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Modern Theory of Thermoelectricity$

Veljko Zlatić and René Monnier

Print publication date: 2014

Print ISBN-13: 9780198705413

Published to British Academy Scholarship Online: June 2014

DOI: 10.1093/acprof:oso/9780198705413.001.0001

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(p.241) Appendix E Scaling

(p.241) Appendix E Scaling

Source:
Modern Theory of Thermoelectricity
Publisher:
Oxford University Press

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<|ϵk|<D|δD|, and a “high-energy” sector of width |δD|. This is sketched in Fig. E.1, where the chemical potential μ separates the occupied from the unoccupied states.

Appendix E Scaling

Fig. E.1 The high-energy particle and hole states that are removed by reducing the band width by |δD| are indicated by shaded rectangles. The occupied and unoccupied electron states are indicated in dark and light shading, respectively.

To derive the effective Hamiltonian, we write the Hilbert space of the full model, H, as a direct sum of three subspaces H0, H1, and H2 and represent the exact many-body wavefunction of the system by the three components ψ={ψ0,ψ1,ψ2}. The wavefunction ψ1H1 describes a low-energy state, such that there are no conduction (p.242) electrons excited close to the upper band edge, D|δD|<ϵk<D, or holes excited close to the lower band edge, D<ϵk<D+|δD|; ψ0H0 describes a state with at least one hole in the lower band edge; and ψ2H2 describes a state with at least one conduction electron in the upper band edge. The full Hilbert space is spanned by H0,H1, and H2, which, however, are not invariant subspaces of HK=Hc+Hsd, because Hsd can transfer a state from H1 to H0 or H2, or from H0 to H2, and vice versa. The Schrödinger equation HKψ=Eψ can be written in matrix form

(E.1)
H00H01H02H10H11H12H20H21H22ψ0ψ1ψ2=Eψ0ψ1ψ2,

and for the ψ1 component we get

(E.2)
H11ψ1+H12ψ2+H10ψ0=Eψ1,

where H11 contains Hc and the part of Hsd that does not transfer ψ1 outside H1. The two terms H12ψ2+H10ψ0 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 ψ1 is obtained by eliminating ψ0 and ψ2 from Eq. (E.2) using the exact expressions

(E.3)
ψ0=1EH00H01ψ1+1EH00H02ψ2

and

(E.4)
ψ2=1EH22H21ψ1+1EH22H20ψ0.

This yields

(E.5)
ψ0=1EH00H01+H021EH22H21ψ1+H021EH22H20ψ0.

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

(E.6)
ψ0=(1EH00H01+1EH00H021EH22H21+1EH00H021EH22H201EH00H01+)ψ1,

Keeping only the lowest-order corrections in J, we approximate (EH00)1H01ψ1(EHc)1H01ψ1H01/|D|ψ1 and (EH22)1H21ψ1(EHc)1H21ψ1H21/|D|ψ1, so that each successive term in the series (E.6) has one more power of 1/D. Repeating the same procedure for ψ2, substituting the ensuing series together (p.243) with ψ0 into (E.2 ), and keeping only the first-order term in 1/|D| yields an effective Schrödinger equation for ψ1:

(E.7)
(H11+ΔH)ψ1=Eψ1

where

(E.8)
ΔH=H101EH0H01+H121EH0H21.

A detailed picture of the action of ΔH is obtained if we introduce the explicit band energies ϵk. 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, |ϵk|D|δD|. This intermediate state is propagated by (EH00)1, after which H10 lets the hole relax to a state with, for example, wavevector k and (low) energy |ϵk|D|δD|. The net effect is a scattering event with a scattering vector q=kk in the subspace H1. The second term in Eq. (E.8) involves an intermediate state with an excited electron at energy D|δD|ϵqD and a hole at energy |ϵk|<D|δD|. The propagation of this configuration is described by (EH22)1. Finally, H12 removes the electron with wavevector q from the upper band edge and creates an electron with energy |ϵk|<D|δD|, so that the total process again amounts to a scattering event in the subspace H1, with scattering vector q=kk. The effective model with a reduced bandwidth is defined by summing over all such scattering events.

