Dynamic spin-correlation function near the antiferromagnetic quantum phase transition of heavy fermions

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Dynamic spin-correlation function near the

antiferromagnetic quantum phase transition of heavy

fermions

C. Pépin, M. Lavagna

To cite this version:

C. Pépin, M. Lavagna. Dynamic spin-correlation function near the antiferromagnetic quantum phase

transition of heavy fermions. Physical Review B: Condensed Matter and Materials Physics

(1998-2015), American Physical Society, 1999, 59 (4), pp.2591 - 2598. �10.1103/PhysRevB.59.2591�.

�hal-01896148�

Dynamic spin-correlation function near the antiferromagnetic

quantum phase transition of heavy fermions

C. Pe´pin*and M. Lavagna†

Commissariat a` l’Energie Atomique, De´partement de Recherche Fondamentale sur la Matie`re Condense´e/SPSMS, 17, rue des Martyrs, 38054 Grenoble Cedex 9, France

~Received 31 March 1998!

The dynamical spin susceptibility is studied in the magnetically disordered phase of heavy-Fermion systems near the antiferromagnetic quantum phase transition. In the framework of the S51/2 Kondo lattice model, we introduce a perturbative expansion treating the spin and Kondo-like degrees of freedom on an equal footing. The dynamical spin susceptibility displays a two-component behavior in agreement with the inelastic neutron scattering ~INS! experiments performed in CeCu6,Ce12xLaxRu2Si2, and UPt3: a quasielastic q-independent

peak as in a Fermi-liquid theory, and a strongly q-dependent inelastic peak typical of a non-Fermi-liquid behavior. Very strikingly, the position of the inelastic peak is found to be pushed to zero at the antiferromag-netic transition. The picture is consistent with the neutron cross sections observed in INS experiments. @S0163-1829~99!13603-8#

I. INTRODUCTION

One of the most striking properties of heavy fermion compounds discovered these last years is the existence of a quantum phase transition1,2driven by composition change~at

xC50.1 in CeCu62xAux and xC50.08 in Ce12xLaxRu2Si2),

pressure or magnetic field. It has been largely discussed in various theoretical approaches.5,6 Important insight is pro-vided by the evolution of the low-temperature neutron cross section measured by inelastic neutron scattering ~INS! ex-periments when getting closer to the magnetic instability. The experiments performed in pure compounds CeCu6 and

CeRu2Si2 by Regnault et al.3and Aeppli et al.4have shown

the presence of two distinct contributions to the dynamic magnetic structure factor: a q-independent quasielastic com-ponent, and a strongly q-dependent inelastic peak with a maximum at the value v_max of the frequency. The former corresponds to localized excitations of Kondo-type while the latter peaked at some wave vector Q is believed to be asso-ciated with intersite magnetic correlations due to Ruderman-Kittel-Kasuya-Yosida ~RKKY! interactions. The frequency width of the quasielastic and inelastic peaks respectively de-fine the single-site and intersite relaxation rateGSSandGIS. Such features have also been observed in UPt3 and called

as ‘‘slow’’ and ‘‘fast’’ components by Bernhoeft and Lonzarich.7Later on, INS experiments have been performed with varying compositions as in Ce12xLaxRu2Si2.8 It has

been observed a narrowing of the single-site relaxation rate

GSS when getting closer to the magnetic transition. At the same time, both the position of the inelastic peak vmax and

the intersite relaxation rate GIS drastically decrease when getting near the magnetic instability. Table I reports the val-ues of GSS, GIS, andvmaxfor the different compounds.

