REMOVING THE DIVERGENCE AT THE KONDO TEMPERATURE BY MEANS OF SELF-CONSISTENT PERTURBATION THEORY

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REMOVING THE DIVERGENCE AT THE KONDO TEMPERATURE BY MEANS OF

SELF-CONSISTENT PERTURBATION THEORY

R. Mattuck, L. Hansen, C. Cheung

To cite this version:

R. Mattuck, L. Hansen, C. Cheung. REMOVING THE DIVERGENCE AT THE KONDO TEMPER- ATURE BY MEANS OF SELF-CONSISTENT PERTURBATION THEORY. Journal de Physique Colloques, 1971, 32 (C1), pp.C1-432-C1-435. �10.1051/jphyscol:19711151�. �jpa-00213971�

REMOVING THE DIVERGENCE AT THE KONDO TEMPERATURE BY MEANS OF SELF-CONSISTENT PERTURBATION THEORY

R. D . MATTUCK and L. HANSEN

Physics Laboratory I, H. C. 0rsted Institute, University of Copenhagen, Denmark and C. Y. CHEUNG

Physics Department, College militaire royal de Saint-Jean, Saint-Jean, Quebec, Canada

Résumé. — Dans les précédentes applications de la théorie de perturbation au problème de Kondo, on trouve une divergence à la température de Kondo 7k. Nous montrons que cette divergence est illusoire et provient de l'emploi de propagateurs électroniques nus. Lorsqu'on renormalise ces propagateurs de manière self-consistante, on ne trouve plus de divergence à TK, et la théorie de perturbation est valide jusqu'aux très basses températures. Cependant, la divergence à 0 °K subsiste, indiquant la présence d'un état lié. La résistance calculée s'écrit sous la forme de Hamann.

Abstract. — Previous applications of perturbation theory to the Kondo problem have shown a divergence at the Kondo temperature, 7k. We show here that this divergence is spurious and comes from the use of bare electron propa- gators in the calculation. When the propagators are self-consistently clothed, there is no divergence at 7k, and pertur- bation theory is valid all the way down to just above zero temperature. However, there is a divergence at T = 0, indicating a bound state there. Our resistance has the Hamann form.

I. Introduction. — There has been much discus- sion concerning the convergence of the perturbation expansion and existence of a bound state in the Kondo problem. Abrikosov's partial sum [1, 2, 3] over the leading logarithmic terms («parquet diagrams») showed a divergence in the s — d scattering amplitude (vertex part) r at the Kondo temperature, TK. This implied breakdown of perturbation theory and tran- sition to a bound state under !TK. On the other hand, Hamann's decoupled equations-of-motion analysis [4] showed no divergence at T^K. Ground state calcu- lations [5] indicate a bound state at T — 0.

Since any decoupling is equivalent to a partial sum [6], the implication is that Hamann has in effect summed over something more than just parquets.

In accordance with this idea, we have utilized a sug- gestion by Doniach [7] and extended the perturbation sum to include non-parquets by self-consistently clothing the j-electron propagators [8, 9]. This removes the divergence in F at T^K, showing that there is no breakdown of perturbation theory or bound state starting in at T^K. Perturbation theory is valid all the way down to just above zero temperature. However, there is still a divergence, hence a bound state, at T = 0. This is in agreement with results obtained by Nozieres [10], using a diagram method different from ours (his method involves unlinked diagrams). The resistance is essentially the same as that calculated by Hamann. Clothing the rf-electron has only a small effect on this result, but giving the rf-energy level a finite width causes considerable changes.

II. General expressions for scattering amplitude and resistance. — We use the Kondo Hamiltonian [5], except that we replace the impurity spin operator by its second-quantized form [11] :

& = £ S^k cl„ C^kx + Y,^sd 4 ) C^ifi - TT-j- X

fat H L 1'

x S (°«'«-S|»'<i)cVcⁱYC^taCj^j,; (1)

here J = s — d coupling constant, N = number of atoms, sk, ed are the bare s, d electron energies relative to the Fermi energy, and <sx,a, 2 Sr / ! are the Pauli matrices for s, d electrons. In contrast to the Kondo 3€, which has exactly one J-electron in the spin \ case, our 36 also includes states with 0 and 2 cf-electrons.

