ANDERSON ORTHOGONALITY DUE TO LOCAL ELECTRON CORRELATION

HAL Id: jpa-00217843

https://hal.archives-ouvertes.fr/jpa-00217843

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.

ANDERSON ORTHOGONALITY DUE TO LOCAL

ELECTRON CORRELATION

H. Kaga, K. Yosida

To cite this version:

JOURNAL DE PHYSIQUE

Colloque C6, supplément au n° 8, Tome 39, août 1978, page C6-846

ANDERSON ORTHOGONALITY DUE TO LOCAL ELECTRON CORRELATION

H. Raga** and K. Yosida**

t Department of Physics, Niigata University, Niigata 950-21, Japan

Résumé.- On montre que la catastrophe d'orthogonalité due à la corrélation électronique locale U ap-paraît dans le modèle asymétrique d'Anderson, mais pas dans le modèle symétrique.

Abstract.- The orthogonality catastrophe due to the local electron correlation U is shown to exist in the asymmetric but not symmetric Anderson model

Does the Anderson orthogonality catastrophe /I/ arise in the presence of local electron-electron interaction ? If it does, what is responsible for it, its local character or the localization of char-ge (or spin) ? We show that the orthogonality catas-trophe results only in the asymmetric case but not in the symmetric cas of the Anderson model, and that the many-body orthogonality index can also be expressed in terms of change in local (d-) electron number / 2 / .

Separating out the mean-field Coulomb poten-tial U<n,>a, a, from perturbation the general asym-metric Anderson model is written :

H

o "

I

e

k V

a

k 0

+ V

l

(a

k0

a

da

+

Wka^

ka k

+ e

d

I

a

da

a

da "

U <

V o

2

(1)

H' = U ( nd +- < nd>0) ( nd +- < nd>0) (2)

where e, = e. + U<n,>_, e. is the d-level in the d d u O a

absence of U and <n,>„ the number of d-electrons in d O

the absence of H1. We study the overlap integral

between the two ground states, |f > = > without and l ^ = > with H', by a perturbation method : <f |y> = lim <Y |s(e|<F >

= lim exp[C(t)] (3)

Here C(t) E <S<t)>. is the connected part of the S-matrix.

In order to investigate the possibility of the orthogonality catastrophe we are interested, among various connected terms in each order in C(t) _

Permanent address

Present address : Department of Physics and Materials Research Laboratory, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801 USA

** The Institute for Solid State Physics, The Uni-versity of Tokyo, Japan

only in the divergent ones for t-*», which turn out to be at most logarithmic. Detailed examination tells us that logarithmic divergence appears as (In t) only in some particular connected diagrams of fourth and higher orders ; second- and third-order connected diagrams give no divergence. Figu-re 1 shows the two typical examples (a), (b) of such renormalized diagrams in time-representation, in which both vertex and self-energy corrections are assumed to have been made.

mm |HfMj

(o) (b)

Fig. 1 : The two basic renormalized divergent con-nected diagrams (a),(b). The halves divided by dot-ted lines are the diagrams that give the dominant contributions for change in the localized d-elec-tron number.

These two divergent contributions C (t) and C, (t) a b

are :

C ( t ) = (UU)

2

f d t ( d t [ d t [ a t EG?

E

G?

3

G?

1(

G?ri +

Jo ' o 'o >o

CG?

2

Gg

1

G?,G°,1

r

f ™

A

| V } » l n t (4)

C ( t ) 4 HHA I

2 { o

V } l n t (5)

b TT(A2+e.2)

I

d

J > >

where G.. = G ( t . - t . ) a* = a* (0) r00 (0) G°(t) = - | g°(u)e"x a , tdto, t > 0 ( t < 0 ) (6) 17 J

o,(-~)

" l o J - m

Wl+ Wg- Wp

+

W gO(w) = -iA sgnu

(w-E~)

'+A~

A

= r p v 2 , p is the density of s-electron states at the Fermi energy E ~ , from which all energies are measured. There are two other divergent diagrams

(c), (d) (not shown) obtained by reversing the electron-lines (6) to the holes-lines(+) and vi-

ce-versa for down-spin ( J . ) in figure 1 (a) and (b) and four diagrams constructed by interchanging up- (+) and down-spins (+) in (a)c(d). Putting toge- ther the total divergent contribution C(t) amounts to :

which becomes an effective interaction constant of real dimensionless quantity except the sign. Equa- tions (3), (9) and (10) demonstrate the following results. (i) In the general asymmetric case, where ~ ( U , E )

#

0, the ground state overlap <S(t)> vani-

d 0

shes for t-rm ; (ii) in the case of the so-called electron-hole symmetry, cd = cod + U/2 = 0 where

0

~ ~ = O f o r U = O a n d ~ = ~ ~ + U / 2 = O f o r U + O d d we have y(U,O) = 0 and thus no orthogonality catas-

trophe exists.

We calcutate change in the d-electron loca- lization due to HI, 6n = <n _d >

-

<n _{d o} >

.

'

= lim (-i&)rdtlIt1dt

E

~

1 + ~ 0 ~ 0 ~ 0

~

~

;]

G

~

T1+T+O -OD -0a 2 Ti 12 2T T2 21 IT

& Y 2 ~ ! ; 1

(1 1)

We notice in figure 1 that the diagrams for the two terms of 6~(~)corres~ond exactly to the halves of the diagrams Ca(t) and Cb(t) if we put r, T' on the central dotted lines ; in fact, the diagrams of 6ii(2) a11 appear in the C(t) diagrams. We obtain

According as whether E ~ > O or Ed<O, we have SE(~'>O or 6~(~)<0 ; thus the effective coupling constant is given by -~(U,E~) and this is attractive for cd>O and repulsive for cd<O, which is physically natural. From eqs. (9) and (12)

We expect that the Anderson orthogonality catastro- phe holds quite generally for any local perturba- tion of electron-electron interaction as well as one-electron scattering potential when the change of local charge is accompanied.

References

/1/ Anderson,P.W., Phys.Rev.Lett. 2(1967)1049 /2/ Kaga,H. and Yosida,K., Prog.Theor.Phys.s(l978)