Any theory aimed to describe the quantum critical phe-nomena in heavy-fermion compounds should account for the so-quoted behavior of the dynamical spin susceptibility. We start from the Kondo-lattice model, which is believed to de-scribe the physics of these systems. We refer to the recent

paper of Tsunetsugu et al.9 for a review of the model. As already pointed out by Doniach in his initial paper,10 the main features result of the competition between the Kondo effect and the RKKY interactions among spins mediated by the conduction electrons. Most of the theories developed so far11–15 agree with the existence of a hybridization gap, which splits the Abrikosov-Suhl or Kondo resonance formed at the Fermi level. The role of the interband transitions has been outlined for long in order to explain the inelastic com-ponent of the dynamical spin susceptibility. For instance, the theories based on a 1/N expansion16–20~where N is simulta-neously the degeneracy of the conduction electrons and of the spin channels! predict a maximum of x

9

(kF,v)/v at

vmaxof the order of the indirect hybridization gap.21

How-ever, the 1/N-expansion theories present serious drawbacks:

~i! the spin-fluctuation effects are automatically ruled out

since the RKKY interactions only occur at the following order in 1/N2_,22 _{~ii! they then fail to describe any magnetic}

instability and hence the quantum critical phenomena men-tioned above, and~iii! the predictions forvmaxand the

asso-ciated relaxation rate cannot account for the experimental observations near the magnetic instability. An improvement brought by Doniach23 consists to consider the 1/N2 correc-tions in an instantaneous approximation: it gives back the ladder diagram contribution to the dynamical spin suscepti-bility and then accounts for the spin-fluctuation effects. Other approaches were proposed in Refs. 24–26. But still the predictions for the frequency dependence of the dynamic

TABLE I. Values of the single-site and intersite relaxation rates GSSandGISand position of the inelastic peakvmaxextracted from

the inelastic neutron scattering ~INS! measurements performed in CeCu6~Ref. 3! and Ce12xLaxRu2Si2at x50 and x50.075 ~Ref. 8!.

GSS~meV! GIS~meV! vmax~meV!

CeCu6 0.42 0.2 0.25

CeRu2Si2 2.0 0.75 1.2

Ce12xLaxRu2Si2 1.4 0.2 0.2

PRB 59

magnetic structure factor presents a gap of the order of the hybridization gap whatever the value of the interaction is. On the other hand, in front of the difficulties encountered when starting from microscopic descriptions, various phenomeno-logical models ~as the duality model of Kuramoto and Miyake27,28 and Ref. 7! have been introduced to describe both the spin fluctuation and the itinerant electron aspects with some successful predictions as the weak antiferromag-netism of these systems.

In this paper, we develop a systematic approach to the Kondo-lattice model for S51/2(N52) in which the Kondo-like and the spin degrees of freedom are treated on an equal footing. The presented approach shows some similarities with earlier works.11,15But while Refs. 11 and 15 essentially describe the phase diagram of the Kondo lattice at a mean-field level, we focus on the effects of spin fluctuations in the magnetically disordered phase hence bringing the spin-fluctuation and the Kondo-effect theories together. The saddle-point results and the Gaussian fluctuations in the charge channel are consistent with the standard 1/N theories. In addition, the Gaussian fluctuations in the spin channel restore the spin-fluctuation effects that were missing in the 1/N expansion. The general expression of the dynamical spin susceptibility that we derive reproduces some of the features postulated in the phenomenological models. It presents a two-component behavior: a quasielastic component superim-posed on an inelastic peak with renormalized values of the relaxation rates, susceptibilities andvmax. In a very striking

way,vmaxis pushed to zero and the inelastic mode becomes

soft at the antiferromagnetic phase transition with vanishing relaxation rate. Predictions are quantitatively compared with experimental results. The quasielastic peak is typical of a Fermi liquid while the other mode breaks the Fermi-liquid description. Our approach might offer new prospects for the study of the quantum critical phenomena in the vicinity of the antiferromagnetic phase transition.