The Hamiltonian H11+ΔH operates in a restricted Hilbert space with the cutoff D|δD| 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, kBT. 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 (EH00)1H01(EHc)1H01 and (EH22)1H21(EHc)1H21, that is, we neglected the terms of order J2 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+=JJ± are different from the longitudinal one Jz. The interaction term is then

(E.9)
Hsd=k,kJ+S+ck,ck,JSck,ck,JzSz(ck,ck,ck,ck,),

(p.244) where S+ and S are the local spin-raising and lowering operators. We now calculate ΔH to lowest order in 1/D.

Appendix E Scaling

Fig. E.2 (A) and (B) illustrate the renormalization of Jz due to the spin-flip terms in ΔH. (A) depicts two consecutive spin-flip processes that first create and then remove an down-spin electron close to the upper band edge. (B) shows the contribution to ΔH that first creates and then removes an down-spin hole in the lower band edge. (C) and (D) illustrate the renormalization of J+. (C) shows the removal of a k-electron and the creation of a q-electron by the terms proportional to J. Eventually, the intermediate q-electron is removed and a k-electron is created, by the terms proportional to Jz. (D) shows the creation of a k-electron and a q-hole by the terms proportional to Jz, and the subsequent annihilation of the intermediate q-hole with the k-electron by the terms proportional to J.

The contribution to ΔH due to two consecutive spin-flip processes, such that a conduction electron with spin ↑ and wavevector k is scattered first into a high-energy intermediate state q and then to a final state k, is illustrated by the diagram in Fig. E.2 (A). This contribution renormalizes Jz and is defined by the expression

(E.10)
J+JSck,c,1EHcqS+cq,ck,,

where the q- and q-summations run over the states in the band edges. At low temperatures, TD, the band edges are unoccupied in the initial state, so that we must have c,cq,=δq,q, which gives

(E.11)
(J+J)(SS+)1E(E0+ϵqϵk)ck,ck,,

where E0 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) ρ0, yields for the spin S=12 model,

(E.12)
J+Jρ0|δD|ΔED+ϵk(1/2Sz)ck,ck,,

(p.245) where we have used SS+=1/2Sz. We have also introduced the renormalized energy of the system, measured relative to that of the unperturbed ground state, ΔE=EE0, and made the approximation ϵqD, which holds for |δD|D.

The contribution to ΔH 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

(E.13)
J+Jρ0|δD|ΔEDϵk(1/2+Sz)ck,ck,=J+Jρ0|δD|ΔEDϵk(1/2+Sz)ck,ck,,

where we have used S+S=1/2+Sz 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 k is given by E0ϵq+ϵk, with ϵqD. 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 H11 due to the spin-flip processes starting from the up-spin states.

Another contribution to ΔH 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 sz(0)Sz term in Hsd, 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 δ(Jz)=δ|Jz| in the (antiferromagnetic) coupling constant, due to the reduction of the band edges by δD:

(E.14)
δ|Jz|=J+Jρ0|δD|1ΔED+ϵk+1ΔEDϵk.

The terms in ΔH that scatter a conduction electron or a hole from an initial state k to a final state k, via a high-energy intermediate state q, are shown graphically in Fig. E.2 (C, D). They are given by the terms

(E.15)
12J+Jzρ0|δD|ΔED+ϵkS+ck,ck,and12J+Jzρ0|δD|ΔEDϵkS+ck,ck,

where we have used SzS+=S+/2 and S+Sz=S+/2. The operators in these expressions coincide with the J+S+s(0) term in Hsd and their coefficient gives the correction to J+. There are also similar processes that scatter an electron or hole from an initial state k to a final state k and modify J. Summing all these terms allows us to write

(E.16)
δ|J±|=J±Jzρ0|δD|1ΔED+ϵk+1ΔEDϵk'.

