LOCALIZED SPIN FLUCTUATIONS IN METALS

JOURNAL DE PHYSIQUE

Colloque C I, supple'ment au no 2-3, Tome 32, Fe'urier-Mars 1971, page C 1 - 19

LOCALIZED SPIN FLUCTUATIONS IN METALS (*) J. R. SCHRIEFFER, W. E. EVENSON (**) and S. Q. WANG (***) Department of Physics University of Pennsylvania Philadelphia, Pennsylvania 19104

R6sum6. - On considkre les idCes principales et les rksultats de plusieurs theories des fluctuations de spin. L'approxi- mation des phases alkatoires decrit bien le problkme quand le couplage d'echange U est assez faible. Cependant, elle surestime la partie collective des fluctuations, et dans les alliages diluCs elle a une instabilitk non physique pour U Uc.

Pour U - Uc, les couplages non lintaires entre les fluctuations de spin suppriment l'instabilite. Ces effets anharmoniques ont kt6 trait& par les methodes de la moyenne fonctionnelle et de la sommation de diagrammes, et ils produisent une variation graduelle des proprietts du systkme quand on augmente Ujusqu'a Ue et au-dela. On considere aussi une approche compltmentaire a celles-la dans laquelle on etudie 1'Cvolution temporelle des fluctuations de spin.

Abstract.

The concepts and results of several theories of localized spin fluctuations are contrasted. The random phase approximation-qualitatively correct for weak exchange coupling U-overemphasizes the collective nature of the fluctuations and in dilute alloys leads to an unphysical instability for U > UC. For U - Ue, nonlinear interactions between spin fluctuations suppress the instability. The se anharmonic effects have been treated through the functional average and the diagram summation schemes, and they lead to a smooth varjation of the system properties as U increases through Uc.

A complementary treatment in which we study the time-evolution of spin fluctuations is also considered.

I. Introduction.

In the theoretical study of ferromagnetism in metals one has traditionally used two opposite limiting cases as points of departure : the Heisenberg model and the itinerant model [I].

The theory of magnetic impurities in metals has procee- ded similarly from two limits : the strong-coupling limit, i . e. the s-d exchange model, and the weak- coupling limit, i. e. the Hartree-Fock approximation.

We will consider here some recent progress on this problem, with a view to elucidating the essential phy- sical concepts that apply in the intermediate coupling region.

Considering the two limits themselves first, we recall that the s-d exchange model for antiferroma- gnetic coupling exhibits the Kondo quenching of the localized moment below a characteristic tempe- rature, T, [2, 31. In the Hartree-Fock approximation [4-81 for weak coupling, on the other hand, the impu- rity effectively adds another state to the band, the cr virtual level

which contributes an exchange- enhanced Pauli term to the susceptibility. If one conti- nues to use the Hartree-Fock approximation for stronger coupling, two degenerate manifolds of single-particle states corresponding t o spin-up and spin-down states of the impurity appear beyond a critical coupling strength. The heuristic approach of ave- raging over these two manifolds leads to a transition from enhanced Pauli t o Curie-like local susceptibility on passing through the critical coupling regime [9].

The Hartree-Fock approximation, being a mean field approach, neglects fluctuations of the local magnetization about its mean value. In the interme- diate coupling regime, a large number of low-lying fluctuating modes contribute substantially to the entropy of the system. These modes appear to domi- nate the dynamics of the system and must be consi-

(*)

Work supported in part ^by the National Science Foun- dation and the Advanced Research Projects Agency.

(**)

Present Address

Department of Physics, Brigham Young University, Provo, Utah 84601.

(*

*

Present Address

Department of Physics, University of Oregon, Eugene, Oregon

dered ab initio. These fluctuations were first treated in the random phase approximation (RPA) by Lederer and Mills [lo]. The local susceptibility, XzA, ^{is repre-}

sented by the diagrams of figure 1, where the solid

RPA

FIG. 1. - The spin fluctuation propagator, xloc, within the

random phase approximation. The solid lines represent zero- order Green's functions, g, for the virtual level, and the dotted

lines are the Coulomb interactions, U.

lines represent zero-order Green's functions, g, for the virtual level and the dotted lines are the Coulomb interactions, u.x:LA can be thought of as the spin fluctuation propagator and is represented as a wavy line. The RPA retains only quadratric terms in the expansion of the free energy density in powers of the fluctuation amplitudes. The energies of these harmonic RPA modes tend t o zero at the critical coupling strength given by the Hartree-Fock approxi- mation, so that the free energy of the system has an unphysical divergence there.

