SOME PROPERTIES OF WOLFF'S MODEL OF A MAGNETIC IMPURITY IN A METAL

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SOME PROPERTIES OF WOLFF’S MODEL OF A

MAGNETIC IMPURITY IN A METAL

P. Schlottmann

To cite this version:

JOURNAL DE PHYSIQUE

Colloque C6, supplément au n° 8, Tome 39, août 1978, page C6-844

SOME PROPERTIES OF WOLFF'S MODEL OF A MAGNETIC IMPURITY IN A METAL

P. Schlottmann

Institut fur Theoretisohe Physik, Freie Universitdt Berlin

1000 Berlin 33, Arnimallee 3, Germany

Résumé.- Nous obtenons l'identité de Ward pour la fonction de vertex et la fonction de Green d'un électron à l'aide de la méthode des équations de mouvement. Cette relation nous donne une équation intégrale linéaire par rapport à la fonction de Green et nous permet d'obtenir la solution du problème à la limite des temps longs.

Abstract.- Using an equation of motion method we obtained the Ward relation between the vertex function and the one—electron Green's function. This relation leads to a linear in-tegral equation for the Green's function and yields the solution of the problem in the long-time limit.

Magnetic impurities having a defined spin

in a nonmagnetic metal have been usually described

by the Anderson model and the Kondo-Hamiltonian.

At low temperatures these models can be described

by a Fermi liquid theory

l\-1j,

i.e. the magnetic

moment (defined at high T) is screened by the

elec-tron gas for sufficiently small T. The Wolff model

/3/ provides an alternative, but much simpler,

des-cription of this low temperature behavior. The

model Hamiltonian for an one-orbital impurity at

the origin is

H = J e, C

+

C, - J E' n + U(n, - <n >)

f k Tea tea

L

<r a ^ + t

ko

a

(n

+

- <n

+

>) (1)

where C. is the creation operator for an electron

with momentum k and spin a, n is the number

ope-rator for an electron of spin

a

at the impurity

site and U is the Coulomb repulsion of electrons

on the impurity. We will consider a kinetic energy

linear in the momentum near the Fermi level, i.e.

e = v (k-kp) where k_ and v_ are the Fermi

momen-tum and velocity Tespectively. E' = E - U <n >+

0 * J- a B ~a

2

characterizes the energy difference of the impurity

levels with respect to the Fermi energy. Here, B

is the external magnetic field and a takes the

va-lues + 1.

The localized impurity electrons may

dif-fuse into the metal due to the overlap between the

Wannier states which is the same as on the regular

lattice. This is the main difference with respect

to Anderson's model where the overlap with the

im-purity electrons is considered as an additional

adjustable parameter.

We derive here the Ward identity relating the vertex function and the one-electron propaga-tor within the long-time limit. We define the following auxiliary three-time correlation func-tions

Faa,(t;p,T) =-i<T{CCT(t)ppa,(T)C*(0)}> (2) where Co = Y, C, and p „ = Y, C, C, . is

essen-ce ka pa '•kTca k+pa

tially a boson-operator /4/. Considering the time evolution with respect to T we obtain

E

•£

-

V

F 3

F

ac

(t

5P'

T)= u

^q W ^ ' ^

+

+ i

\_S(i) -

6(-t - T ) H G

a

(t)

Li 37 " V O Vo

(t;

P'

T)

"

V

*l

qP

a

ff(t

"'

T> 0 )

Here Ga(t) = -i< T{Cff(t)C*(o)}> . Laplace

transfor-ming equation (3) with respect to T we have an algebraic system of equations, which solved for F„ ,(t;Z..) = Y F ,(t;q,Z.) yields era' m '•q aa' n m

' ^ . v - v ^ o ^ - a

(I

P

Z~F]

L ~

U

l> Zm-v

F

P J J

and F (t;Z„) = F (t;Z- ) U Y - (5) a-aK ' m' acr m' [L ^-Vpp J v ' Here, Z = 2H imT are the thermodynamical Bose poles.

The vertex function can be obtained by La place transforming equation (4) with respect to the time t (6 = 1/1)

-0 J dtc"*1^,:;^) = %(Z„.>UeffCVlX'VV

"WW" <

6

>

Defining a n e f f e c t i v e ~ o t e n t i a l The s u s c e p t i b i l i t y i s given by t h e e f f e c t i v e p o t e n t i a l s Duo, and i s i d e n t i c a l t o the RPA (random phase approximation) r e s u l t . Following Yamada's / I / we o b t a i n f o r t h e v e r t e x a n a l y s i s of t h e Anderson model we s e p a r a t e even

1 ( 8 ) and odd p e r t u r b a t i o n o r d e r s i n U and w r i t e ( z n , z n + ~ ; - z a ) =

