PHYSICAL REVIE% B VOLUME 23, NUMBER 11 1 JUNE 1981

Kondo lattice: Renormalization study using a new pseudofermion representation'

Vitor Rocha VieiraThe Department of Physics and The James Frarick 1nstitute, The University of Chicago, Chicago, 1]lirIois 60637

(Received 18 December 1979; revised manuscript received 20 November 1980)

A new field-theory method for spin2

is developed using the Berezin and Marinov path-

integral formalism. This method does not have the usual troubles of Abrikosov's pseudofer-mion representation. In particular, the linked graph theorem is valid. This method is then ap-

plied to the study of the Kondo lattice. In the disordered state the effective coupling satisfiesthe usual renormalization-group equation for the Kondo problem. In the ferromagnetic state it

displays the competition between Kondo effect and magnetic ordering, Using the first-orderrenormalization-group equation, Doniach's criterion based on comparison of the binding ener-gies of the Kondo singlet and the Ruderman-Kittel-Yosida ferromagnetic state is recovered. In

second order the feedback from the Kondo effect on the magnetization enhances the Kondo ef-fect and suppresses magnetic order.

INTRODUCTION

The Kondo lattice is thc generalization of the Kon-do problem to the case in which there is a spin ateach lattice site interacting with the conduction elec-trons. Its more interesting feature is probably thecompetition between magnetic ordering and Kondoeffect. Its study is relevant for mixed-valent systemsin certain limiting conditions. Recently, Jullien,Fields, and Doniach' have considered a (one-dimensional) 1D analog of the Kondo lattice, "theKondo necklace. "

In thc field-theoretic treatment of the Kondo prob-lem, one usually uses Abrikosov's' pseudofermionrepresentation for spins. However, the application ofthis method is limited by the fact that the linkedgraph theorem is not valid, as was shown by Keiter. 3

%e have developed an alternative method usingthe Berezin and Marinov" path integral formalism.%e ~rote down thc spin propagator for this methodand derived the perturbation-theory rules for the in-teraction with an external field and for the Kondo lat-tice.

Although we will consider later only the zero-temperature Kondo lattice, we have presented thefinite-temperature formalism because it is easier toderive and for possible future generalization to finitetemperature.

This method is then applied to the Kondo lattice.Probably in application to real systems thc more in-teresting cases are spatially inhomogeneous solutionsfor the spontaneous magnetizations. Ho~ever, inthis study of the competition between magnetic or-dering and Kondo effect, we have considered the fer-

rornagnetic state, for simplicity. %e have derived thezero-temperature renormalization-group equation forthe effective coupling in the disordered and in theferromagnetic states and discussed its solutions.

In Sec. II, we derived the finite-temperature fieldtheory for spin —,, in Sec. III, we obtained the mean-

field theory for the Kondo lattice, and in Secs. IVand V, we derived the zero-temperaturerenormalization-group equations for the disorderedand ferromagnetic states, respectively. Finally in Sec.VI, we listed the conclusions.

II. FIELD THEORY FOR SPIN2

Usually, in the field-theoretic treatments of theKondo problem, onc uses the Abrikosov's pseudofcr-mion representation2 for the spin operators. Howev-er, as we will discuss later, this method has some in-convenicnces. In particular, as shown by Keiter, ' thelinked graph theorem is invalidated, restricting theapplication of' this method.

In this section, we develop an alternative treat-ment, which does not have these problems, usingBerezin and Marinov's path integral formalism. 4 Thespin operators are r'eprescntcd by

S„(t)-—'iel f)(t)] (t—) (r im =1, 2, 3) (2.1)

in which 1;(t) are Grassmann number fields. Asshown by Berezin and Marinov, 4 a path integral ex-pression for the partition function can bc.written, di-viding the interval [0, P] in W subintervals, as a W-

fold Grassmann number integralI

N N N

Z=Tr(e 'st '+)~= lim 3~tz Jl exp X t' (i)(—1)' 'sgn(j —/)1;(j) ——X&(g(/)) Q d'f(k)23N/2

i,j 1 N; 1 k 1

(2.2)

6043

6044 VITOR ROCHA VIEIRA 23

(2.3)

in which the action S =Sp+S~, has. a free term givenby

Sp —i t dE

dg(r)dE

(2.4)

—M IM OJ (2.5)

in which d'f=d(3dgqd(( and N is even.The factor sgn(j —i) implies antiperiodic boundary

conditions', i.e., f(i+N) -—((i). Therefore, theFourier transform of ((r) is defined on the FermiMatsubara frequencies «(=(2p+1)4r/P, p =0, +1,+2, . . . .

In the continuum limit, we arrive at the followingexpression for the partition function:

Z-J e-s5))

having assumed in the last expression that A is alongthe Z direction.

