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VOLUME 85, NUMBER 5 PHYSICAL REVIEW LETTERS 31 JULY 2000 Coherence Scale of the Kondo Lattice S. Burdin, 1,2 A. Georges, 3 and D. R. Grempel 1,4 1 Département de Recherche Fondamentale sur la Matière Condensée, SPSMS, CEA-Grenoble, 38054 Grenoble Cedex 9, France 2 Institut Laue-Langevin, B.P. 156, 38042 Grenoble Cedex 9, France 3 CNRS-Laboratoire de Physique Théorique, Ecole Normale Supérieure, 24 Rue Lhomond 75005 Paris, France 4 Service de Physique de l’Etat Condensé, CEA-Saclay, 91191 Gif-sur-Yvette Cedex, France (Received 6 April 2000) It is shown that the large-N approach yields two energy scales for the Kondo lattice model. The single-impurity Kondo temperature, T K , signals the onset of local singlet formation, while Fermi-liquid coherence sets in only below a lower scale, T . At low conduction electron density n c (“exhaustion” limit), the ratio T T K is much smaller than unity, and is shown to depend only on n c and not on the Kondo coupling. The physical meaning of these two scales is demonstrated by computing several quantities as a function of n c and temperature. PACS numbers: 71.27.+a, 71.10.Fd, 71.20.Eh How coherent quasiparticles form in the Kondo lattice has been a long-standing issue. For a single impurity, there is a single scale T K below which the local moment is screened and a local Fermi-liquid picture applies. T K can be defined, e.g., as the scale at which the effective Kondo coupling becomes large. All physical quantities (e.g., specific heat, susceptibility) obey scaling properties as a function of T T K in the limit where both T and T K are smaller than the high-energy cutoff (bandwidth, denoted 2D in the following). In contrast, for a Kondo lattice, one may suspect that the physics is no longer governed by a single scale. Indeed, while magnetic moments can still be screened locally for temperatures lower than the single- impurity Kondo scale T K , the formation of a Fermi-liquid regime (with coherent quasiparticles and a “large” Fermi surface encompassing both conduction electrons and the localized spins) is a global phenomenon requiring coher- ence over the whole lattice. If at all possible, it could be associated with a much lower coherence temperature T . The situation is reminiscent of strong-coupling super- conductivity in which local pair formation may occur at a much higher scale than T c , the scale at which long-range order sets in. As was originally pointed out by Nozières [1], this issue becomes especially relevant when few conduction electrons are available to screen the local spins, i.e., in the limit of low concentration n c ø 1. In this “exhaustion” regime, two possibilities arise: (i) either magnetic order- ing wins over Kondo screening or (ii) a paramagnetic Fermi-liquid state still manages to form, but with a much suppressed coherence scale T ø T K . Nozières has suggested in Ref. [2] that T n c D for strong coupling J K D ¿ 1 (where T K J K ), while T T 2 K D for weak coupling J K ø D. Recently, several studies [3–8] have addressed this issue using dynamical mean-field theory (DMFT) [9]. The conclusion was that two scales are indeed present in the Kondo lattice, with the coherence scale with T ø T K in the exhaustion regime. Because the DMFT equations require a numerical treatment, no detailed analytical insight into these two scales was obtained, even though the validity of the estimate [2] T ~ T 2 K D was questioned [8]. In this Letter, we solve this problem using the slave- boson approach, in the form of a controlled large-N solu- tion of the SUN Kondo lattice model. This approach has been extensively used in the past twenty years [10–15]. Surprisingly, the issue of the coherence scale and the tem- perature dependence of physical quantities has not been discussed in detail in the exhaustion limit n c ø 1 (see, however, Ref. [16]). We find that the large-N approach provides a remarkably simple and physically transparent realization of the two-scale scenario described above. Fur- thermore, because of its simplicity, it allows for an ana- lytical calculation of the coherence scale, which is found to disagree (for weak coupling) with Nozières’ estimate in Ref. [2] (while it agrees with it at strong coupling). We also calculate the temperature dependence of several physi- cal quantities and find remarkable agreement with the more sophisticated (and demanding) DMFT approach. We consider the Kondo lattice model: H X ks e k c 1 ks c ks 1 J K N X i ss 0 S ss 0 i c 1 i s 0 c i s . (1) In this expression, the spin symmetry has been extended to SUN s 1,..., N and the local spins will be consid- ered in the fermionic representation: S ss 0 i f 1 i s f i s 0 2 d ss 0 2, with the constraint P s f 1 i s f i s N 2. Standard methods [10] are used to solve this model at large N : a bo- son field B i t (conjugate to the amplitude P s f 1 i s c i s ) is introduced in order to decouple the Kondo interaction, and the constraint is implemented through a Lagrange multi- plier field l i t . For N `, a saddle point is found at which the Bose field condenses B i t r and the La- grange multiplier takes a uniform, static value i l i t l. Two quasiparticle bands v 6 k are formed, solutions of v1lv1m2e k 2 r 2 0. Changing variables to v 6 k , the three saddle-point equations can be cast in the compact form: 1048 0031-9007 00 85(5) 1048(4)$15.00 © 2000 The American Physical Society

