process again amounts to a scattering event in the subspace ${\mathcal{H}}_{1}$, with scattering vector $\mathbf{\text{q}}={\mathbf{\text{k}}}^{\mathrm{\prime}}-\mathbf{\text{k}}$. The effective model with a reduced bandwidth is defined by summing over all such scattering events.

The Hamiltonian
${H}_{11}+\mathrm{\Delta}H$ operates in a restricted Hilbert space with the cutoff
$D-|\mathrm{\delta}D|$ but its low-energy eigenvalues coincide with the eigenvalues of the initial model with the cutoff *D*. For the scaling laws to hold, the initial and the final state of the system should neither have excited electrons at the upper band edge nor excited holes at the lower band edge, which limits the lowest effective bandwidth to the value given by the thermal energy of the system,
${k}_{B}T$. Furthermore, the fact that the effective antiferromagnetic couplings increase when the bandwidth decreases leads to an additional constraint. In the derivation of the scaling equations, we made the approximations
$(E-{H}_{00}{)}^{-1}{H}_{01}\simeq (E-{H}_{c}{)}^{-1}{H}_{01}$ and
$(E-{H}_{22}{)}^{-1}{H}_{21}\simeq (E-{H}_{c}{)}^{-1}{H}_{21}$, that is, we neglected the terms of order
${J}^{2}$ and higher. Thus, the reduction of the bandwidth has to terminate once the coupling constants become of order unity.

We now show that the effective model derived by this procedure has the same form as the initial model and derive the scaling equation for the effective coupling constants for the Kondo and Coqblin–Schrieffer models.

# E.2 Poor man’s scaling for the Kondo model

Following Anderson (1970) (for details see also Hewson, 1993), we consider a generalized Kondo model in which the transverse couplings ${J}_{+}={J}_{-}\equiv {J}_{\pm}$ are different from the longitudinal one ${J}_{z}$. The interaction term is then

(p.244) where ${S}^{+}$ and ${S}^{-}$ are the local spin-raising and lowering operators. We now calculate $\mathrm{\Delta}H$ to lowest order in $1/D$.

The contribution to $\mathrm{\Delta}H$ due to two consecutive spin-flip processes, such that a conduction electron with spin ↑ and wavevector $\mathbf{\text{k}}$ is scattered first into a high-energy intermediate state $\mathbf{\text{q}}\downarrow $ and then to a final state ${\mathbf{\text{k}}}^{\prime}\uparrow $, is illustrated by the diagram in Fig. E.2 (A). This contribution renormalizes $-{J}_{z}$ and is defined by the expression

where the $\mathbf{\text{q}}$- and ${\mathbf{\text{q}}}^{\mathrm{\prime}}$-summations run over the states in the band edges. At low temperatures, $T\ll D$, the band edges are unoccupied in the initial state, so that we must have ${c}_{\mathbf{\text{q}},\downarrow}{c}_{\mathbf{\text{q}}{\text{}}^{\mathbf{\prime}},\downarrow}^{\u2020}={\mathrm{\delta}}_{\mathbf{\text{q}},\text{q}{\text{}}^{\mathbf{\prime}}}$, which gives

where ${E}_{0}$ is the energy of the initial state. Carrying out the summation over the states in the upper band edge, assuming a constant density of states (DOS) ${\mathrm{\rho}}_{0}$, yields for the spin $S=\frac{1}{2}$ model,

(p.245) where we have used ${S}^{-}{S}^{+}=1/2-{S}_{z}$. We have also introduced the renormalized energy of the system, measured relative to that of the unperturbed ground state, $\mathrm{\Delta}E=E-{E}_{0}$, and made the approximation ${\u03f5}_{\mathbf{\text{q}}}\simeq D$, which holds for $|\mathrm{\delta}D|\ll D$.