For a free spin, the propagator takes the simpleform

g(«(«() = I/i «( (2.14)

From Eqs. (2.11) and (2.14) we see that the propa-gator with and without a magnetic field satisfyDyson's equation with a self-energy given by

'S(m I fr/m Hq (2.15)

As usual in field theory we will single out the qua-dratic part of the action and use it to develop a per-turbation theory. The different correlation functionsare best calculated adding a source term. to the action,and taking functional derivatives with respect tothose sources. The fundamental identity to be used1s

and an interaction term given by

IPSi- J X(((r)) dr

and

(2.6)Jtexp(-, )rA g+frrt)dg= JdetA exp( —,pre (q)

(2.16)

S]- lim, i, Q d'$ (k)~ 2 k

(2.7)

Using Eq. (2.1) the Hamiltonian for the interactionwith a magnetic field is represented by

X--A S--H„(—i /2). „,.~, ( )r~. ( r)

giving the following contribution to the action:

S(r = ——,' X(((—«()(—iH, e„( ) f,„(«()

(2.8)

(2.9)

From Eqs. , (2.5) and (2.9) we conclude immediate-

ly that the spin propagator

g,.( ) -- (g(( )(.(- ))must have its inverse given by

g(m («() = ( «(g(m (Hr er(m

(2.10)

(2.11)

o)2 Ho) 01 1 2

2 2—HOJ OJ 0

I 40 oJ + H0 0

(2.13)

representing the precession of a spin in a magneticfield. We could have guessed physically this expres-sion. Proceeding backwards, one would then write

Eq. (2.5), fixing the expression for S((. The advan-

tage of the Berezin and Marinov's derivation is that itleads automatically to the antiperiodic boundary con-ditions.

Inverting the matrix in Eq. (2.11) we arrive finallyat

g(m(«() =, , (8(m«(~+a(mH«(+H(Hm)(2. 12)1 1

lN cv +H

in which the matrix A is antisymmetric, and f and riare multicomponent Grassmann number fields, asdiscussed in the Appendix. From this expression,defining the expectation value of an operator as

Jl exp( —,' )rrt ()( ) d]

~ ~ ~

Jl exp(T~(rrl ]) df(2.17)

it is easy to verify that

&z, , r, , ) =&,—,,', ,(2.18)

(Cs 4 )(4&rC&43) cps( t4(3 ~r3((+(4( p +(4((+(3(

and so on. The structure of these equations showsthat Wick's theorem is valid. The linked graphtheorem is also valid. These expressions have somefeatures in common with fermion field theories andothers with boson field theories. They display the oc-currence of alternating signs, according to the parityof the permutation, as in a fermion theory and, be-cause the fields are real, it is possible to contract anypair of fields, as in a boson field theory. Whenevaluating the average of 2n fields, there are(2n —1)!!possible terms as in a real boson theory,contrary to the n! terms, that we can get when weaverage n + n complex Fermi fields. Terms differingonly in a permutation of two fields have a differentpermutation sign. From Eq. (2.1) it follows that thefields belonging to the same vertex are multiplied by

2 6„( which is also antisymmetric under the permuta-

tion of I, m. We use a single diagram for these two

terms, and drop the factor 2. In the "spin side" ofthe vertex we write then only e,~ . To keep track ofall the minus signs we adopt the following two con-

KONDQ LATTICE: RENORMALIZATION STUDY USING A NE%. . .

FIG. 1, Averaging over the fields all contractions are pos-sible. Several terms of the average are associated with thesame diagram. In first order, the diagrams represent 1 and2 terms of that average, giving a total of (4-1)!!=3 terms.

ventions already used in Eqs. (2.5) and (2.10):(i) The propagator G~ (ru) is defined as —(f~(co)&& (,„(-co) ) with the fields in this order and (ii) in theHamiltonian the fields are in the opposite order. Asa consequence, in a loop, the contraction made upfrom the first and last fields have them in the wrongorder, giving rise to a minus sign. Loops contributethen with a minus sign as it is usual in fermiontheories. %e represent the propagator by a dottedline with the arrow running from the second argu-ment to the first. From its definition, it is antisym-metric under the combined interchange of l, m andchange of sign of co.

As an example, in Figs. 1 and 2, we show, respec-tively, the first- and second-order diagrams (both con-nected and disconnected) for the propagator, togetherwith the number of different terms of the averagesrepresented by them.

In Fig. 2, we considered separately the last two setsof four terms. They are both necessary to obtain aproperly antisymmetrized propagator. However, in a

loop they give the same result arid could be con-sidered together. In a loop the direction of the arrowis immaterial because, changing it, all the frequenciesin that loop change sign, but the order of the multi-

plication of the ~,&

's is also reversed, compensatingany change in sign.

~III%( )

/

v

For a spin in the presence of a magnetic field wedefine the "magnetic field side" of the vertex as 0„.This rule, 'together with the 6 ~ rule for the spin sideof the vertex, gives directly Eq. (2.15) for the self-energy.

%'e derive now some results that will be usefullater. Using the propagator for A 0 0, as shown in

Fig. 3(a), the magnetization in the presence of amagnetic field is

~r 'srlm X Bml()PH H,

P „~ 2 H(2.19)

as expected. Using the propagator for A-O, asshown in Fig. 3(b) we evaluate the free spin suscepti-bility

(2.20)

obtaining the Curie law as usual. The time-dependent (r it) correlation functions in the pres-ence of a field A also agree with what one would ob-tain using elementary quantum mechanics.

%e will consider now the Kondo lattice, in whichthere is a spin at each lattice site, interacting with theconduction electrons, The Hamiltonian is

(2.21)

and it has the same structure as the Hamiltonian forthe spin in a magnetic field. %'e can therefore direct-

ly apply all the results that we have derived aboutaverages over spins.

Because the Kondo lattice is translationally invari-

ant, it is convenient to draw all diagrams in momen-tum space.

The perturbation theory rules for the Kondo latticeare as follows: (a) For the spin (dotted lines) use thek-independent propagator of Eqs. (2.12) and (2.13).For the electrons (solid lines) use

G.s' (k, o)) = [iso —(s-„—p, )]g.s+-,' h cr.s (2.22)

(a, p=I, 2) .