An early attempt to remove the RPA instability was carried out by Suhl and coworkers [ll-131. They determined x,,, by self-consistently dressing the virtual level propagator with spin fluctuations as shown in figure 2. Roughly speaking, as x,,, increases near the RPA instability it self-consistently increases the self- energy thereby decreasing G and saturating the insta- bility. While this approach is appealing, in the strong coupling limit it gives a Curie constant which is much smaller than that for spin 3. I n addition Hamann [14] has shown that in this theory TK varies exponen- tially as (U/T)'rather than the proper form U/T [15], where r is the virtual level width. Btal-Monod and

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1971105

C 1 - 2 0

R. SCHRIEFFER, W. E. EVENSON AND S.

WANG

-=o+(-J

loc

FIG.

in the renormalized random phase approximation.

Mills [16] have shown that in the intermediate-coupling regime vertex corrections such as shown in figure 3 are at least as important as the self-energy corrections of figure 2. They conclude that it is apparently impos- sible to obtain accurate results by performing a partial summation of the most divergent diagrams in each order, since the next to leading divergent diagrams in the next higher order give contributions compa- rable to the leading divergence in a given order. Thus, the ordinary diagrammatic techniques appear to be of limited value in this case.

FIG.

Some vertex corrections

Recent attempts to circumvent these difficulties have turned to a functional integral formulation of the problem [17-211. In this scheme the two-body interactions are exactly transformed away in favor of averaging over a random

magnetic

field acting on the virtual level. This technique has the advantage that large amplitude fluctuations and interactions between fluctuations can be handled in a natural way.

Certain approximations to the functional integral can be connected with subsets of diagrams, although the functional integral method appears to be able to sum wide classes of physically relevant diagrams that are intractable by conventional diagram methods.

11. Spin Fluctuations in the Functional Integral Method.

We will consider the Anderson model [5] of a magnetic impurity, writing the Hamiltonian as R

X, + ^{XI where}

and

The notation is as in [5] and [19] ; Ed, =

+ ^U/2.

The partition function can be written exactly as [I91

x exp[Tr log (1 - K ) ] , (3) where

is the local random

magnetic

field. Z, is the par- tition function far %,, and K is a matrix in frequency (v) and spin ( a ) spaces whose matrix elements are

Here c = ( 2 npU)l/' and gz is the zero-order virtual level Green's function, which is, for a Lorentzian level of width r

zN(0) ( V 12,

g,"

1 , ^{w, = (2 n} + ^{1) z} . ( 6 )

iw, -

+ i p r sgn on

To determine 2, we must now evaluate both Tr log (1

K ) and the integrals in (3). Various approximation schemes depend on the size of the coupling strength U / n r .

1. THE LOCAL SUSCEPTIBILITY.

It can easily be shown that the local dynamic susceptibility has the form.

which agrees with the expression for the static suscep- tibility given in [19], if v = 0.

2. THE RANDOM PHASE APPROXIMATION. - For the weak coupling case, U / d ' < 1, it is natural to retain only the K 2 term in Tr log (1

K ) (Eq. 3). Thus Tr log ( 1 - K)

, 1 ^{Tr K'}

^{cZ 1 q v / t y IZ , (8)} where the

polarization bubble >> q, is defined by

This approximation to Tr log is illustrated in figure 4a,

where the dashed lines represent the r: s and the bubble

is qv. Keiter has derived a general expression for

q v [22], which, in a magnetic field B, I SZ, I 4 P I '

and the symmetric case, reduces to [18].

LOCALIZED SPIN FLUCTUATIONS IN METALS C 1 - 2 1

FIG. 4.

(a) Diagram for Tr K2.

(b) Sixth-order diagram for free energy in RPA.

On performing the <,-integrals in (3) one obtains

-

z = exp[- pFRpA] ^{= exp} - log (1 - c2 qv/n> .

20 [ ^{= o} I

The free energy is given as a sum of contributions from independent modes. In conventional diagram language log (1 - c2 qv/z) is the bubble sum for the interaction H I , shown in figure 4b.

Using (8) one finds a local susceptibility of the form

which has the same diagrammatic form as figure 1.

For I Qv j 4 p r y

As 2 U / n r approaches unity, XLtA(~) diverges, signa- ling the breakdown of this approximation. To remove this unphysical divergence, one must include inter- action between the fluctuation modes [23]. In fact, to obtain even the proper term to order U / T in

~ , ~ , ( v ) , one must consider first-order interactions between the modes, since the expression for x,,,(v) contains a factor 1/U, and one has to retain terms up to Un+' in the density matrix to obtain ~ , , ( v ) correct to order Un.

