EFFECTIVE EXCHANGE INTERACTIONS AND MAGNETIC GROUND STATE OF STRONGLY CORRELATED ELECTRONS

(1)

HAL Id: jpa-00217796

https://hal.archives-ouvertes.fr/jpa-00217796

Submitted on 1 Jan 1978

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

EFFECTIVE EXCHANGE INTERACTIONS AND

MAGNETIC GROUND STATE OF STRONGLY

CORRELATED ELECTRONS

J. Spalek, A. Oleś, K. Chao

To cite this version:

(2)

JOURNAL D€ PHYSIQUE

Colloque C6, supplkment au

no

8,

Tome

39,

aolit 1978, page ~ 6 - 7 7 0

E F F E C T I V E EXCHANGE INTERACTIONS AND MAGNETIC GROUND STATE OF STRONGLY

CORRELATED ELECTRONS

.l. S p a l e k , A.M. 01e&* and K.A. ~ h a o ~

Physics Department, ImperiaZColZege, London,

England

I n s t i t u t e of Physics, Jage

l

Zonian University, Cracow,

PO

land

ll

Physics Department, Linkoping University, Sueden

R6sumd.- Nous a p p l i q u o n s une n o u v e l l e t r a n s f o r m a t i o n c a n o n i q u e 1 deux s y s t c m e s :

i / e l e c t r o n s d a n s une bande d t r o i t e , e t ii/ i m p u r e t 6 magndtique. C e t t e t r a n s f o r m a t i o n

nous donne u n H a m i l t o n i e n m a g n l t i q u e pour chacun d e c e s s y s t s m e s ; une mdthode d e f o n c - t i o n s d e Green a v e c d6couplage c o n s e r v a n t l e moment, nous permet a l o r s d ' o b t e n i r l e u r s p r o p r i l t d s . Nous o b t e n o n s e n t r e a u t r e s une l o i d e Curie-Weiss pour l a s u s c e p t i b i l i t i 4 d e l ' i m p u r e t l , e t prouvons qu'une bande 1 m o i t i d p l e i n e

-

c o r r e s p o n d a n t 1 un i s o l a n t d e Mott

-

a un $ t a t f o n d a m e n t a l a n t i f e r r o m a g n 6 t i q u e d e H e i s e n b e r g .

A b s t r a c t . - A new c a n o n i c a l t r a n s f o r m a t i o n i s a p p l i e d t o t h e s y s t e m s of e l e c t r o n s i n a narrow band and m a g n e t i c i m p u r i t y , which e n a b l e u s t o d e r i v e e f f e c t i v e H a m i l t o n i a n s f o r t h o s e s y s t e m s . F u r t h e r we s t u d y t h e i r p r o p e r t i e s v i a s i n g l e - p a r t i c l e Green f u n c t i o n s w i t h t h e moment c o n s e r v i n g d e c o u p l i n g . The most i n t e r e s t i n g r e s u l t s o b t a i n e d a r e t h e Curie-Weiss law f o r t h e i m p u r i t y s u s c e p t i b i l i t y and a proof t h a t f o r a p a r t i c u l a r c a s e o f a h a l f - f i l l e d band. C o r r e s p o n d i n g t o t h e Mott i n s u l a t o r , t h e ground s t a t e i s t h a t of H e i n s e n b e r g a n t i f e r r o m a g n e t ( i . e . t h e b a n d w i d t h of q u a i s p a r t i c l e s t a t e s v a n i s h e s ) .

I . THE EFFECTIVE INTERACTIONS.- 1 . 1 . THE CANONICAL means t h a t t h e hopping w i t h f o r m a t i o n o f doubly

TRANSFORMATION.- We decompose t h e H i l b e r t s p a c e of occupied s i t e i s i m p r o b a b l e , e x c e p t i n v i r t u a l H a l m i t o n i a n H i n t o s u b s p a c e s ( e . g . by p r o j e c t i o n t r a n s i t i o n s . I n s u c h a s i t u a t i o n t h e b a r e band i s w i t h o p e r a t o r Pi f o r i - t h s u b s p a c e ) s u c h t h a t t h e s p l i t i n t o subbands due t o e l e c t r o n c o r r e l a t i o n s mean d i f f e r e n c e ' b e t w e e n e n e r g y l e v e l s w i t h i n e a c h and f o r a number of e l e c t r o n s p e r atom n < 1 o n l y

\

s u b s p a c e i s much s m a l l e r t h a n t h e c o r r e s p o n d i n g t h e l o w e s t one i s i m p o r t a n t . I t i s d e s c r i b e d by one between them. Then a p p l y i n g a c a n o n i c a l t r a n s - t h e e f f e c t i v e H a m i l t o n i a n

f o r m a t i o n t o H we remove from t h e system h i g h l y _I _l

_-

improbable t r a n s i t i o n s between s u b s p a c e s and r e - H a t c i j > b i a b j a + T(K-J-

7

J)

1 "

i a v j a ' + J

Z

"

< i j >

p l a c e them by t h e v i r t u a l p r o c e s s e s which l e a d t o

<ii,,$,i.zj

,

a u S (2) a h i g h e r o r d e r c o u p l i n g w i t h i n a g i v e n s u b s p a c e _{where b . = a . ( I - n .} _{) ,}_via=b$bia, _{and t h e e f f e c -}

( f o r d e t a i l s s e e / I / ) . l a 15 1-0

t i v e exchange c o n s t a n t J 2(t+V)2/(U-K+J)-J. The INTERACTIONS IN A NARROW BAND.- We f i r s t con- n e x t , h i g h e r subband c o n t a i n s d o u b l y o c c u p i e d s i -

sider a narrow with all interac- t e s . I t is n e g l i g i b l e i f we remain i n t h e r a n g e of

t i o n s between t h e ' n e a r e s t n e i g h b o u r s < i j > i n c l u d e d temperatures

2zl l

the llubbard ( c . f . a l s o / 2 , 4 / ) . model (K=J=V=O) we r e c o v e r known r e s u l t s / l / .

