SIMPLE RESULTS FOR THE SUSCEPTIBILITY OF A SPIN-1/2 KONDO SYSTEM

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SIMPLE RESULTS FOR THE SUSCEPTIBILITY OF A SPIN-1/2 KONDO SYSTEM

P. Schlottmann

To cite this version:

P. Schlottmann. SIMPLE RESULTS FOR THE SUSCEPTIBILITY OF A SPIN-1/2 KONDO SYSTEM. Journal de Physique Colloques, 1978, 39 (C6), pp.C6-1486-C6-1492.

�10.1051/jphyscol:19786591�. �jpa-00218084�

JOURNAL D E PHYSIQUE Colloque C6, supplkment au no 8, Tome 39, aotit 1978, page C6-1486

P. Schlottmann

I n s t i t u t fUr Theoretische Physik, Freie Universitiit Berlin, 1000 Berlin 33, ArnimaZZee 3, Germany

Rdsum6.- On considsre la susceptibilitd statique et le temps de relaxation d'une impuretd Kondc de spin 112. On distingue trois diffdrents cas : (a) pour les tempdratures trSs basses l'impuretd forme un complexe nonmagndtique avec les dlectrons de conduction, (b) pour les hautes tempdratures le spin d'impuretd devient libre et (c) un rdgime intermddiaire s'dtablit entre ces deux limites pour des tempdratures autour de TK. Le comportement P basse et haute tempdratures peut gtre alors ddcrit d'une fason simple. Mais on a besoin de mdthodes sophistiqudes pour ddcrire le rdgime intermddiaire.

Ces traitements que nous avons r6examinds ici conduisent finalement P des rdsultats simples.

Abstract.- The static susceptibility and the relaxation rate of a spin-112 Kondo impurity are dis- cussed. Three different physical situations must be distinguished : (a) at low temperatures the im- purity and the conduction electrons form a nonmagnetic complex, (b) at high temperatures the impuri- ty spin is asymptotically free and (c) a crossover regime between these two limits for temperatures around TK. The low and high temperature behavior have a relatively simple description, while sophis- ticated theories are needed to treat the crossover situation. These treatments, reviewed here, final- ly lead to simple results, i.e. a Curie-Weiss law for the susceptibility.

1. INTRODUCTION.- A magnetic impurity in a metal matrix is usually described by the s-d-Hamiltonian

where the impurity is idealized by a spin S = 112 and B represents the impurity Zeeman energy. J is the exchange coupling constant ; it is positive for antiferromagnetic coupling and negative for ferro- magnetic coupling. The thermodynamics of the impu- rity is given by the static susceptibility and the relaxation rate of the spin. Perturbation theory for these quantities /I/ for B = 0

yields the well known logarithmic Kondo divergen- ces which -1e present in all orders of perturba- tion. Here p is the density of states at the Fermi level and D is a cut-off for the electronic excita- tions. The infrared behavior arises from the elec- tron-hole excitations generated by flips of the im- purity spin. At T = 0 the probability of creating an excitation of energy w is proportional to I/w, and the probability of an excitation with energy larger than w is proportional to RnD/w, which di- verges when w + 0. For T < Tk = De -I/

I

JP

1

^{the tor-}

rection term in the perturbation series is larger than the main term. Abrikosov /2/ summed the most divergent terms of the vertex function, i.e. those with the maximum number of logarithms in every or-

der

This expression indicates that the perturbation se- ries converges for ferromagnetic coupling, while for antiferromagnetic coupling the series does not converge for w and T < Tk. An attempt to eliminate the divergences within the most divergent approxi- mation is the Suhl-Nagaoka theory 131. This theory, however, fails to explain the low-energy behavior

(&I and T < T ), where nonleading divergences beco- k

me dominant.

