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THE DECIMATION METHOD AND ANDERSON LOCALIZATION
D. Weaire, C. Lambert
To cite this version:
D. Weaire, C. Lambert. THE DECIMATION METHOD AND ANDERSON LOCALIZATION. Jour-
nal de Physique Colloques, 1981, 42 (C4), pp.C4-47-C4-49. �10.1051/jphyscol:1981406�. �jpa-00220710�
JOURNAL DE PHYSIQUE
CoZZoque C4, supptbment au nolO, Tome 4 2 , octobre 1981
THE DECIMATION METHOD AND ANDERSON LOCALIZATION
D. Weaire and C.J. Lambert
Physics Department, University CoZZege, DubZin 4 , Ireland.
A b s t r a c t We d i s c u s s t h e a p p l i c a t i o n o f t h e d e c i m a t i o n method t o t h e problem o f Anderson l o c a l i z a t i o n . I n o n e d i m e n s i o n , t h e r e n o r m a l i z e d i n t e r a c t i o n s d e f i n e d i n t h i s method a r e f o u n d t o s c a l e t o w a r d s weak c o u p l i n g ( i n d i c a t i v e of l o c a l i z e d s t a t e s ) f o r a l l v a l u e s of t h e s t r e n g t h o f ( d i a g o n a l ) d i s o r d e r . I n two d i m e n s i o n s , t h e i n t e r a c t i o n s a p p e a r t o s c a l e t o w a r d w e a k l s t r o n g c o u p l i n g f o r l o w l h i g h d i s o r d e r . However t h e l a t t e r r e s u l t s r e - main p u z z l i n g , i n t h a t no c l e a r c o n n e c t i o n between t h e s c a l i n g b e h a v i o u r and t h e l o c a l i z a t i o n l e n g t h h a s emerged.
I n t r o d u c t i o n I n r e c e n t y e a r s v a r i o u s s c a l i n g t h e o r i e s o f l o c a l i z a t i o n h a v e been p r o p o s e d ( 1 - 3 ) . T h e s e i n c l u d e r e n o r m a l i z a t i o n i n k - s p a c e
( 1 ) a n d on t h e l i n e a r c h a i n g e n e r a t e d by t h e r e c u r s i o n method ( Z ) , a s w e l l a s t h e o r i e s b a s e d on s c a l i n g i n r e a l s p a c e ( 3 , 4 ) . Of t h e s e t h e most n o t a b l e i s t h a t w h i c h l e d Abrahams e t a 1 ( 4 ) t o a s s e r t t h a t a l l s t a t e s a r e l o c a l i z e d i n two d i m e n s i o n s . The"b1ock-spin" t r a n s f o r m a t i o n which t h e y u s e d is a u n i t a r y t r a n s f o r m a t i o n from t h e o r i g i n a l (Ander- s o n ) H a m i l t o n i a n t o o n e which i s e x p r e s s e d w i t h r e s p e c t t o a b a s i s s e t of e i g e n f u n c t i o n s o n b l o c k s o f sites, The b l o c k s a r e i n t u r n
,
com- b i n d i n t o l a r g e r b l o c k s , and s o on. I n t h e n u m e r i c a l r e a l i z a t i o n of s u c h a scheme i t i s c o n v e n i e n t t o r e t a i n o n l y t h o s e e i g e n f u n c t i o n s whose e i g e n v a l u e s l i e w i t h i n some r a n g e ( 5 ) o r , t a k i n g t h i s approx- i m a t i o n i n i t s most e x t r e m e form, o n l y t h e o n e which i s c l o s e s t t o a c h o s e n e n e r g y ( 6 ) . L e e ' s e a r l y work ( 5 ) , i n t h i s s p i r i t , seemed t o c o n f l i c t w i t h t h e p r e d i c t i o n s of Abrahams e t a l , b u t h i s more r e c e n t work ( u n p u b l i s h e d ) i s more c o n s i s t e n t w i t h them.An a l t e r n a t i v e r e a l s p a c e method is o f f e r e d by t h e d e c i m a t i o n t e c h n i q u e ( 7 - l o ) , i n which a s u b - l a t t i c e is removed a t e a c h s t a g e . T h i s r e s u l t s i n a r a t h e r d i f f e r e n t t y p e of s c a l i n g h e h a v i o u r ( s e e T a b l e 1)
Block Method D e c i m a t i o n Table Number of b a s i s f u n c t i o n s
p e r s i t e P r o p e r t i e s o f
r e n o r m a l i z e d Range of i n t e r a c t i o n s
L a r g e N e a r e s t
Neighbour Range
I n t h e f o l l o w i n g s e c t i o n , we d i s c u s s some r e s u l t s o f t h e a p p l i - c a t i o n of t h e d e c i m a t i o n method t o t h e l o c a l i z a t i o n problem.
