FINITE SIZE SCALING APPROACH TO ANDERSON LOCALIZATION

(1)

HAL Id: jpa-00220709

https://hal.archives-ouvertes.fr/jpa-00220709

Submitted on 1 Jan 1981

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are published 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.

FINITE SIZE SCALING APPROACH TO ANDERSON LOCALIZATION

J. Pichard, G. Sarma

To cite this version:

J. Pichard, G. Sarma. FINITE SIZE SCALING APPROACH TO ANDERSON LOCALIZATION.

Journal de Physique Colloques, 1981, 42 (C4), pp.C4-37-C4-45. �10.1051/jphyscol:1981405�. �jpa-

00220709�

(2)

JOURNAL DE PHYSIQUE

CoZZoque 64, suppl6ment au nOIO, Tome 42, octobre

1981

FINITE SIZE SCALING APPROACH TO ANDERSON LOCALIZATION

J . L . Pichard and G. Sarma

Service de Physique du SoZide e t de ~Gsonance Magngtique, CEN-SacZay,

B.P.

N02,

91190

Gif-sur-Yvette, France

A b s t r a c t .

-

The Anderson l o c a l i z a t i o n problem i s s t u d i e d by a new s c a l i n g t h e o r y of f i n i t e systems which has been proved t o be powerful f o r d i f f e r e n t phase t r a n s i t i o n problems [ 11.

I n t h r e e dimensions, t h e r e i s a t r a n s i t i o n a t a c r i t i c a l v a l u e Wc of t h e d i - s o r d e r parameter from a r e g i o n of e x p o n e n t i a l l y l o c a l i z e d s t a t e s t o a r e g i o n of

extended

s t a t e s . When W d e c r e a s e s t o Wc, we f i n d a v a l u e e q u a l t o 0 . 6 6 f o r t h e c r i t i c a l exponent v urelative t o t h e divergence of t h e l o c a l i z a t i o n l e n g t h I n two dimensions, a c r i t i c a l v a l u e Wc of t h e d i s o r d e r parameter s e p a r a t e s a r e g i o n of e x p o n e n t i a l l y l o c a l i z e d s t a t e s (W

>

Wc) from a r e g i o n of

"quasi extended" s t a t e s

which a r e

non square s m a b l e

and f a l l o f f a s ~ - n ( ~ ) . We g i v e t h e v a r i a t i o n of t h e exponent n(W) i n t h e whole "quasi extended" r e g i o n . On t h e o t h e r hand, we show t h a t t h e divergence of t h e l o c a l i z a t i o n l e n g t h when W d e c r e a s e s t o Wc i s now c o n t r o l l e d by an

essential singularity.

The behaviour of t h e dimensionless oonductance i s given i n a l l t h e c a s e s . I n p a r t i c u l a r , i n t h e two dimensional wehk d i s o r d e r phase, i t i s shown t o obey

lL

-2n(W

a

power law decay \$

v e r s u s t h e " s i z e " L.

I. INTRODUCTION.

-

The f i r s t p a r t of t h i s paper w i l l be devoted t o a review of t h e method and t h e r e s u l t s t h a t we d e s c r i b e d i n two previous p u b l i c a t i o n s 121, 131. I n a second p a r t , we w i l l e x t r a c t from t h e s e r e s u l t s t h e behaviour of t h e zero tempera- t u r e conductance of two o r t h r e e dimensional d i s o r d e r e d systems i n t h e absence of any i n e l a s t i c p r o c e s s e s . A comparison w i l l be made w i t h o t h e r t h e o r i e s .

The b a s i s of our method l i e s i n t h e s t u d y of t o p o l o g i c a l l y one dimensional systems, taken i n f i n i t e i n one d i r e c t i o n , as a f u n c t i o n of t h e f i n i t e v a l u e R of t h e

(d-1) remaining t r a n s v e r s e d i r e c t i o n s . Such a n approach

never provides a d i r e c t simu- l a t i o n

of t h e d-dimensional i n f i n i t e l a t t i c e , b u t allows t o p r e d i c t i t s behaviour thanks a

scaling theory

whose parameter i s p r e c i s e l y t h e f i n i t e t r a n s v e r s e l e n g t h R.