The singularity in the vertex function sug- gests the existence of anomalous matrix elements, in analogy to the theory of superconductivity. The anomalous pairing corresponds to the singlet and triplet states of the impurity spin coupled to the conduction electron spin density at the impurity site. This description is valid at very low tempe- ratures and it is the basis for Yoshida's ground- state theory /4/ of the Kondo problem. The ground- state is the singlet state with a pairing energy of the order of Tk. The system is nonmagnetic having a finite susceptibility,

xo

% I/Tk. Since the spin up and down states of the impurity are mixed in the singlet state, the impurity spin rellxation rate 1/T does not vanish and is of the order of Tk. The

1

strong coupling between the impurity and the con- duction electrons also allows a F e d liquid des- cription of the low temperature K~ndo~properties /5,6/.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19786591

In summary, at high temperatures the system behaves essentially like a free spin and at low tem- peratures like a singlet state. The two limits re- present very different physical situations ; each situation can be described by a relatively simple theory. For temperatures around Tk there is a cross over between these distinct qualitative behaviors.

The main problem consists in treating this cross- over region, where no simple physical picture helps to choose a convenient basis of states. Taking the free spin or the singlet state as the starting ba- sis perturbation treatments break down for T T

k and fail to describe the crossover. Sophisticated theoretical treatments are required to interpolate between the high and low temperature Kondo behavior.

These treatments finally lead to simple results ; i.e. a Curie-Weiss law for the static susceptibili- ty.

The experimental situation is in agreement with this qualitative picture. Bulk susceptibility measurements 171, as well as accurate ~Zssbauer hyperfine field data /8/ for Fe impurities in Cu can be fitted by a Curie-Weiss law ; at low tempe- ratures an a-b~' behavior for X is observed, as

0

expected from a Fermi liquid theory. The impurity spin relaxation rate has been measured by neutron scattering / 9 / and by magnetic resonance of near neighbor Cu-nuclei /lo/. At low temperatures the impurity life-time is finite, decreases gradually when the temperature is raised and approaches a Korringa-law (TI x T % (JQ)-~, with strongly enhan- ced coupling constant) at high T.

The theoretical methods to describe the cross- over region are either in the spirit of the long- time approximation or applications of renormaliza- tion group techniques. In section 2 we give a brief discussion of the basic ideas of these lines of approach, which lead to the resonant level model and to Wilson's numerical renormalization. In sec- tion 3 we sketch some selfconsistent approximation schemes, which are based on simple physical ideas and are able to give a qualitative interpolation between the two limits. We do not always follow the cronological order in this review. Some conclu- ding remarks follow in section 4.

2. a) RENORMALIZATION GROUP METHODS.- The renorma- lization group is a transformation procedure in which an effective Hamiltonian is constructed by eliminating electronic states from the system. In

other words, for B = 0 the system is determinedby J and D ; if the cut-off parameter is reduced to D(<D),

..

the technique provides the renormalized cou-

-.

-

pling constant J(D), which is obtained by integra- ting the equation / l l, 121

d5/d fin

DID

= jzp

- $

^{J3p2 + 0(J4) (5)}

-.

with the initial condition J(D) = J. The right hand side of equation (5) is determined by perturbation theory and is valid only if Jp

-

<< 1. Keeping only the second order termwe recover equation (4) for

-

^-.

w = D and

r

⁼ J, Equation ( 5 ) is the scaling law of the system. By reducing the cut-off parameter the system is transformed and eventually reaches a fi- xed point when tends to zero. The fixed points are given by the zeroes of equation (5) ; there is the trivial solution J* = 0, which is the fixed point for ferromagnetic coupling. In this case the conclusion derived from equation (4) is not modi- fied by nonleading divergences : For ferromagnetic coupling the impurity behaves asymptotically as a free spin. For antiferromagnetic coupling equation

- -

(5) shows that J grows when D is reduced ; when jp reaches the order of unity, terms of all orders in

ip

must be included in equation (5). There are two possibilities for the fixed point : equation (5) has a root at some finite J or there is no e root for finite

5.