For simplicity we will measure the electron energiesfrom the Fermi surface and ~rite simply e instead ofs —p, . (b) Frequency and momentum are conservedat each vertex. (c) For both spin and electron loops

/

V

4ggfs ~Pl

FIG. 2, In second order, the diagrams represent 1, 2, 2,2, 4, and 4 terms of the average over the fields, giving a to-tal of (6 —1)!!= 15 terms. FIG. 3. {a) Magnetization loop. {b) Susceptibility bubble.

VITOR ROCHA VIEIRA

introduce a factor —1. (d) The "spin side" of thevertex is e,i, which should be looked at as the rcomponent of a vector matrix with indices I, m. The"electron side" of the vertex is o'&. One shouldmultiply these matrices in the order of their positionon the spin or electron lines. For a loop this impliesa trace. A diagram of order n is multiplied by (iJ)"(e) Each diagram must be divided by its symmetryfactor. (f) We sum over the independent frequenciesand momenta using (.1/P) X„and(1/&) X-„

jtd k/ (2e)~, respectively.

For simplicity we shall assume a constant densityof states po and a flat band for —D & ~ & D.

Because the linked graph theorem is valid, we only

have to consider linked diagrams.Most of the times the symmetry factor is 1. Usual-

ly the factor 1/n! coming from the exponential can-

. cels against permutations of the vertices and the fac-tor 1/2" coming from Eq. (2.1) is canceled becauseeach diagram represents different equivalent terms ofthe average with respect to the spins. The exceptionsare diagrams with spin loops in which a factor —is

left and the free-energy diagrams. Changing thedirection of the arro~ on a spin loop does not changethe value of the diagram, as discussed. If this opera-tion produces a new diagram (for diagrams with spe-cial symmetry, like the zeroth-order spin-suscep-tibility bubble it does not), they can be added up,eliminating the factor —.

%e have stated the finite-temperature perturba-tion-theory rules. The zero-temperature rules follow

directly from these making the usual rotation from

imaginary to real frequencies and using the usual i 8prescription. The spin propagator has a pole at co 0and the contour on the complex plane passes throughit, giving a contribution of one-half the residue atthat pole.

Looking at Eq. (2.1), we can compare this field-

theory formalism with the pseudofermion method ofA,brikosov' ~here the spin operators are representedby

(2.23)

in which S & are spin S matrices. The introduction ofcomplex fields with creation and destruction opera-tors for each magnetic quantum number introducesfictitious states, which have to be frozen out. Thisis done providing the pseudofermions with an energy~. The vacuum state is annihilated by the spinoperators, and therefore the fictitious states arefrozen out using the "normalization" e~"/(2S + 1) inthe limit A, ~. However, as pointed out by Keiter, 3

this procedure invalidates the linked graph theorem.In the method presented here, instead of complex

fields we use real fermion fields and no fictitiousstates are introduced. Therefore, we do not have tointroduce any energy A. and no X ~~ limit is taken.The symmetry factors are computed differently andthe matrices S'& are replaced by e,~ . In this form,this method is valid only for spin —,. The linked

graph theorem is valid.As an example, we calculate the first nonzero con-

tribution to the spin self-energy in the absence of anyfield, i.e., the second-order diagram sho~n in Fig. 4.

1 1 1 1 1g(m («&) = (&J) &r(p&spmrresrrso

V P I CO) 0) ICtl2 a2 I (QI2 Ol( + CO)ki' k2 ~1&2

2 I Psi Pa2-J' X 1 —tanh tanhf/2 2 2 le (0] a2)

k(, k2

(2.24)

g'"(co) = —4i(Jpo)'aolnD/(s)( (2.25)

FIG. 4. Second-order diagram for the spin self-energy.

Symmetrizing this expression in k&, k2, we verify

that this is an odd function of eo, which taking intoaccount the symmetry of 51, agrees with the general

property of antisymmetry of the propagator. Thelinearly divergent term vanishes. In Abrikosov'smethod (X & 0) it does not vanish and this implies arenormahzatkon of A, .

With logarithmic accuracy Eq. (2.24) reduces to

I

displaying the occurrence of logarithms, a characteris-tic feature of these problems.

III. MEAN-FIELD THEORY

In this section we develop the mean-field theoryfor the ferromagnetic state of the Kondo lattice. 'For convenience, we add and subtract to the Hamil-tonian for this problem a mean-field Hamiltonianrepresenting the interaction of the spins and the con-duction electrons with internai magnetic fields A andh, respectively. These fields describe the effect ofthe electron and spin magnetizations on the spins andelectrons, respectively. They have to be evaluated ina self-consistent manner and should not be confused

23 KONDO LATTICE: RENORMALIZATION STUDY USING A NEW. . . 6047

V

FIG. 5. Hartree-Fock approximation is defined by the re-

quirement that the expectation value of H& —Hmf vanishes.FIG. 6. String of bubbles for the spin susceptibility in the

Hartree-Fock approximation.

with any external applied fields. We have then

X=X0+Hg=(XO+H f)+(XQ X f) (3.1)

small compared to the bandwidth (T && D):1

m = —hp() (3.5)

H -21J Im (3.2)

(3.3)

in which m and M are the electron and spin magneti-zations, respectively, in agreement with what we havesaid.

In mean-field theory, the transition occurs in-

dependently of the sign of J. For J & 0, spins andelectrons tend to align in the same direction (fer-romagnetic coupling) and for J & 0, they tend toalign in opposite directions (antiferromagnetic cou-pling) showing a tendency towards a "locally-spin-compensated" state.