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Page 1: Coherence Scale of the Kondo Lattice

VOLUME 85, NUMBER 5 P H Y S I C A L R E V I E W L E T T E R S 31 JULY 2000

Coherence Scale of the Kondo Lattice

S. Burdin,1,2 A. Georges,3 and D. R. Grempel1,4

1Département de Recherche Fondamentale sur la Matière Condensée, SPSMS, CEA-Grenoble, 38054 Grenoble Cedex 9, France2Institut Laue-Langevin, B.P. 156, 38042 Grenoble Cedex 9, France

3CNRS-Laboratoire de Physique Théorique, Ecole Normale Supérieure, 24 Rue Lhomond 75005 Paris, France4Service de Physique de l’Etat Condensé, CEA-Saclay, 91191 Gif-sur-Yvette Cedex, France

(Received 6 April 2000)

It is shown that the large-N approach yields two energy scales for the Kondo lattice model. Thesingle-impurity Kondo temperature, TK , signals the onset of local singlet formation, while Fermi-liquidcoherence sets in only below a lower scale, T. At low conduction electron density nc (“exhaustion”limit), the ratio TTK is much smaller than unity, and is shown to depend only on nc and not onthe Kondo coupling. The physical meaning of these two scales is demonstrated by computing severalquantities as a function of nc and temperature.

PACS numbers: 71.27.+a, 71.10.Fd, 71.20.Eh

How coherent quasiparticles form in the Kondo latticehas been a long-standing issue. For a single impurity,there is a single scale TK below which the local momentis screened and a local Fermi-liquid picture applies. TK

can be defined, e.g., as the scale at which the effectiveKondo coupling becomes large. All physical quantities(e.g., specific heat, susceptibility) obey scaling propertiesas a function of TTK in the limit where both T and TK aresmaller than the high-energy cutoff (bandwidth, denoted2D in the following). In contrast, for a Kondo lattice, onemay suspect that the physics is no longer governed by asingle scale. Indeed, while magnetic moments can still bescreened locally for temperatures lower than the single-impurity Kondo scale TK , the formation of a Fermi-liquidregime (with coherent quasiparticles and a “large” Fermisurface encompassing both conduction electrons and thelocalized spins) is a global phenomenon requiring coher-ence over the whole lattice. If at all possible, it couldbe associated with a much lower coherence temperatureT. The situation is reminiscent of strong-coupling super-conductivity in which local pair formation may occur at amuch higher scale than Tc, the scale at which long-rangeorder sets in.

As was originally pointed out by Nozières [1], thisissue becomes especially relevant when few conductionelectrons are available to screen the local spins, i.e., in thelimit of low concentration nc ø 1. In this “exhaustion”regime, two possibilities arise: (i) either magnetic order-ing wins over Kondo screening or (ii) a paramagneticFermi-liquid state still manages to form, but with a muchsuppressed coherence scale T ø TK . Nozières hassuggested in Ref. [2] that T ncD for strong couplingJKD ¿ 1 (where TK JK ), while T T2

KD forweak coupling JK ø D. Recently, several studies [3–8]have addressed this issue using dynamical mean-fieldtheory (DMFT) [9]. The conclusion was that two scalesare indeed present in the Kondo lattice, with the coherencescale with T ø TK in the exhaustion regime. Becausethe DMFT equations require a numerical treatment, no

1048 0031-90070085(5)1048(4)$15.00

detailed analytical insight into these two scales wasobtained, even though the validity of the estimate [2]T ~ T2

KD was questioned [8].In this Letter, we solve this problem using the slave-

boson approach, in the form of a controlled large-N solu-tion of the SUN Kondo lattice model. This approach hasbeen extensively used in the past twenty years [10–15].Surprisingly, the issue of the coherence scale and the tem-perature dependence of physical quantities has not beendiscussed in detail in the exhaustion limit nc ø 1 (see,however, Ref. [16]). We find that the large-N approachprovides a remarkably simple and physically transparentrealization of the two-scale scenario described above. Fur-thermore, because of its simplicity, it allows for an ana-lytical calculation of the coherence scale, which is foundto disagree (for weak coupling) with Nozières’ estimatein Ref. [2] (while it agrees with it at strong coupling). Wealso calculate the temperature dependence of several physi-cal quantities and find remarkable agreement with the moresophisticated (and demanding) DMFT approach.