The first case has been discus- sed by Abrikosov and Migdal /11/ and yields a power dependence for the susceptibility at low T. In the second case

Sp

goes to infinity when 6 tends to ze- ro and the ground-state of the system is a singlet.

It was Wilson's /13/ main achievement to show that the fixed point for antiferromagnetic coupling is J* = confirming thus Yoshida's groundstate theo- ry 141. The impurity spin is strongly coupled to the electron gas. Starting the renormalization from the free spin limit an infinite number of skeleton diagrams is needed for the crossover. This can be achieved only by numerical renormalization. It should be mentioned that the correct characteristic energy (Kondo temperature), T: =

DG

exp(-1/Jp), is obtained by integrating equation (5).

We want to review briefly Wilson's 113,141 renormalization procedure, which consists of three steps. First, the scattering of the electrons with the impurity is reduced to a nearest neighbor hop- ping on a semiinfinite chain with the impurity lo- cated at the origin. Second, the subsystem consis- ting of the states at the impurity site (a singlet, a triplet and two doublets with energies

-

T; 3 J,

JOURNAL DE PHYSIQUE

+

-

4 1 J and 0 , r e s p e c t i v e l y ) and t h e s t a t e s of t h e f i r s t few s i t e s of t h e c h a i n is d i a g o n a l i z e d . Only a s e t of low-lying s t a t e s i s k e p t and a g a i n diago- n a l i z e d t o g e t h e r with t h e n e x t few s i t e s of t h e c h a i n . T h i s procedure i s r e p e a t e d u n t i l a s t a t i o - nary s i t u a t i o n b u i l t s up. Due t o t h e l a t t i c e formu- l a t i c n , t h e p a r i t y of t h e wavefunction p l a y s a fun- damental r o l e and one h a s t o d i s t i n g u i s h between so- l u t i o n s o b t a i n e d from an even and odd number of s i - t e s . I n t h e t h i r d s t e p t h e s o l u t i o n s from c h a i n s w i t h an even and odd number of s i t e s a r e i d e n t i f i e d w i t h t h e two simple p h y s i c a l l i m i t s , i . e . t h e asym- p t o t i c a l l y f r e e s p i n and t h e s i n g l e t s t a t e . There a r e two s t a t i o n a r y s i t u a t i o n s : t h e f i r s t one i s almost s t a t i o n a r y and b u i l d s up a f t e r a few i t e r a - t i o n s ; i t i s s e p a r a t e d from t h e second one (which corresponds t o t h e f i x e d p o i n t ) by a c r o s s o v e r r e - gion. This crossover s e p a r a t e s t h e weak c o u p l i n g from t h e s t r o n g coupling r e g i o n . For small i n i t i a l coupling J, t h e system i s s e e n t o have u n i v e r s a l s c a l i n g p r o p e r t i e s , which depend o n l y on t h e ener- gy s c a l e Tk. As mentioned above t h e main g o a l of X

t h i s method was t o show t h a t t h e f i x e d p o i n t f o r

-

a n t i f e r r o m a g n e t i c coupling i s J = and t o give an e s s e n t i a l l y e x a c t i n t e r p o l a t i o n between t h e asymp- t o t i c freedom (weakly coupled s p i n ) and t h e i n f r a - r e d s l a v e r y ( s t r o n g l y coupled s p i n ) . The c a l c u l a t e d s u s c e p t i b i l i t y i s roughly f i t t e d by a Curie-Weiss law f o r temperatures l a r g e r t h a n 0.5 T

k '

2. b) THE LONG-TIM? APPROXIMATION.- The long-time approximation i s based on NoziZre and de Dominicis' s o l u t i o n 1151 of t h e X-ray t h r e s h o l d problem. The essence of t h i s approximation l i e s i n t h e d i s t i n c - t i o n between s p i n - f l i p s c a t t e r i n g , caused by t h e S and S -terms i n e q u a t i o n ( I ) , and t h e spin-non-