With t h e h e l p of

OseZedec's theorem

on random m a t r i x p r o d u c t s , we d e f i n e d and c a l c u l a t e d i n [ 2 ] t h e

reZavrmt

l o c a l i z a t i o n l e n g t h Lmax f o r " s t r i p s t ' and "bars", l e a d i n g r e s p e c t i v e l y t o two o r t h r e e dimensional systems. These c a l c u l a t i o n s were made f o r a t i g h t b i n d i n g model a t zero energy (band c e n t r e ) w i t h "diagonal" d i s o r d e r c h a r a c t e r i z e d by a r e c t a n g u l a r d i s t r f b u t i o n of width W of t h e random p o t e n t i a l .

11. THE OSELEDEC'S THEOREM AND ITS IMPLICATIONS.

-

For d e f i n i t e n e s s , l e t us c o n s i d e r on an i n f i n i t e s t r i p of width R t h e Anderson model r e p r e s e n t e d i n t h e t i g h t - b i n d i n g approximation by t h e S c h r 6 d i n g e r e q u a t i o n

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981405

(3)

JOURNAL DE PHYSIQUE

where sn i s d i s t r i b u t e d randomly between - ~ / 2 and ~ / 2 . The l a t e r a l boundary condi- t i o n s a r k taken t o be p e r i o d i c ($n,m+R = I ) ~ , ~ ) . When n goes t o i n f i n i t y , t h e v e c t o r

%(E), whose 2R components a r e :

n i s o b t a i n e d from a n i n i t i a l v e c t o r Ao(E) by a p p l y i n g t h e m a t r i x product hdn=

II

Ti, where Ti i s a r e a l (2Rx 29,) t r a n s f e r m a t r i x . i= 1

These t r a n s f o r m a t i o n s and t h e i r product a r e s y m p l e c t i c , i . e . : t h e i r eigen- v a l u e s a r e p a i r s whose elements a r e t h e i n v e r s e of each o t h e r . ~ s e l e d d c ' s theorem s t a t e s [ 4 1 t h a t

a matrix r e&sts,

d e f i n e d by :

*

^1,211

^,

r

⁼^{l i m}(Mn Mn) n+

-

where

*

denotes a m a t r i x t r a n s p o s i t i o n .

I f we c a l l exp y l , . . . , e x p ygR t h e e i g e n v a l u e s of

r

and v l ,

...,

^{V ~ R}t h e corresponding e i g e n s p a c e s , t h e n t h e p r o p e r t y :

l i m

;

1 kn ~ ~ = Mya ~ u ~ ~

n +-

h o l d s f o r any v e c t o r u E va. These ya a r e the"Lyapounov c h a r a c t e r i s t i c exponents"

(L.C.E.) of t h e random m a t r i x p r o d u c t

%.

T h i s i m p l i e s , f o r any system a s a s t r i p o r a b a r , t h a t any a r b i t r a r y v e c t o r ho(E) i s a l i n e a r combination of a s e t of v e c t o r s whose asymptotic behaviour i s g i v e n by t h e d i f f e r e n t L.C.E. ya. The

physical

wave- f u n c t i o n s , however, a r e

necessarily

a l i n e a r combination of only R f u n c t i o n s decaying a s exp

- Y a l ~ I ,

where appear t h e only R p o s i t i v e L.C.E. (We r e c a l l h e r e t h a t t h e 2%

L.C.E. go by p a i r s of o p p o s i t e v a l u e s ) . I t i s obvious t h a t f o r l a r g e Z and f o r any

f i n i t e

v a l u e of R , t h e decay a t i t i f i n i t y i s t h e n c o n t r o l l e d by t h e

singZe term

c o r r e s ponding t o

the smaZZest

L.C.E. ymin = ( L ~ ~ ~ ) - ~ . Thus

hx

i s indeed t h e r e l e v a n t

localization Zength

f o r " s t r i p s " and "bars".

111. THE LOCALIZATION LENGTH

ha,. -

The d i f f e r e n t L.C.E. have been c a l c u l a t e d w i t h t h e h e l p of a method d e s c r i b e d i n t h e case of e r g o d i c problem of h a m i l t o n i a n and d i s s i p a t i v e dynamical systems [ 5 ] , [ 6 ] . The numerical p r o c e s s i s based on a mathema- t i c a l method which r e l a t e s t h e sum of t h e f i r s t 6 L.C.E. t o t h e asymptotic divergence of a volume d e f i n e d by b orthogonal u n i t v e c t o r s . Contrary t o o t h e r "simulation"

methods, we emphasize t h a t t h e s e c a l c u l a t i o n s can be performed

t o any fixed accuracy.