II. PRESENTATION OF THE APPROACH

We consider the Kondo-lattice model~KLM! constituted by a periodic array of Kondo impurities with an average number of conduction electrons per site nc,1. In the grand canonical ensemble, the Hamiltonian is written as

H5

(

ks «k c_k†_scks1J

(

i Si•

(

ss8cis † _t ss₈cis8 2mNS

S

1 NS

(

ks c_k†_scks2nc

D

~1!

in which S_i represents the spin (S51/2) of the impurities distributed on the sites~in number NS) of a periodic lattice;

c_k†_s is the creation operator of the conduction electron of momentum k, spin quantum numbers characterized by the energye_k52(_{^i, j&}t_{i j}exp(ik_{•Ri j}) and the chemical potential

m;t are the Pauli matrices (tx,ty,tz) and t0 the unit ma-trix; J is the antiferromagnetic Kondo interaction (J.0).

We use the Abrikosov pseudofermion representation of the spin Si:Si5(ss8fis

† _t

ss₈fis8. The projection into the physical subspace is implemented by a local constraint

Q_i5

(

s fis 1_f

is2150. ~2!

A Lagrange multiplier li is introduced to enforce the local constraint Q_i. Since @Q_i,H#50, l_i is time independent.

In this representation, the partition function of the KLM can be expressed as a functional integral over the coherent states of the fermion fields

Z5

E

Dc_isDf_isdl_iexp

F

2

E

0

b

L~t!dt

G

, ~3!

where the LagrangianL(t) is given by

L~t!5L0~t!1H0~t!1HJ~t!, L0~t!5

(

is c_i†_s]_tci_s1 fis † _] tfi_s, H0~t!5

(

ks ek c_k†_scks 2mNS

S

1 NS

(

k_s c_k†_scks2nc

D

1

(

i li Qi, HJ~t!5J

(

i Sf i•Sci, with S_c i5(ss8cis † _t ss₈cis8and Sf i5Si.

We perform a Hubbard-Stratonovich transformation on the Kondo interaction term HJ(t). Since more than one field is implied in the transformation, an uncertainty is left on the way of decoupling. We propose to remove it in the following way. First, we note that HJ(t) may also be written as

HJ~t!52 3J 8

(

i nf c_inc f_i1 J 2

(

i Sf c_i•Sc f_i, ~4! where S_{f c} i5(ss8fis † _t ss₈cis8and nf c_i5(ss8fis † _t ss₈ 0 c_is8

~re-spectively, Sc f_i and nc f_i their Hermitian conjugate!.

The Kondo interaction term is then given by any linear combination of J(iSf i•Sci~with a weighting factor x) and of the term appearing in the right-hand side of Eq. ~4! @with a weighting factor (12x)#. x is chosen so as to recover the usual results obtained within the slave-boson theories. One can check that this is the case for x51/3. The Kondo inter-action term is then given by

HJ~t!5JS

(

i ~Sfi•Sci1Sf ci•Sc fi!2JC

(

i

nf c_inc f_i, ~5! with JS5J/4 and JC5J/3.

Performing a generalized Hubbard-Stratonovich transfor-mation on the partition function Z makes the fields Fi,Fi*

~for charge! andjf_i,jc_i appear~omitting the fields associated

to Sf c_i,Sc f_i). We get Z5

E

dFidFi*djf_idjc_iDcisDfisdliexp

F

2

E

0 b L

8

~t!dt

G

, ~6!

with L

8

~t!5L0~t!1H0~t!1HJ

8

~t! H_J

8

~t!5

(

iss8 ~cis † _f is † _! 3

S

2JS ijf_i•tss8 JCFi*t_ss8 0 J_CF_it_ss 8 0 _2J Sijc_i•tss8

D

S

cis8 fis8

D

1JC

(

i Fi *_Fi1JS

(

i jfi .jc_i. A. Saddle point

The saddle-point solution is obtained for space- and time-independent fieldsF0,l0, jf₀, andjc₀. In the magnetically disordered regime (jf₀5jc₀50), it leads to renormalized bands a and b as schematized in Fig. 1. Noting s₀(*)

5JCF0

(_*)_and«

f5l0,aks

† _u0

_&

_andb

ks

† _u0

_&

_{are the eigenstates}

of

G₀21s~k,t!5

S

]t1«k s0 *

s0 ]t1«f

D

, ~7!

with, respectively, the eigenenergies (]_t1E_k2) and (]_t

1Ek1). In the notations xk5«k2«f, yk65Ek62«f, and Dk

5

A

xk 2_14s 0 2 , we get y_k65~xk6Dk!/2. ~8!