We h a v e f u r t h e r e x t e n d e d t h e method t o a d o u b l y g e n e r a t e (1=1,2) band w i t h t h e b a r e Hamil- t o n i a n / I /

1'

"

I

n i a n j a ~

-

t .

X i * S j +

v

c ~ j ~ i - a a ~ a

a j a

H

<$JP < I J > i j l t i j l a ; i a a I j o + 'I ilnli+nli++

+

+ h.c.1 + J a i + a i + a j + a j + a < l J > ( 1 )

'

' 2 nli+nl.i+ +

Ea

" a i a " ~ i a (3) il

~ h ~ last two terms are c o n t r i b u t i o n to hopping from where U. a r e i n t r a a t o m i c Coulomb i n t e r a c t i o n s f o r

t h e i n t e r a c t i o n between e l e c t r o n s . We make t h e ca- t h e e l e c t r o n s o n t h e same U1 and o n d i f f e r e n t U2 n o n i c a l t r a n s f o r m a t i o n i n t h e c a s e I ~ + v ( < < u - K o r b i t a l s . The l a s t t e r m i n (3) c o n t a i n s a l s o t h e

(3)

Ising part of the intraatomic exchange with exchan-

ge constant J=U2-U3 (the more realistic cases are

under study). The bare degenerate band splits into

many subbands and there is

a

ferromagnetic ground

state with orbital ordering for n=I 131, and anti-

ferromagnetic ordering with Hund's rule intraatomic

state for n=2. The effect of Hund's rule is weake-

ned by the intrasubband hopping as n deviates from

n-2.

1.3. INTERACTION BETWEEN IMPURITY AND A HOST.- We

treat the Wolff model with a magnetic imouritv at

0-th site /5/

as an example. The coupling between the impurity

and the host is specified by yand the double pri-

med sum in 141 excludes the terms either i-0 or

j = O .

Now,if the impurity

is close to the Fermi

level

and U>>

I E ~ - E ~ I ' L

t, we can consider the

E

d

level plus conduction band as a one subspace while

the doubly occupied impurity level with energy

U+ed formsanother. The result is similar to that

of Schrieffer and Wolff /6/ but with the singularity

in their original approach being removed 171. Fur-

thermore, we have obtained the condition of forma-

tion of localized moments within the moment-conser-

ving decoupling 181. Our results indicate that with

approaching the bottom

of

the conduction band

the localized moment is almost saturated.

Now consider the case in which the locali-

zed moment is most likely to exist, namely if

1

u + E ~ - E ~ J >>

1

t

1 l

t

l

<<U.Accordingly

w e remove

from /4/ the hopping processes with fortnation of

empty impurity site. We then obtain a Hamiltonian

The conduction electrons in the &chity of the im-

purity are thus polarized antiparallel to its mo-

ment. In the mean field approximation this gives

a

susceptibility of Curie-Weiss form with Q=2S (S,+l)

(

I+Y)~/C~E~(U+E~)}<O.

2. ITINERANT ELECTRONS VS. MOTT INSULATOR.- We

decoupling for the single-particle Green functions

181. Leaving the discussion of the phase diagram

to the paper /4/ we consider here only one parti-

cular feature of the solution for the Hubbard model.

In the antiferromegnetic phase the lowest

subspace described by /2/ splits into two subbands

of quasiparticle states with energies

with

where

-a

W1

=

'

S

'

S

.

> + m n.

>

-

-p,

and S

=

m

l

mo lo

=

<babia>,

2

=

?/t, nu

=

<niu>,

and /m,i/ is

a pair of nearest neighbours. The two-particle cor-

relation functions in

(6)

are calculated from the

Roth procedure /8/ with the result

<s:sT'

> =

-

S

'

/

I-n/-l,

and <nmania>

=

n n

-

s2/

1-n/2//

l-n/-l.

U

-a

The correlation functions s % / ~ ~ + w ~ ~ + O

as n+l. But

from

(

12) we see that in this case the bandwidth of

quasiparticle states vanishes and additionally, the

equation for magnetization reduces to the mean

field result for the Heisenberg antiferromagnet,

as one would intuitively expect for the Mott insu-

'l,

lator. Furthermore, when J

#

0

the degeneracy in

energy of the ground state for all magnetic phases

with n+1 /8,9/ is removed.

References

/I/ Chao, K.A., SpaZek,

J. and Oleb

A . . M . ,

J.Phys.

C10 (1977) L271; Phys. L e t t . W (1977) 163;

-

Phys.Stat;Solidi

(b)84

(1977) 747; Phys. Rev.

B, in press.

/2/ Spazek,

J. and Ole4

A.M., Physics B

B

(1977)

375. /4/

Spalek, J., Ole4,

A.M. and Chao,

K.A.,

in pre-

paration.

/5/