X Y

f l i p s c a t t e r i n g caused by t h e S -term. Following Anderson e t a l . 1161 we i n t r o d u c e d i f f e r e n t cou- p l i n g c o n s t a n t s f o r t h e s e p r o c e s s e s , J and J

1 1 1 '

r e s p e c t i v e l y . The spin-non-flip i n t e r a c t i o n ( J )

I I

f o r a given s p i n d i r e c t i o n corresponds t o an ordina- r y p o t e n t i a l s c a t t e r i n g and can b,e t r e a t e d e x a c t l y , w h i l e t h e s p i n - f l i p terms a r e considered a s t h e p e r t u r b a t i o n . There a r e many ways t o f o r m u l a t e t h e long-time approximation ; t h e most e l e g a n t way i s t h e b o s o n i z a t i o n technique 1171 which we b r i e f l y d e s c r i b e h e r e .

Since a c o n t a c t p o t e n t i a l only i n v o l v e s s- wave s c a t t e r i n g , t h e e l e c t r o n - h o l e e x c i t a t i o n s of

t h e system a r e d e s c r i b e d by density-wave o p e r a t o r s

which obey boson commutation r u l e s i f we confine o u r s e l v e s t o t h e low-lying e x c i t a t i o n s . Here k o i s a momentum cut-off of t h e o r d e r of t h e Fermi momen- tum. The k i n e t i c energy of t h e e l e c t r o n gas and t h e spin-non-flip i n t e r a c t i o n a r e e a s i l y expressed i n terms of t h e s e o p e r a t o r s . I n o r d e r t o t r a n s c r i b e t h e s p i n - f l i p s c a t t e r i n g we need t h e boson represen- t a t i o n of fermions f o r t h e Wannier s t a t e s a t t h e im- p u r i t y s i t e 1171

This t r a n s c r i p t i o n is e x a c t only f o r an i n f i n i t e bandwidth (ko ^{+ m )} and i f a charge o p e r a t o r i s in- cluded t o reproduce t h e c o r r e c t anticommutation re- l a t i o n s . The l a t t e r drawback seems t o b e only f o r - mal, w h i l e t h e f i n i t e k,, r e p r e s e n t s an approxima- t i o n t o t h e problem. We i n t r o d u c e s p i n - d e n s i t y waves

%

= J2(bk+ 1

-

bk+) and charge-density waves

ak = n(bk+ 1 + bkJ-). Since t h e i n t e r a c t i o n terms de- pend only on s p i n - d e n s i t y o p e r a t o r s , t h e spin-den- s i t y waves decouple from t h e charge-density waves, which appear only i n t h e k i n e t i c energy. This cor- responds t o t h e i n t u i t i v e n o t i o n t h a t t h e s p i n dy- namics should g e n e r a t e only s p i n d e n s i t y e x c i t a t i o n s and n o t d i s t o r t t h e charge d i s t r i b u t i o n . Keeping only t h e s p i n - d e n s i t y e x c i t a t i o n s , t h e Hamiltonian t a k e s t h e form

The f i r s t two terms correspond t o harmonic o s c i l l a - t o r s d i s p l a c e d from t h e i r z e r o p o i n t s by a s m a l l amount p r o p o r t i o n a l t o J p . The c a n o n i c a l t r a n s f o r -

mation

II

produces an a d d i t i o n a l s h i f t , p r o p o r t i o n a l t o a , from t h e z e r o p o i n t . S remains i n v a r i a n t under t h e

+ z +

1 +

t r a n s f o r m a t i o n and U S U = S exp{% X _E(ak-%)).