However, for ed = 0, the only effect of these extra states is to introduce an extra factor of 2 when calcu- lating the self-energy [1].

We first look at the s — d vertex part, T. If just parquets are included, this is given by figure 1. Letting

w ^ - y y "K Y) y w

Ps ~

vO

y

h

K

--

A

s

' y K> A A. X X r~\ H\

FIG. 1. — The s—d scattering amplitude r in parquet approxi- mation. K is the s—d pair bubble.

r = T° + T'ff.S and summing over only ladder parquets we find :

r'(com) = (7/2 N) [1 + K - J K2T1, (2) where a particle-hole term and a bare interaction term have been omitted for brevity (see ref. [9], eq. 32.

Note that J i n the present paper is called JTT in ref. [9]).

Here i<Cis the s — d pair bubble (see Fig. 1) given by : K(co^m) = ^ 7 I \ E *G(fc, k', - co^s) x

KfK * COS

x \G(d, ms + com) . (3) Because the G's here are in general clothed, it makes things simpler if we introduce spectral representations for them :

£ iG(k, k', os) = \ + X do' As{(a,) ,

k,k' J - o o CO — 1C0S

iG(d, cos) = do/ -T*B> , (4) J — oo tU K US

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19711151

REMOVING THE DIVERGENCE AT THE KONDO TEMPERATURE C 1 - ⁴³³

where the s and d spectial densities are given by : AS(of) = - 1 Im [ i ~ ( k , k', of f id)] ,

n ^k,kl

Ado') = 1 Im [iG(d, or + id)] (5) where, for example, G(d, w' + id) is G(d, w3 analy- tically continued from io, to just above the real axis.

Introducing (4) in (3) and carrying out the frequency sums [g] yields

J + m

K(om) =

m j

where f(w) is the Fermi function.

If instead of just ladders we include all parquets, we find instead of (2) :

This expression is just a generalization of Silverstein and Duke's solution of the linearized integral equa- tions for the vertex part [3].

The S-electron self-energy, C,, and resistance, R, may be found from r' with the aid of figure 2. After

FIG. 2. - The S-electron self-energy C, (called E, in the text) in terms of r.

doing the spin sums and introducing the spectral form of the quantity (3) KT' :

where

+ id) T1(w' + id)

l

where g(o) is the Bose function. The resistance may be obtained from the imaginary part of Z,(v + ^id),

e. g. :

+ m

Im Z,(v + ^{i6) = n}

1

- m

by means of the expression :

R = (3 cN/e2 pvi) Im C,(O + id) .

Hence we find :

Our procedure will always be to evaluate K first, in some approximation, use it to calculate r r , obtain AKr from KT', then find R from (12).

111. Calculation using bare S and d electrons. -

When both s and d electrons are bare, (5) yields Ad(wt) = 6(wr), for zd = 0, and A,(ol) = p(or), where p(?') is the density of states for the bare conduc- tion electrons, taken to be a constant = p, in a band of width 2 D. Placing this in (6) gives for K

'" tanh ( 4 2 T) K O K(x + i6) = 3 _{4 N}

5

JP n2

- 1

[ ]

4 N x 2 + ^{(2 7)'}

where we have set the Boltzman constant equal to 1.

For W = 0, this diverges logarithmically as T -+ 0.

