978 T. A. CALLCOTT AND A. U. MAc RAE 178

Ni and the cesium layer, or does it reflect a sharpeningof a single-resonance frequency that allows us to resolvepeaks in the density of initial states for the interbandtransition that accompanies the plasma excitation?

(3) Is the ordering of the cesium layer, reflected inthe LKKD measurements, important to the observedstructure in the PED? Data taken on another surfaceof Ni may provide further information on this point.

(4) What is the mechanism by which the cesiumlayer enhances the absorption?

These and other such questions may prove usefulfor guiding further discussion and experiments, but areless important than experiments designed to check thecentral elements of the proposed explanation. These are(a) that the surface plasma excitation exists, (b) thatit provides an dlicient optical-absorption mechanism in

the surface region, and (c) that its character can bemodified by changing the surface conditions and inparticular by depositing cesium on the surface. Itshould be possible to confirm the existence of the surfaceplasma excitation by measuring the energy losses ofrelatively slow electrons ((1000 eV volts) reflectedfrom thin Ni films. Such measurements have beenused extensively in the past to study surface plasmaexcitations. "~ If the plasma excitation contributes toabsorption, and its character varies with surface con-dition, the magnitude of structure in the opticalconductivity that has been associated with the anom-alous peak at —4.5 eV should also vary with surfacecondition, and, in particular, with the deposition ofcesium.

~ L. K. Jordon and E. J. Scheibner, Surface Sci. 10, 373 (1968).

PHYSICAL REVIEW VOLUME 178, NUMBER 3 15 FEBRUAR Y 1969

Self-Consistent Solution of the Anderson Model*

ALBA THEUMANNt

Belfer Graduate School of Sconce, Feshioe University, Nem Fork, New Fork 10033(Received 29 May 1968)

The Anderson model is studied by means of retarded double-time Green's functions in the limit U ~ oo .Using a truncation procedure and solving self-consistently for the averages, an integral equation for thetransition matrix is derived. The exact solution to this equation can be found by analytic methods and isvery sIIT1I@I to the solution to the s-d exchange model. The known perturbation result for high temperaturesis also obtained from our integral equation.

INTRODUCTION

ONE time ago a model Hamiltonian was introduced

~ ~

~

~

~

~

~

~ ~

~

by Anderson to describe a localized state of an im-purity in a metal. ' Rim' went beyond the Hartree-Pockapproximation used by Anderson and looked for asolution by means of the double-time Green's-functionmethod. ' However, the solution found by Kim isessentially equivalent to perturbation theory anddiverges at the Fermi surface at low temperatures.

Our self-consistent treatment of Anderson's Hamil-tonian is very similar to that used by Bloomfield andHamann' in the solution of Nagaoka's' equations forthe s-d exchange model.

In Sec. I we look for the equations of motion of thedouble time, retarded Green's functions, and we solve

~ Supported by U. S. Air Force 0$ce of Scientific ResearchGrant No. 1075-66.

t Present address. Institutt for Teoretisk Fysikk, N. T. H.Trondheim, Norway.' P. W. Anderson, Phys. Rev. 124, 41 (1961).' D. J. Rim, Phys. Rev. 146, 455 (1966).' D. N. Zubarev, Usp. Fiz. Nauk?1, 71 (1960) /English transl. :soviet Phys. —Usp. 3, 320 (196olj.' Ph. E. BloomMd and D. R. Hamann, Phys. Rev. 164, 856(1967).We thank Professor Horwitz for pointing out this referencebefore its publication.' Y. Nagaoka, Phys. Rev. 138, A1112 (1965).

for the one-particle Green's functions for conductionelectrons, after making the system of equations finiteby a truncation procedure. We find our equations to bevery similar to Nagaoka's equations in the limit U ~ .We derive an integral equation for the transition matrix,of the same type as that discussed in Refs. 6 and 7, andwe solve the integral equation with the method ofBloomfield and Hamann. We show how the known per-turbational result can be obtained after expanding the tmatrix in a power series in pV / ~

es~, valid for T)Tx.In Sec. II we calculate an approximate expression for

the low-temperature resistivity, finding a result veryclose to that of Hamann' except for the presence of aterm of order V' due to the finite lifetime of the elec-trons at the d level.