Let us note U_k†_s the matrix transforming the initial basis (c_k†_sf_k†_s) to the eigenbasis (a_k†_sb_k†_s). The Hamiltonian being Hermitian, the matrix Uks is unitary: UksUk_s

† _5U k_s † Uks 51. In the notation Uk_s † ₅

S

2vk uk uk vk

D

, we have uk5 2s0/ yk2

A

11~s0/yk2! 25 1 2

F

11 xk Dk

G

, ~9! vk5 1

A

11~s0/yk2! 25 1 2

F

12 xk Dk

G

.

The saddle-point equations together with the conservation of the number of conduction electrons are written as

s05 1 NS JC

(

ks ukvknF~Ek2!, 15 1 NS

(

ks u_k2nF~Ek2!, nc5 1 NS

(

ks vk 2_n F~Ek2!. ~10! Their resolution leads to

uyFu5D exp@22/~r0JC!#,

2r0s0 2

/uyFu51,

m50, ~11!

where yF5m2«F and r0 is the bare density of states of

conduction electrons (r051/2D for a flat band!. Noting y 5E2«F, the density of states at the energy E is r(E)

5r0(11s0

2_/y2_{). If n}

c,1, the chemical potential is located

just below the upper edge of the a band. The system is metallic. The density of states at the Fermi level is strongly enhanced towards the bare density of states of conduction electrons:r(EF)/r05(11s0

2

/y_F2);1/(2r0uyFu). That

corre-sponds to the flat part of theaband in Fig. 1. It is associated to the formation of a Kondo or Abrikosov-Suhl resonance pinned at the Fermi level resulting of the Kondo effect. The low-lying excitations are quasiparticles of large effective mass m* as observed in heavy-Fermion systems. Also note the presence of a hybridization gap between thea and theb bands. The direct gap of value 2s0 is much larger than the

indirect gap equal to 2uy_Fu. The saddle-point solution trans-poses to N52 the large-N results obtained within the slave-boson mean-field theories.

B. Gaussian fluctuations

We now consider the Gaussian fluctuations around the saddle-point solution. Following Read and Newns,17we take advantage of the local U~1! gauge transformation of the La-grangian L

8

(t)

Fi→riexp~iui!,

fi→ fi

8

exp~iui!,

li→li81i]ui/]t.

We use the radial gauge in which the modulus of both fields

Fi andFi*are the radial field ri, and their phaseui ~via its time derivative! is incorporated into the Lagrange multiplier FIG. 1. Energy versus wave vector k for the two bandsa and b.

Note the presence of a direct gap of value 2s0and of an indirect

li, which turns out to be a field. Use of the radial instead of the Cartesian gauge bypasses the familiar complications of infrared divergences associated with unphysical Goldstone bosons. We let the fields fluctuate away from their saddle-point values: ri5r01dri, li5l01dli, jf_i5djf_i, andjc_i

5djc_i. After integrating out the Grassmann variables in the partition function in Eq.~6!, we get

Z5

E

DriDliDjf_iDjc_iexp@2Se f f#, ~12!

where the effective action is

Se f f52

(

k,iv_n ln Det G21~k,ivn! 1b

F

J_C

(

i r_i21J_S

(

i jfi•jci1NS~m n_c2l₀!

G

, with @G21_~iv n!#i jss85

S

@~2ivn2m!di j2ti j#dss82JSijf_i.tss8di j ~s01JCdri!dss8di j ~s01JCdri!dss8di j @2ivn1«f1dli#dss8di j2JSijc_i.tss8di j

D

.

Expanding up to the second order in the Bose fields, one obtains the Gaussian corrections S_{e f f}(2) to the saddle-point effective action S_{e f f}~2!51 b q,i

(

v_n

F

~d r dl!D_C21~q,iv_n!