Both the spins and the electrons feel as a magneticfield the magnetization produced by the others. Thespin magnetization, given by Eq. (2.19), is

M -—, tanhpH/2 (3.4)

and the electron magnetization is, for temperatures

in which BCp is the Hamiltonian for free electrons and

Xa is the Kondo Hamiltonian of Eq. (2.21).The perturbation theory is developed using

3.'~ —X f as the perturbation. The Hartree-Fock ap-proximation is defined by the requirement that its ex-pectation value should vanish. This is shown di-

agrammatically in Fig. 5. We have then

with pp the density of states for one spin direction,which we assumed to be constant.

Iterating Eqs. (3.2)—(3.5) we arrive at the self-consistent equation

H = J'po tanhpH/2

having a nonzero solution only if

T~ —Jpp .1

2

(3.6)

(3.7)

(3.9)

sho~ing that the spins are completely polarized, andthat h and H are, respectively, first and second orderin J.

Although this mean-field theory makes sense, ifwe try to include second-order diagrams, the Kondoeffect begins to come in. For example, evaluatingthe off-diagonal components of the second-order spinself-energy, shown in Fig. 4, we find for zero exter-nal frequency

The transition point can be directly obtained by ex-amining where the normal-state spin susceptibility

x=x'(I- —,'J'pop) ' (3.8)

given by the string of bubbles of Fig. 6 diverges.When T-O, Eqs. (3.2)—(3.5) reduce to

S)p (0)-——(Jpp)' ) da~daq I —tanh —a~+ —tanh—P h P h 2

4 ~j

2 2 2 2 a~—a2+h

t

p h p h pH 2(a( —ag)+ tanh —e1+——tanh —e2 —— tanh(a( —a2)' —H'

P h P h 2H—1 —tanh —e1 ——tanh —e2 ——2 2 2 2 (a) —a))' —H', (3.10)

For very small fields, with logarithmic accuracy,this leads to

I

competition between the magnetic ordering and theKondo effect.

g~'q" (0) - i2(Jpo)'(H +h) lnPD/2 (3.11)

S~V (0) =i 2(Jpo) H lnH/D (3.12)

In Secs. IV and V we will see how to handle the

and for T = 0, it gives, also with logarithmic accuracyIV. RENORMALIZATION-GROUP EQUATION

IN THE DISORDERED STATE

In this section we derive the zero-temperaturerenormalization-group equation satisfied by the effec-

6048 VITOR ROCHA VIEIRA 23

0(&-M} 0

FIG. 7. In the renormalization-group approach, electronstates are eliminated at the ends of the band.

FIG, 9. Vertex and spin self-energy diagrams included insecond order of the renormalization group.

tive coupling, '9 in the absence of any ordering.Physically, we follow Anderson and eliminate elec-

tron states at the ends of the electron band, as shownschematically in Fig. 7, and we see what is the effectof this procedure on the effective coupling. As weintegrate out more electron states we follow its evolu-tion, and try to understand its behavior.

It happens that the most divergent diagrams for theKondo lattice are the same diagrams for the Kondoproblem. The fact that the Kondo lattice is transla-tionally invariant plays no special role. The essentialdifference between the two problems is that we canhave a spin magnetization producing a competitionbetween ordering and Kondo effect, as we will see inSec. V. In this section we consider the disorderedstate.

In the lowest order in J, only the diagrams shownin Fig. 8 are'considered. These diagrams can be ob-tained one from the other changing the direction ofthe spin arrow, as discussed before. When the exter-nal frequencies go to zero, their contribution to the

effective coupling is J'pp de e for T =0, with theelectron energy restricted to the "energy shell" in-tegrated out, i.e., with D(1 —gi) & ~e~ (D. If wehad decided to integrate over the whole electronband, we would have obtained a divergent result.

In lowest order in J, the renormalization-groupequation for the effective coupling is then

dJeffp2(Jeff )2 (4.1)

dl

This equation coincides with the usual renormali-zation-group equation for the Kondo problem, ' be-cause the spin propagator is k independent.

As shown by several authors, ~ if we start with asmall negative value, the effective coupling slowly ap-proaches zero. If, instead, we start with a small posi-tive value, the effective coupling will grow.

We can go beyond this approximation, and obtainthe renormalization-group equation for the effectivecoupling in second order in J. Following Fowler and

d -1+—1 dSI dec)

(4.2)

we obtain from the self-energy diagram of Fig. 9, us-ing Eq. (2.25)

d =1 —4(Jpp)2I (4.3)

From the vertex diagrams of Figs. 8 and 9, it fol-lows that the vertex is corrected by

I'=1+2(Jpo)i+2(Jpp)'I . (4.4)

From Eqs. (4.3) and (4.4), we arrive at the follow-ing renormalization-group equation for the effectivecoupling J'"=J dI'

y (Jcffp )=2(J"'p )'tl —(J'"'po) 1 .

dl(4.5)

It is interesting to note that, although the coeffi-cients of the second-order terms in Eqs. (4.3) and(4.4) do not coincide' with those obtained for S = —, ,

Zawadowski or Abrikosov and Migdal, ' this can bedone including the vertex and the spin self-energy di-agrams shown in Fig. 9. In these diagrams, as in theprevious ones, the existence of the lattice plays againno role, because of the k independence of the spinpropagator.

There is one point worth noting: when the fre-quency transfer Leo from the spin to the electronlines vanishes, the spin lines in the vertex diagram ofFig. 9 carry the same frequency, giving rise to the ap-pearance of a double pole in the evaluation of that di-agram. Transforming the sum over Fermi-Matsubarafrequencies into an integral in the complex plane andusing the residue theorem to evaluate it, we obtainone term proportional to PSq„p, which blows up asT 0. What happens is that the vertex diagram ofFig. 10 contains the spin-susceptibility bubble, which,as already seen in Eq. (2.20), also has the samePSg p behavior canceling the unwanted piece of thevertex diagram of Fig. 9.