I. DERIVATION OF AN INTEGRAL EQUATIONFOR THE TRANSITION MATRIX

Anderson's model Hamiltonian is

H=Q esas, tas, +Q end, td, +Udttdtdstdck,e

+V Q (as,td, +d,tas,), (1)k, e

4 D. Falk and M. Fowler, Phys. Rev. 158, 567 (1967).~ D. R. Hamann, Phys. Rev. 158, 570 (1967).

SOLUTION OF THF AN DERSON MODEL

+VX &(caa a'd« tlcaat')&

+U«caa c'dcdtI caat')), (g)

where aI„and a&, are the creation and annihilation The three new equations are

operators of electrons in the band with spin s (s= t' or l);d,t and d, are the creation and annihilation operators ofelectrons at the impurity site; U is the Coulomb repul- =—V((dc~dcdt

Iaatt))

sion between electrons at the impurity; and V is thematrix element that connects the impurity and band + Z (( a a' .c t I

at' )&

states, assumed to be a real constant for simplicity in

this work.We introduce the retarded double-time Green's

functions'

GQB(t)= —i(&A(t),B(0)}), t)0=0 t(0

(~—ea )((dc'oa cdt I oat'))

(2) = —V p ((ca.'«a cdt I caat'))+ V«dc'dcdt I aat'))

+VZ &(dc'o'«.

tlat»')&,

(9)

oa

«A IB)&=-2m

G (t)eics+ia)cdt

+V 2 «dc'o. acta t I oat'))

where ( ) indicates statistical average and the curlybrackets indicate an anticommutator. The Fouriertransform. is de6ned as

= (I/2~)(~l)~aa —V P &(. c'«a't I o»')&

and the average+V((dc'dade luat')). (10)

where

(BA)= —2 f(&u) Im((A I B))„+cade), (4) We can now decouple the Green's functions appearingin the right-hand side of (8)-(10) in the usual way as

f(cu)= 1/(e"fr+1).

We take the Boltzmann constant k= T. As is usual,we obtain a chain of equations of motion for the Green'sfunctions:

(~—., )&&oa., Ia»t)&= (1/2~)haa. +V(&dt I o»t)&

((cta t'o.cd t I oat'))—(cuba a'ca.a)«d t I oat')),((cta c'dact. rloat')) —(oa a'dc)((a. tlccat'», (11)«dc'o'«-t

Ioat'»=(dc'oa c&«o-t I oat'».

We also have the Hermiticity condition

(~—«)((dt Io»')&= V 2 &&o.~ Iaaf'&&

wheregg =dgtdg,

( —"—U)«mad t Io»')&

Because of the absence of a magnetic 6eld, there is no

+U« d ~ tyy 5distinction between averages of operators with spin upand down, therefore we drop the spin indices in thefollowing. Thus,

(nc) = &tat) = (rt), (datuaa) = (d ttuat)= (d aa).

= —V E ((o.c'dcdt I o»'))

+V P &(dc'ct. cdt I o»')&

Introducing Eq. (11) in Eqs. (5)—(10), we can solvefor ((aa laat)):

«'I '»=G'()+V Z ((dctdco t I oat )). (7) 4a 1 t(c0)+— (12)

2s. ce—ea 2s. (ce—ea)(co—ea.)In the equation for ((ncdc I aatt)), three new functions

appear, but we cannot truncate them without splitting where t(ce) = &(d I d )) is the transition matrix' whichtwo d operators with spin up and down, hence breaking satisfies the condition t(ru+ib) = [t(ca iJ)], an—d isthe correlation at the impurity. given by

1+[U/R(ra)] & (e)+ V[p(2«+ U ce)+p(ce)]}—t(ca) = V' (13)

ce—«—Vah(cu) [U/R(co)] V. a[E—(2«+ U—~)—If'(ce)]—[U/g(ce)] Vag(~) [p(2«+ U ~)+p(~)]8 This method of decoupling was introduced by Daniel C. Mattis (unpublished).9 D. Langreth, Phys. Rev. 150, 516 (1966).

980 ALBA THEUMANN

In the usual approximation of a square density of states,

A(co) =Q ~A(cd &ib) =p df~Wzzrp ~

n co—$&zb(14)

where p is the density of states at the Fermi surface and D is the band width. In addition,

1 1K(2«+U ~) K(~—)=~ LE (a~ta.)j

2Nd+ U—07—6' CO—6P~

1 1F(2«+ U cd)+F—(co) =P (dtac, )~ +

E26d+ U c0 6p Gl Cg~

R(co) =M Eg U+ V'A(2«+ U—cd) —2V'A(cd)In the limit U —+ ~,

lim K(2«+U cd) = lim—F(2«+ U co) =0,—+~CO +~co

(15)

(16)

and the expression for t(cd) becomes

lim-"R(c0)

1—(n) —VF(ra)t(cd) = V'

c0—«—V'A(co) —V'K(cd)+ V'A(co)F(cd)

The averages that appear in (15) and (16) are evaluated through (4) in terms of the functions

1Gg(~) =P ((a„laz')) =— Pl+A((a)t(co)],

fl 2% Go —II'

1 1 1&(~)

V 2m (o—eI,

These equations are very similar to Nagaoka's (2.24) and (2.25).'