I

-

1

Xo = 1

F ( 'even + 'odd ) ' 'even

-

I - ( U / V ~ ) Z '

Here, Z =IIi (2n+l)T a r e thermodynamical fermion _n (U/vF) ₍₁₈₎

p o l e s . 'odd' 1 -(U/vF) 2

The t h r e e - l e g v e r t e x e q u a t i o n

(8)

c o r r e s - We have now t h a t f o r b o t h t h e Anderson and t h e ponds t o t h e s c a t t e r i n g of one fermion q u a s i p a r t i c l e Wolff models t h e s p e c i f i c h e a t can be w r i t t e n a s w i t h the emission of a Bose e x c i t a t i o n . I n analogy _(yo = 2n2/3vF)

t o t h e electron-phonon i n t e r a c t i o n we can d e f i n e 6Cv = Yo 'even (19) t h e p o l a r i z a t i o n through t h e Dyson e q u a t i o n f o r t h e and t h e s u s c e p t i b i l i t y t o s p e c i f i c h e a t r a t i o a s e l e c t r o n - h o l e e x c i t a t i o n s . The p o l a r i z a t i o n i s gi- yoxT/6CV = 1 +

g

ven by t h e e x p r e s s i o n odd 'even

For t h e c a s e of t h e Wolff model i t reduces t o na (z,) = T

In

G~ ( Z J ~ ( z n , z n + % ; - ~ ) G ~ ( Z ~ + Z , ) =

I 1

+

U/v,.

= ( 'P zm-vFP ) T

In

C G ~ ( Z ~ ) - G ~ ( Z ~ + Z . ~ ) ~ (9)

The thermodynamic p r o p e r t i e s were a l r e a d y only the asymptotic part of Go(Z ) f o r large will o b t a i n e d by M a t t i s 141 through t h e e x c i t a t i o n spec- s u r v i v e t h e summation over t h e thermodynamical p o l e s . trum of t h e bosonized Hamiltonian. The p r e s e n t r e - Hence the dressed polarization is equal to the bare s u l t ~ a r e com~lementary t o t h o s e o b t a i n e d by M a t t i s one / 4 / and Fogedby 151. The d e t a i l s of t h e c a l c u l a t i o n

C 1

no

( z $

=

(

1

-=---

)

T

1

{ G ~ ~ ( Z )-G,,("~Q) = a r e published elsewhere 161.

P 'm V ~ P n

References Here, we have a complete c a n c e l l a t i o n between v e r t e x

and s e l f - e n e r g y c o n t r i b u t i o n s . / I / Yamada, K., Prog. Theor. Phys.

53

(1975) 970 We write D y s o n ~ s equation f o r the effective 121 NoziPres, P . , J. Low Temp. Phys.

17

(1974) 31 p o t e n t i a l s Duo, between e l e c t r o n s w i t h s p i n o=

2

131

Wolff, P.A., Phys. Rev.

124

(1961) 1030 D++ = UII-D-+

,

D-- = UII+D+-

,

D+-=D-+=U+UII-D-- (1 1) / 4 / M a t t i s , D.C., Ann. Phys. (N.Y.)

89

(1975) 45 They y i e l d t h e f o l l o w i n g e f f e c t i v e i n t e r a c t i o n / 5 / Fogedby, H . C . , J. Phys. C

10

(1977) 2869

( 1 2 ) / 6 / S c h l o t t m a n n , P . , P h y s . R e v . B ~ ( 1 9 7 8 ) i n D++(Z.~) = D--(ZJ = UII(Z.dUeff(Z$ pr-int

.

and D+-(Z.m> = D-+(Z,$ = Ueff

(Zd

(1 31 where Ueff(Zm) i s d e f i n e d by e q u a t i o n ( 7 ) . - With t h e a i d of t h e v e r t e x and t h e e f f e c t i v e p o t e n t i a l we can c a l c u l a t e t h e e l e c t r o n s e l f e n e r g y

1

= - T

lm

Go(ZQ++Z.$ru(Z,, Zn+zm;-ZJDao<ZJ (14) Using Dyson's e q u a t i o n we o b t a i n a n i n t e g r a l equa- t i o n f o r

I

G i A ( z d Go(z,) = 1

-

T ~ , ( z ~ ( f + O ~ ( Z , ~ Z , , J m (15) where Goq (Z,d i s given by

Goo(Za> = Goo(Zn.)/C 1 + E'oGuo(Zn)) (16) Goo i s t h e Green's gunction corresponding t o t h e one-body s c a t t e r i n g problem o f f t h e p o t e n t i a l E ' For small r e a l e n e r g i e s and zero temperature we have Go(.) = G ~ ~ ( . ) exp { i n ( U / v ~ ) 2

_{x w + ~ : ,}