We consider the Kondo lattice model:

H Xks

ekc1kscks 1

JK

N

Xiss0

Sss0

i c1is0cis . (1)

In this expression, the spin symmetry has been extended toSUN s 1, . . . , N and the local spins will be consid-ered in the fermionic representation: Sss0

i f1isfis0 2

dss02, with the constraintP

s f1isfis N2. Standard

methods [10] are used to solve this model at large N : a bo-son field Bit (conjugate to the amplitude

Ps f1

iscis) isintroduced in order to decouple the Kondo interaction, andthe constraint is implemented through a Lagrange multi-plier field lit. For N `, a saddle point is found atwhich the Bose field condenses Bit r and the La-grange multiplier takes a uniform, static value ilit l. Two quasiparticle bands v

6k are formed, solutions of

v 1 l v 1 m 2 ek 2 r2 0. Changing variablesto v

6k , the three saddle-point equations can be cast in the

compact form:

© 2000 The American Physical Society

Page 2: Coherence Scale of the Kondo Lattice

VOLUME 85, NUMBER 5 P H Y S I C A L R E V I E W L E T T E R S 31 JULY 2000

Ω2

1JK

,12

,nc

2

æ

Z 1`

2`dv nFv

3 r0

µv 1 m 2

r2

v 1 l

3

Ω1

v 1 l,

r2

v 1 l2 , 1

æ. (2)

Here, nF is the Fermi function, nc2 is the conductionelectron density per spin color, and r0e

Pk de 2

ek is the noninteracting density of states.In the large-N approach, the onset of Kondo screening

is signaled by a phase transition at a critical temperatureTK , below which rT becomes nonzero. The equationfor TK is 2JK

Rde r0e tanhe 2 m02TK e 2

m0, with m0 being the noninteracting chemical potential(at T TK ). This equation coincides with that for thesingle-impurity case: Kondo screening of individual lo-cal moments starts taking place in this approach preciselyat the single-impurity Kondo scale. We have derived anexplicit expression for TK in the weak-coupling regimeJK ø D:

TK De21JK r0eF q

1 2 eFD2 FK nc , (3)

FK nc exp

µZ D2eF

2D1eF

dv

jvj

r0eF 1 v 2 r0eF2r0eF

∂,

(4)

where eF is the noninteracting Fermi level [given bync2

ReF

2D de r0e]. This expression is valid for aneven density of states (DOS) r02e r0e whichvanishes outside the interval 2D , e , 1D. The factorFK , equal to unity for a constant density of states, variessmoothly with nc (or eF). In contrast, the prefactor1 2 eFD212 vanishes in the low-density limitnc ! 0, and suppresses TK .

We now consider the low-temperature limit T ø TK , inwhich the large-N approach leads to an extremely simpleFermi-liquid picture [10]. It is somewhat oversimplifiedin that the conduction electron self-energy Sck, v r2v 1 l is purely real (finite lifetime effects are ab-sent at N `) and momentum independent. Even so,it captures some of the most important features of theproblem. The zero-frequency shift Scv 0, T 0 rT 02lT 0 is precisely such that the Fermi sur-face has a large volume containing both conduction elec-trons and local spins. Indeed, by adding the last twosaddle-point equations in (2), one obtains m 2 Scv 0, T 0 e

.F , where e

.F is the noninteracting value of

the Fermi level corresponding to nc 1 12 fermions perspin color. As detailed below, all physical quantities atT 0 are directly related to a single energy scale, propor-tional to the boson condensation amplitude T r2T 0D. It is possible to derive an analytical expression forT in the weak-coupling limit, which is valid for a general(bounded) density of states and arbitrary density nc. This

expression, which has apparently not been reported before,reads

T De21JKr0eF 1 1 eFDDeF

DFnc , (5)

Fnc exp

µZ DeF

2D1eF

dv

jvj

r0eF 1 v 2 r0eFr0eF

∂,

(6)

where DeF e.F 2 eF . As FK , F varies smoothly

with nc [17]. The total density of states at theFermi level rv 0 rcc0 1 rff0 is given byr0 r0e.