(19)

(20)

(21)

Q (acta„)= —2 f(cd) ImG„(co+ib)da&, (22)

(dtac, )= (aktd) = —2 f(cd) ImI'g(co+ib)dco (23)

Introducing (20)—(23) in (15) and (16), we finally solve for K(&o) and F(co) in terms of t(co):

1F(a)+ib) =—p

V D(24)

K(co+i b) =p

while t(&o) satisfies the integral equation

f(k)Li+X([—'b)z(p —iS)$d~,

n a&—$+ib

(25)

z(a+ib) = V'~ 1—(n) —pf(~)

t($ i8)d)—D (u —)+i8

f(t)co «+zzrpV' pV'— —d$ 2izrpV'p — i(g zb)d]

~. (26—)

n cd—$+zb n (o—)+ib

jliary functiWe de~ne the aux( + b) 1—2zzrpt(&+

1+2zzrpt(rd[~(„+ib)]*=&("

DF RSo MODELLUT&o~ pF THE A 98i

(27)

ih), Eq. (26) becomesand in terms of f(rd+ib, q.

(a i = — zzr Vz(2(n) 1—)—pV'&+zb) —I

rd ~a+zzrpf(t)

i) a) $+z—b

o)—ed,+i~pV2 —pV'f")

[~(~+;b)]*d~ I(28

i) CO—(+zb

er of d electrons with gage number o e e8) becomes

f(~). [O(&+ b)] & .

D

V2 pVz—.+~ ~ Jaxis. T ere tlonfs(c0)=f(r0+zb

f(z0+ib) =~

co—eg —p d za

axis. The retarded function z0-y v ueo axis. T eresa is . 9 w ile the advancesatisfies Eq. (29), w i e

(29)

71 b Hamann,s-d exchange mode y1 1 h

'ved from the s- excilar to t eco(

' d't ol t'othis

1exca nonlinear singu n a

the following wweconsi erh analytic prop

omopaper for

The averand Eq. (2

V2p(rd —z = — —V—&) =(— 2 V2

D frd —eg —impV2 —pV

i) rd $ i b— — (30)

=11—r

(a—ih)r' ' duetot er'

h behavior of P(cross section a

ole at co= e~,g

without modifying ephysical properties o o

We take the limit

f(()i)M $ zb

pV' and~

zg~ ~ ~,wit

'h the condition

. [4(k—zb)]*4 ~, (33rd —$ ib——D

lexctions of the comp exthe following functionsWe introduce t e ovariables s:

(31)

4'(~+zb)

f(&)

i) a& $+ib—=~1—r

=const(g

nd 30) can ben' . (31), Eqs. (29) and (In the limit of Eq.

written as

x(.)=1—r f(E)1

i)s—(

4'(h)dkipz(S) = 1—Zzrr —r

)('($)d(,~,(s)=1+z~r—r

(34)

(35)

(36)

r &&+ )1" i), ( )i) co—)+ih

har ed Particles (Inter-tatistical Mechanics of Charg' er-

"N. I. Muskhehshvilj. ,ho8, Ltd. , Gronsnge,

ebtth hand lower half-plane, u

ched from a othe cut is approach

ALBA THF. UMANN 178

it'(~) = X+(~)/ pi+(~)

0"(~)=X-(~)/s ~(~) .It is important to notice that

Lz i+(~)]"=z~(~)& s i-(~)

From (37), (38), and the identity

(37)

(38)

=P wis b(/d —&),co $+ib

we can prove that

F(z) = q/i(z) s/g(z) —LX(z)]'

is analytic on the whole s plane and

lim F(z) =s'F'.

Hence

in mind, Eqs. (32) and (33) can be written in terms ofthe functions

The approximation

D f(() (~2+~2T2) //2

P— df lnn co $ (D+c0)

becomes exact in the limit T-+ 0.To 6nd a function q/i(z) that is sectionally holo-

morphic of Gnite degree at in6nity and satis6es theboundary condition (40) is called the homogeneousHilbert problem.

H(co) is a complex function of the real variable /d, andin both limits H(&D) = 1. X is the change in the phaseof H(ru) when &a travels from D to —+D.

