THE KONDO PROBLEM : IS FINAL VICTORY JUST AROUND THE CORNER ?

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THE KONDO PROBLEM : IS FINAL VICTORY JUST AROUND THE CORNER ?

H. Suhl

To cite this version:

H. Suhl. THE KONDO PROBLEM : IS FINAL VICTORY JUST AROUND THE CORNER ?. Jour-

nal de Physique Colloques, 1971, 32 (C1), pp.C1-421-C1-423. �10.1051/jphyscol:19711149�. �jpa-

00213968�

EFFET KONDO ET FLUCTUATIONS DE SP/N.

THE KONDO PROBLEM :

IS FINAL VICTORY JUST AROUND THE CORNER ? (*)

by H. SUHL

Physics Department, University of California, Sam Diego, La Jolla, Calif., 92037

Résumé. — Les méthodes mathématiques utilisées pour résoudre le problème posé par les impuretés paramagné- tiques dans les métaux sont passées en revue et discutées d'un point de vue physique. Des succès théoriques limités obtenus récemment sont examinés et confrontés aux questions importantes qui restent posées.

Abstract. — A survey is given in physical terms of the mathematical methods that have been brought to vear on the problem of paramagnetic impurities in metals. Recent limited successes of the theory are reviewed and measured against the serious questions that remain open.

Even in a review talk such as this, it is hardly necessary to recapitulate what the Kondo problem is : the aggregate of questions raised by the discovery of Kondo in 1964 that the resistivity of a metallic alloy with paramagnetic impurities, when calculated to third order in the coupling constant of the conduction electron spin density to the localized spin, diverges as log T as the temperature T tends to the absolute zero.

The underlying model in this calculation is the so called S-d model in which the interaction takes the form H

= 3(r)S.s(r)dr (Usually / is considered to be of very short range, so that this coupling is replaced by /£2S.s(0) where lis the atomic volume).

Since Kondo's initial work, successive waves of theorists have attacked this problem, but although substantial inroads have been made, is still represents a very considerable challenge.

The first efforts to improve the Kondo result were attempts to sum up an infinite class of perturbation terms, selecting in each order only the contribution with the highest power of log T. Strict adherence to this procedure led Abricosov [1] to the conclusion that the effective scattering amplitude that figures in the calculation of the resistivity should be proportioned 11 H— log — I where s is some cutoff energy, of order of the Fermi energy. This does not help much ; on the contrary : it moves the divergence up from T = 0 to T = T

~ e exp - (e/J), the so called Kondo temperature. However, the Abricosov result suggests that at and below T

one is dealing with a resonant scattering process, and that what is missing is an imaginary part in the denominator of Abricosov's result.

As it stands, the result is inconsistent with the requirement of unitarity of the S-matrix. A treatment that is consistent with unitary was suggested by the present author [2]. When reexpressed in perturbative language it amounts to inclusion of less-than predo- minantly divergent terms in each order of the deve- lopment in powers of coupling strength. Naturally a

(*) Sponsored in part by the Air Force Office of Scientific Research, U. S. Air Force, Grant Number AF-AFOSR-610-67.

theory that respects unitarity does not give infinities.

Instead the resistivity approaches its unitarity limit as T falls below T

. The precise relation of the S- matrix treatment to diagramatic perturbation theory was later established by Brenig & Goetze [3], and by Keiter [4].

Concurrently, Nagaoka [5] developed a Green's function theory of the Kondo problem, proposing a decoupling procedure for terminating the infinite hierarchy of Green's function equations of motion.

Later, it was demonstrated by the Diisseldorf and Cologne groups [6] that at least for the calculation of transport quantities, the results were entirely equiva- lent to those of S-matrix theory.

Although the above three procedures have recently come under attack, and for very good reasons that I will presently enumerate, many practical calculations based on one or other of them have been made in the past and are continuing to be made today. I would judge that these have been reasonably successful for transport quantities, but less so where thermodynamic quantities are concerned.

Suhl and Wong made computer calculations of resistivity and of the anomalous thermopowers observ- ed in dilute alloys, in reasonably good correspondence with experiments, at least down to temperatures not too far below T

. Hamman and Bloomfield [8] and Zittartz and Mueller Hartmann [9] were able to obtain solutions in closed form by assuming specific density of states functions. The closed form leads to results for transport quantities that depend on T only through log T. Most of the measurements are consistent with this around and above 3°

; well below T

, however, resistivity measurements seem to give p = p (unitary limit) (1 — (T/0)

) where 0 is a constant, and this result simply does not follow from the simple theories just described. Likewise, the most recent calculations of susceptibility [10] give a Curie constant that vanishes as T -*• 0, but not fast enough to save the susceptibility from becoming infinite as T =*• 0. Rather general theoretical considerations and perhaps some of the measurements, suggest that it should remain finite.