Inserting (13) in (2) we see that T' diverges at the temperature

TK, = 3 D exp(- 4 N/3 I ^{J I} p) for J < 0 (14)

; D exp(- 4 NIJp)

TK2 = - for J > 0 , (15)

indicating a bound state for either sign of J. On the other hand, if the full parquet expression (7) is used, we obtain

T K = + D e x p ( - N / I J I p ) f o r J < 0 , (16) i. e., a bound state just for antiferromagnetic coupling [l, 21. Evaluating R in this last case with the aid of (13), (7), (9) and (12), with Ad(&) = S(&), yields R = (97-c~~ cl32 e2 v: N) ^X

which is just Abrikosov's result, and diverges at T,.

IV. Self-consistently clothed S-electron, bare d.

- Our approximation for the S-propagator in the self-consistent calculation is shown in figure 3. Inser- ting this into the K-bubble (see Fig. 1) yields figure 4, which is an equation for K in terms of Z,. But by

FIG. 3. -Approximation for S-electron propagator used in the self-consistent calculation.

FIG. 4. - Equation for the S-dpair bubble in the self-consistent calculation.

figure 2, C, is a function of T, and by figure 1, T is a function of K, so what we have is a set of three equa- tions to be solved self-consistently for K, C, and T.

It is easiest to solve first for K. From (5) and figure 3, the spectral representation for the S-electron is

AS(ot) = p(of) - np2 Im Z,(wt + is) . (18) In deriving this, we have used the Nagaoka-Hamann approximation 112, 41 :

