HAL Id: jpa-00225732
https://hal.archives-ouvertes.fr/jpa-00225732
Submitted on 1 Jan 1986
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.
HARMONIC EXCITATIONS IN QUASICRYSTALS
J. Luck
To cite this version:
J. Luck. HARMONIC EXCITATIONS IN QUASICRYSTALS. Journal de Physique Colloques, 1986, 47 (C3), pp.C3-205-C3-210. �10.1051/jphyscol:1986321�. �jpa-00225732�
JOURNAL DE PHYSIQUE
Colloque C3, supplement au n07, Tome 47, juillet 1986
HARMONIC EXCITATIONS IN QUASICRYSTALS
J.M. LUCK
Service de Physique ThBorique, CEA-Saclay, F-91191 Gif-sur-Yvette Cedex, France
RdsumQ - O n Q t u d i e l e s e x c i t a t i o n s harmoniques d e s q u a s i c r i s t a u x (phonons) s u r un modSle unidimensiocnel simple. Le s p e c t r e de c e modlle e s t un ensemble de Cantor, dont on d i s c u t e l e s p r o p r i d t & s d ' a u t o - s i m i l a r i t s . Les modes pro- p r e s s o n t t o u j o u r s " c r i t i q u e s " , c ' e s t - 2 - d i r e n i Ctendus n i l o c a l i s & .
A b s t r a c t - The harmonic e x c i t a t i o n s (phonons) o f q u a s i c r y s t a l s a r e s t u d i e d i n a simple one-dimensional model. The spectrum i s a Cantor s e t , which e x h i b i t s s e l f s i m i l a r i t y p r o p e r t i e s . The e i g e n s t a t e s a r e g e n e r i c a l l y " c r i t i c a l " , i .e.
n e i t h e r extended nor l o c a l i z e d .
I - INTRODUCTION
I n t h i s communication, I w i l l summarize a r e c e n t work i n c o l l a b o r a t i o n w i t h D i m i t r i P e t r i t i s [ I ] . We have chosen t o u s e t h e well-known p r o j e c t i o n method [2-51 t o gene- r a t e an almost p e r i o d i c t i l i n g of t h e l i n e with two t y p e s of t i l e s , namely s h o r t and long segments, o f r e g p e c t i v e l e n g t h s s = s i n 8 and c = c o s 8, where 0 i s t h e a n g l e between t h e x-axis and t h e s t r i p used t o c o n s t r u c t t h e t i l i n g ( s e e F i g u r e 1 ) . ye
s h a l l r e s t r i c t o u r s e l v e s t o t h e model d e f i n e d by t h e golden mean : t a n 8 = 7-I = -(A-1).
2 The harmonic e x c i t a t i o n s (phonons, t i g h t - b i n d i n g e l e c t r o n i c modes, e t c . ) a r e assumed
t o be d e s c r i b e d by a Laplace o p e r a t o r i n v o l v i n g o n l y n e a r e s t n e i g h b o r i n t e r a c t i o n s :
where t h e c o u p l i n g s X a r e a t t a c h e d t o l a t t i c e bonds. We d e f i n e them a s depending only on t h e bond lengphs, namely A = X f o r s h o r t bonds (Rn=s) and An = X f o r long bonds (P. = c ) . Choosing un'its 2uch ghat X = 1 , we keep As = $ < I a s $ f r e e parameter. n
The e q u a t i o n obeyed by e i g e n s t a t e s o f A r e a d s
where z d e n o t e s t h e squared eigenfrequency w L i n reduced u n i t s . S e c t i o n s I1 and 111 of t h i s c o m u n i c a t i o n a r e devoted t o t h e spectrum and t h e e i g e n s t a t e s of Eq.(2), r e s p e c t i v e l y .
11. - THE SPECTRUM
It w i l l be convenient t o r e w r i t e Eq.(2) i n terms of new dynamical v a r i a b l e s Q which
l i v e on bonds, and a r e d e f i n e d by n'
-1 (3)
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986321
C3-206
F i g . l - Construction of a one-dimensional q u a s i c r y s t a l by t h e p r o j e c t i o n method.
Eq.(Z) can then be r e c a s t i n t h e following matrix form :
and hence
(4)
(5) w h e r e Tn i s a p r o d u c t o f (n-1) e l e m e n t a r y 2 x 2 transfer m a t r i c e s .