S

d r dl

D

1~djf z _dj c z_!D S i21_~q,iv n!

S

djf z djc z

D

1~djf1djc1!D'21S ~q,ivn!

S

dj2f djc2

D

1~dj2f djc2!D'21S ~q,ivn!

S

djf1 djc1

DG

, ~13!

where the boson propagators split into the following charge and longitudinal spin parts

DC21~q,ivn!5

S

JC@12JC$w¯2~q,ivn!1¯wm~q,ivn!%# 2JCw¯1~q,ivn!

2JCw¯1~q,ivn! 2w¯f f~q,ivn!

D

, ~14! D_Si21~q,iv_n!5

S

J_S2wi_{f f}~q,iv_n! JS@11JSwc f i _~q,iv n!# JS@11JSwif c~q,ivn!# JS 2_w cc i _~q,iv n!

D

and equivalent expression for the transverse spin part D'21_S (q,iv_n). The expression of the different bubbles are given in the Appendix. The charge-boson propagator DC(q,ivn) associated to the Kondo effect is equivalent to that obtained in the 1/N-expansion theories. The originality of the approach is to simultaneously derive the spin propagator D_Si21(q,iv_n) and

D'21_S (q,iv_n) associated to the spin-fluctuation effects. Note that in the magnetically-disordered phase, the charge and spin contributions in Se f f are totally decoupled.

III. DYNAMICAL SPIN SUSCEPTIBILITY

The next step is to consider the dynamical spin susceptibility. For that purpose, we study the linear response M_f to the source term 22Sf•B ~we consider B colinear to the z axis!. The effect on the partition function expressed in Eq. ~6! is to change the Hamiltonian H_J

8

(t) to H_J

8

B(t),

H_J

8

B~t!5

(

iss8 ~cis † _f is † _!

S

2JSijf_i•tss8 JCFi*t_ss8 0 J_CF_it_ss 8 0

(

a5x,y,z ~2JSijci a_2Bd az!.tssa ₈

D

S

cis8 fis8

D

1JC

(

i Fi *F_i1J_S

(

i jfi .j_c i. ~15!

Introducing the change of variablesj_c

i

a_5j

c_i

a_2iB/J

S, we connect the f magnetization and the f f dynamical spin susceptibility to the Hubbard Stratonovich fieldsjf_i,

M_fz521 b ]ln Z ]Bz 5i

^

jfi z

_&

, ~16! xabf f52 1 b ]2_{ln Z} ]Ba]Bb52

^

jfi a_j f_i b

_&

₁

_^

_j f_i a

_&^

_j f_i b

_&

_.

Using the expression~14! for the boson propagator D_Si21(q), we get for the longitudinal spin susceptibility

xif f~q,ivn!5

wif f~q,ivn!

12J_S2

F

wi_{f f}~q,iv_n!w_cci ~q,iv_n!2wi2_{f c}~q,iv_n!2 2

J_Swif c~q,ivn!

G

~17!

and equivalent expression for the transverse spin susceptibil-ityx'f f(q,ivn). The diagrammatic representation of Eq.~17! is reported in Fig. 2. The different bubbles wf f(q,ivn),

wcc(q,ivn), andwf c(q,ivn) are evaluated from the expres-sions of the Green’s functions

Gf f~k,ivn!5uk 2_G aa~k,ivn!1vk 2_G bb~k,ivn!, Gcc~k,ivn!5vk 2_G aa~k,ivn!1uk 2_G bb~k,ivn! ~18! Gc f~k,ivn!5Gf c~k,ivn!

52ukvk@Gaa~k,ivn!2Gbb~k,ivn!#,

where G_aa(k,ivn) and Gbb(k,ivn) are the Green’s func-tions associated to the eigenstatesa_k†_su0

&

andb_k†_su0

&

. In the low-frequency limit, one can easily check that the dynamical spin susceptibility may be written as

xf f~q,ivn!5

xaa~q,ivn!1x¯ab~q,ivn!

