the decimation method and anderson localization

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.

Abstract We d i s c u s s 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 problem of Anderson l o c a l i z a t i o n . I n o n e dimension, 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 found 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 of ( 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 toward 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.

Introduction 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 of l o c a l i z a t i o n have been proposed (1-3). These 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-space ( 1 ) and 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 Of t h e s e t h e 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 ) . most n o t a b l e i s t h a t which l e d Abrahams e t a 1 ( 4 ) t o a s s e r t t h a t a l l The"b1ock-spin" t r a n s f o r m a t i o n s t a t e s a r e l o c a l i z e d i n two dimensions. 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 (Anders o n ) H a m i l t o n i a n t o one 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 of sites, The b l o c k s a r e i n t u r n , comb 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 is 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 approxi m a t i o n i n i t s most extreme form, o n l y t h e one which i s c l o s e s t t o a chosen 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 each 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 Decimation Table Number of b a s i s f u n c t i o n s P r o p e r t i e s of Large per s i t e renormalized N e a r e s t Range of i n t e r a c t i o n s Neighbour Range In 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 Anders 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 The o f f - d i a g o n a l e l e m e n t s 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 ) . a r e of magnitude 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

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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 subspace according 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 remove an e n t i r e s u b l a t t i c e . The number of 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 is t h e dimension. 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 whether 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 one 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 of 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 dimension ( I r ) and two-dimensional s q u a r e A l s o shown lattice ( 0 ) f o r comparison a r e i n v e r s e localization lengths ( 0 ) c a l c u l a t e d f o r t h e 2d c a s e by Yoshino and Okazaki ( 1 2 ) .

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We have 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 of one and two 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 dimensions. 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 found t o behave a s ~ a ( ~ 1; ) , exp ~ - a (n>'J'i j (4) a f t e r n decimations. Here a i s an i n v e r s e decav 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 of e i g e n s t a t e s a t t h e chosen 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 dimension (10) 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 and samples of up I n t h e l i m i t of 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 ) t o lo8 s i t e s . was found 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 expected correspondence does indeed hold i n t h i s case. 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 extrapolated 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 alternatives. 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 disprove 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).

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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 Nitzan A,, Freed F., Cohen M.H., Phys. Rev. B15 (1977) 4476. Stein J., Krey u., Solid State Comm, 27 (1978) 1405. 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., Weai~eD., 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 ~ouchesSummerSchool (1978), 111, eds. Bahm R., Maynard R., Toulouse G. (North-Holland). (12) Yoshino S., Okazaki M., 3. Phys. Soc. Jap. 43 (1977) 415.

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