F Zc, with Zc the conduction electron quasi-particle residue 1Zc 1 2 ≠Scv≠vjv0 1 1

e.F 2 eF2r2

0 . In the weak-coupling limit, all physi-cal quantities at T 0 are directly related to r0 r0e.

F DeF2TD, and are thus renormalized by theratio TD. For instance, the f-electron susceptibilityand the specific heat coefficient (per spin color), aregiven by xf ~ r0 and g p2r03, respectively.The Drude weight may also be computed with the resultDR ~ TDr0e.

F DeF2.The physical content of these expressions and of

Eqs. (3)–(6) is that two different energy scales are rele-vant for the Kondo lattice model: one TK is associatedwith the onset of local Kondo screening; the other T isassociated with Fermi-liquid coherence and the behaviorof physical quantities at T 0. These two scales havethe same exponential dependence on JKD at weakcoupling, but very different dependence on the conductionelectron density in the exhaustion limit nc ø 1, in whichT ø TK . This is in qualitative agreement with Nozièresproposal in [2], but not with his estimate TTK ~ TKD(which would thus depend on the coupling). We findthat the ratio TTK depends only on nc in this limit, ina manner which reflects the behavior of the bare DOSr0e at the bottom of the band. For a DOS vanishing ase 1 Da , Eqs. (3)–(6) yield TTK ~ n

1211ac .

Figure 1 displays the nc dependence of TK and of the in-verse of the f-electron susceptibility, obtained by solvingthe large-N equations, both for a single impurity and forthe lattice. It is seen that TK and 1xfT 0 are identi-cal energy scales for the single impurity case, but havevery different density dependence for the lattice model(TTK ~ n

13c at low density for the semicircular DOS

used here). Near nc 1, TK falls below 1x , reflectingthe vanishing x for the Kondo insulator. These curves areremarkably similar to those obtained by Tahvildar-Zadehet al. in their quantum Monte Carlo studies of the peri-odic Anderson model in infinite dimensions [5] (see also[5–8]). In the inset of Fig. 1, we plot the dimensionlessnumber TKxf0 as a function of lnTKD. This numbergoes to a universal value at weak coupling in the single-impurity case, while it has intrinsic density dependencefor the lattice. This shows that TK is not the appropriatelow-temperature scale, especially for nc ! 0. The insetof Fig. 2 shows the effective mass ratio mm (computed

1049

Page 3: Coherence Scale of the Kondo Lattice

VOLUME 85, NUMBER 5 P H Y S I C A L R E V I E W L E T T E R S 31 JULY 2000

-3 -2 -10.2

0.3

0.4

log(TK/D)

TK χ

(0)

(Im

p)

0.0

0.5

1.0

1.5

TK χ(0) (latt)

0.0 0.2 0.4 0.6 0.8 1.00.0

0.1

0.2

nc

TK/D

0.0

0.1

0.2

1/(4 χ (0) D)

FIG. 1. Solid line: density dependence of TK (defined as theslave-boson condensation temperature). Squares: 14xf T 0 for a single impurity. Dashed line: 14xf T 0 for thelattice. JKD 0.75. Inset: TK dependence of TKxf 0 for theimpurity (left scale, solid symbols) and the lattice (right scale,open symbols) for nc 0.1 and 1.0 (impurity) and nc 0.1,0.5, and 0.9 (lattice) from top to bottom. All plots are for asemicircular DOS.

from the specific heat) and the Drude weight, as a func-tion of nc. This vanishes in the limits nc ! 0 and nc ! 1.The effective mass m is proportional to e11JKr0eF , witha density-dependent prefactor which diverges as nc ! 0and vanishes at half-filling.

We have also studied the behavior of several physicalquantities as a function of temperature, by solving Eq. (2)numerically. Figure 2 shows the product TKxT for theKondo lattice, for several values of nc, as a function ofTTK . All of the curves merge at TTK 1, where theboson decondenses and xf reaches the free spin value[xf T 14T for T . TK ]. No universal scaling func-tion of TTK describes the temperature dependence in

0.0 0.2 0.4 0.6 0.8 1.00.0

0.1

0.2

0.3

nc

m/m

*

0.00

0.02

0.04

0.06

DR /D

*R

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

TK χ

(T)

T/TK

FIG. 2. T dependence of the f-electron susceptibility for thelattice for nc 0.01, 0.2, 0.5, 0.8, and 0.9, from top to bottom.Inset: Inverse effective mass (left scale, solid line) and Drudefactor (right scale, dashed line) as functions of density. In allplots, JKD 0.75.