The spectral density of the spin fluctuations is given by the imaginary part of the continuation of

x,,, as a function of iSZv to the real SZ-axis. In the critical regime, 2 U / n r - 1, one has, using (13),

Thus, the characteristic frequency of spin fluctuations is reduced from /3T by a factor of (1 - ^{2 Ulnr),}

which is the analog of critical slowing down near continuous phase transitions. As we will see mode- mode interactions tend to reduce the slowing down effect.

3. THE QUARTIC APPROXIMATION. - The lowest approximation for mode-mode coupIing is given by retaining the K4 term in Tr log (I - K ) :

since Tr K 3 = 0 for B = 0. Tr K4 is represented in figure 5.

I

va

FIG.

Diagram for Tr

Because of the random phases of the r: ^{s, one}

might expect the terms of the form 1 tV, l2 I c,, ^j2

and ( tV l4 to dominate the terms in which the four frequencies are unpaired. As we will discuss below, this may not be the case in the interesting regime 2 U / d 2 1. If, however, we keep only these paired- frequency terms, we have

Tr log (1 - K) c2 qv 1 5, 1'

v

The coefficients A,,, and Bv are discussed in the appen- dix. They vary with frequency on the scale of

To gain insight we first consider the simple case, A,,, = A for I SZ, I and ( Q , ( < pr ^{and A,, = 0}

otherwise. Also B, = A/4 for I BV I < p r and Bv

0 otherwise. Because of the large number (pT/n 9 1) of modes that contribute to 1 A,, 1 5, 12,

~ ' 3 0

the fluctuations of the individual I t,. I2 terms will largely cancel in C. Therefore, we replace

v'

including the case v

v'. Thus Tr log (1 - K) e

where

C 1 - 22 J. R. SCHRIEFFER, W. E. EVENSON AND S. Q. WANG and we have neglected < 15,12 > compared to

c ^{0, < I tVl2 > .}

Using (18), the self-consistency condition deter- mining < 1 5, ]' > ^is

where we have used the approximation (10) for q v and have taken 2 U / n r 1. Replacing the last sum by an integral, we obtain

For small frequencies A,,. has the limiting value 1/71 (ljf33, which we take for A. Then the prefactor of the log in (22) is (2 U/nS)'/8 -- 118. Since D is approximately 0.22 as (1

2 U/nT) passes through zero, a comparison of Eq. (12) (RPA) and (20) (Quar- tic) shows that the RPA divergence has been circum- vented and the critical slowing down has been strongly reduced.

So far we have treated all modes on an equal footing.

In the critical regime the v

0 fluctuation has the largest amplitude, so it seems proper to treat this particular mode more carefully than the others. To do this we define, as in [19], K = KO + K,, where

KUU'

~ n n ' =

-

^g,b arm,

- (23) If we let K'

K,(1 - KO)-', then

Tr log (1 - K)

Tr log (1 - KO) + Tr log (1 - K') .

(24) Tr log(1

KO) is precisely the static approximation for Tr log (1

K), which has been evaluated previously [18, 19, 211. We then consider Tr log(] - K') in the quartic approximation of Eq. (161, but now the various coefficients will be denoted by q:, A:,, B:, and they depend on 5,. These coefficients are discussed in the appendix. Now we repeat the self-consistent calcu- lation of (17). There will, of course, be a self-consistent set < 1 5, l 2 > ,+o for every value of 5,. However, because of the computational complexities and the expectation that the lo-dependence of Tr log (1 - K') is relatively weak compared with that of Tr log (1 - KO), we evaluate < I tv l2 > only for to

0 and use this set throughout the calculation.

The detailed numerical analysis has been carried out including the frequency dependences of A:,. and B:,. However, the quartic terms Bi 1 5, 14, were treated exactly rather than by (17). The resulting Xz,(~) is plotted in figure 6 for several values of 2 U / n r and of p r . In figure 7 we show Xz,(~) as a function of B r f o r several values of 2 U/nT with the corresponding curves from RPA and the static approximation [IS,

191 shown for comparison.

, ---

-- - R P A

11

FIG. 6 . - The dynamic local susceptibility, xloc(v), in units of &/r, calculated in quartic approximation and in RPA. The

curves are labeled with B r ^values.

Qvmr116 Appmrlmotlon

15 Slotlc Appr0xlmot~on

FIG. 7. -The static local susceptibility, xloc(0), in units of as a function of PI'. Curves for quartic approximation,

static approximation, and RPA are compared.

