L]~TT]~RE AL NUOVO CIMENTO VOL. 23, N. 10 4 Novembre 1978

The Anderson Local ization and Statistical Fluctuations.

E, C]/AMOUN a n d R. ABoU-CHACRA

Faculty o] Science, Lebanese University, Hadeth - Beirut, Lebanon

(ricevuto 1'11 Agosto 1978)

The mathematical complexity of the Anderson localization (1) has led many authors to use various approximate methods (3). These approximate methods can be divided roughly into two classes: the theoretical approximate theories and the direct numerical calculations. The numerical methods include the recent works of LICCIARDELLO and T~OULESS (3), WEArer, WILLIAMS and SRIVASTAVA (4.5). Such direct calculations imply the handling of information about a lattice sample. This information concerns mainly the eigenvalues and the eigenvectors, which require considerable computing storage and computing t ime for large systems. Thus, and despite the recent improvement by YOSHINO and OKAZAK: (~), we arc still limited to relatively small samples; and the extrapolation of certain localization parameters to infinite-size samples is not obvious (5). It is noticed that such small lattice samples lead to fluctuations in the numerical values of the various localization parameters. Such fluctuations could, apart from masking the position of the mobility edge, give the impression that some intermediate states exist between extended and exponentially localized.

We report here certain numerical observations concerning the eigenvectors obtained from finite systems, which show the difficulties involved in judging the localization of individual states. Then, an attempt to solve these difficulties is given by taking the average of the localization parameter over many equivalent samples.

Consider a lattice sample of N atoms, the corresponding t tamil tonian involving nearest neighbour interaction only is given by

H = E ~ili>(il § V E li><j[, i it

where we have one basis function [i} per site i. In the second sum i and j are nearest neighbours and V is the constant interaction taken to be uni ty in the calculations. The ei are the site energies and considered as independent random variables with a

(1) P. W. ANDERSON: Phys. Rev., 109, 1492 (1958). (2) D. C'. LICCARDELLO a n d E. N. E c o N o ) I o u : Phys. Rev. B, 11, 3697 (1975). (a) D. C. LICCARDELLO a nd D..]-. THOULESS: J. Phys. C, 11, 925 (1978). (4) D. WEA:RE a n d A. R. WILLIAMS: J. Phys. C, 10, 1239 (1977). (~) D. WEAIRE a n d V. SRIVASTAVA: J. Phys. C, 10, 4309 (1977). (~) S. YOSHINO a n d M. OKAZAKI: Solid State Commun., 20, 81 (1976).

367

~ 6 ~ E. CtIAMOUN and R. ABOU-CIIACRA

fiat distribution between =L W/2. With this above Hamiltonian the time-independent SchrSdinger equation leads to finding the eigenvalues and eigenstates of an N • N matrix. The original Anderson criterion of localization has been interpreted and put in the participation ratio criterion by B~LL and DEAN (7). The participation ratio of an eigenstate y~ is (N~j) -1, where aj is obtained from ~+ by

N 21

i=l i=l

with a~ being the amplitude at the site i of the eigenstate ~j. For extended states ~j vanishes like N -1 whereas it is finite for localized states, and it tends to one for a state completely localized on one sits. We are going to use the numerical values of ~ to determine the localized states and compare our results with those obtained by other workers.

We have used the Housholder technique (8) in our calculations to determine the eigenvalues and eigenstates of square lattices and diamond lattices for different values of the number of atoms N. The eigenvalues E~ obtained, in the energy range • (ZV ~ W/2), are numbered from the top end of the band to the bottom end. So, we refer to these eigenvalues, and their eigenstates, by their numbers unless otherwise is necessary. Periodic boundary conditions in all directions were assumed, and the constant V was taken to be unity. Consider two matrices of order N and the values of the e~ in each of these matrices taken randomly with the flat probability distribution of width W. So the set of the ei figuring in the first matr ix is not identical to that figuring in the second matrix, although they are all uniformly distributed between =L W/2. We could have many matrices like that, obtained for the same N and the same W, such matrices (or samples) are referred to later as statistically equivalent. The eigen- values for such samples lie in the same range =L (ZV~- W/2), and it was found nu-

o 1 l 1 l l l l

5 10

@

o 5 10

Fig . 1. - A plot s h o w i n g the s i g n of t h e e i g e n v e e t o r Y~L c o r r e s p o n d i n g to E_~ 4.65 for a s q u a r e l a t t i c e w i t h N = 100 a n d 1V = 1.5. T h e two k i n d s of s h a d i n g re fe r to t he s i g n of t he a m p l i t u d e , t he cross g i v e s the pos i t ion of the p e a k . The f igures (a) a n d (b) a r e for two s t a t i s t i c a l l y e q u i v a l e n t s a m p l e s w h e r e r162 i s r e sp ec t i ve ly equa l to 0.039 a n d 0.034.

(T) P . DEAN a n d R. J . BELL: Discuss . Faraday So t . , 50, 55 (1970). (s) J . ~r WILKINSON: The Algebraic .Eige~vatue Problem (London , 1963).