Method and R e s u l t s The S c h r d d i n g e r e q u a t i o n f o r t h e f a m i l i a r Ander- s o n H a m i l t o n i a n may b e w r i t t e n
N
4
where t h e d i a g o n a l e l e m e n t s Hii a r e randomly d i s t r i b u t e d ( u s u a l l y w i t h a r e c t a n g u l a r d i s t r i b u t i o n of w i d t h W ) . The o f f - d i a g o n a l e l e m e n t s H . . a r e o f m a g n i t u d e u n i t y when i and j c o r r e s p o n d t o t h e n e a r e s t n e i g h - l J b o u r s of a g i v e n l a t t i c e and z e r o o t h e r w i s e .
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981406
JOURNAL DE PHYSIQUE
S i t e N may b e e l i m i n a t e d , by p r o j e c t i n g H o n t o a ( ~ - 1 ) - d i m e n s i o n a l s u b s p a c e a c c o r d i n g t o
I n t h i s manner, as r e q u i r e d by t h e d e c i m a t i o n method, we may re- move an e n t i r e s u b l a t t i c e . The number o f s i t e s i s t h e n h a l v e d and t h e s c a l e of l e n g t h i n c r e a s e d by 2 l I d , where d i s t h e d i m e n s i o n . The p r o - c e d u r e i s t h e n r e p e a t e d many t i m e s , t o s e e w h e t h e r t h e r e n o r m a l i z e d H a m i l t o n i a n t e n d s t o w a r d s o n e i n which o f f - d i a g o n a l e l e m e n t s v a n i s h
( i n d i c a t i v e o f l o c a l i z a t i o n ) .
F i g u r e 1.: E x t r a p o l a t e d i n v e r s e decay l e n g t h a f o r one d i m e n s i o n ( I r ) and two-dimensional s q u a r e l a t t i c e ( 0 )
.
A l s o shown f o r comparison a r e i n v e r s e l o c a l i z a t i o n l e n g t h s ( 0 ) c a l c u l a t e d f o r t h e 2d c a s e by Yoshino a n d Okazaki ( 1 2 ) .We h a v e c a r r i e d o u t t h e above p r o c e d u r e i n t h e c a s e o f o n e and two d i m e n s i o n s . I n g e n e r a l ; t h e r e n o r m a l i z e d i n t e r a c t i o n s between s i t e s i and j , s e p a r a t e d by a d i s t a n c e r+, a r e f o u n d t o behave a s
~ a ( ~ ) , ~
1;
exp - a (n>'J 'i j ( 4 )a f t e r n d e c i m a t i o n s . Here a i s an i n v e r s e d e c a v l e n g t h which w e might e x p e c t t o b e r e l a t e d t o t h e i n v e r s e l o c a l i z a t i o n i e n g t h a o f
e i g e n s t a t e s a t t h e c h o s e n e n e r g y ( 9 ) . The most r e a s o n a b l e a n n a t z would seem t o b e
lim ( n )
a = n+m a ( 5
I n o n e d i m e n s i o n ( 1 0 ) a ( " ) was computed a t t h e band c e n t r e (E=O) f o r a v a r i e t y of w i d t h s W r a n g i n g from ~ = 9 0 - ~ t o W = 30 a n d s a m p l e s o f up t o
l o 8
s i t e s . I n t h e l i m i t o f l a r g e n , t h e p a r a m e t e r a d e f i n e d by ( 5 ) was f o u n d t o b e f i n i t e f o r a l l W . Comparison w i t h e x a c t f o r m u l a e f o r t h e l o c a l i z a t i o n l e n g t h i n t h e h i g h and low d i s o r d e r l i m i t s ( 1 1 ) i n -d i c a t e d t h a t t h e e x p e c t e d c o r r e s p o n d e n c e d o e s i n d e e d h o l d i n t h i s c a s e . The r e s u l t s f o r a a r e i n c l u d e d i n F i g . 1.