The p r e c i s i o n a t t a i n s 1% f o r a product of 104 m a t r i c e s . We have a l s o checked t h a t t h e L.C.E. go by p a i r s of o p p o s i t e v a l u e s .

The f i g u r e s (1) and ( 2 ) show t h e v a r i a t i o n of t h e l o c a l i z a t i o n l e n g t h ,L v e r s u s t h e width R of a s t r i p o r t h e s i d e R of a b a r a t t h e band c e n t r e ( E = O ) f o r d i f f e r e n t v a l u e s of t h e d i s o r d e r parameter W.

I V . SCALING STUDY OF THE LOCALIZED REGION.

IV.a) Three dimensional systems.

-

The f i g u r e 2 shows t h e v a r i a t i o n of Lmax v e r s u s t h e s i d e R of a b a r f o r d i f f e r e n t v a l u e s of W. For W > We, Lmax converges f o r i n c r e a - s i n g R towards a f i n i t e l i m i t wbich i s t h e

ZocaZization Zength 5,

of t h e

i n f i n i t e

Lattice.

Our a n a l y s i s o f t h i s r e g i o n i s based upon t h e fundamental

seazing m s a t z :

(4)

Figure 1 : ,,L versus the width

R

^ofa

s t r i p

Figure 2 : Lmax versus of the _{s i d e}

R

of a

baa.

(5)

JOURNAL DE PHYSIQUE

A t t h e t r a n s i t i o n , where &(W) becomes i n f i n i t e , t h e a n s a t z (1) obviously i m p l i e s a l i n e a r growth of Lmax v e r s u s R, which g i v e s t h e c r i t i c a l v a l u e of t h e d i s o r d e r parameter Wc 19 t 0.5. Our method f o r t h e d e t e r m i n a t i o n of t h e c r i t i c a l behavior of

cao

c o n s i s t s i n t h e r e s e a r c h of which p o s t u l a t e d divergence w i l l a c t u a l l y g i v e t h e s c a l i n g function f d e f i n e d by ( 1 ) . We have looked f o r a divergence of t h e form:

Ern -

^(w-w,)-', s o a s t o map t h e r a t i o s Lmax(R,W)/crn(W) of t h e l o c a l i z e d r e g i o n i n t o t h e s i n g l e f u n c t i o n f ( x ) . This requirement i s s a t i s f i e d with

v -

^0.66 ^?0.02 i n agreement w i t h t h e r e s u l t of S t e i n and Krey [71. The s c a l i n g f u n c t i o n f i s shown i n f i g u r e 3 .

1V.b) Two dimensional systems. - The f i g u r e 1 shows t h e v a r i a t i o n of Lmax v e r s u s t h e width R of a s t r i p f o r d i f f e r e n t v a l u e s of W. For W > Wc (Wc

-

^5.95

⁺

0.05). Lmax a g a i n converges towards a f i n i t e l i m i t f o r i n c r e a s i n g R, corresponding t o exponen- t i a l Z y l o c a l i z e d s t a t e s . But h e r e , we f a i l e d i n a l l our t r i a l s f o r f i n d i n g a s c a l i n g f u n c t i o n f ( x ) w i t h a power law divergence of

&.

This l e d us t o look f o r an e s s e n t i a l s i n g u l a r i t y of t h e form 5,

-

exp(w-wc)-9. I n such a c a s e , a new s c a l i n g assumption has t o be p o s t u l a t e d

[a].

T h i s t u r n s o u t t o be s u c c e s s f u l and g i v e s a v a l u e of 9

-

^0.5

⁺

0.1. The s c a l i n g f u n c t i o n i s shown i n f i g u r e 4.

V . QUANTITATIVE ANALYSIS OF THE EXTENDED ( o r "quasi extended") REGION

For W < Wc, L m a x d i v e r g e s v e r s u s i n c r e a s i n g R , which i m p l i e s t h a t we cannot d e f i n e a f i n i t e l o c a l i s a t i o n l e n g t h f o r t h e i n f i n i t e l a t t i c e . The problem i s t o r e l a t e t h i s t o t h e a c t u a l asymptotic behaviour of t h e wave f u n c t i o n i n t h e i n f i n i t e l a t t i c e .