k>O

Choosing a = 1

-

fi t h e e x p o n e n t i a l f a c t o r multi- p l y i n g S+ and S- i s of t h e type of e q u a t i o n ( 7 ) . Hence, i t i s p o s s i b l e t o d e f i n e new s p i n l e s s f e r - mion q u a s i p a r t i c l e s b u i l t up from t h e s p i n - d e n s i t y o p e r a t o r s , which we denote by c (c'). The s p i n den+

k k

s i t y o p e r a t o r s a r e expressed i n terms of t h e new fermions by a r e l a t i o n s i m i l a r t o e q u a t i o n ( 6 ) . I n >

terms of t h e new fermion o p e r a t o r s the Hamiltonian can be w r i t t e n a s /18.19/

+ 1 + 1

~ ( d d - ?) ( c c

-

(10)

where V = J

a,

^{yp = f i}

-

1

-

J p l f i and

* ?

1 I I

c+ = E c i s t h e Wannier s t a t e of t h e new fermions k .k

a t t h e lmpurity s i t e . Here we made use of t h e ana- logy between t h e fermion and the s p i n 112 a l g e b r a and r e p l a c e d t h e s p i n o p e r a t o r s by fermion opera- t o r s (s-t-t d ;

s++

di ;

sZ

= did - L ) . I ~ y = o

2

e q u a t i o n (10) reduces t o a r e s o n a n t l e v e l l o c a t e d a t an energy B below t h e Fermi l e v e l w i t h a width of A = rpv2 ( t h e Toulouse l i m i t / 16,201)

.

The sus- c e p t i b i l i t y i s f i n i t e a t z e r o temperature, c o r r e s - ponding t o a nonmagnetic g r o u n d s t a t e , and g r a d u a l l y goes over t o a Curie-Weiss behavior when t h e tempe- r a t u r e i s r a i s e d . Hence, t h e r e s o n a n t l e v e l i s a b l e t o d e s c r i b e t h e c r o s s o v e r between t h e s i n g l e t s t a t e and t h e a s y m p t o t i c a l l y f r e e s p i n . The Toulouse li- mit corresponds t o a l a r g e spin-non-flip coupling c o n s t a n t . Since t h e r e n o r m a l i z a t i o n group y i e l d s a growing coupling c o n s t a n t when t h e bandwidth i s reduced, t h e system should s c a l e through t h e Tou- l o u s e l i m i t and e q u a t i o n (10) f o r y = 0 should b e t h e s o l u t i o n of t h e Kondo problem. U n f o r t u n a t e l y t h i s s c a l i n g behavior could n o t b e proven, and Wilson's r e n o r m a l i z a t i o n seems to i n d i c a t e t h a t t h e r e s o n a n t l e v e l d e s c r i p t i o n of t h e Kondo e f f e c t i s only q u a l i t a t i v e l y c o r r e c t 1141.

The thermodynamics of t h e Kondo-Hamiltonian i n t h e long-time l i m i t i s e q u i v a l e n t t o t h a t of a c l a s s i c a l one-dimensional Coulomb gas of p o s i t i v e and n e g a t i v e charged r o d s a l t e r n a t i n g l y d i s t r i b u - t e d on a r i n g . S c h o t t e and S c h o t t e /21/ used t h i s c l a s s i c a l analogy f o r Monte-Carlo c a l c u l a t i o n s of t h e s u s c e p t i b i l i t y . T h e i r r e s u l t s a r e a l s o f i t t e d by a Curie-Weiss law.

3. SELFCONSISTENT APPROXIMATION SCHEbES.- Up t o h e r e we have mentioned Wilson's a s y m p t o t i c a l l y e x a c t numerical s o l u t i o n , t h e Monte-Carlo c a l c u l a - t i o n s u s i n g t h e one-dimensional Coulomb gas analo- gy and t h e rough s i m i l a r i t y t o a r e s o n a n t l e v e l . I n t h i s s e c t i o n we review two s e l f c o n s i s t e n t appro- ximation scWmes, which d e s c r i b e t h e asymptotical- l y f r e e s p i n , t h e s i n g l e t s t a t e and t h e c r o s s o v e r r e g i o n f o r s m a l l Kondo couplings.