Defining

X.FIG. 8.- Vertex diagrams included in first order of the re-

normalization group.FIG. 10. Diagram used to cancel the static piece of the

vertex diagram of Fig. 9.

KONDO LATTICE: RENORMALIZATION STUDY USING A NE%. . .

when using the pseudofermion method2 of Abriko-

sov, the final equation (4.5) satisfied by the effectivecoupling is the same, as it should. That those coeffi-cients do not coincide is not surprising if we recallthat the spin-propagator matrix has different dimen-sions and factors multiplying the diagrams areevaluated differently in the two methods.

The solution to Eq. (4.5) does not diverge even forJ'"po )0. In this case it slowly approaches one. '

From Wilson's" numerical nonperturbativercnormalization-group transformation, we know thatthe effective coupling does not saturate, but divergesinstead. In this sense the first-order renormalization-

group equation is better than the second one, . In thedisordered state one can obtain qualitatively correctresults by using the first-order renormalization-groupequation until the effective coupling becomes one,and then invoking the "Toulouse limit. "" For theordered state, however, wc have to consider thesecond-order renormalization-group equation, be-cause the feedback between magnetic ordering andKondo effect docs not occur in first order, %e ex-

pect that qualitative features will not depend on theparticular value for the effective coupling (0.5, forexample) at which we choose to stop the scaling.

V. RENORMALIZATION-GROUP EQUATIONIN THE ORDERED STATE

For the Kondo lattice, a spin magnetization is pos-sible. In this section we try to include this featureand to understand the competition between magneticordering and Kondo effect.

The problem is handled exactly in the same way asin Sec. IV. The basic difference is that we usc nowthe propagators of Eqs. (2.12) and (2.22) in the pres-ence of the internal magnetic fields. Using again thevertex diagrams of Fig. 8, and looking for correctionsto the vertex, i.e., terms of the form 0'&e,i, we findtwo contributions: one coming from the SI and 8 &

terms of the spin and electron propagators, respec-tively, and another from the H, ~,~ and h, o-'& terms.The renormalization-group equation for the effectivecoupling is then given by

D I} (t'od

1 2(.J, (if»+a)(if»)'+if»(h/2 H)di f}D "—D p [(if»+a) —(h/2) )if»(f» +H )

f

= 2(J' p«) D da —, tanh —a ———tanheff 2 I} 1 /3 h PH 1

f

This equation gives directly Eq. (4.1) if we set H = h = 0.From now on we will consider the case J ) 0 (antiferromagnetic coupling), in which H and h are antiparallel.

In the limit T 0, a function like tanh(P/2)(a —h/2) becomes a step function, and is no longer continuous ata- h/2. This fact implies that, when using a flat band for the conduction electrons, we arrive at differentrenormalization-group equations in the two cases h/2 & D and h/2 & D:

d ( Jeff p )

dl eff 2"'""-D+h/2+H'D+h/2+H '" 2"e

(5.2)

(5.3)

If we had decided to use a Lorentzian, i.c.,p(a) =D/rr(a'+D'), for example, we would have

arrived at a single renormalization-group equation,smoothly interpolating between these two. Thephysical results would be the same.

These equations can be interpreted in two differentways. In the first interpretation, as in the first appli-

cations of the renormalization group, the cutoff Dvaries, as a consequence of the elimination of theelectron degrees of freedom. In this case D =Doe '

and the fields H and h are constant (at this level ofapproximation). In the second interpretation, as in

the late developments of the renormalization group,field operators, energies, etc. , are rescaled to sct thecutoff at its initial value. In this case D is constant

and the fields H and A grow exponentially; i.e., theyare proportional to e'. Looking at Eqs. (5.2) and

(5.3), we see that the relevant quantities are in factH/D and h/D, and not H, h, and D, separately. Wewill follow the second interpretation. As a conse-quence Eqs. (5.2) and (5.3) are therefore comple-mented by

(5.4)

(s.s)

in this approximation in perturbation theory.Equations (5.2) and (5.3) are easily integrated giving

6050 VITOR ROCHA VIEIRA 23

Jeffpo=1

Jeff )

D + ho/2+Ho h—21n for —(D(J' pp)p De '+hp/2+Hp 2e

D + ho/2+Ho —De '+ hp/2+Hp—21nDe '+hp/2+Hp

for —)Dh

2

(5.6)

(5.7)

with Ho and ho, the initial values of H and, h, respec-tively.

Looking at these equations, we immediately con-clude that the presence of the fields has the effect ofslowing down the growth of the effective coupling.Moreover, it follows from Eq. (5.7) that, as pointedout by Abrikosov, 6 only the field H, produced by theelectrons and acting on the spins sets the final limiton the growth of the effective coupling. If H is ini-

tially sufficiently large, i.e., if

Ho )D exp[-1/2(Jpp)p] (5.8)

then J'"po does not diverge, as shown in Fig. 11, ob-tained using a computer, for the half-filled case, inwhich D po 0.5.

The discontinuous change in slope atI lnD/(hp/2), i.e., at the point in which h/2reaches D, is due to the fact that we have set T -0,and used a flat band, as discussed.