1050

Fig. 2, in contrast to the single-impurity case. PlottingxT xT 0 does not restore scaling, since qualita-tive differences in the T dependence are seen for differ-ent densities. Figure 3(a) shows the T dependence of theentropy. A linear behavior ST gT is found at lowtemperature for all densities nc fi 1. The slope g ~ r0decreases with increasing density as does the temperaturescale T

F up to which ST is linear. TF is of the order of

T øTK at low nc, while it can be estimated by compar-ing gT to e2TK T as T

F TKj ln1 2 ncj ø TK T

for nc ! 1. For nc 1, TF is a better estimate of the co-

herence scale than T itself.At low densities, after a steep initial rise, ST re-

mains close to ln2 up to TK . This can be interpreted interms of the strong-coupling picture discussed below. Athigher densities, most of the variation takes place in therange T

F , T , TK . The specific heat CT , shown inFig. 3(b), has a two-peak structure [18]: the peak at TK

signals the onset of Kondo screening and appears in thismean-field description as a discontinuity of CT . The sec-ond peak, at T

F , signals the Fermi-liquid heavy-fermionregime. As nc increases, weight is gradually transferred tothe high-temperature peak until the low-temperature peakdisappears completely in the Kondo insulator limit.

In the strong-coupling limit JKD ¿ 1, the large-Nresults support the physical picture proposed by Lacroix[19] and further discussed in [2]. In this picture, the con-duction electrons bind to nc spins, forming singlets belowT JK TK , the binding energy of a singlet. The re-maining 1 2 nc “bachelor spins” behave as itinerant fer-mions subject to a constraint of no double occupancy. Thehopping integral of the resulting effective infinite-U Hub-bard model is teff 2t2. The (holelike) sign of thismatrix element implies that these 1 2 nc fermions have aFermi surface volume corresponding precisely to nc 1 1particles. In the exhaustion limit nc ø 1, one has effec-tively a weakly doped U ` Hubbard model. SolvingEq. (2) at strong coupling yields a quasiparticle residue

0.0 0.5 1.00.0

0.4

0.8

1.2

1.6(b)

C(T

)/kB

nc=0.07

nc=0.3

nc=0.15

nc=0.8

nc=0.99

T/TK

0.0 0.5 1.00.0

0.2

0.4

0.6

0.8

1.0(a)

nc=0.07

nc=0.15

nc=0.3

nc=0.8

nc=0.99

S/k

B

T/TK

FIG. 3. Entropy (a) and specific heat (b) for the lattice modelfor several values of the density. JKD 0.75.

Page 4: Coherence Scale of the Kondo Lattice

VOLUME 85, NUMBER 5 P H Y S I C A L R E V I E W L E T T E R S 31 JULY 2000

-1.0 -0.5 0.0 0.5 1.00.0

0.2

0.4

0.6

0.8

1.0n c(

ε)

ε/D

-1.0 -0.5 0.0 0.5 1.00.0

0.2

0.4

0.6

0.8

D Γ

(ω)

ω/D

FIG. 4. nce for TTK 1.0 (solid line), 0.5 (dashed line),and 0.005 (dotted line). nc 0.15. Inset: spectral density Gvfor TTK 1 (solid line), 0.5 (dashed line), and 0.25 (dottedline). For T , TK there is a d-function peak in the gap notshown for clarity. nc 0.2. In all plots JKD 0.75.

Zc ~ nc, and hence a coherence scale T ncD corre-sponding to a Brinkman-Rice estimate for this doped Mottinsulator [2]. Notice, however, that at finite N this uniformsolution may become unstable to magnetism or phase sepa-ration of localized singlets and unscreened spins. This pic-ture sheds some light on the T dependence of the entropyand specific heat in the exhaustion limit reported above.As the system goes through TK , it loses the magnetic en-tropy nc ln2 of the nc bound spins. The remaining en-tropy 1 2 nc ln2, is lost below T. The two peaksof unequal weight in the specific heat reflect these pro-cesses [20].