I n two d i m e n s i o n s ( 8 ) o u r c a l c u l a t i o n s t o d a t e have n o t r e s u l t e d i n s u c h a p l e a s i n g consistency w i t h o t h e r e s t i m a t e s of a . F i g . 1 shows t h e r e s u l t s f o r a 512 s i t e s y s t e m , u s i n g a l i n e a r e x t r a p o l a t i o n
in n-l. The limit of a is zero, to within the error estimate, up to W=6, which has been previously identified as the Critical disorder for the Anderson transition or a quasi-transition (4)
.
However,above this value of W there is a large discrepancy between the extra- polated a and the more direct calculation of localization lengths by Yoshino and Okazaki (12).
Failure to understand the origin of this discrepancy has impeded further progress with the method, in which the next logical step would be to
discard
the weakest part of the long-range interaction, analogous to the approximation used by Lee (5). There are various possibilities for an explanation. We have checked that direct diagonalization gives results close to those of Yoshino and Okazaki for large disorder,so their results may be trusted. There remain essentially two alter- natives. Either the decay length is not after all simply related to the Localization length or our calculation is rendered unreliable by finite size and incorrect extrapolation.Localization theory is full of plausible but not quite provable propositions of the same kind as (5). Tn this case, as in others, doubt is not unreasonable but the proposition is as awkward to dis- prove as it is to prove ! Tt seems best to keep an open mind and pursue the numerical analysis further. The discrepancy of Fig 2 is so large as to make one sceptical of the chances of a reconciliation of the two approaches, based on more extensive calculations. On the other hand, just such a surprising size-dependence has emerged in the recent block-method numerical studies of Lee (unpublished).
Acknowledgements This work was carried out during the tenure (C.J.L.) of a Department of Education Fellowship and also benefited from an National Board for Science and Technology Grant.
References
(1) Nitzan A,, Freed
F.,
Cohen M.H., Phys. Rev. B15 (1977) 4476.(2) Stein J., Krey
u.,
Solid State Comm,27
(1978) 1405.(3) Licciardello D.C., Thouless D.J., Phys. Rev. Letts.
35
(1975) 1475.(4) Abrahams E., Anderson P.W., Licciardello D.C., Ramakrishnan T.V.
Phys. Rev. Lett
-
42 (1979) 673.(5) Lee P.A., Phys. Rev. Lett.
42
(1979) 1492.(6) Domany E., Sarker S., Phys. Rev. B20 (1979) 4726.
(7) Aoki H., Solid State Comm. 31(197r999.
(8) Lambert C.J., W e a i ~ e D., P h E . Stat. Sol. (6)
101
(1980) 591.(9) Aoki H., J. Phys. C . 13 (1980) 3369.
(10) Lambert C.J., Phys. ~ x t . 78A (1980) 471.
(11) Thouless D.J., Les ~ouchesSummer School (1978), 111, eds. Bahm R., Maynard R., Toulouse G. (North-Holland).
(12) Yoshino S., Okazaki M., 3. Phys. Soc. Jap.