In this order of approximation the internal fields,to be determined self-consistently, change only dueto the rescaling made to keep the bandwidth D con-stant. Without this rescaling they could haveremained the same. As we have seen before, in

mean-field theory Ho- J po. Therefore, the criteriongiven by Eq. (5.8), leads us to Doniach's" criterionbased on the comparison of the binding energies of aKondo singlet and of a Ruderman-Kittel-Yosida fer-

romagnetic state. In Fig. 12 we show the behavior ofthese two energies as a function of Jpa. Contrary towhat happens in the Kondo problem, where the Kon-do effect always wins, if Jpp ( (Jpp)", the magneti-zation wins over the Kondo effect and suppresses itsdivergence. For the half-filled case (Jpp)" =0.19.

The present derivation of Doniach's criterion, be-ing based only on the use of the renormalizationgroup, in first-order perturbation theory, is quite gen-eral, and therefore it provides us with a justificationfor its use beyond the d -1 Doniach model.

%'e have seen that the presence of the spontaneousmagnetization tends to suppress the Kondo effect.However, this approximation does not include anyfeedback from the Kondo effect on the magnetiza-tion, or on the growth of the effective coupling. Wewill consider now the next approximation, in whichthis competition can be observed.

In second order in J, we consider, as before, thediagrams of Fig. 9. For zero fields, we have alreadyseen that, when the frequency transfer d co from thespin to the electron lines vanishes, the vertex di-

agram of Fig, 9 had a static piece diverging for T 0.We argued then that the diagram of Fig; 10 could beused to cancel it. We adopt the same procedure here.

0.06—

QoP50

1st order

ItJPGj =0025

0.04

Pp

0.I25—0.02 .

0.000 0,05 O. IO

Jp

0,15 20.20

(~~,)"0.25

'0 5 l0

FIG. 11, Behavior of Jpo under the renormalization

group, in first order. The presence of a spontaneous mag-

netization tends to suppress the Kondo effect. The discon-

tinuous change in slope is due to the use of a flat band.

FIG. 12. Comparison of the binding energies of a Kondosinglet (cg. J ) and of a Ruderman-Kittel-Yosida ferromag-

0netic state (~ e ), in Doniach's criterion. Contrary towhat happens in the Kondo problem, for Jpo& (Jpo), themagnetization wins over the Kondo effect. For the half-

filled case (Jpo) =0.19;

KONDO LATTICE: RENORMALIZATION STUDY USING A NE%. . . 6051

The second-order correction to the vertex is then:1

P '

P P g 1 x' —xH+H2I'~' = (Jpo)2 I daf de2 —I —tanh —a) +—tanh —a2 ——L4 2 2 2 2 x (x+H)(x —H)'

P A P+ —tanh —e~ + ——tanh —e2 ——2 2 2 2

r r

I PH 2x3 —10H'x+4H' P I I ' h~ —tanh + (s.9)2x(x+H)(x —H)' 2 cosh'PH/2 x H —H —H

in which x -a~ —a2+2h/2. When H, h 0, Eq. (5.9) reduces toI

Pai P~2I'2' =2(Jpo)'J Jl deaf da2—I —tanh tanhgJ 2 2 (e, —e, )'

leading directly to the second term of Eq. (4.4).Similarly using the (33) component of the self-energy diagram of Fig. 9, we have the following second-order

correction

d"'=(Jpo)' I~~ I deide2 ——I —tanh —sf+ —tanh —e2 ——4-0 al 2 2 2 2 2

+ —, tanh —e~ + ——tanh —e2 —— tanh, + H H (5.11)P h P A PHt

(x —H)'

in which x =af —e2+2A/2, as before. When Hh 0, Eq. (S.ll) reduces to the result obtained using (2.24) andleads to Eq. (4.3).

Using Eqs. (5.9) and (5.11) if we take the limit H, h 0, before the limit T 0 we arrive at the usualrenormalization-group equation for the effective coupling, given by Eq. (4.5). If, however, the limit T 0 is tak-en before, the final renormalization-group equation does not reduce to Eq. (4.5) when we let H, h 0. Thisequation is

d(Jeff )Jeff )2

d/ D + h/2+H

+2(J" )'D -3 1

D —hl2+H 2D —2h/2+ H

for h/2 (D, and

3i

2D+2A/2+H II

2D —2A/2+H + 42H D +h/2+ H 2H D —h/2+ H 2H

(5.12)

d(J""p )=2 J"" )'D ' +D + h /2+ H D+ h/2+ H—

4

2 J ff )3D 3I

2D+2A/2+H 2I

2D+2A/22H —2D+2A/2+H H —2D+2A/2

for h/2 & D.The magnetic field H, is also affected by the renormalization-group transformation. Its variation is determined

by the off-diagonal component of the self-energy diagram of Fig. 9. Using Eqs. (3.10) and (S.4) ~e finally arriveat

(2D+H)' f h

(D + H)' (A/2)'—=H —2(Jeffpo)~D X

h/2+Dln for —)D (s.ls)

6052 VITOR ROCHA VIEIRA

0.50 0.50

Jp,0.25

O.i0

Hp

0.05

I I I Ii

I I I

2nd order

( Jp,), =0.2,5

Po

0.25-0 0,2 0.4

0. . . , I

0 0.2 0.4

00 0.03 0.06

FIG. 15. Same as in Fig, 14. In this case there is stillsome spontaneous magnetization when we stopped the itera-tion procedure.