It is clear from figures 6 and 7 that the RPA insta- bility has been suppressed ; however (22) overesti- mates this suppression compared to the more detailed quartic calculation. The strong decrease in xzC(v) for small v which is evidenced in figure 6 is consistent with our separate treatment of v = 0 terms in the detailed calculation.

To gain insight into the additional diagrams summed in the quartic approximation we note that if the quartic terms, Eq. (16), are expanded down from the exponent of exp Tr log (1 - K), the resultant &,-integrals are of the form

Thus, each factor of 2 x I t, 1' contributes a factor (1 - c2pv/n)-l, the n ! entering as a counting factor.

Since 2 n < I tv l2 > S 1 for 2 U / n r 1 and v 5 p r ,

we see from (7) that each factor of I tVl2 can be iden-

tified with a free spin fluctuation propagator XF:,"(~).

LOCALIZED SPIN FLUCTUATIONS IN METALS (21-23 The general quartic interaction corresponds to a

scattering of a pair of spin fluctuations as indicated in figure 8a, while the paired-frequency terms lead

v v' u

v'tm vtrn v

FIG. 8. - (a) Diagram for the general quartic interaction.

(b) Some low-order diagrams with the general quartic inter- action.

to

elastic

scattering only, i. e. m = 0 (or v = v').

The general diagram in the quartic approximation is obtained by hooking together these 4-vertices in any possible way. A few low-order diagrams are shown in figure 8b. Diagrams of this form have recently been investigated in another formalism by BCal- Monod, Hurault and Maki 1241. As in our present discussion while all quartic interactions are formally included in the theory, they explicitly treat only paired- frequency vertices. In a treatment similar to the sim- plified example above, they approximate the 4-vertices by a constant with a frequency cutoff and evaluate the functional integral using essentially the mean field factorization of (17), with results corresponding to (22).

There is no a priori reason to believe that the inelastic vertices (m

0) of figure 8 are significantly less important than the m = 0 vertices if one is near the instability. In fact, the arguments of BCal-Monod and Mills 1161 suggest that a proper treatment of the intermediate coupling regime requires the inclusion of all higher order vertices as well. At first sight this appears impossible to achieve. However, if one argues that the frequency dependence in the problem is dominated by the spin fluctuation propagator, i. e.

by the K' terms in Tr log(1 - K), and that the fre quency dependences of the n-point vertices can be adequately modeled by their low-frequency values with cutoffs at frequency p r , the problem can be conveniently cast in the time language. While this assumption is roughly correct for the Kondo regime 2 U/nP % 1, it may well fail in the intermediate cou-

pling regime since the critical slowing down is sharply reduced by the mode-mode coupling so that the cha- racteristic frequency for 2 U/nT

1 may still be of order Dr. Thus, a detailed understanding of the intermediate coupling regime may require inclusion of the specific frequency dependence of the n-point vertices.

4. THE TIME-DOMAIN APPROACH. - If we accept the above point of view, namely that for 2 U / n r p 1 the dynamics of the problem are dominated by the strong frequency dependence of the K2 terms, repla- cing the n-point vertices for n > 2 by their zero- frequency limits, then the free energy functional is given by

where F, is the free energy in the static approximation [IS, 19, 211 with its t2 terms (and 2,) removed, and FRpA is given in (1 1).

This approximation corresponds physically to the idea that the anharmonic effects relax sufficiently rapidly that they can be treated as in local thermo- dynamic equilibrium, while the harmonic effects are nonlocal in time. Since the time for an electron to decay off the localized state is of order I/PF, it is clear that anharmonic effects decrease rapidly for frequencies greater than PI'. Thus, approximation (26) is appropriate only for functions t(z) smoothed on a time scale of order I/pr. This point of view is very similar to that taken by Anderson and Yuval 125, 261 and by Hamann 120, 27, 281. The main diffe- rence between (26) and their approach is that by using the Nozieres-DeDominicis solution of the x-ray pro- blem 1291 they implicitly include some time dependence in the anharmonic terms but FRpA is treated correctly only for small v.

In order to smooth the functions C(T) on the time- scale l//lr, we divide the time interval [0, 11 into N ⁼ l/q domains each of width q 1 / P r and take 5(r3 at the center of the 2-th interval, say, as repre- sentative of <(z) throughout that interval. We express /3FRPA [t(r)] in the form

=

1: ^dr 1 dzf <(r) 5k') 9 ( r - 7') , (27)

0

where

and

as derived in the appendix. Now the partition function is

C 1 - 2 4 J. R. SCHRIEFFER, W. E. EVENSON AND S. Q. WANG We can find the susceptibility x,,,(z) from

x exp ( ^{-q Z VFI(TZ) - n ? ' I: &}

~ ~ - v ]

I 11'

In order to actually calculate the susceptibility from (31) we must perform the N - pr-dimensional integral numerically. Preliminary results are quali- tatively similar to the quartic approximation discussed above with the expected suppression of the RPA insta- bility. More detailed work is in progress on the eva- luation of (31) over the whole range of coupling strengths, and the results will be published elsewhere.