I n t h e whole r e g i o n W < Wc, we f i n d t h a t : Lmax

-

^a(W)^R^{and Lmax}

-

^b(W) ^R²

r e s p e c t i v e l y f o r s t r i p s and b a r s . The n e c e s s i t y of such behaviours can be t h e o r e t i - caZly shown i n a weak d i s o r d e r ( o r weak s c a t t e r i n g ) t r e a t m e n t .

It i s now n a t u r a l t o a s s i g n some " u n i v e r s a l " meaning t o :

R 1

lim - = - R2 1

a(w) o r l i m

-

=

-

& - + a Lmax ~ + mLmax b (W)

r e s p e c t i v e l y . These two q u a n t i t i e s a r e indeed given by t h e f o l l o w i n g g e o m e t r i c a l i n v a r i a n t :

These i n t e g r a l s a r e d e f i n e d on a very long s t r i p ( o r b a r ) and t h e boundary c o n d i t i o n s a r e f i x e d i n t h e middle, s o a s t o g e t t h e exp(-;kL) decay i n both d i -

\ Lmax'

r e c t i o n s . Because of t h e p e r i o d i c l a t e r a l boundary c o n d i t i o n s , t h e f l u x has t o be taken only through t h e upper and lower s i d e s ( o r f a c e s ) of a s t r i p ( o r a b a r ) , and

(2) indeed g i v e s

- -

a(W) ( o r 1

- rn).

¹ Now, i f we assume t h a t I$(R,w)

I

i n t h e i n f i n i t e l a t t i c e f a l l s o f f i s o t r o p i c a l l y , t h e i d e n t i f i c a t i o n of (2) w i t h t h e i n f i n i t e l a t t i - c e l i m i t g i v e s

1 d

a(W) - n R ;iii l o g l $ ( ~ )

I

i n two dimensions I - Zn I2

$

^logl$(R)

I

i n t h r e e dimensions b (W)

(6)

F i g u r e 3 : The s c a l i n g f u n c t i o n f f o r t h r e e dimensional systems W

2

_Wc.

F i g u r e 4 : The s c a l i n g f u n c t i o n f o r two dimensional systems W Wc.

(7)

JOURNAL DE PHYSIQUE

V . a ) T h e t h r e e dimensional system. - I n t e g r a t i n g (4) g i v e s t h e following

asymptotic

behaviour :

I$(R,w)

I

I+co(w)

I

exp 2Tlb(W) 1

T h i s shows t h a t i n t h r e e d i m e n s i o n s , t h e d i s o r d e r f o r W < Wc i s

insuffGeient

f o r l o c a l i z i n g t h e s t a t e s , I$(R,w)

I decreasing

towards a

f i n i t e

l i m i t I I J J ~ ( W )

I

^when

I$(R,W)

I

i s c o n s t r a i n e d t o b e e q u a l t o u n i t y i n some f i n i t e r e g i o n of t h e system.

The l i m i t i s g i v e n by : l $ C O ( ~ ) I = exp

-

& w h e r e c i s some c o n s t a n t . A non z e r o v a l u e of

1$001

i s c h a r a c t e r i s t i c of

extended s t a t e s .

The f i g u r e 5 shows a s i t should be t h a t

l ~

goes t o one i n t h e

~ ~ l

l i m i t of v a n i s h i n g d i s o r d e r and t o z e r o when W reaches We. So ($,(w)

I

appears t o us a s a n a t u r a l "order parameter" i n t h e r e g i o n of extended s t a t e s . To our knowledge, such a concept has n o t been proposed t i l l now.

