Perturbative Analysis of the Ising Model on the Koch Carpet

Revista Brasileira de Física, Vol. 15, nQ3, 1985 Perturbative Analysis of the Ising Model on the Koch Carpet R.F.S. ANDRADE Instituto de Ffsiw, Universidade Federal da Bahia, 40000, Salvador, BA, Brasil and S.R. SALINAS Instituto de Ffsim, UUniversidaade de São Paulo, Caixa Postal 20516, Sáo Paulo, 01498, SP, Brasil Recebido em 28 de outubro de 1985 Weuseaperturbativeschemetoconsideran Ising modelona l a t t i c e d e f i n e d as the c a r t e s i a n product o f a Koch curve b y a 1 i n e a r chain. To l e a d i n g o r d e r , we w r i t e t h e s i n g u l a r p a r t o f t h e f r e e energy i n terms o f c e r t a i n s p i n - s p i n c o r r e l a t i o n s on t h e I s i n g square l a t t i c e . There i s no change i n t h e c r i t i c a 1 exponent a s s o c i a t e d w i t h t h e s p e c i f i c hea t Abma . I.INTRODUCTION There a r e many questions r e g a r d i n g t h e e f f e c t o f t h e Hausdorff diniensional i t y D on t h e c r i t i c a 1 behavior o f s p i n models fractal lattices. d e f i ned on I t was f i r s t c o n j e c t u r e d t h a t i t p l a y s t h e same r o l e as the Euclidean dimension d, as i n the case o f E = 4 - d expansions used i n t h e renormalization- group scheme. T h i s has been checked by t h e ap- p l i c a t i o n o f approximate and a l s o exact decimation procedures t o s e v e r a l models. The r e s u l t s o b t a i n e d so f a r i n d i c a t e t h a t t h e c r i t i c a 1 behavior may depend on D, b u t a l s o depends on the sionality, so- called spectral dimen- DS, and on o t h e r p r o p e r t i e s o f t h e f r a c t a l l a t t i c e s , a s t h e branching c h a r a c t e r and t h e c o n n e c t i v i t y 1-4 . Despite t h e elegance o f the decimation procedures, which q u i t e adequate f o r t a k i n g care o f t h e s e l f - s i m i l a r i t y of the are fractal s t r u c t u r e s , we b e l i e v e t h e r e i s s t i l l room f o r a d d i t i o n a l exact as w e l l as approximate analyses o f t h e c r i t i c a 1 behavior. We have r e c e n t l y used the t r a n s f e r m a t r i x formal ism t o study t h e I s i n g model on a Koch curve 5 . From t h e exact r e s u l t s i n zero f i e l d , we show t h a t t h e r e is s i t i o n , and the f r a c t a l d i m e n s i o n a l i t y p l a y s a t r i v i a l r o l e . no tranIn the present paper we consider an I s i n g model on a l a t t i c e d e f i n e d as a c a r - t e s i a n product o f a Koch curve by a l i n e a r chain. The s i o n a l i t y o f t h i s a n i s o t r o p i c Koch c a r p e t i s D = 1 + fractal dimen- If log4/1og3. we walk a l o n g t h e x d i r e c t i o n we have a Koch curve. Along t h e y d i r e c t i o n , however, t h e r e i s j u s t a simple l i n e a r c h a i n o f spins. This model n o n - t r i v i a l because t h e r e a r e two k i n d s o f nearest- n e i g h b o r tions: interac- ( i ) nearest- neighbor p a i r s o f spins along x and y chairis, w i t h li+j-R-kl is = 1, i n t e r a c t w i t h an exchange parameter 'i,j (ii) J ; nearest- neighbor p a i r s o f spins introduced by t h e Koch curve, that is, o f t h e form Si .S i n t e r a c t w i t h an exchange parameter JE,. ,J i + 2 , j 9 I t i s easy t o f o r m u l a t e t h e p a r t i t i o n f u n c t i o n o f t h e Ising model on t h e a n i s o t r o p i c Koch c a r p e t according t o thestandardprocedures t o consider two-dimensional s t a t i s t i c a l problems. I n p a r t i c u l a r , w r i t e a transfer matrix, but the analysis o f the largest we eigenvalue becomes a n