T H E A N D E I t S O N L O C A L I Z A T I O N A N D S T A T I S T I C A L F L U C T U A T I O N S ~

merical ly tha t t h e y are not the same for all samples but an e igenvalue E~. changes sl ightly f rom one sample to an equiva len t one. These f luctuat ions are due ma in ly to t he finite N, and the most i m p o r t a n t f luctuations seem to affect the eigenvectors .

Two kinds of f luctuat ions are observed: the posi t ion of the peak of the e igenvector and the shape of the e igenvector change f rom one sample to an equiva len t one. Fo r instance we give in fig. 1 for a square la t t ice of N -- 100 and I V = 1.5 the posit ion of the peak of t he c igenvector ~'~ for two different and equiva len t samples. The value of the peak is also different in the two cases. This behaviour is no t par t icular to this eigenstate, bu t i t was noticed, wi th var ious degrees, for different eigenstatcs and dif- ferent sample sizes. Calculations were carr ied also on N = 144 and 57- -225 systems. The pa rame te r ~ g iven by (1) was ca lcu la ted for each eigenstate y~. I t is not iced tha t for e igenvalues E i well into the region of localized eigenstates the pa rame te r ~ is not pract ica l ly affected by the f luctuat ions ment ioned above. But for the s ta tes in the region of the mobi l i ty edge, ~j varies by up to 20% from one sample to another sta- t i s t ical ly equiva len t .

The same analysis was done on the d iamond la t t ice wi th N = 216. The fluctua- t ions are s t ronger here than those of the square la t t ice . In order to analyse the ex-

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v

I 1.2

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0

0 0 0 0 0

0 O0 0 0 0

b) o

X X X X x X

X a)

XX X X X X

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I 2.8:

F i g . 2. - T h e r e s u l t s o f <log lai!>av o b t a i n e d f r o m t h e e i g e n v e c t o r y~j, f o r a d i a m o n d l a t t i c e , a n d c o r r e s p o n d i n g to E i ~_ 4.3 w i t h N = 216 a n d IV = 5. T h e a v e r a g e i s t a k e n h e r e o v e r s i t e s a t d i s - t a n c e r f r o m t h e p e a k of YJi" F o r t h e t w o e q u i v a l e n t s a m p l e s (a) a n d (b) t h e e x p o n e n t i a l l o c a l i z a - t i o n i s c l e a r i n t h e f i r s t s a m p l e a n d ~j i s r e s p e c t i v e l y e q u a l to 0.07 a n d 0 .061 ,

ponent ia l decay of the cigenstates we plot as an example in fig. 2 the logar i thm of the geometr ical average <Log lail>av, in te rms of r, where r is the dis tance f rom the peak of the e igensta te (9,o), and for two equ iva len t systems wi th N = 216, W = 5 and E~_ ~ 4.3. There is a f luctuat ion in ~s here of about 15~ .

An a t t e m p t was made also to smooth the f luctuat ions of the ~j; this was done numerical ly by consider ing the values of ~ obta ined from different equiva len t samples

(o) B . J . LAST a n d D . J . THOVLESS: J . P h y s . C, 7, 699 (1974) . (~o) E . CHA,~OVN a n d R . ABOL~-CIIACRA: Lef t . N u o v o C imen to , 22, 274 (1978) .

370 ~:. CHAMOUN and R. ABOU-CHAC~A

0.12

<5.> 0.10

0.08

0.06

0.04

0.02

a) / !

x /

x /

b)

/ /

/ /

/ /

I I / I I I i I I I

0 1 2 3 4 5 6 0 1 2 3 4 5 E E

F i g . 3. - T im r e s u l t s of <~j> as a f u n c t i o n of E, a r e p l o t t e d i n t h e t w o c a s e s : a) t h e s q u a r e l a t t i c e w i t h N = 100, W = 5 ; b) t h e d i a m o n d l a t t i c e w i t h N = 2 1 6 , W = 6 .

and then taking the average of these values (gj>. In our calculations the average <~.> was taken over five equivalent samples. Figure 3 gives the variation of (~> for the square and diamond lattice in terms of E. By comparison of these figures with similar ones obtained by LICCIARDELLO and THOUL]~SS (3), we notice that the variation of (~j> in terms of E is smoother which is an advantage if these curves are to determine the mobili ty edge E c. Here again (3), we do not notice a sudden change of <~j) in terms of E, which could be at t r ibuted to intermediate states between extended and expo- nentially localized. These results in fig. 3 are also in agreement with those obtained by Wv, AI1~E and SRIVASTAVA (s) which were calculated for larger-size systems.

The above calculations give us an insight into the difficulties encountered in inter- preting numerical calculations. Thus, we conclude tha t the localization of a state might be wrongly judged from the results of a numerical calculation on a finite sample. This error is greater the closer this state is to the mobility edge. The method of averaging over equivalent samples seems to be giving better results than previously found but it would be interesting if the distribution of ~j can be found, then a weighted average may be considered. However we feel that the problem of fluctuations due to the finite size of samples is still at the heart of the numerical calculations, and still needs a careful and detailed analysis before any decisive conclusion could be reached about the reli- ability of the numerical methods.