-

. -

I n t e g r a t i n g (3) g i v e s

I $ ( R , ~ >

1 -

^R-~(') w i t h n(w) =

-

_n_a(w>1

This r e s u l t proves a

power Zau decay

of t h e wave f u n c t i o n s i n t h e

whole "quasi extended" region

(W

<

Wc). The knowledge of t h e numbers a(W) allows us t o p l o t n(W) v e r s u s W ( f i g u r e 6 ) . A t t h e t r a n s i t i o n , we f i n d ?j(Wc) = 1 ^a I n W e whose r e g i o n W < W,,thewave f u n c t i o n s a r e

non square s m a b Z e ,

which l e d us t o c a l l them

"quas7,-esterrded".

g i g u r e 5 : l + C O ( ~ )

1

v e r s u s W (d = 3) W < Wc.

(8)

Figure 6 : The exponent q(W) characteristic of the power-law decay of the

"quasi extended"

states and of the conductance. d = 2 W

<

^Wc.

VI.

RESULTS FOR THE CONDUCTIVITY.

-

For any topologically one dimensional system, it was shown above that *he wave functions decay as exp -yaJZI in

R independent

"Oseledec subspaces", thus giving

R trrmsmissim c o e f f i c i e n t s

Ta(Z) = exp

-

^2ya

1

^Z

I .

These Oseledec subspaces correspond to the

"channel" concept

of Anderson et a1 191.

The generalization of Landauer's formula [ l o ] gives the dimensionless conductance

It is obvious that for large Z and for any

i n i t e

value of R, this reduces to the

single term

corresponding to y min = (Lmax)-

.f .

V1.a) The

Zocakized

region.

-

The infinite lattice limit obviously gives g(Z) = exp

- -

22

i n any dimension.

Em

V1.b) The three dimensional

extended

region.

-

^g(Z,R) ⁼

b (W) R2

Taking the limit L

* ^kR,

^k³^{1 ,}

^R

^{-t m,} ^one^{g a t s}^g(Z,R) ⁼

- ^,

whence the con-

ductivity 2 2

(figure 7)

(9)

JOURNAL DE PHYSIQUE

It is interesting to note that b(W -Wc) is of the order of 0.05 which is close to Mott's prediction on the value of the minimum metallic conductivity [ I l l . However, for a definite conclusion about the minimum metallic conductivity, we have to look more carefully to the behaviour of b(W) close to W

.

Figure 7 : d = 3 W < W the conductivity multiplied by

-T

versus W.

e

V 1 . c ) The two dimensional

"quasi extendedtr

region.

-

For the strip, 2

z

g(Z,R)

-

^exp

^- ^&.

In the same way as for the wave function's decay, this implies the

seazing law

g(Z,R)

since

R

must disappean in the infinite lattice limit. This result, which gives a

vanishing

conductance for Z -t a, has been recently derived in a different way by Mott and Kaveh [ 1 2 ] , and a first order expansion in powers of q(W) gives the well known "logarithmic correction" approximation [ 131.

(10)

REFERENCES

1. B. D e r r i d a , L. de Seze and J. Vannimenus, I n t e r n a t i o n a l Conference on Disordered Systems and L o c a l i z a t i o n Rome, May 13-15, 1981

2. J.L. P i c h a r d and G . Sarma, J. Phys.

e,

L127 (1981) 3. J.L. P i c h a r d and G . Sarma, J. Phys. C , t o b e p u b l i s h e d 4. V.B, Oseledec (1968) Trans. Moscow Math. Soc.

19,

197-231

5. B e n e t t i n G. and Galgani L. (1979) I n t r i n s i c S t o c h a s t i c i t y i n Plasmas p. 93, G. Lava1 and D. G r e s i l l o n e d i t o r s .

6. I. Shimada and T . Nagashima (1979) Prog. Theor. Phys.

61,

1605 7. J. S t e i n and U . Krey, Z e i t . tiir Phys.

z,

287 (1979)

8. C.J. Hamer and M.N. B a r b e r , J . Phys.

e,

259 (1980)

9. P.W. Anderson, D.J. T h o u l e s s , E. Abrahams and D.S. F i s h e r , Phys. Rev.

E,

8 , 3529 (1981)

1 0 . R . Landauer, P h i l o s . Mag.

2,

863 (1970) 11. N.F. Mott, P h i l o s . Mag, t o be p u b l i s h e d 1 2 N.F. Mott and M. Kaveh, P r e p r i n t 1981

13.E. Abrahams, P.W. Anderson, D.C. L i c c i a r d e l l o and T.V. Ramakrishnan, Phys. Rev.

L e t t . - 42, 673 (1979).