Putting (18) in (6) with A, = 6 ( ~ ) yields :

~~~n + m

K(om) = Ko(wm) -

1

- m

We may find Im Z, from (1 l), (9) with A, = a(&).

This gives

Equations (20) for K in terms of C,, (21) for C, in terms of T, and (2) or (7) for T in terms of K are the three equations to be solved self-consistently. They may be reduced to a single self-consistent equation for K by inserting (21) in (20), and then putting (2) or (7) into the result. By all rights, this procedure should yield a formidable integral equation for K.

But because of the lucky relation

[g(wf) + $1 ^{[f(ot) -} +l ^{= -} ' ^{4 ,}

things simplify, and we miraculously obtain the alge- braic equation :

K ⁼ KO - (3 JP2 n2/16 N) KT', (22) where r' is given by (2) or (7). This has the low- temperature solution :

K % a - + ^X + 3 b[X2 + c(Jnp/2 N)2]% , (23) where X = - KO + a + (318) d(Jnpl2 N)', and (a, b, c , d ) = ( - + , 1, 1, l ) f o r J < O , o r ( 2 , - 1, 3, - 1) for J > 0, when (2) is used, and = (- 3, 1, 318, 3) when (7) is used. For w = 0 and T + 0, K no longer diverges as it did in (13) but instead goes to a. Putting (23) into (2) or (7), we find that T' ( o = 0) no longer diverges at TK but instead diverges logarithmically a t T = O :

r' (CO = 0) u(l/p) In (012 T) . (24) This indicates breakdown of perturbation theory and a bound state just at T = 0.

What has happened here is that the divergence a t TK has been removed by the ⁽⁽ negative feedback ⁾⁾ in the self-consistency loop [l31 : when K in (2) or (7) increases, so does r ' . When T' tends to become large, by (21) so does C,. But when Z, increases, because of the minus sign in front of the second term on the right of (20), this makes K smaller, thus reducing T'.

The divergence of T at T = 0 is weak enough so that there is no corresponding divergence in the resis- tance, which is obtained by putting (23) into (12) with A, = 6(&) :

where

( l J l p/2 N) In (012 T;) = - a - (318) d(Jpn12 N ) ~ . This is essentially Hamann's result [4].

V. Self-consistently clothed ^S, and clothed d. - Now we clothe the d-electron as well as the S. Starting with an expression analogous to figure 2 for the d- electron self-energy, C,, it may be shown that Im C, may be written directly in terms of the spectral density and self-energy for the S-electron :

X { 1 + tanh (42 T) tanh [(W - &)l2 T] ) . (26) This was evaluated using Im C,, from (21) with K from (23) and A, from (18). From this we could cons- truct A,(w) and found it to be roughly a Lorentzian of width W, = Th(T), where h(T) varies from -- 0.1 at T = TL down to 0 at T = 0. Hence, A, is so narrow that when inserted into K in (6) or R in (12) it is for all T nearly a 6-function relative to the other factors, meaning that the d-electron is so lightly clothed that it looks essentially bare. Hence, clothing the d-electron causes no essential change in the resistance result (25).

VI. Effect of giving the d-level a finite width. - It is known that the resistance formula (25) agrees well with experiment except in the limit T + 0. At T ⁼ 0, it comes up with negative infinite slope, whereas experiment shows negative finite slope, or, at higher concentrations, positive slope with a low temperature maximum. In an attempt to improve agreement with experiment in this region, we are investigating the effect of giving the d-electron energy level a small finite width coming, for example, from interactions between impurities.

The expressions developed in section I1 are valid for the case of finite d-width provided we interpret Ad(&) now not as a temperature-dependent spectral density coming from the S - d interaction in (1) but rather as a temperature-independent broadening function coming from external sources.

Consider now the self-consistent K-equation in figure 4. All d-propagators in this equation are now considered to have energy width W. In the limit T % W, Ad(&) will be a 6-function in comparison to the Fermi and Bose functions, so we will just get the results of section IV. For lower temperatures, we can get an idea of the effect by using the broadened A, in the first diagram on the right of figure 4, but trea- ting A, as a 8-function for two lines in the second diagram : for the d-line on the right edge of this diagram, and also where it occurs in the return loop of C, (see Fig. 2). This should be reasonable in the temperature region where the second diagram is small compared to the first (for (Jp/2 N) -- 0.1, an

REMOVING THE DIVERGENCE AT THE KONDO TEMPERATURE C 1 - ⁴³⁵

estimate based on (22), (23) shows this to be valid better salutjon is required in order to draw any definite for T down to about (0.1) TK). With this approxima- conclusion.

tion, it is easily shown that the self-consistent equation

for K (22), is still valid but with K, replaced by (6), _R with A, = p ( o ) and Ad(&) of width ^{W .} The result for

resistance is :

where UT

F = (T;)~/[T' _- + ( ~ / 2 ) ~ ] . _- W ^TK

FIG. 5. - Effect of giving the d-level a finite width. Curve a : The behavior of this function is sketched qualita- d-level width = 0. curve ^{b :} d - 1 ~ ~ ~ 1 width = T~ is the Kondo

tively in figure 5. In the region where it is expected temperature. Note that curve b is based on (27) which is not

to have some validity, i. e., for T > W, we see that valid for T < ^W.

the resistance curve is starting to become more hori-

zontal, as does the experimental curve. In the T < W The authors wish to acknowledge the help of Ulf Lar- region, R goes to zero, in disagreement with experi- sen, who suggested the idea of an externally broadened ment. However, the approximation of treating the d-level and participated in some of the calcuIations A, as a 8-function is not valid in this region, so a on it.

References

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[3] SILVERSTEIN (S.) and DUKE (C.), Phys. Rev., 1967, [9] CHEUNG (C.) and MATTUCK (R.), Phys. Rev., 1970,

161, 456 and 470. November.

[4] HAMANN (D.), Phys. Rev., 1967, 158, 570. [l01 NOZI~RES (P.), unpublished.

[S] KONDO (J.), c( Solid State Physics B, ed. F. Seitz [l11 TAKANO (F.) and OGAWA (T.), Puog. Theou. Phys., et al., Academic Press, 1969, 23, 184. 1966, 35, 343.

[6] THEUMANN (A.) and MATTUCK (R.), t o be published. [l21 NAGAOKA (Y.), Phys. Rev., 1965, 138, A 1112.

[7] DONIACH (S.), Phys. Rev., 1968, 173, 599. [l31 SUHL (H.), Phys. Rev. Left., 1967, 19, 442.