o n - t r i v i a l problem, probably w i t h o u t an a n a l y t i c a l s o l u t i o n . We then r e s o r t t o a p e r t u r b a t i v e scheme, i n t r o d u c e d by Barber t o study t h e I s i n g model on a square l a t t i c e w i t h nearest and n e x t - n e a r e s t n e i g h bor i n t e r a c t i o n s , which g i v e s t h e leading term o f t h e 161 f free energy f o r I J / J I < < i . l t should b e m e n t i o n e d t h a t i t i s e q u a l l y easy F to a p p l y t h i s scheme d i r e c t l y , w i t h o u t r e f e r e n c e t o t h e t r a n s f e r m a t r i x method. The f r e e energy i s w r i t t e n , as i n t h e work o f Barber, i n o f t h e f r e e energy o f t h e corresponding I s i n g model on the terms underlying simple q u a d r a t i c l a t t i c e , t h a t i s , w i t h JF = 0, p l u s a c o r r e c t i o n term i n v o l v i n g t h e s p i n - s p i n c o r r e l a t i o n f u n c t i o n s r e l a t e d t o t h e terms pending on J de- F' (S A Unl i k e t h e case o f t h e s u p e r - a n t i f e r r o m a g n e t i c considered by Barber, i n the present case t h e c r i t i c a l s h i f t e d b u t t h e c r i t i c a 1 index F) m o d e l temperature is u a s s o c i a t e d w i t h t h e s p e c i f i c heat r e - mains unchanged. As we a r e n o t aware o f o t h e r a p p l i c a t i o n s of Baber's scheme, we a l s o checked t h i s procedure i n t h e case o f t h e e x a c t l y s o l u b l e I s i n g model on t h e Union-Jack l a t t i c e . WI then have s t r o n g indi- cations t h a t the f r a c t a l dimensionality o f the anisotropic Koch c a r p e t p l a y s a t r i v i a l r o l e , which i s r e s t r i c t e d t o t h e s c a l i n g o f the c o r r e - l a t i o n l e n g t h along t h e x d i r e c t i o n . T h i s paper i s organized as f o l l o w s . I n s e c t i o n 2 we d e f i n e t h e 1 and discuss, on t h e b a s i s o f t h e t r a n s f e r, m a t r i x d i f f i c u l t i e s t o o b t a i n an exact s o l u t i o n . formulation, the I n section 3 l e a d i n g o r d e r expression o f t h e f r e e energy i n t h e we obtain limiting case 16) = J J ~ / J<< J 1 . I n s e c t i o n 4 we use Barber's procedure t o the c r i t i c a l behavior o f t h e model. F i n a l l y , some are presented i n s e c t i o n anal yze concluding remarks 5. 2. THE FORMULATION OF THE PROBLEM I n zero f i e l d the I s i n g model on t h e a n i s o t r o p i c Koch c a r p e t i s given by the H a m i l t o n i a n whereN=bN, L=bL, g=g(L,i) =L(4{-2), a n d N indicates s t e p i n the c o n s t r u c t i o n o f the c a r p e t . The f i r s t sum in the N- th eq. (2.1) de- s c r i b e s the nearest neighbor I s i n g i n t e r a c t i o n s which a r e homeomorphic t o t h e I s i n g square l a t t i c e w i t h N2 s i t e s . The second sum takes care o f the non- periodic i n t e r a c t i o n s a l l o n g the x - d i r e c t i o n o f t h e mode1,which a r e responsible f o r t h e f r a c t a l c h a r a c t e r . The model d e s c r i bed by eq. (2.1) i s homeomorphic t o t h e I s i n g model on a square l a t t i c e pZus n o n - p e r i o d i c t h i r d - n e i g h b o r i n t e r a c t i o n s . We can then use t h e w e l l known techniques developed mensional s p i n problems, However, t o analyze two-di- t h i s homeomorphism, wh i c h p l a n a r l a t t i c e s w i t h h i g h e r o r d e r neighbor i n t e r a c t i o n s fractal and models, a l r e a d y i n d i c a t e s the k i n d o f d i f f i c u l t i e s i n the relates solution the problem. Indeed, the presence o f c r o s s i n g bonds i n p l a