In the limit of strong coupling Anderson and Yuval and Hamann [20, 25-28] have argued that the domi- nant paths {(z) corresponds to correlated hopping between the free energy minima corresponding to spin-up and spin-down states, which leads to the Kondo quenching of the magnetic moment at low temperatures.

111. Conclusion.

We have seen that coupling between the fluctuation modes drastically alters the

The detailed evaluations of these sums are compli- cated, and the results are very long. The explicit expressions are given in the appendix of the Ph. D.

thesis of S. Q. Wang 1301.

The coefficients A:,, and B: are the same as A,,, and Bv with g replaced by g [I91 (Cdu

5,, - ocg,).

2. THE APPROX~MATION

( ~ ( t ) . - q(t) is defined in (29) as the Fourier transform of pv. If we use the definition (9) for q,, then the expression for q(t) becomes

cp(t) = - - 1 g: g;+, e-'*~"

2

In the symmetric case, Ed, = 0, we need only consi- der 0 < t < 112, since

local susceptibility of the random approxi-

mation by eliminating the unphysical divergence of

= F: ^{icon +} i p r s g n mn ⁼ - gu(- t) , (A4) the static susceptibility and by greatly reducing the

slowing down of the fluctuations.

While this effect is already obtained within the quartic approximation, it is likely that a quantitative treatment requires inclusion of high order interactions between the modes. If the characteristic fluctuation frequencies in the intermediate coupling regime were small, one could treat the problem in the time domain as has been done for the strong-coupling (Kondo) regime 120, 25-28]. Judging from the quartic approxi- mation, the characteristic fluctuation frequency may remain of order PI' for intermediate coupling. By comparing the results of the frequency and time domain approaches, one can expect to obtain insight into the validity of these approximation schemes in this regime.

IV. Appendix.

1. THE COEFFICIENTS IN THE

QUARTIC APPROXIMATION. - The coefficients Avy, and Bv of the quartic approximation, Eq. (16), are sums of Green' s functions from Tr K4 :

and for 0 < t < 1, gu(l

t ) = ga(t). Using the Pois- son summation formula,

where f(o) is the Fermi function, and a' refers to sgn Im o. So gu(O+) = - +. When t 9 l/pr, the only significant contributions to the integral (A5) occur for o < p r , so

gU(t)

- e-u't ,f(m) d o = -

¹

p r sin nt - ^(A61

We can combine this result with that for gu(Of) and the symmetries of gu(t) in a simple approximation for gu(t),

1 < ^t < ^{1 :}

-

sgn t g7(t) "

J p 2 r

sln

nt + 4 (A71 The approximation (29) for cp(t) then follows imme- diately if we insert this expression into (A3).

References

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N. Y., 1966. WILLIAMS (H. J.), COREZWIT (E.) and SHERWOOD

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Letters, 1970, 24, 225.

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[21] KEITER (H.), Phys. Rev. B, 1970, to be published.

[22] In Keiter [21] Eq. A6 put

0 and

+

(i

to obtain an exsression valid for general values

of PI- and

1231 If one writes the interaction as

(U/4) (ndt - ndL)'

+ (U/4) (ndt t ndl)' and replaces

by to leave H invarient, as Miihlschlegel [I71 and Hamann [20] have done, one can linearize the interaction with two variables x and y. By retai- ning quadratic terms in x and y one obtains the conventional RPA with a divergence at 2 U/nT

2. The precise location of the singu- larity is not of great importance since in either scheme the divergence is eliminated by the mode- mode interactions.

[24] BEAL-MONOD (M. T.), HURAULT (J. P.) and MAKI (K.), to be published.

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Letters, 1969, 23, 89.

1261 YUVAL (G.) and ANDERSON (Pi W.), Phys. Rev. B, 1970, 1, 1522.

1271 HAMANN (D. R.), Phys. Rev. B, 1970 to be published.

[28] ANDERSON (P. W.), YUVAL (G.) and HAMANN (D. R.), Phys. Rev. B, 1970, 1, 4464.

[29] NOZI~RES (P.) and

DOMINICIS (C. T.), Phys. Rev., 1969, 178, 1097.

1301 WANG (S. Q.), Unpublished Ph. D. thesis, University

of Pennsylvania, 1970.