The contribution to $\mathrm{\Delta}H$ due to two consecutive spin-flip processes that create and remove an up-spin hole in the lower band edge is given by the diagram in Fig. E.2 (B). We have

where we have used ${S}^{+}{S}^{-}=1/2+{S}_{z}$ and have taken into account that the energy of the intermediate state with an electron removed from the lower band edge and an electron added to the state $\mathbf{\text{k}}{\text{}}^{\mathbf{\prime}}$ is given by ${E}_{0}-{\u03f5}_{\mathbf{\text{q}}}+{\u03f5}_{\mathbf{\text{k}}{\text{}}^{\mathbf{\prime}}}$, with ${\u03f5}_{\mathbf{\text{q}}}\simeq -D$. In the Eq. (E.13 ), we have also brought the fermionic operators to normal order. The operator expressions (E.12) and (E.13) give the correction to ${H}_{11}$ due to the spin-flip processes starting from the up-spin states.

Another contribution to
$\mathrm{\Delta}H$ that has the same combination of spin and fermionic operators as in Eqs. (E.12) and (E.13 ), except for the sign, is obtained when considering the spin-flip processes that start from the down-spin states. Summing all these spin-flip processes yields an operator expression that coincides with the
${s}_{z}(0){S}_{z}$ term in
${H}_{sd}$, where we have introduced the *z*-component of the conduction-electron spin at the impurity site (0). The coefficient of that operator gives the change
$\mathrm{\delta}(-{J}_{z})=\mathrm{\delta}|{J}_{z}|$ in the (antiferromagnetic) coupling constant, due to the reduction of the band edges by
$\mathrm{\delta}D$:

The terms in $\mathrm{\Delta}H$ that scatter a conduction electron or a hole from an initial state $\mathbf{\text{k}}\uparrow $ to a final state $\mathbf{\text{k}}{\text{}}^{\mathbf{\prime}}\downarrow $, via a high-energy intermediate state $\mathbf{\text{q}}\downarrow $, are shown graphically in Fig. E.2 (C, D). They are given by the terms

where we have used ${S}_{z}{S}^{+}={S}^{+}/2$ and ${S}^{+}{S}_{z}=-{S}^{+}/2$. The operators in these expressions coincide with the ${J}^{+}{S}^{+}{s}^{-}(0)$ term in ${H}_{sd}$ and their coefficient gives the correction to ${J}^{+}$. There are also similar processes that scatter an electron or hole from an initial state $\mathbf{\text{k}}{\text{}}^{\mathbf{\prime}}\downarrow $ to a final state $\mathbf{\text{k}}\uparrow $ and modify ${J}^{-}$. Summing all these terms allows us to write

This derivation shows that the effective Hamiltonian ${H}_{11}+\mathrm{\Delta}H$ defined in the reduced Hilbert space has the same form as the initial Kondo Hamiltonian and that the low-energy excitations will be unchanged provided the coupling constants are rescaled in agreement with Eqs. (E.14) and (E.16).

(p.246)
At low temperature, the energy of the excitations relative to the ground state is much smaller than the bandwidth *D*, and the energy dependence of the coupling constants can be neglected. We can also neglect
${\u03f5}_{\mathbf{\text{k}}}$ and
${\u03f5}_{\mathbf{\text{k}}{\text{}}^{\mathbf{\prime}}}$ relative to *D*, since we are only interested in the scattering of conduction electrons near the Fermi surface. Thus, the reduction of the Hilbert space leads to the scaling equations

which have to be solved for a given initial condition ${J}_{\pm}({D}_{0})={J}_{\pm}^{0}$ and ${J}_{z}({D}_{0})={J}_{z}^{0}$. Note that $\mathrm{\delta}D$ is negative, since it corresponds to the reduction of the bandwidth, and the factor of two is a consequence of the twofold degeneracy of the spin-1/2 state. Integration of the differential scaling law yields the trajectories ${J}_{\pm}(D)$ and ${J}_{z}(D)$ along which the low-energy excitations of the model remain invariant with respect to the changes of the bandwidth. The effective couplings contain the information about the effect of the high-energy states on the low-energy dynamics, and, as long as the scaling laws hold, the correlation functions of the effective model can be calculated by lowest-order perturbation theory.