n a r precludes t h e establishment o f exact s o l u t i o n s by a l l known methods. the present case t h i s i s e a s i l y seen i f we discuss the of lattices ma i n In steps towards t h e s o l u t i o n i n termc; o f t h e t r a n s f e r m a t r i x method. We r e f e r t o the c l a s s i c a l treatments 7-9 f o r f u r t h e r discussions and d e t a i l s o f the calculations. L e t P be t h e t r a n s f e r m a t r i x which takes i n t o account a l l i n t e r a c t i o n s i n a row a l o n g t h e x d i r e c t i o n and a l s o between spins i n adj a c e n t rows (y and y+l) f = . Theri the f r e e energy per s p i n , f k ~ T -log(Tr N' P ) N + c a , i s given by k ~ T -log X max ' N where X i s t h e l a r g e s t eigenvalue o f P. I n o r d e r t o f i n d max convenient t o w r i t e P i n terms o f a s e t o f 2N m a t r i c e s (2.2) Xmax of it is order 2Nx2N (see r e f . 8 f o r a c h a r a c t e r i z a t i o n o f t h i s s e t and i s p r o p e r t i e d , I f we c a l 1 P I lattice, t h e t r a n s f e r m a t r i x o f t h e simple i t i s easy t o see t h a t i t d i f f e r s terms w i t h a f o u r - f o l d product of '?J i s based on t h e f a c t t h a t each term from P by the I s i n g square presence of m a t r i c e s . The d i a g o n a l i z a t i o n o f P uuv r = exp(-i u v I ) describes a s i m i l a r i t y t r a n s f o r m a t i o n between two w e l l - d e f i n e d r e p r e s e n t a t i o n s o f the matrices u r 1-i' T h i s t r a n s f o r m a t i o n may a l s o be represented by i n t h e 2N-dirnensional space o f t h e m a t r i c e s 1-iv by t h e product o f severa1 terms o f t h e t y p e define p I U r uv' a rotation Thus, i f Pjr i s g i v e n u. it is pos s i b 1e to as the product o f t h e corresponding terms a l i z a t i o n of p I u The diagonuva i s then an easy problem which leads t o t h e eigenvalues o f P I . However, the presence o f f o u r - f o l d products i n eq. impossi b l e t o r e l a t e P t o a 2 M N m a t r i x p . u c t s do n o t d e s c r i b e r o t a t i o n s among t h e (2.5) m a k e s i t Indeed, t h e f o u r - f o l d prod- rP, since the s i m i l a r i t y trans- f o r m a t i o n g i v e s r i s e t o t h e presence of t h r e e - f o l d products o f t h e which a r e l i n e a r l y independent of t h e r 1-i r's 'S. Due t o t h i s s p e c i f i c d i f f i c u l t y (which w i l l a l s o appsar i f we choose another method) , s o l u t i o n s o f p l a n a r models wi t h crossirig bonds, and hence of f r a c t a l l a t t i c e s w i t h Hausdorff dimensions D > 2, remain t o be found. So, i n t h e n e x t s e c t i o n we r e s t r i c t t o a formulation i n the l i m i t (E( 5 3. THE LIMIT OF SMALL JF lei treatment (JF/J(C< 1 . Also, we a r e f o r c e d t o r e s o r t t o t h e p e r t u r b a t i v e scheme introduced by Barber. ihen our s t i l l << I, we may w r i t e t o l e a d i n g o r d e r where Let us assume t h a t i t i s n o t necessary t o consider h i g h e r o r d e r terrns i n eq. (3.1) t o d e t e c t p o s s i b l e changes i n t h e c r i t i c a l model w i t h respect t o t h e simple square l a t t i c e . p t = P ( C O SBJE) ~ -NO and use eqs . (3.1) and (3.2) , b e h a v i o r o f the I f we c a l 1 , (3.3) we have t o l e a d i n g o r d e r s i nce (2 s i n h BJ) N/2 To f i r s t o r d e r i n w = tanh BJ6, the eigenvalues o f P 1 a r e g i v e n where XI and matrix P I IX I I a r e t h e eigenvalues and eigenvectors of t h e transfer o f t h e sirnple I s i n g square l a t t i c e . The f r e e energy per s p i n i s g i v e n by f = iim N- - -logbr 