We f i r s t review t h e approximation scheme f o r t h e dynamical and s t a t i c s u s c e p t i b i l i t y by ~ E t z e

e t a l . /22/. We focus our a t t e n t i o n on t h e i m p u r i t y e x c i t a t i o n spectrum, ~ " ( w )

,

which determines t h e s t a t i c s u s c e p t i b i l i t y through

Assuming t h a t t h e impurity dynamics i s e s s e n t i a l l y a r e l a x a t i o n phenomenon we i n t r o d u c e a r e l a x a t i o n k e r n e l o r n o i s e spectrum, N(w), d e f i n e d by t h e f o l - lowing e q u a t i o n

With t h i s e x p r e s s i o n t h e c o n d i t i o n ( 1 1) i s automa- t i c a l l y f u l f i l l e d f o r any N(w) having t h e c o r r e c t a n a l y t i c a l p r o p e r t i e s . The f l u c t u a t i o n - d i s s i p a t i o n theorem g i v e s a r e l a t i o n connecting t h e absorb t i o n spectrum xt'(w) /w w i t h t h e s p i n f l u c t u a t i o n s

<(SZ

-

< s Z > 1 2 > . I n t h e absence of a magnetic f i e l d

s2

=

1

and we have

z 4

I/x,

⁼ 4T + 2

J*

^R [eofh

6 ^- 3

x 1 ' ( ~ ) / x 0 (13) For g i v e n r e l a x a t i o n k e r n e l , N(w), e q u a t i o n (13) provides a t r a n s c e n d e n t a l e q u a t i o n which determines X,. Up t o h e r e t h e f o r m u l a t i o n i s e x a c t ; now we have t o f i n d a s u i t a b l e approximation f o r N(w). I f , f o r i n s t a n c e , N(w) = i x o / T 1 t h e s p i n decays expo- n e n t i a l l y i n time. It i s assumed t h a t N(w) h a s l e s s s t r u c t u r e t h a n ~ ( w ) and hence i t i s a more conve- n i e n t c a n d i d a t e f o r approximations.

The s p i n r e l a x e s through s p i n f l i p s . I n lowest o r d e r i n t h e c o u p l i n g c o n s t a n t s we have N(w) = i a ( J 0 1 2 ) ~ . N(w)/xo y i e l d s t h e Korringa r e l a - x a t i o n r a t e 1/T

1

= RT(JP)* f o r Xo = 1/4T. A s p i n -

1

f l i p g e n e r a t e s an i n f i n i t e number of e l e c t r o n - h o l e e x c i t a t i o n s i n t h e e l e c t r o n gas ; each e x c i t a t i o n y i e l d s a Kondo l o g a r i t h m ( s e e e q u a t i o n ( 3 ) ) . I n our f i r s t approximation f o r N(w) we considered one s p i n - f l i p w i t h a l l t h e accompanying e x c i t a t i o n s . T e c h n i c a l l y t h i s means t h a t N(w) i s e v a l u a t e d i n second o r d e r i n IT. and t o a l l o r d e r s ( e x a c t l y ) i n Jll. The r e s u l t i s s i m i l a r t o t h e X-ray t h r e s h o l d a b s o r b t i o n spectrum

J P

~ ~ ' l ( w ~ + ( 2 r r r ) ~ ~

11

(14) I n t h i s approximation t h e s p i n - f l i p s a r e indepen- d e n t from each o t h e r and they do n o t i n t e r f e r ; we can c o n s i d e r i t a s t h e random phase approximation f o r t h e s p i n f l i p s . I n t h e e v a l u a t i o n of e x p r e s s i o n (14) i t i s assumed t h a t t h e s p i n i s p e r f e c t l y s t a - b l e a f t e r t h e s p i n - f l i p . This i s n o t a r e a l i s t i c assumption, s i n c e t h e s p i n c o n t i n u e s i n t e r a c t i n g w i t h t h e e l e c t r o n gas through t h e s-d-Hamiltonian.