The blame may be placed either :

1) on the inadequacies of the approximations of the Series summation, S-matrix or Greens function methods, or

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19711149

C 1 - 422 H. SUHL 2) on the inadequacies of the S-d model itself. As for l), the central approximation is restriction to one- electron and one-hole intermediate states in the per- turbation series or the unitarity relation. This approxi- mation automatically disregards a number of effects that may be quite important. For example, it amounts to allowing autocorrelation of the localized spin to extend over infinite times. This is clearly unphysical : even if the local S is regarded as fixed in magnitude, its direction becomes uncorrelated as time increases. As for 2), it should be remembered that in the cc true

Hamiltonian, explicitely spin dependent forces of the type S. s(0) do not occur. For example in the Anderson model, the electron in the localized state does not flip its spin directly. Rather, the localized spin is destroyed and then rebuilt with opposite orientation. It is true that if the localized state is taken to be infinitely sharp the disappearance is confined to a very short time interval, but even then, as indicated at the end of this paper, it may have important consequences. In addi- tion, the local state must have some width, and then some long-lived component of the state with zero local spin must be reckoned with [l l].

As for the S-d model, a very suggestive attempt at its solution was recently made by Anderson and Yuval.

They temporarily abandon term-by-term rotational invariance and write down the cc time )>-dependent perturbation series for the partition function, taking as part of the unperturbed Hamiltonian the longitu- dinal part JSs(0) of H 1 which between spin flips has the appearance of a highly localized Zeemann field constant in time. They then hypothesise that the time interval between flips is long on a timescale set by the Fermi energy. They are then permitted to use the asymptotic form (valid for large times) [l21 of the electron propagator in a constant field. This approxima- tion enables them to establish an equivalence between the Kondo problem and a certain one dimensional many body problem. From the structure of their result they conclude (in collaboration with Hamann) that the spin susceptibility should go to a finite values as T -+ 0. No proof is given, however, that the time interval between spin flips is, in fact, large enough to permit making the above mentioned approximation.

Methods aimed at a direct evaluation of quantities (such as partition function, resistivity, etc.) from a Hamiltonian more fundamental than the S-d model are known as spin fluctuation methods. A case in point is the susceptibility of the Anderson model 1131. A straight application of diagramatic methods leads to an infinity in the response to a uniform magnetic field, as the parameters of the model (interaction energy U on the local site, position of the virtual one-electron level, and the coupling V of the local site to the band states) enter the critical region characterized by existence of a local moment. This infinity can be removed by calculating the electron and hole propa- gators self-consistently together with the magnetic response. The result [l41 is a susceptibility curve that simulates fairly closely a Curie law at high tempera- tures (in that region of the parameter space in which magnetic moment formation is favored), but flattens out to a constant value a T

0. Similarly the resisti- vity can be calculated on this basis [l51 and flattens out towards T = 0. However, it was subsequently

shown by Hamann that the flattening out occurs at a temperature proportional to exp - (&/a2 instead of exp - ( E / J ) as expected from the Kondo effect, it being supposed that J is of order V2/U, in accordance with the Schrieffer-Wolff transformation of the Anderson model into the Wolff model.

Another type of spin fluctuation theory is the so called functional integral method. Since this is the subject of talks at this conference by the formost advocates of this method 1161, I will only sketch it briefly : it is possible to rewrite the partition function for the Anderson model as a problem in which the repulsion between up- and down-spin electrons at the local site is mediated by a random time varying Zeemann field, provided a Gaussian average is finally taken over all possible random fields. Aside from various plausible results in the non-magnetic regime, interesting conclusions can also be drawn in the magne- tic range, particularly in the Kondo range. If once again the crucial assumption is made that the variation in spin direction is slow, an essentially closed form for the partition sum as functional of the random field l(z)

the time or (temperature)-') can be obtained.

In the magnetic regime it takes the form exp - F(0 where the functional F(5) has an absolute minimum at l ⁼

a constant, to, corresponding to the Hartree Fock solution. However these isolated two points in function space contribute little to the partition sum ; one covers much more of function space if one allows

t to switch every now and then between plus and minus to. Hamann has shown that the contribution to the total partition sum from N such switches repro- duces the Nth term in the Anderson-Yaval perturba- tion series. Thus the Anderson model plus the hypo- thesis of slow switching gives the S-d model. This is, of course, no proof that the slow switching assumption is valid.