12J_S2x_aa~q,iv_n!x¯_ab~q,iv_n! ~19!

for both the longitudinal and the transverse parts.

xaa~q,ivn!5 1 b

(

k nF~Ek2!2nF~Ek21q! iv_n2E_k2_1q1E_k2 , x ¯_ab~q,iv n!5 1 b

(

k ~uk 2 vk1q 2 _1v k 2_u k1q 2 _! 3nF~Ek 2_!2n F~Ek1_1q! iv_n2E_k1_1q1E_k2 .

Equation ~19! constitutes the main result of the paper from which the whole physical discussion on the q and v depen-dence of the dynamical spin susceptibility follows and com-parison with experiments is made.

IV. PHYSICAL DISCUSSION

From Eq. ~19!, one can see that the dynamical spin sus-ceptibility is made of two contributions xintra(q,ivn) and

xinter(q,ivn),

xf f~q,ivn!5xintra~q,ivn!1xinter~q,ivn! ~20!

with xintra~q,iv_n!5 x_aa~q,ivn! 12J_S2x aa~q,ivn!x¯ab~q,ivn! , ~21! xinter~q,ivn!5 x ¯_ab~q,ivn! 12J_S2x aa~q,ivn!x¯ab~q,ivn! . ~22!

xintra(q,ivn) and xinter(q,ivn), respectively, represent the renormalized particle-hole pair excitations within the lower

a band, and from the lowerato the upperbband. The latter expression is reminiscent of the behavior proposed by Bern-hoeft and Lonzarich7 to explain the neutron scattering ob-served in UPt3 with the existence of both a ‘‘slow’’ and a

‘‘fast’’ component inx

9

(q,v)/vdue to spin-orbit coupling. Also in a phenomenological way, the same type of feature has been suggested in the duality model developed by Kura-moto and Miyake.27 To our knowledge, the proposed ap-proach provides the first microscopic derivation from the Kondo-lattice model of such a behavior. The bare intraband susceptibility x

aa(q,v) is well approximated by a

Lorentz-ian

x_aa21_~q,v!5r

aa~q!21

S

12i_Gv

0~q!

D

, ~23! where r_aa5x_aa

8

(q,0) and G0(q) is the relaxation rate of

orderuyFu5TK. This corresponds to the Lindhard continuum of the intraband particle-hole pair excitations x_aa

9

(q,v)Þ0 as reported in Fig. 3. In the same way, we propose to sche-matize the low-frequency behavior v!v0(q) of the bare

interband susceptibility by x ¯ ab

8

,21_~q,v!5r ab~q!21

F

12_vv 0~q!

G

, ~24! where r_ab5x¯ ab

8

(q,0) and v₀(q) is a characteristic fre-quency scale of the interband transitions. The value ofv₀(q) is strongly structure dependent. In the simple case of a cubic band structureek522t(cos kx1cos ky1cos kz)~tight-binding scheme including nearest-neighbor hopping!, we find a weakly wave-vector-dependent frequency around q5Q of order of v052uyFu/(r0JC). The latter result does not stand for more complicated band structures as obtained by de Haas–van Alphen studies29 combined with band-structure calculations in heavy fermion compounds. In the following, FIG. 2. Diagrammatic representation of Eq.~17! for the

we will leave v0(q) as a parameter. Figure 3 reports the

continuum of interband particle-hole excitations ¯x_ab

9

Þ0. Due to the presence of the hybridization gap in the density of states, the latter continuum displays a gap equal to 2s₀, the value of the direct gap at q50, and 2uyFu, the value of the indirect gap at q5Q ~close to k_F). More precisely, we have shown x ¯ ab

9

~0,v!54r0 s0 2 v

A

v2_24s 0 2 at 2s0,v,D, ~25! x ¯ ab

9

~Q,v!52r0 1 11v2/~2s0!2 at 2uyFu,v,2D.