From (40) we have

H (ao) = E(ru)/D((o),

where X(cu) is real and

ImD((u)phase of H(&s) = tg '

ReD((a)

F(z) = «(z) z i(z) —LX(z)]'=+F' (39)

Solving for s/q(z) in terms of z/i(z) and after a littlealgebra, we Gnd that

D(cu) =(ReX )'—(ImX )'+s'Z'

+i(2 ReX ImX ).

sr~(c0) X+((0)X (au)+ir'I"=H(co),

pi (&o) (X )'+ Fir' becausef(60) & 1.

(41) Thus the phase of H(s&) is always between &ir/2 andthe index X=O. The fundamental solution" qi(z) isgiven by

wheref(~)

d$.n co—f&ib

X~(co)= 1—I'

The integral appearing in (41) is very well known': n lnH(g)s/i(z) = (1+iirF) exp~ — d$

~

. (44)2~i n z P )f(g) t' (0 ) D+/d is'

d$= —e~ s& ~pin w —, (42)n &u $&ib k —2iriT) 2irT 2 It is easy to see that (44) is a solution evaluating

qi(~Rib) and taking logarithms of both sides of (40).We can determine now if/"(/d) from (37) and find that

where 0'(z) is the digamma function. "0'(z) lnz for

~s

~&&1,

'I/(~) = —c—2 In2, c=0.5772.

From (43) it follows that

(40)ReD(co) = (ReX)'+ ir'F'LI —fi(&)])s iFiLI—fi(~)])0

The approximation

(i02+z.2Ti) I/2

Re2miTJ 2z T

is the right asymptotic expansion for ~&&T.Therefore, we approximate X~(~) in (41) by

' f(h)X ( )=1—FP dalai Ff( )

n~ —5-(~2+ir2TR)1/2

1+I' ln Aim Ff(co) . (43)(D+co)

~ Handbook of Mathematket Fgnctkes, edited by M. Abramo-vitz and I. A. Stegun (U. S. Department of Commerce, NationalBureau of Standards, %ashington, D. C., 1964}.

X~((u) 1 n lnH(g)exp dp . (43)

(1+i~F) 2~i

A perturbational solution can be obtained from (26)if we realize that t(ru+ib)~l '. Therefore, we canreplace

V'(1—(e))t($ i b) = V'((d

~dt))—

eg ib— —and obtain

t(ru+ib)~

~ ~

f(k)~V'(1—(I)) 1—pV'n au $+ib $—eg ——ibJ

~

~

f(&)co—~g+2x p V'—pl' d( . (46)

z) Cd —$+1b

178 SOLUTION OF THE ANDERSON MODEL 983

D

Xexp! P&2~i

1ln!H(() I d&! . (50)

a&—$

II. RESISTIVITY From (43),

This result coincides with Eq. (3.8) of Dworin" in (1—is F) X+((o)the limit U —+ ~. It was also obtained by Kim, except 0"(~)=

~2/2 I/2 X ~ 2 ~2@2 1/t2I II(~)!-c/I

for the second term in the numerator. This termoriginates in the self-consistent evaluation of theaverages (a~td) that were neglected in Ref. 2.

The resistivity is given by

R= rr/cree'r, (47)

((g2+s.2T2) I/2

Xg(ru)~1+ F ln a&r Ff((u)(D+~)

where r, the scattering lifetime of conduction electronsat energy &T from the Fermi level, is

1/r = —A Imt((y+ib)=+A Re(1/2s p) [1—Ps((u)]. (48)

biff(~)l . «I, (49)c0 )+i—8

For low concentration the constant A is proportionalto the concentration of impurities.

It can be proved that (45) can be written'

(1+s'Fm) '" Xp((u)0 "(~)=

1+is F ([X+(ru)]'+s'F2) '"D

Xexp!I 2~i

CO2+X2T2=—ln &is Ff(c0) for

Ic0

I «D, (51)2 7r2TK2

where we introduced the parameter ~T~= De-'fr.This coincides with the expression for the Rondo

temperature, ' with the correspondence J=pV'/~q. Itmay be noticed that the effective antiferromagneticcoupling found by other authors is J=2pV'/ez, in thelimit U —+ ~.'~"

We can approximate 7 by the value it takes at theFermi surface, and therefore we need to evaluate theintegral in the exponential of (50) for ra= 0.