C6- 1490 ^{JOURNAL DE PHYSIQUE}

In our second approximation we included a relaxa- tion ansatz, similar to equation (12), for the spiw flip operators appearing in N(w).

Expression (14) already shows the main diffe- rence between ferromagnetic (J < 0) and antiferro- magnetic (J > 0) coupling. For J < 0 the relaxation function N"(w) is reduced below the Korringa value by the Kondo divergences. The spin relaxation rate is proportional to the temperature, but smaller than Korringa's rate. The impurity is weakly cou- pled to the electron gas for all T and the suscep- tibility follows a Curie law, xoT = const. For

J > 0, on the other hand, the function NrT(w) is en-

hanced above the Korringa value. In this case the interference of the electron-hole excitations is constructive and favors the relaxation. Equation (13) provides a finite susceptibility and a finite life-time of the impujty at zero temperature, cor- responding to a nonmagnetic groundstate. When the temperature is raised the nonmagnetic impurity com- plex gradually breaks up and transforms to an al- most free magnetic moment. The susceptibility is described by a Curie-Weiss law and is in quantita- tive agreement with Schottes Monte-Carlo calcula- tions /21/. The relaxation rate grows with tempera- ture and approaches the Korringa behavior for high T.

This method uses the strong physical connec- tion between the dynamic and static properties of the impurity. A finite susceptibility at T = 0 is only possible with a finite relaxation rate and vice versa. The method gives the same qualitative behavior as Wilson's renormalization and has the advantage of easy extension to nonzero magnetic fields. Here, a perturbation expression for N(w) yields the complete frequency and temperature de- pendence of ~(w). Some drawbacks are the consequen- ces of the perturbational basis of the theory : (a) the rotational invariance of the s-d-Hamiltonian is broken due to the preferential treatment of J (b) the relaxation function shows a cusp in the li-

II'

mit T, w + 0 and does not reach the unitarity bound, (c) a reasonable specific heat could not be obtai- ned and (d) the resonant level is not correctly in- cluded as a limit. The latter is not really a draw- back since a theory for small couplings should not necessarily be valid for strong coupling parame- ters.

We now review a simple calculation procedure 1181 within the long-time approximation, which re-

produces the resonant level limit as well as the ferromagnetic one (J < 0), which avoids most of the above drawbacks and which can be extended to nonzero magnetic fields. For this purpose we consi- der the transformed expression of the s-d-Hamilto- nian, equation (10). If y = 0 equation (10) reduces to a resonant level located at the Fermi level with a width of A = .rrpv2. For V = 0 equation (10) essen- tially yields the X-ray threshold problem. We treat the resonant level as the unperturbed system and the y-term as the perturbation. We take into ac- count only the leading logarithmic contributions;

but such that both special cases (V = 0 and y = 0) are correctly included. The procedure has several advantages : (a) the width of the resonant level acts like an infrared cut-off, (b) the free energy varies with T~ at low temperatures and (c) the per- turbation expansion for y > 0 is an oscillating se- ries, which converges and can be summed up. This is similar to a perturbation theory with respect to J in the ferromagnetic case, where nonleading con- tributions are known to play a secondary role. In other words, instead of trying to build up a finite impurity life-time (or susceptibility) at T = 0 by perturbation in J (equations (2) and ( 3 ) ) , we start with a finite impurity line-width.

We have shown /18/ that the invariant cou- pling associated with the four fermion interaction y is not renormalized in leading logarithmical or- der, since the bubbles with parallel and antiparal- lel lines cancel each other at the Fermi surface.