Far from the antiferromagnetic wave vector Q 5(p,p,p), xf f(q,v) is dominated by the intraband transi-tions. In the low-frequency limit, the frequency dependence of x_intra

9

(q,v) can be approximate to a Lorentzian

xf f

9

~q,v!'xintra

9

~q,v!5v xintra8 ~q!Gintra~q! v2_1G intra~q!2 ~26! with Gintra~q!5G0~q!@12I~q!#, ~27! xintra8 ~q!5 raa~q! ~12I~q!!.

I(q)5J_S2x_aa

8

(q,0)¯x

8

_ab(q,0). One has x_aa

8

(0,0)5r_aa(0)

5r(EF) and x_ab

8

(0,0)5r0. The contribution expressed in

Eq. ~26! is consistent with the standard Fermi-liquid theory.

Note that the product Gintra(q)xintra8 (q)5raa(q)G0(q) is

independent of I.

Oppositely, at the antiferromagnetic wave vector Q,

xf f(q,v) is driven by the interband contribution and we get

xf f

9

~Q,v!'xinter

9

~Q,v!5v Ix_inter

8

Ginter ~v2vmax!21Ginter 2 ~28! with vmax5v0~12I!, Ginter5v0 2 ~12I!/G0, ~29! xinter

8

5rab/~12I!,

wherev0, rab, G0 and I are the values ofv0(q), rab(q),

andG₀(q) and I(q) at q5Q. The role of the interband tran-sitions have already been pointed out in previous works.21 However while the previous studies conclude to the presence of an inelastic peak at finite value of the frequency related to the hybridization gap whatever the interaction J is, we em-phasize that the renormalization of ¯x_ab(Q,v) into

xinter(Q,v) leads to a noteworthy renormalization of the interband gap. Due to the damping introduced by intraband transitions, x_inter

9

(Q,v) takes a finite value at frequency much smaller than the hybridization gap. Both the relaxation rate Ginter vanishes and the susceptibilityxinter

8

diverges at the antiferromagnetic transition with again the product

Ginterxinter

8

independent of I. Remarkably, the valuevmaxof

the maximum of x_inter

9

(Q,v)/v is at the same time pushed to zero. Such a behavior has been effectively observed in

Ce12xLaxRu2Si2 ~Ref. 8! with a reduction of Ginter and

vmax, respectively, by a factor 4 and 6 when x goes from 0 to

0.075 so when getting closer to the magnetic instability oc-curing at x50.08. In order to make the comparison more quantitative, we propose to deduce the values ofv0 and (1

2I) from the experimental data using the Eqs. ~29!: v0

5G0Ginter/vmax and (12I)5vmax 2

/(G0Ginter). Table II re-ports the results starting from the experimental values of

G0,Ginter,vmax ~respectively, noted GSS,GIS,v0 in

experi-mental papers! extracted from the INS results obtained in

CeCu6 ~Ref. 3! and Ce12xLaxRu2Si2 at x50 and x50.075

~Ref. 8!. The predictions for v0 and (12I) in these

com-pounds seem reasonable. The Stoner enhancement factor (1

2I) decreases in Ce12xLaxRu2Si2 from x50 to x

50.075. (12I) of CeCu6 is intermediate between those

two systems.

V. CONCLUSION

In this paper, we have set up an approach of the S51/2 Kondo-lattice model that enlarges the standard 1/N expan-sion theories up on the spin-fluctuation effects. The latter effects are proved to be essential for the behavior of the FIG. 3. Continuum of the intraband and interband electron-hole

pair excitationsx_aa9 (q,v)Þ0 and x_ab9 (q,v)Þ0. Note the presence of a gap in the interband transitions equal to the indirect gap of value 2uyFu at q5kF, and to the direct gap of value 2s0at q50.

TABLE II. Predicted values of the characteristic frequency scale v0for the interband transitions and the Stoner enhancement factor

(12I) from the INS data on GSS,GIS, andv0~respectively, noted

G0, Ginter, andvmax in experimental papers! for the same three

compounds as in Table I. Note that Ce12xLaxRu2Si2at x50.075 is

very close to the antiferromagnetic instability while the Stoner en-hancement factor (12I) for CeCu6is intermediate between that of

the two concentrations x50 and x50.075 of Ce12xLaxRu2Si2.