Finally I would like to indulge in a speculation :that the reason for the existence of the Kondo effect even at arbitrary weak coupling J, is in the neglect of the energy dependence of J. By the nature of its approxima- tions, the Schrieffer-Wolff transformation discards this energy dependence. As already mentioned, the true Hamiltonian has no spin dependent forces. In a more physical model, like that of Anderson, the spin flip amplitude therefore has no energy independent c( Bornterm w. Rather the flip happens via the interme- diate step of either completely emptying or doubly occupying the virtual level, as in the unitarity diagram shown in figure 1, in which the dotted line is the elec- tron in the virtual level and the solid line a conduction electron. Only the conversion amplitude c has a Born

FIG. 1. - In the Anderson model, the role of the Born term for spin flip scattering is played by a two-stage energy depen- dent process involving annihilation and re-creation of the

local spin.

THE KONDO PROBLEM : IS FINAL VICTORY JUST AROUND THE CORNER ? C 1 - 423 term, which is simply V. In principle c, z, and the non-

spin flip amplitude t should be calculated from the simultaneous unitarity equations that they satisfy.

This is a very sustantial undertaking. However, as a first attempt let us regard the c< driving terms c2/energy denominator in the equation for z (Fig. 1) as simply given, with c replaced by its Bo value V.

Neglecting the width of the virtual level completely we get one pole term (virtual level doubly occupied in the intermediate state) which for very large U is outside the band, and an inelastic cut (virtual level empty, with two conduction electrons in the intermediate state). If

is within a symmetric band, so that - ed is also in the band, this cut starts at z = -

and goes as far as - cd + ^{W, where} W is the width of the band (fig. 2). Thus the amplitude for << real

conversion of a

FIG. 2. - The inelastic cut (heavy portion of the energy scale) inathe scattering amplitude caused by the conversion process local electron has its threshold above the Fermi level.

But as is well known from scattering theory such an inelastic process has repercussions on the phaseshift a t all energies including at the Fermi level z = 0. A pre- liminary calculation shows that if z(x + ^is) and z(x - i ~ ) are to be boundary values of an analytic function z(z) as the real energy axis is approached from above and below, then this << long range effect >>

of the inelastic process forces one to subtract from the Refer [l] ABRICOSOV (A. A.), Physics, 1965, 11, 5.

[2] SUHL (H.), in Rendiconti de la Scuola Internazionale di Fisica

Enrico Fermi

1966, Academic Press, London, 1967.

131 BRENIG and GOETZE, Zeitschr. fur Physik, 1968, 217,

1 0 0 100.

[4] KEITER, --.

Zeitschr. _LL. -- fiiv Physik, 1968, 213, 466; 1968, [5] NAGAOKA (Y.), Phys. Rev., 1965,138 A, 11 12 ; Progr.

of Theor. Phys., 1967, 37, 13.

[6] ZMTARTZ (H.), Zeitschr. fur Physik, 1968, 217, 43.

[7] SWL & WONG, Physics, 1967, 111, 17.

[S] HAMANN & BLOOMFIELD, Phys. Rev., 1967, 164, 856.

value at z ⁼ 0. The logarithmic T dependence thus disappears a t small z. Alternatively, we may near z ⁼ 0 use the old solutions of the S-matrix theory, but replace J by J,,,, where

- - 1 - -

cot [ ^{P(O) J} ;l 6"'") - dx]

Jeff

X P ( ~ )

Here p(x) is the state density in the band (with norma- lized edges a t x = + 1. Y(x) is the imaginary part of the phaseshift, which is zero below the threshold - E,, and above threshold is given by

exp - 4 6" = 1 - 4 n2p(x) (tanh e) ^p2(x)

(V very small), where p2(x) is the state density in the inelastic range. In formula (l), to stress the character of the effect, the spin non-flip amplitude has been discarded entirely. (Also, a certain end-point singularity in the dispersion integral over 6" has been neglected.

This is of no importance, however.) Inclusion o f t does not give a qualitative change of the result, the validity of which cannot be fully judged untiH a full self consistent treatment of z, t and c becomes available.

In conclusion one might say, in answer to the ques- tion in the title : perhaps victory is just around the corner, but will probably remain so for some time to come.

[9] ZITTARTZ & MULLER-HARTMANN, Zeitschr. fir Physik, 1968, 212, 380.

[l01 TING, in press ; BRENIG & GOETZE, in press.

[l11 ANDERSON & YUVAL, Phys. Rev. Letters, 1969,23, 89.

[l21 NOZIERES & DE DOMINICI, Phys. Rev., 1969, 178, 1097.

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[l31 ANDERSON (P. W.), Physics Rev., 1961, 124, 41.

[l41 SUHL (H.), Phys. Rev. Letters, 1967, 19, 442.

[l51 LEVINE, RAMAKRISHNAN & WEINER, Phys. Rev. Let- ters, 1968, 29, 1370.

[l61 HAMANN (D. R.), Phys. Rev. Letters, 1969, 23, 95.

[l71 WONG, EVENSON & SCHRIEFFER, Phys. Rev. Letters,

1969, 23, 92.