From (40) and (51),

64cr'f'(c0) )—c/2

!!&(~)I=I 1—(in' [(ra'+ s'T')/s'Tx'] 14r'[1+f2(ca)]) 'i'

We approximate ln!H(au) I by64X4

h I&(~)I=—f'(~ T)s hI1—

f In' [(aP+s'T')/s'T/r']+8s') '/

(52)

(53)

Equation (53) gives the exact expression for T=0; it is also exact in the region of co for which f(cd, T)= 1 or 0, andit coincides with an expansion of (52) in powers of

Iln(T'/Trc')

I

' at the Fermi surface. Therefore we can approxi-mate the integral

because

1h I&(~) I-«=-kP

D 64 4 q[f((,T)——:]ln! 1—

(in~[(P+ s.2T2)/s ~Txm]+8s. m) &]

f'(~) = [f(~)—s]'+ I:f(~)—s]+-.',64 4 if((,T)—lI(T) —s Re lnl 1—

n k (ln [(P+s T~)/s2T/r ]+8s~)2/ (~gThe integral in (55) can easily be performed if we approximate [see Ref. 7, Eq. (3.4)]

f(k T) ' f(h 0)—$&s.iT

Introducing (56) in (57),

D 64m 4 2)d)!1(T)=+- l !

1—4 ( (in2[()2+s.2T2)/s. 2Tx2]+8sm) 2j P+s.2TR

"L.Dorwin, Phys. Rev. 164, 818 (1967). %'e thank Dr. L. Dmorin for this observation.'4 T. R. Schrie8er and P. A. Wolf, Phys. Rev. 149, 491 (1966)."D. J. Scalapino, Phys. Rev. Letters 16, 937 (1966)."M. Fihich and G. Hoxwitz Phys. Rev. 168, $08 {1968).

(54)

(55)

(56)

(57)

ALBA THEUMANN 178

The integration is now straightforward, and theleading terms in an expansion in !ln(T /T22)! ' andI' are

~4I(T)~222r41'2+—

3 !ln(T2/Tx2)! ' (58)

X~(0) 1' ln(T/Tx) a-,'i2r1'. (60)

From (48), (50), and (58)—(60) we obtain for thelow-temperature resistivity

m A 1n(T/Tx)R~—ne' 22rp Dn'(T/Tx)+57r'/4]'"

r 1+-1r(2r—1)

2 Dn'(T/Tx)+sir'/2]'"I

+Ol !. (61)I ln2(T/Tx) i

We showed in Sec. I how the perturbative solutionfor T)Tx is obtained from (26), so we can recover theknown result

m V' 2r in(T/T4r)R~—Ane2 !ed! 21 (nT2/T )x+9 r /224

m 2r 1 ln(T/Tx)——F T& Trr. (62)ne' 2 p 1n2(T/Tx)+92r2/4

The first two terms in Eq. (61) coincide with theprevious result found by Hamann. ' The third termdepends on the finite width of the d level, and it is acharacteristic feature of Anderson's model.

I(T)((1 for low temperatures. We obtain from (40)and (51)

I &(~)I'"[X 2(~)+2r2T2]—1l2—

(~)!2+~2@2)1/2

i tr—Im(X+(4e)) 2

Xexp —tg '! !, (59)2 ERe(X+(40))2+2r2F2)

III. CONCLUSIONS

Using retarded, double-time Green's functions, wefound a self-consistent solution to Anderson's model inthe limit U —+ ao. The solution obtained difI'ers fromthe corresponding solution of the s-d exchange modelfound by Bloomfield and Hamann4 by the presence ofan imaginary term of order I'=pl 2/! 44!, due to thebroadening of the d level by the interaction with theband.

The discrepancy between the two models was alreadymentioned, in different approximations, by Dworin"and Horwitz and Fibich. "

We found that the properties of the system varysmoothly with temperature and instabilities do notoccur when T&T~.

Xagaoka's low-temperature solution for the s-d ex-change model, that shows a pole in the scattering crosssection of the band electrons, was also found by Takanoand Ogawa, " using a different decoupling procedure.We do not think that is possible to find that type ofsolution for Anderson's model.

An interesting feature of our equations is that if welook at Eq. (15) for the particular value of U= —244that situates both levels symmetrically above and belowthe Fermi surface, the logarithmic anomaly vanishes. "On the other hand, there are no problems when we takethe limit U —+ .

We evaluated the resistivity for low temperatures,and it reaches a finite maximum value at T=O, de-creasing monotonically when the temperature increases.The leading logarithmic term in an expansion in

! In(T/Tx)! ' coincides with the result of the s-dexchange model.

ACKNOWLEDGMENTS

I would like to express my appreciation to ProfessorD. C. Mattis for suggesting the problem and veryhelpful discussions. I am also grateful to ProfessorG. Horwitz and Dr. G. Lucas for their interest andvaluable comments.

"F.Takano and T. Ogawa, Progr. Theoret. Phys. (Kyoto) 3P,348 (1966).

» We thank Dr. D. R. Hamann for this observation and forpointing out Ref. 17.