We display in Fig. 4 the distribution function of theconduction electrons ncek c1

k ck. Very close to TK ,ncek has the shape of a Fermi function centered aroundthe noninteracting Fermi level eF , with a small thermalbroadening of order TK (øD in weak coupling). As Tis reduced below TK , weight is transferred to scales oforder e

.F , the interacting Fermi level associated with the

large Fermi surface. The feature at eF broadens as T isdecreased. When Fermi-liquid coherence establishes atT T

F , a discontinuity (of amplitude Zc) develops ate e

.F . Finally, we note that, in the large-N approach,

the Kondo lattice model can be exactly mapped ontoan effective single-impurity model coupled to a self-consistent bath of electrons. This mapping holds moregenerally for any approach in which the conductionelectron self-energy depends only on frequency, suchas dynamical mean-field theory [9]. For a semicircularDOS, the Green’s function of the self-consistent bath isG0ivn ivn 1 m 2 D2Gcivn421. The inset ofFig. 4 displays the continuous evolution of the spectral

density Gv 2ImG0v 1 i01p as T is reducedbelow TK . Above TK , it coincides with the noninteractingDOS r0, while the bath is split into two bands below TK .The width of the lower band depends on nc and is smallfor nc ø 1, leading to the low coherence scale. Theexistence of a sharp gap separating the two bands is anartifact of the large-N limit (except at nc 1). In morerealistic treatments it is replaced by a pseudogap [5,7,8].

In conclusion, we have reconsidered the time-honoredlarge-N approach to the Kondo lattice model, with spe-cial emphasis on the exhaustion limit of low electron den-sity. We showed that two energy scales appear for whichwe have obtained explicit analytic expressions: the Kondoscale associated with the onset of local Kondo screeningand a much lower scale associated with Fermi-liquid co-herence. Physical quantities reflect the crucial role playedby these two scales. In this approach, magnetic ordering issuppressed by quantum fluctuations: more elaborate treat-ments such as DMFT must be used to assess whether thecoherence scale can actually be reached or magnetic or-dering sets in at a higher temperature.

Useful discussions with B. Coqblin, M. Jarrell, A. Jerez,G. Kotliar, C. Lacroix, P. Nozières, and A. Sengupta aregratefully acknowledged. We thank the Newton Institute,where part of this work was performed, for its hospitality.

[1] P. Nozières, Ann. Phys. (Paris) 10, 19 (1985).[2] P. Nozières, Eur. Phys. B 6, 447 (1998).[3] M. Jarrell et al., Phys. Rev. Lett. 70, 1670 (1993).[4] M. Jarrell, Phys. Rev. B 51, 7429 (1995).[5] A. N. Tahvildar-Zadeh et al., Phys. Rev. B 55, 3332 (1997);

Phys. Rev. Lett. 80, 5168 (1998).[6] A. N. Tahvildar-Zadeh et al., Phys. Rev. B 60, 10 782

(1999).[7] N. S. Vidhyadhiraja et al., cond-mat /9905408.[8] Th. Pruschke et al., cond-mat/0001357.[9] A. Georges et al., Rev. Mod. Phys. 68, 13 (1996).

[10] See, e.g., D. M. Newns and N. Read, Adv. Phys. 36, 799(1987); A. C. Hewson, The Kondo Problem to Heavy Fer-mions (Cambridge University Press, Cambridge, England,1993).

[11] N. Read et al., Phys. Rev. B 30, 3841 (1984).[12] T. M. Rice and K. Ueda, Phys. Rev. Lett. 55, 995 (1985).[13] A. J. Millis and P. A. Lee, Phys. Rev. B 35, 3394 (1987).[14] A. Auerbach and K. Levin, Phys. Rev. Lett. 57, 877 (1986).[15] P. Coleman, Phys. Rev. B 35, 5072 (1987).[16] C. Lacroix and M. Cyrot, Phys. Rev. B 20, 1969 (1979);

B. Coqblin et al., Physica (Amsterdam) 282B, 50 (2000).[17] A similar prefactor was obtained in the context of a three-

band model. M. Grilli et al., Phys. Rev. B 42, 329 (1990).[18] A similar behavior was observed in DMFT calculations.

M. Jarrell (private communication).[19] C. Lacroix, Solid. State. Commun. 54, 991 (1985).[20] In the exhaustion limit, TK far exceeds the kinetic energy

of the f electrons: this is reminiscent of the formation ofthe Zhang-Rice singlet [F. C. Zhang and T. M. Rice, Phys.Rev. B 37, 3759 (1988)].

1051