P0

FIG. 13. Renormaliz'ation-group flow, in H po and Jpospace, having used mean-field theory for the initial values ofthe spontaneous magnetizations,

This renormalization-group equation can be com-pared to a similar result obtained by Anderson andYuval for the renormalization of the magnetic field in

the Kondo problem. "In this renormalization-group treatment we have

looked for the divergent corrections to the variousquantities in which we are interested. This is the rea-son why the field h changes only due to the rescalingthat we have made to keep the bandwidth D constant,However, in this order of perturbation theory thereare other nondivergent corrections which make theirappearance in the renormalization group as initial

transients, the effects of which amount to finitecorrections to the initial parameters (like the shift ofT, from its mean-field value in a fermmagnet).Thus, in principle, we may have Ho and AO qualita-tively different from their mean-field values, but notqualitatively. One of the quantities which was notconsidered in this study was electron damping. Itwould be affected by the'second-order electron self-energy, which is not divergent. Ho~ever, the inclu-

sion of electron damping should only modify the fiatelectron density of states, rounding it up, and would

not modify the physical results we have obtained.Looking at Eqs. (5.12)—(5.15) one can see how the

competition between the Kondo effect and the mag-netization works. The second-order term in dH/di isnegative and so it tends to decrease H, is oppositionto the first-order term, which tends to increase H.When H is very small, d(J'"rpo)/dl gets big andtherefore J'"'po tends to increase faster. This has theeffect of making the second-order term in dH/dleven more important. One concludes then that, if His sufficiently small, H will decrease and J,"will in-

crease; i.e., the Kondo effect wins over the magneti-zation. This situation is shown in Fig. 13, obtainedusing a computer. For the initial values of H and h

we have used the mean-field values of Eq. (3.9). InFigs. 14 and 15, we have selected two of the curvesof Fig. 13, and displayed J and H separately as afunction of I, the renormalization-group parameter.In both cases we have stopped the renormalization-group transformation when Jp0 reached the value 0,5.

In the case of Fig. 14, the Kondo effect has won

already, and H went down to zero when we stopped,and in Fig. 15 there is still some magnetization left.

However, if the initial value of H is sufficientlylarge then the magnetization wins over the Kondo ef-fect and the situation is qualitatively the same as in

the first-order approximation. This can happen, forthe antiferromagnetic state in which po in Eq. (3.5) is

replaced by the electron susceptibility Xo(Q) at the

0,50 I ~ 1 I1

I I I

-1Oe04X IO l i i s

11 I I

I I I1

I I I I 0.20i

. I I I

Jp,0.25—

2nd order

(Jp ),=0.05

Hp,

0.02xl0

2nd order

( Jp, },=0.05

J'p

0.25

---1st order—2nd order

Hp

0.10

( Jp, )a=0.05ho =0.05H, =0.01

0 025I

0.50

-1Ox 10 i

0 0.25 0,50 0 I 1 I I I I

0 5I I I I 1 I 1 I I

10 0 5

FIG. 14. Behavior of Jp0 and Hp0 under the renormali-

zation group. In this case the spontaneous magnetizationhas already been renormalized to zero when we stopped theiteration procedure.

FIG. 16. Enhancing the electron magnetization, resultsobtained in first and second order become qualitatively thesa fne.

23 KONDO LATTICE: RENORMALIZATION STUDY USING A NEW. . . 6053

wave vector Q of the spin-density wave. This partic-ular value of Q is the one which maximizes Xo(Q),and therefore is necessarily larger than po. Thus theinitial value of H may be considerably larger thanJ2p

In Fig. 16, we have used the same initial values forJ and h as for in Fig. 14, but have increased the ini-tial value of 0 by a factor of 4. The results using thefirst- and second-order renormalization-group equa-tions are qualitatively the same.

ordering wins and the situation is qualitatively thesame we had using the first-order renormalization-group equations.

In future work, we intend to study more extensive-ly the renormalization flows for different couplingsand molecular fields and extend the calculations tofinite temperature.

ACKNOWLEDGMENTS

VI. CONCLUSIONS

We have developed a new field-theory method forspin

2using Berezin and Marinov's path integral for-

mulation. As a consequence of this formulation, an-tiperiodic boundary conditions in time and Fermi-Matsubara frequencies should be used. The spin pro-pagator in the presence of an external magnetic fieldwas written down. The finite-temperature perturba-tion-theory rules for the case of a magnetic field andfor the Kondo lattice were derived. The linked graphtheorem is valid. This is the biggest advantage overAbrikosov's pseudofqrmion representation, whoseapplication has some restrictions because it violatesthe linked graph theorem.

We applied then this method to the study of theKondo lattice. Duc to the k independence of thespin propagator, in the disordered state the effectivecoupling satisfies the usual zero-temperaturerenormalization-group equation for the Kondo prob-lem. Because this field-theory method is differentfrom Abrikosov's pseudofermion representation inseveral aspects, although the final equation. is thesame, some of the intermediate quantities were dif-ferent.

Going beyond mean-field theory, the competitionbetween Kondo effect and magnetic ordering must betaken into account. This was done deriving therenormalization-group equations in the presence ofinternal self-consistent fields acting on thc spins andon the electrons.

In first-order perturbation theory the electron mag-netization, when sufficiently strong, sets a limit onthe growth of the effective coupling. Our criterioncoincides with Doniach's argument using the bindingenergies of the Kondo singlet and of the Ruderman-Kittel- Yosida ferromagnetic state.