G0

~meV! ~meV!Ginter ~meV!vmax v~meV! (12I)u0uded. ded.

CeCu6 0.42 0.2 0.25 0.34 0.74

CeRu2Si2 2.0 0.75 1.2 1.25 0.96

Ce12xLaxRu2Si2 1.4 0.2 0.2 1.4 0.14

dynamical spin susceptibility near the magnetic phase tran-sition. Our approach provides a microscopic derivation of the main features assumed in the phenomenological models of heavy Fermions as the duality model. We predict a two-component behavior of the dynamical spin susceptibility: a quasielastic peak typical of the Fermi-liquid excitations, and an inelastic peak at a valuevmaxof the frequency, which is

strongly renormalized due to spin-fluctuation effects. Out-standingly well, the frequency of the inelastic peak is pushed to zero at the antiferromagnetic transition at the same time as the frequency width vanishes. The results have been com-pared to the inelastic neutron scattering experiment data with reasonable predictions for the Stoner enhancement factor

(12I) and the characteristic frequency v0 of the interband

contribution to the susceptibility. Obviously, more experi-ments are needed for a systematic test. The issue is important since it may have implications for the quantum critical phe-nomena around the antiferromagnetic critical point. Work is currently in progress in that direction and will be presented in a forthcoming paper. We expect the two underlined modes to have different effects on the critical behavior with, on the one hand, the first ‘‘intraband’’ mode acting as a paramag-non mode as in the Hertz-Moriya-Millis theory,5 and on the other hand, additional effects brought by the second ‘‘inter-band’’ mode.

ACKNOWLEDGMENTS

We would like to thank G.G. Lonzarich, N.R. Bernhoeft, G.J. McMullan, L.P. Regnault, J. Flouquet, S. Raymond, P. Brison, and K. Miyake for very helpful discussions.

APPENDIX

The expressions of the different bubbles appearing in the expression of the boson propagators@cf. Eq. ~14!# are given

here~with i51, 2, m or f f )

w

¯_{i~q,ivn!5wi~q,ivn!1wi~2q,2ivn!,}

w1~q,ivn!52 1 b ks,iv

(

_n G_{c f} 0 s _~k1q,iv n1ivn!Gsf f₀~k,ivn!, w2~q,ivn!52 1 b ks,iv

(

_n G_cc 0 s _~k1q,iv n1ivn!Gsf f₀~k,ivn!, wm~q,ivn!52 1 b ks,iv

(

_n G_{c f} 0 s _~k1q,iv n1ivn!Gc fs₀~k,ivn!, wif f~q,ivn!52 1 b ks,iv

(

_n

Gsf f₀~k1q,ivn1ivn!Gsf f₀~k,ivn!,

wcc i _~q,iv n!52 1 b ks,iv

(

_n G_cc 0 s _~k1q,iv n1ivn!Gccs₀~k,ivn!, wf c i _~q,iv n!52 1 b ks,iv

(

_n G_{f c} 0 s _~k1q,iv n1ivn!Gsf c₀~k,ivn!, w'f f~q,ivn!52 1 b ks,iv

(

_n G_{f f} 0 ↑ _~k1q,iv n1ivn!G↓f f₀~k,ivn!, w'cc~q,ivn!52 1 b ks,iv

(

_n G_cc 0 ↑ _~k1q,iv n1ivn!Gcc↓ ₀~k,ivn!, w'f c~q,ivn!52 1 b k_s,iv

(

_n G_{f c} 0 ↑ _~k1q,iv n1ivn!Gf c↓₀~k,ivn!, ~A1! where G_cc 0 s _(k,iv

n), Gsf f₀(k,ivn), and Gsf c₀(k,ivn) are the Green’s functions at the saddle-point level obtained by in-versing the matrix G₀s(k,t) defined in Eq. ~7!.

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