2 N PY = k ~ T -iog 2 N [iFx (I w < i ~ x ~ ~ m ~N ,~ x > ) ] (3.7) where t h e s u p e r s c r i p t max r e f e r s t o t h e l a r g e s t eigenvalue and i t s c o r responding e i g e n v e c t o r . I n t h e l i m i t N i s g i v e n by + w, the expectation value o f Q h' A k iim G " I Q ~ I C " >= where <S.t,j > S llk, ~ " l f Pr i a rr r*ixYx> = I > (3.8) I a2,1 i s a s p i n - s p i n c o r r e l a t i o n , on t h e u n d e r l y i n g I s i n g square l a t t ice, between s i t e s (.i,j) and (k,R) where f 51 . Thus we have i s t h e f r e e energy o f t h e I s i n g square l a t t i c e . cosh I f we d e f i n e 2W and c a l 1 K and E t h e complete e l l i p t i c i n t e g r a l s o f t h e f i r s t andsecond k i n d r e s p e c t i v e l y , w i t h modulus k, i t i s easy t o o b t a i n t h e expression This c o r r e l a t i o n i s continuous as a f u n c t i o n o f T, w i t h a d e r i v a t i v e which becomes s i n g u l a r a t t h e c r i t i c a 1 temperature, given by k = l , o f I s i n g model on t h e square l a t t i c e . the I n t h e f o l l o w i n g s e c t i o n w e use eqs. (3.9) and (3.1 1) t o perform a d e t a i l e d a n a l y s i s i n t h e neighborhood o f t h e c r i t i c a 1 temperature. 4. THE CRITICAL BEHAVIOR I n t h e neighborhood o f t h e c r i t i c a 1 temperature, w r i t e t h e f o l l o w i n g asymptotic expressions f o r t h e s i n g u l a r t h e f r e e energy and t h e s p i n - s p i n c o r r e l a t i o n f u n c t i o n model on a simple square l a t t i c e , and of T'?, we can parts the of Ising where j = ~ / Tk , and B j = j-j C , with j C = J / k B T C . Inserting these ex- pressions into eq. ( 3 . 9 ) , we have to leading order the singular part of the free energy where wc = tanh (€iC). The central idea of üarber's method is the assumptionthatthe dependence of the singular part of f with respect to A j ' = j + q is the same as the dependence of the singular part of flwith respect to A j , where q is a function of the small parameter u. For small deviations i t i s enough to suppose a 1 inear dependence of q, as wel 1 as of the cri ti cal exponent a, wi th respect to w. Of we wri te 17 = Aw and a = Bw , we should have which can be written as (4.5) Comparing with eq. (4.3) we have A = -2 f i 371 and B = O , (4 .6) whích índícates that the critical exponent a keeps the same value as in the Ising model on the square lattice. There is just a shift critical temperature, which is given by if we make wC = €iC, and T C in the is the critical temperature of the Ising 9 0 square lattice. I t is now appropr iate to make some comments concerning the difference between Barber's and our own results. To this purpose, let us consider an Ising model on a square lattice, with first and second Due neighbor interactions, as in the case of the SAF model of üarber. t o t h e absence o f a term p r o p o r t i o n a l t o bj , i n eq. (4.3) log(Ajl) have n o t found any change i n t h e v a l u e o f t h e c r i t i c a 1 exponerit we .Terms o f t h i s type, however, do appear i n theSAF model because i n t h i s case the dominant c o r r e c t i o n i s a f o u r - s p i n c o r r e l a t i o n f u n c t i o n whichspl i t s i n t o a square o f two- spin c o r r e l a t i o n s , each one o f them p r o p o r t i o n a l t o A, J log I ~ j l .I t with shoud be n o t i c e d t h a t these tt-rms d o n o t appear i n the case o f t h e present model due t o t h e form o f Indeed, terms o f the type ( l - k 2 ) ~ 2 ,which,for to bj 1oglAj I)', a term T - eq . (3.1 1) . Tc, a r e p r o p o r t i o n a l cancel o u t i n t h i s equation i n d e n t i c a l l y . We a r e n o t aware whether t h i s s i t u a t i o n a l s o occurs f o r a11 c o r r e l a t i o n s a t longer distances. I f t h i s happens as described above, t h e models belonging t o t h i s category w i l l have t h e same c r i t i c a 1 behavior as t h e simple square lsing lattice. 