It c a n b e s h o w n f r o m the g e o m e t r y o f the m o d e l t h a t the s e q u e n c e o f m a t r i c e s TF , w h e r e FL a r e the F i b o n a c c i n u m b e r s , d e f i n e d b y F = 0 , F ^ l , a n d the r e c u r s i o n r e l a - tion ^ L - F L - J + F L ^ , o b e y a n e x a c t t h r e e - t e r m r e c u r s i o n :
(l?)
and h e n c e their n o r m a l i z e d t r a c e s x = i T r Tj-T s a t i s f y
(7)
T h i s l a s t p r o p e r t y h a s b e e n used b y several a u t h o r s [ 6 - 1 0 ] to s t u d y S c h r o d i n g e r e q u a - tions w i t h a l m o s t p e r i o d i c p o t e n t i a l s . I n the p r e s e n t s i t u a t i o n , t h i s r e c u r s i o n h a s a simple g e o m e t r i c a l o r i g i n , a n d t h e r e f o r e h o l d s for all v a l u e s o f z and p .
The study o f t h e mapping (7) shows t h a t t h e spectrum of A i s g e n e r i c a l l y a Cantor s e t . F i g u r e 2 shows a p l o t of t h e i n t e g r a t e d d e n s i t y of s t a h e s B ( z ) f o r p=1/2 : i t s Cantor s t r u c t u r e i s q u i t e a p p a r e n t . We have shown t h a t ( 7 ) h a s more q u a n t i t a t i v e con- sequences, namely t h e e x i s t e n c e of power-law s i n g u l a r i t i e s , modulated by p e r i o d i c
H I4
1.
-/ /
'
Bt . I R ( . I . I . l
0. 3. 6. z
Fig.2 - P l o t of t h e i n t e g r a t e d d e n s i t y of s t a t e s H(z) f o r p=1/2 ( f u l l c u r v e ) , and af t h e u n d e r l y i n g average l a t t i c e (dashed c u r v e ) .
a m p l i t u d e s , i n H(z) a t each gap edge. I n t h e simple c a s e o f t h e upper bound of t h e spectrum zma,, o u r r e s u l t r e a d s
where A = 0.427174, and t h e period of P i s A = 2.252999. F i g u r e 3 i l l u s t r a . t e s t h i s behavior.
111 - THE EIGENSTATES
The n a t u r e of t h e e i g e n s t a t e s of a l m o s t p e r i o d i c e q u a t i o n s i s known a s a d i f f i c u l t s u b j e c t . I n t h e p r e s e n t c a s e , we have n e v e r t h e l e s s been a b l e t o c h a r a c t e r i z e t h e i r e s s e n t i a l f e a t u r e s .
Our main r e s u l t i s t h a t t h e eigenmodes a r e n e i t h e r extended nor l o c a l i z e d . The absence of l o c a l i z a t i o n i n t h e type o f model we c o n s i d e r h a s been proven r i g o r o u s l y by Delyon and P e t r i t i s [ I l l ; we s h a l l n o t develop t h e i r argument p r e s e n t l y . The e x i s t e n c e of c o n v e n t i o n a l extended s t a t e s (quasi-Bloch-waves) i s a mare s u b t l e ques- t i o n , which may depend on t h e d i o p h a n t i n e p r o p e r t i e s o f t h e s l o p e t a n 8 1121. A heuristic s c a l i n g a n a l y s i s of t h e mapping (7) l e a d s u s t o t h e f o l l o w i n g conclusion : t h e e i g e n s t a t e s have a c h a r a c t e r i s t d c l e n g t h &(w2), beyond which t h e y l o s e t h e memory of t h e i r i n i t i a l phase. T h i s "memory l e n g t h " t h e r e f o r e demarcates two regimes:
a t s c a l e s s m a l l e r t h a n 5 , a n e i g e n f u n c t i o n behaves mare o r l e s s l i k e a plane wave ; a t s c a l e s l a r g e r than 5 , i t s behavior i s " c r i t i c a l " , and c h a r a c t e r i z e d by wild v a r i a - t i o n s i n small r e g i o n s , f a r a p a r t from each o t h e r , on t o p of a v e r y small background.
Moreover, we have argued t h a t t ( w 2 ) i s f i n i t e f o r a l l non-zero f r e q u e n c i e s , but
C3-208 JOURNAL DE PHYSIQUE
d i v e r g e s w i t h a n e s s e n t i a l s i n g u l a r i t y a t small frequency :
F i g . 3 - Log-Log p l o t of 1-H(z) a g a i n s t (am x-z) f o r p=1/2, showing t h e s c a l i n g behavior of t h e spectrum around i t s upper %ound.