This derivation shows that the effective Hamiltonian H11+ΔH 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 ϵk and ϵk 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

(E.17)
d|J±|dlnD=2ρ0JzJ±andd|Jz|dlnD=2ρ0J±2,

which have to be solved for a given initial condition J±(D0)=J±0 and Jz(D0)=Jz0. Note that δD 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±(D) and Jz(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±dJ±=JzdJz. Integrating from the initial to the final cutoff yields the scaling invariant [J±(D)]2[Jz(D)]2=(J±0)2(Jz0)2. For each initial condition, the invariant flow of the coupling constant in the (J±,Jz)-plane is described by a hyperbola, specified by the value (J±0)2(Jz0)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±=Jz=J, the scaling equation reads

(E.18)
d|J|dlnD=2ρ0J2=2ρ0|J|2,

and the initial condition is J(D0)=J0. Integrating from the initial to the final cutoff, with a constant DOS ρ0, gives the scaling invariant:

(E.19)
lnD12ρ0|J|=lnD012ρ0|J0|ln(kBTK),

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

(E.20)
kBTK=D0exp12|J0|ρ0.

(p.247) The divergence of J as DkBTK restricts the validity of the scaling law to D/kB>max{T,TK}. Taking DkBT, where the proportionality constant is of the order of unity, we obtain, for a given scaling trajectory defined by TK, a temperature-dependent coupling constant

(E.21)
2ρ0|J(T)|=1ln(T/TK),

which increases as the temperature is reduced. For a given trajectory, the product 2ρ0|J(T)|ln(T/TK) 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.

Appendix E Scaling

Fig. E.3 Diagrams giving the renormalization of J0. Full lines describe the propagation of band states and dashed lines the propagation of the localized states.

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 Nm and NM, 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 ΔH 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+Δ|. The renormalization of the coupling constants is obtained by summing the states close to the band edges over the wavevectors 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):

(E.22)
δ(ρ0|Jmm|)δD=Nm(ρ0Jmm)2DNM(ρ0JmM)2D+Δ,δ(ρ0|JMM|)δD=Nm(ρ0JmM)2DNM(ρ0JMM)2D+Δ,E.22δ(ρ0|JmM|)δD=Nm(ρ0JmM)(ρ0Jmm)DNM(ρ0JMM)(ρ0JmM)D+Δ,

which show that the couplings grow when D changes by δD. Assuming a symmetric exchange coupling, Jmm=JMM=JmM=J(D), we obtain the scaling equation1

(E.23)
δ(ρ0|J|)δD=Nm(ρ0J)2DNM(ρ0J)2D+Δ=Nm(ρ0|J|)2DNM(ρ0|J|)2D+Δ

which can be integrated from D0 to D to yield the scaling invariant

(E.24)
1ρ0|J(D)|NmlnDNMlnD+Δ=1ρ0|J0|NmlnD0NMlnD0+Δ.

The Kondo temperature of the CF-split Coqblin–Schrieffer model is again defined by kBTK=D for the value of D at which J diverges, which leads to

(E.25)
kBTKD0NmkBTK+ΔD0+ΔNM=exp1ρ0|J0|.

For small splitting, such that ΔkBTK, the result is the same as the Kondo temperature of a (Nn+NM)-fold-degenerate model:

(E.26)
kBTKH=D0exp1(Nm+NM)ρ0|J0|.

(p.249) For large splitting, ΔkBTK, 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 ΔD0, (E.25) gives TK, 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 ΔkBTkBTK, we find

(E.27)
kBTKD0=D0ΔNM/Nmexp1Nmρ0|J0|,

where, in the cases of Ce and Yb compounds, the ratio D0/Δ is of order 10 or more. This large enhancement of TK 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=kBT gives the scaling trajectory of the Coqblin–Schrieffer model:

(E.28)
ρ0|J(T)|=lnTTKNmkBT+ΔkBTK+ΔNM1,

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 TK.

At high temperature, kBTΔ, the effective coupling follows the same trajectory as in the case of a (NM+NM)-fold-degenerate level with an effective high-temperature Kondo scale TKH. If kBTKHΔ, the scaling trajectory terminates at about kBTΔ, where the system makes a crossover to the strong-coupling regime. However, for kBTKHΔ, which happens in many heavy fermion systems, we can continue the scaling to temperatures below Δ and depopulate the excited CF states. For kBTΔ, 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 Nm-fold-degenerate system.2 Thus, the high- and low-temperature properties of an actual system can be inferred from effective models that are (Nm+Nm)-fold or Nm-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 ( Jmm0=JMM0=JmM0=J0), 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.

(2) But with modified TK given by Eq. (E.27).

(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 Δ. Energy is conserved by the scattering processes. As the width of the band is reduced bellow Δ ( |2D|<Δ), 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).