5. CONCLUSIONS We have s t u d i e d an I s i n g model on a l a t t i c e d e f i n e d by a c a r t e s i a n product o f a Koch curve and a 1 i n e a r c h a i n (which a n i s o t r o p i c Koch carpet). Our main i n t e r e s t resides on we c a 1 1 t h e the appl i c a t i o n o f d i f f e r e n t techniques t o i n v e s t i g a t e t h e i n f l u e n c e o f the f r a c t a l d i m e n s i o n a l i t y on t h e c r i t i c a 1 p r o p e r t i e s o f I s i n g m d e l s on f r a c t a l l a t t i c e s . The exact t r a n s f e r m a t r i x formal ism f o r t h e a n i s o t r o p í c Koch c a r p e t has been discussed and we have p o i n t e d o u t t h a t t h e f o u r - f o l d products o f r presence o f m a t r i c e s makes i t impossible t o o b t a i n an ex- p l i c i t expression f o r t h e l a r g e s t eigenvalue. We then restrict / 1 , and f o l low a procedure treatment t o t h e reg i o n ( J ~ / J<< by Barber t o analyze t h e c r i t i c a 1 behavior o f t h e SAF I s i n g model. s i n g u l a r p a r t o f t h e f r e e energy can be w r i t t e n i n terms energy and a p a r t i c u l a r s p i n - s p i n c o r r e l a t i o n f u n c t i o n , our introduced of The the free is not which d i f f i c u l t t o c a l c u l a t e , f o r the I s i n g model on a simple q u a d r a t i c t i c e . These r e s u l t s , w i t h i n t h e framework o f Barber's procedure, latcould a l s o have been o b t a i n e d w i t h o u t any connection w i t h t h e t r a n s f e r m a t r i x formalism. U n l i k e t h e case o f the SAF model, we d e t e c t a s h i f t c r i t i c a 1 temperature b u t no change i n t h e c r i t i c a 1 exponent in the associated w i t h t h e s p e c i f i c heat. We thus conclude t h a t t h e f r a c t a l d i m e n s i o n a l i t y o f t h e a n i s o t r o p i c Koch c a r p e t has a t r i v i a l effect behavior. T h i s c o n c l u s i o n a l s o holds f o r o t h e r models on the wiht critica1 the same form o f the s p i n - s p i n c o r r e l a t i o n s f u n c t i o n s . Incidentally, t h a t Barber's scheme r e a l l y works i n t h e case o f t h e we checked exactly sol uble I s i n g model on t h e Union Jack l a t t i c e . REFERENCES 1. Y . Gefen, (1980) B.B. Mandelbrot, and A. Aharony, Phys. Rev. L e t t . 45, 8556 . 2. Y . Gefen, A. Aharony, and 0 . 0 . Mandelbrot, J. Phys. A16, 1267 (1983). 3. R. Rammal and G. Tolouse, J. Phys. L e t t . 44, 13 (1983) . 4. M. Suzuki, Prog. Theor. Phys. 69, 65 (1983). 5. R.F.S . Andrade and S .R. 6. M.N. Barber, J. Phys. A22, 679 (1979). Sa 1 inas, J. Phys. A17, 1665 (1984) 7. B. Kaufman, Phys. Rev. 76, 1232 (1949) 8. K. H uang , S t a t i s t i c a Z Mechrmics, 9. R.J. . . John W i 1ey, New Y o r k , 1983. Baxter, ExuctZy SoZved ModeZs i n S t a t i s t i c a Z Mechanics,Academic Press , London, 1982. Usamos um esquema p e r t u r b a t i v o para considerar o modelo de Ising numa rede d e f i n i d a p e l o produto c a r t e s i a n o de uma curva de Koch por uma. cadeia l i n e a r . Em ordem dominante, escrevemos a p a r t e s i n g u l a r da energ i a l i v r e em termos de determinadas correlações spin- spin para o modelo de I s i n g d e f i n i d o numa rede quadrada. Não hã mudanças no e x p o e n t e c r í t i co associado ao c a l o r e s p e c í f i c o .