Although t h e t r a c e mapping ( 7 ) i s i n t i m a t e l y r e l a t e d t o t h e v a l u e t a n 0 = T -1 of t h e s l o p e , we t h i n k t h a t a l l e i g e n s t a t e s a r e " c r i t i c a l " f o r a l l i r r a t i o n a l v a l u e s of t a n 0 having t y p i c a l d i o p h a n t i n e p r o p e r t i e s .
We end t h i s s e c t i o n by two i l l u s t r a t i o n s of t h e s t r a n g e behavior of t h e e i g e n s t a t e s . The f i r s t one c o n s i s t s i n p l o t t i n g a n eigenmode i n i t s phase plane (w,Q), where and Q a r e r e l a t e d through Eq.(3). F i g u r e 4 shows t h e e i g e n s t a t e s number 61 and 620 of a sample of l e n g t h 1000 (each f i g u r e t h e r e f o r e c o n t a i n s 1000 p o i n t s ) . I n t h e f i r s t c a s e , t h e memory l e n g t h i s l a r g e r t h a n 1000, and t h e s t a t e i s c l o s e t o being a p l a n e wave ( e l l i p s e ) . I n t h e second c a s e , 5 i s small, and t h e s t a t e develops a r i c h s t r u c t u r e , both i n ~ o s i t i o n space and i n i t s phase p l a n e .
We have a l s o computed t h e f o l l o w i n g moments a s s o c i a t e d with a given mode
where t h e e i g e n s t a t e cp on a sample of l e n g t h N i normalized i n such a way t h a t p = I . 1 Localized e i g e n s t a t e s would have ua I. N1-'f', w h i l e extended ones would g i v e r i s e t o f i n i t e 11% a s N -t -. F i g u r e 5 shows a p l o t o f t h e moment p, f o r a l l eigen- modes of a sample of l e n g t h N = 800 . A v e r y r i c h s e l f s i m i l a r s t r u c t u r e a p p e a r s , which i s r e l a t e d t o t h a t of t h e d e n s i t y of s t a t e s ( s e e F i g u r e s 2-3). The l e t t e r s o n F i g u r e s 2 , 3 , 5 a r e a c o n s i s t e n t n o t a t i o n f o r r c e r t a i n sequence of gaps. The t h r e e f o l d f i n e s t r u c t u r e of each peak of pm can be explained i n terms of a p a r t i c u l a r s i x - c y c l e of t h e t r a c e mapping (7). The r e a d e r i s r e f e r r e d t o o u r p u b l i c a t i o n [ I ] f o r a more d e t a i l e d d i s c u s s i o n of t h e s e phenomena.
F i g . 4 a s i z e N
sample
Fig.4b - Same a s f i g u r e 4a for the e i g e n s t a t e number 620.
JOURNAL DE PHYSIQUE
F i g . 5 - P l o t o f t h e moment d e f i n e d i n Eq.(lO) f o r a l l e i g e n s t a t e s o f a sample of s i z e N = 8 0 0 s i t e s , o r d e r e d a c c o r d i n g t o i n c r e a s i n g f s o q u e n c i e s .
REFERENCES
111 J . M . Luck and D. P e t r i t i s , J . S t a t . Phys. 42, 289 (1986).
[ 2 ] P. Kramer and R. N e r i , Acta. C r y s t . e, 5m (1984).
131 M. Duneau a n d A. K a t z , Phys. Rev. L e t t . 54, 2688 (1985).
[ 4 ] R.K.P. Zia and W.J. D a l l a s , J . Phys. h 1 8 T ~ 3 4 1 (1985).
151 V. E l s e r , Phys. Rev. E, 4892 ( 1 9 8 5 ) F
[ 6 1 M. Kohmoto, L.P. Kadanoff and C. Tang, Phys. Rev. L e t t . - 50, 1870 (1983).
171 !4. Kohmoto, Phys. Rev. L e t t . 51, 1198 ( 1 9 8 3 ) .
L81 1. Kobmoto and Y. Oono, Phys. L e t t . s, 1 4 5 (1984).
191 S. O s t l u n d , R. P a n d i t , D. Rand, H.J. S c h e l l n h u b e r and E.D. S i g g i a , Phys. Rev.
L e t t . 5 0 , 1873 (1983).
[ 101 S. 0 s t Z n d and R. P a n d i t , Phys. Rev. E , 1394 ( 1 9 8 4 ) . [ I 1 1 F. Delyon and D. P e t r i t i s , Corn. Math. Phys. ( i n p r e s s ) . [ I 2 1 B. Simon, Adv. Appl. ?lath. 3, 463 ( 1 9 8 2 ) .