This is a consequence of the underlying X-ray thre- shold analogy. The resonance width A, however, is renormalized by the interaction y. The renormaliza- tion factor is given by the derivative of the spin-

+ +

flip (comutator) correlation function <<c d;d c>>

w with respect to the bare spin-flip correlation func- tion. This derivative obeys multiplicative renorma- lization in leading order, in analogy to the X-ray threshold spectral function. The renormalization factor is then obtained from perturbation theory by integrating the Lie differential equation of the

renormalization group.

An alternative way to calculate the renorma- lization factor is presented here. Ion leading order it is also given by the X-ray spectral (corntator) correlation function,

x

^{(Z )}

,

convoluted with

X-ray m the d-propagator Gd

where Zm = iw = i2nTm and Zn = iw = iaT(Zn+l) are

m n

the thermodynamical Bose and Fermi poles, respecti- vely. By analytical continuation into the complex energy plane we obtain the X-ray spectral (anticor mutator) correlation function

where Bx is the incomplete Beta function with x = Z.rrT/D.(The discontinuity of this function on the real energy axis is related through the fluc- tuation-dissipation theorem to that of the spinflip commutator correlation obtained by multiplicative renormalization). Both methods yield the same re- normalization factor for small yp. For small Jp the results differ by non-logarithmical contributions in the perturbation expansion, which are not taken into account by the multiplicative renormalization, but are necessary to reproduce the X-ray spectral function in the limit A -+ 0.

The renormalization factor, equation (16), should have been calculated with the renormalized resonance width Q(Z), instead of A. This transforms equation (15) into an integral equation for Q(Z).

Within leading logarithmic accuracy, however, we have the following selfconsitent equation for Q(Z),

(JIJ = J l small)

D 2Jp a-iZ Q-iZ

n

= J~ZT{ (=)

r

^{( I-sp+} -1

/r

^{(JP+ -1}

-

2~rT 2nT

(Q-~Z)/~RT} (17)

Here I' is the gamma function. Within the present approximation we obtain the same characteristic energy as in reference /221, Go = ~(Jp14)~' JP, which is obtained from equation (17) for Z = T = 0

(J > 0 but small). For Z = 0 we have that Q is an increasing function of temperature, which varies as =

n

⁰ +

f

_2JP ⁽

~ ) 3

₀ for T <<

,

^{and as}

Q = T ) at high temperatures. For the ferromagnetic case (J < 0) we obtain

n

= m 2 a T ( ~ ) 2JP, which vanishes for T -+ 0. For 2 2nT

Z >> Q ,T we have

Q =

-

^iz

*

₄ ^{D ~ / - ~}

^- _Il

^{~ ) ~ ~ ~}

Since the invariant coupling is not renorma- lized in leading order, i.e. the vertex and self- energy diagrams cancel each other, the susceptibi- lity is given by

A simplified expression for

x

can be obtained by neglecting the energy dependence of Q. This appro- ximation is in the spirit of leading logarithmic accuracy and is valid if $2 >> T for J > 0. We ob-

0

tain that Q(T) is the relaxation rate of the impu- rity spin and that the static susceptibility is gi- ven by

For J > 0 at high temperatures and in the ferro- magnetic case we reproduce the perturbation theory result for the s-d-Hamiltonian

Hence, the relaxation rate has a finite value at T = 0 (for J > 0) and goes over to the Korringa law at high temperatures.

This simple approximation scheme only rough- ly yields the correct qualitative behavior. However, improvements of the method are required in order to obtain explicit results, i.e. a plot of

xo

^{as a}

function of T.

4. CONCLUSIONS.- The low and high temperature re- gimes in the Kondo problem correspond to simple physical situations. The main difficulty is to find a treatment for the crossover region. Beside Wilson's numerical and asymptotical exact diagona-

lization of the problem, we have reviewed two ap- proximation schemes, which offer a correct qualita- tive description. We believe that a numerical so- lution should not be the ultimate answer to the problem. Efforts should be spent in developing new analytical methods, even approximate, along this line of thinking.

JOURNAL DE PHYSIQUE

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-