# E.3 Analysis of the scaling equations

The scaling trajectories for the transverse and longitudinal couplings of the anisotropic Kondo model are related, as can be seen by eliminating *D* from Eqs. (E.17) and writing the differential scaling law as
${J}_{\pm}\text{}d{J}_{\pm}={J}_{z}\text{}d{J}_{z}$. Integrating from the initial to the final cutoff yields the scaling invariant
$[{J}_{\pm}(D){]}^{2}-[{J}_{z}(D){]}^{2}=({J}_{\pm}^{0}{)}^{2}-({J}_{z}^{0}{)}^{2}$. For each initial condition, the invariant flow of the coupling constant in the
$({J}_{\pm},{J}_{z})$-plane is described by a hyperbola, specified by the value
$({J}_{\pm}^{0}{)}^{2}-({J}_{z}^{0}{)}^{2}$. In the case of an antiferromagnetic coupling, the reduction of *D* to arbitrary small values results in an infinitely large coupling.

In the particular case of the isotropic Kondo model, with ${J}_{\pm}={J}_{z}=J$, the scaling equation reads

and the initial condition is $J({D}_{0})={J}_{0}$. Integrating from the initial to the final cutoff, with a constant DOS ${\mathrm{\rho}}_{0}$, gives the scaling invariant:

where we see that the Kondo temperature is defined by the value of *D* at which the effective coupling diverges. This leads to

(p.247)
The divergence of *J* as
$D\to {k}_{B}{T}_{K}$ restricts the validity of the scaling law to
$D/{k}_{\text{B}}>\text{max}\{T,{T}_{\text{K}}\}$. Taking
$D\propto {k}_{B}T$, where the proportionality constant is of the order of unity, we obtain, for a given scaling trajectory defined by
${T}_{K}$, a temperature-dependent coupling constant

which increases as the temperature is reduced. For a given trajectory, the product $2{\mathrm{\rho}}_{0}|J(T)|\text{ln}(T/{T}_{K})$ is a scaling invariant.

From the analysis of the scaling equation, Anderson (1970) conjectured that the Kondo temperature separates the high-temperature perturbative regime from the low-temperature nonperturbative one. This conjecture has been confirmed by exact calculations, which have shown that the ground state of a Kondo system is characterized by an infinitely large coupling, such that the impurity spin is completely screened by the conduction electrons and the impurity susceptibility is Pauli-like.

# E.4 Poor man’s scaling for the Coqblin–Schrieffer model with crystal field splitting

The coupling constants of the Coqblin–Schrieffer model are renormalized by the scaling procedure in the same way as in the spin-1/2 Kondo model. For simplicity, we consider an impurity with two CF levels separated by an energy Δ; the degeneracies of the lower and upper CF levels are
${N}_{m}$ and
${N}_{M}$, respectively. The elimination of the band edges by the poor man’s scaling introduces a new feature, not present in the case of a single level. Namely, the localized state itself can now change its energy in the scattering event, so that the intermediate states can differ by about Δ. The contribution to
$\mathrm{\Delta}H$ due to the exchange scattering is illustrated in Fig. E.3, where (A) and (B) describe
(p.248)
intermediate electrons and holes with energy of the order of
$|D|$, and (C) and (D) describe the intermediate particles with energy close to
$|D+\mathrm{\Delta}|$. The renormalization of the coupling constants is obtained by summing the states close to the band edges over the wavevectors
$\mathbf{\text{q}}$ and over the internal degrees of freedom *m* or *M*. Making the same approximations as in the case of the Kondo model, and neglecting the renormalization of the potential scattering, we obtain the scaling equations for the effective coupling (Yamada et al., 1984):

which show that the couplings grow when *D* changes by
$\mathrm{\delta}D$. Assuming a symmetric exchange coupling,
${J}_{mm}={J}_{MM}={J}_{mM}=J(D)$, we obtain the scaling equation^{1}

which can be integrated from
${D}_{0}$ to *D* to yield the scaling invariant

The Kondo temperature of the CF-split Coqblin–Schrieffer model is again defined by
${k}_{B}{T}_{K}=D$ for the value of *D* at which *J* diverges, which leads to

For small splitting, such that $\mathrm{\Delta}\ll {k}_{B}{T}_{K}$, the result is the same as the Kondo temperature of a $({N}_{n}+{N}_{M})$-fold-degenerate model:

(p.249) For large splitting, $\mathrm{\Delta}\gg {k}_{B}{T}_{K}$, and temperatures such that the thermal excitations to higher CF states can be neglected, the system behaves as an effective doublet. However, as long as $\mathrm{\Delta}\ll {D}_{0}$, (E.25) gives ${T}_{K}$, which is much enhanced with respect to the Kondo temperature of a model with the degeneracy set by the lowest CF level but without the higher CF levels. For $\mathrm{\Delta}\gg {k}_{B}T\gg {k}_{B}{T}_{K}$, we find

where, in the cases of Ce and Yb compounds, the ratio ${D}_{0}/\mathrm{\Delta}$ is of order 10 or more. This large enhancement of ${T}_{K}$ by the virtual transitions to the excited CF states explains the surprisingly large Kondo temperatures of many heavy fermion systems.

Reducing the cutoff in Eq. (E.24) to $D={k}_{B}T$ gives the scaling trajectory of the Coqblin–Schrieffer model:

which is very useful for a qualitative analysis of experimental data on CF-split Kondo systems. Note, that the lowest-order scaling theory breaks down (the coupling becomes too large for the theory to hold) when the temperature approaches ${T}_{K}$.

At high temperature,
${k}_{B}T\gg \mathrm{\Delta}$, the effective coupling follows the same trajectory as in the case of a
$({N}_{M}+{N}_{M})$-fold-degenerate level with an effective high-temperature Kondo scale
${T}_{K}^{H}$. If
${k}_{B}{T}_{K}^{H}\simeq \mathrm{\Delta}$, the scaling trajectory terminates at about
${k}_{B}T\simeq \mathrm{\Delta}$, where the system makes a crossover to the strong-coupling regime. However, for
${k}_{B}{T}_{K}^{H}\ll \mathrm{\Delta}$, which happens in many heavy fermion systems, we can continue the scaling to temperatures below Δ and depopulate the excited CF states. For
${k}_{B}T\ll \mathrm{\Delta}$, the second term on the right-hand side of Eq. (E.28) can be neglected and
$J(T)$ follows the same scaling trajectory as in a
${N}_{m}$-fold-degenerate system.^{2} Thus, the high- and low-temperature properties of an actual system can be inferred from effective models that are
$({N}_{m}+{N}_{m})$-fold or
${N}_{m}$-fold degenerate and have the CF splitting absorbed into the effective Kondo scale. The correlation functions for such effective models are easy to calculate.^{3}

## Notes:

*Modern Theory of Thermoelectricity*. First Edition. Veljko Zlatić and René Monnier

© Veljko Zlatić and René Monnier 2014. Published in 2014 by Oxford University Press.

(^{1})
It was pointed out by Nozières and Blandin (1980) that, even for a fully symmetric initial Hamiltonian (
${J}_{mm}^{0}={J}_{MM}^{0}={J}_{mM}^{0}={J}^{0}$), the reduction of the bandwidth makes all the coupling constants different; i.e., scaling breaks the initial symmetry of the model. It appears, however, that in the weak-coupling regime the symmetry is *not* badly broken and that it still makes sense to neglect the potential scattering and to consider only the symmetric part of the effective Hamiltonian.

(^{3})
In the unrenormalized model, it is possible for an electron to scatter off the impurity, changing its energy by Δ, and for the impurity to change its energy by
$-\mathrm{\Delta}$. Energy is conserved by the scattering processes. As the width of the band is reduced bellow Δ (
$|2D|<\mathrm{\Delta}$), such processes are not possible, as there are no states differing by Δ in the conduction band. However, we still continue the scaling as if all generalized spin transitions are possible. Thus, it appears that the formal validity of the above procedure is somewhat questionable. A possible justification is that all the second-order processes that we are considering are virtual. Energy has to be conserved only on by the whole fermionic system and the energy released (absorbed) by the impurity is absorbed (released) by the whole conduction band. This view is supported by a calculation based on Yosida’s variational theory which gives the same expression for the Kondo temperature (Yamada et al., 1984).