In second-order perturbation theory, there is afeedback of the Kondo effect on the magnetization.If the electron magnetization is sufficiently small (forexample, if we use mean-field theory for the initialvalues) then the magnetization tends to be supressedand the effective coupling tends to grow. The Kondoeffect suppresses the magnetic ordering. However, ifthe electron magnetization is enhanced, as it can bein the case of antiferromagnetism, then the magnetic

I would like to express my gratitude to ProfessorJohn A. Hertz for his guidance and constructive criti-cism during the progress of this work. I warmlythank him for his continuing advice, patience, andunderstanding throughout my graduate career. Thiswork was supported by NSF Grant No. 77-12637.

APPENDIX

Grassmann numbers''" are characterized by theiranticommutation relations

(A2)

where a is a complex number. A Grassmannnumber is real if ]'= f.

To obtain the differential of a function, we use theusual rules, provided that we keep the relativepositions of the variables. As an example d((~$2)=(d(~)i;, +f~(di;2). To obtain the derivatives, wetake out the differentials to the left or to the right,keeping track of the minus signs. We have, thereforeleft, 5/8(k and right, 8/8)k derivatives.

Integration is defined preserving the usual proper-ties of linearity and invariance under translations. Asa consequence

Jl'I dg=O (A3)

and, by definition, we have then

(A4)

The rule for the change of variables of integrationis similar to the usual one, with a difference, howev-er: instead of multiplying by the Jacobian of the

(A I)

In particular f =0. As a consequence, every func-tion of a finite number of Grassmann numbersreduces to a polynomial.

The analog of the complex conjugation is theoperation defined by

6054 VITOR ROCHA VIEIRA 23

transformation, we divide by it. We have, therefore, The same result can be seen rewriting Eq. (2.12) in

the form

Jtf (r,) d"( = Jtf (((p, )) d"p, ,8p

(As)51 =I'I'9+ +-~i 8 ++-~i 9- (N3)

with d"g = d$ ' ' ' df&.Using the definition of Pfaffian 6'of an antisym-

metric matrix A, one can easily see by inspection that

in which P- are the projectors for precession in thetransversal directions and P' is the projector in thelongitudinal direction. They are defined by

exp —,g~A q d"q=(PA (A6) ~ Im p Slm —t &rima+

Taking the square of this expression and making achange of variables, one shows easily the well-knownidentity ((PA)'=detA. Finally, making a translationin the integration variables in Eq. (A6), one arrives at P" =S' =

Im Im

(N4)

(+(~)= 1 ((](ul) + /(2(al)l (Nl)

in the reference frame in which the magnetic field His along the third direction, it is easy to see from Eq.(2.13) that the only nonvanishing propagators are

1+—, r —+

I Cd —H1

i«)+H(N2)

Jtexp( —,' qrAq+rtrp)d"q , -ddetA exp( —,p, rA 'p, )

(A7)

This is Eq. (2.16) used in the development of theperturbation theory for spin —.

Note added: After the submission of this paper Ihave been able to describe this new field-theoreticmethod for spin —, in a more simplified and trans-parent way.

Defining the fields

Finally, it is important to know how the spin com-ponents S+ and S3, are represented in terms of f+and (, . Using Eqs. (2.1) and (Nl) we have then

S+ = + (f3)+ —(+f3)1

s, = —,'(~,~—

~ ~,) .

This tells us immediately what are the vertices for aparticular problem. As an example, it is a trivial ex-ercise to derive, for the Heisenberg ferromagnet, themean-field theory from a self-consistent Hartree-Fock calculation and the longitudinal and transversal(displaying spin waves) susceptibilities from a RPAcalculation. The elegance of the method and the sim-

plicity of the calculation should be compared with

previous treatments of this problem. "

Presented as a thesis to the Department of Physics, TheUniversity of Chicago, in partial fulfillment of require-

ments for the Ph. D, degree,tOn leave of absence from Centro de Fisica de Materia Con-

densada, Av. Professor Gama Pinto, 2, Lisboa, Portugal.Present address: Institut Max von Laue-Paul Langevin,156 X Centre de Tri, 38042 Grenoble Cedex, France.

'R, Jullien, J. Fields, and S, Doniach, Phys. Rev. Lett. 38,1500 (1977); Phys. Rev. B 16, 4889 (1977),

2A. A. Abrikosov, Physics (N.Y.) 2, 5 (1965),3H. Keiter, Z. Phys. 213, 466 (1968)'„Phys. Lett. A 36, 257

(1971).4F. A. Berezin and M. S, Marinov, Ann. Phys, (N.Y,) 104,

336 (1977); JETP Lett. 21, 320 (1975).5The antiperiodic boundary conditions has also been found

in path integral formulations for electron fields: D, E.

Soper, Phys. Rev. D 18, 4590 (1978); Y. Ohnuki and T.Kashiwa, Prog. Theor. Phys. 60, 548 (1978).

6A. A. Abrikosov, Physics (N.Y.) 2, 61 (1965).7B. Schuh, Z. Phys. B 34, 37 (1979).M. Fowler and A. Zawadowski, Solid State Commun. 9,

471 (1971);A. A. Abrikosov and A. A. Migdal, J, LowTemp. Phys. 3, 519 (1970).

P. W, Anderson, J. Phys. C 3, 2436 (1970).' K. G. Wilson, Rev. Mod, Phys. 47, 773 (1975)."G, Toulouse, C, R. Acad. Sci. 268, -1200 (1969).' S. Doniach, Physica B 91, 231 (1977).'3P. W. Anderson and G. Yuval, J, Phys. C 4, 607 (1970).'4F. A. Berezin, The Method of Second Quantization

(Academic, New York, 1966).'5R. B. Stinchcombe, G. Horwitz, F. Englert, and R. Brout,

Phys. Rev. 130, 155 (1963).