On the validity of scaling theory for the anisotropic ising model

Volume 38A, number 2

PHYSICS

ON

LETTERS

THE VALIDITY OF SCALING THE ANISOTROPIC ISING

17 January, 1972

THEORY MODEL

FOR

I. G. ENTING and J. OITMAA

Departmetd of Physics, Mo~zash Uni~,ersity, Clayton, Victo~'ia 3168, A ust ralia Received 30 November 1971

When scaling theory is applied to successive t e r m s of a perturbation expansion for the anisotropic IsJngmodel, a consistent gap index of 1.75 is predicted. We have recently derived a s e r i e s expansion for this model and p r e s e n t here an analysis of the s e r i e s which suggests that the scaling theory r e s u l t s are not obeyed. In two r e c e n t p a p e r s the p r e s e n t a u t h o r s h a v e o b t a i n e d a high t e m p e r a t u r e s e r i e s f o r t h e z e r o f i e l d s u s c e p t i b i l i t y of an a n i s o t r o p i c I s i n g m o d e l on t h e s i m p l e c u b i c l a t t i c e [1,2]. T h i s q u a n t i t y i s a f u n c t i o n of two i n d e p e n d e n t v a r i a b l e s , t h e t e m p e r a t u r e T and t h e a n i s o t r o p y p a r a m e t e r V,

n

4

5

6

7

8

9 I0

i

i

~

t

t

i

7~ r~ 6

7(2'

s o t h e c r i t i c a l b e h a v i o u r of t h i s m o d e l w o u l d b e e x p e c t e d to b e m o r e c o m p l i c a t e d than the s i m p l e s i n g u l a r i t i e s n o r m a l l y a s s u m e d at the c r i t i c a l point. T h e m e t h o d of a n a l y s i s u s e d in [11 and [2] w a s to e v a l u a t e the s e r i e s at v a r i o u s f i x e d v a l u e s of q, a s s u m e t h a t the r e s u l t i n g s e r i e s h a d a s i m p l e s i n g u l a r i t y , and u s e the s t a n d a r d r a t i o and P a d d a p p r o x i m a n t m e t h o d s to o b t a i n e s t i m a t e s of the c r i t i c a l t e m p e r a t u r e and e x p o n e n t a s f u n c t i o n s of ~o We c o n c l u d e d t h a t t h e v a r i a t i o n of Tc with q n e a r the two d i m e n s i o n a l l i m i t ~ = 0 w a s in d i s a g r e e m e n t with the p r e d i c t i o n s of g e n e r a l i z e d s c a l i n g l a w s . We a l s o c o n c l u d e d t h a t the e x p o n e n t y v a r i e d c o n t i n u o u s l y in c o n t r a s t to t h e d i s c o n t i n u o u s j u m p t h a t i s g e n e r a l l y b e l i e v e d to o c c u r at = 0. W e a r e not c o n c e r n e d in t h i s l e t t e r with t h e v a r i a t i o n of T a l t h o u g h we m e n t i o n in p a s s i n g t h a t t h e r e i s one m o d e l , n a m e l y t h e " e i g h t - v e r t e x m o d e l " w h i c h h a s b e e n s o l v e d r i g o r o u s l y by B a x t e r [3], w h i c h d o e s d i s p l a y a c o n t i n u o u s l y varying exponent. T h e s e r i e s d e r i v e d in [11 and [2] a r e of the form X=

2

I

~2

I

0ll

I

}In

Fig. 1. E s t i m a t e s of exponentsT {r) from ratios of the Xr s e r i e s (open circles) and the Xr seines {full c i r c l e s ) . The scaling r e s u l t s are shown by c r o s s e s .

~ ~ m=0 n =0

b,~n vm wn : ~

~ hnm vmrl '~

m=0 n : 0

(1)

with ~' = t a n h K , w = t a n h ~ , K = J ~,T. T h e v a l u e s of b,t/nfor m + n < 11 and h m n f o r m < 1 1 a r e g i v e n in [1] and [2]. 1Rapaport [4] h a s a r g u e d t h a t f o r s m a l l v a l u e s of r/ t h e s e s e r i e s a r e too s h o r t to y i e l d the c o r r e c t c r i t i c a l b e h a v i o u r by s t r a i g h t f o r w a r d a n a l y s i s . F o l l o w i n g i d e a s of Abe [5] and C o n i g l i o [61 he h a s c o n s i d e r e d the f o l l o w i n g a s y m p t o t i c f o r m f o r the s u s c e p t i b i l i t y n e a r the two d i m e n s i o n a l c r i t i c a l temperature T 2 107

Volume 38A, n u m b e r 2

PHYSICS

(~) X~

~ a v ~];'(T- T2 )-)' (2) n 0 f o r w h i c h s c a l i n g a r g u m e n t s [5.6] s u g g e s t t h a t 7 0.) = ( r + 1)'r w i t h 7 = 1.75. It s h o u l d b e n o t e d t h a t t h i s r e s u l t f o r ?(r) l e a d s to t h e r e l a t i o n Tc(~) - T 2 (~: 771/17'3 w h i c h i s not c o n s i s t e n t w i t h t h e r e s u l t s of 12]. It i s p o s s i b l e to m o d i f y (2) by r e p l a c i n g ( T - T 2) by (K 2 - K ) o r (t' 2 -v ) o r r e p l a c i n g t7 b y w . A l l t h e s e f o r m s h a v e t h e s a m e a s y m p t o t i c l i m i t , i.e. t h e s a m e V(r), b u t o u t s i d e t h e l i m i t i n g r e g i o n T - , T 2. r1--~0, s o m e of t h e s e f o r m s will be better approximations than others. On the b a s i s of (2) R a p a p o r t h a s i n v e s t i g a t e d by P a d d a p p r o x i m a n t s t h e s e r i e s f o r t h e q u a n t i t i e s ~ VX

0o

Xr =

~ hm; t v m (3) 077r1~ 0 m=~ a n d h a s o b t a i n e d e s t i m a t e s o f 7 ( r ) f o r ~" = 1 , 2 , 3 w h i c h a p p e a r to b e in good a g r e e m e n t w i t h t h e s c a l i n g p r e d i c t i o n s . U s i n g t h e r a t i o m e t h o d we h a v e r e e x a m i n e d t h e s e r i e s (3) a n d a l s o t h e series for Xr. = "d;'X¢ ;H'j w=0 Estimates Y (r) II

~

bran v m

(4

17 J a n u a r y 19#2

S i n c e t h e c o r r e c t i o n s to ~ i r) will b e of o r d e r l ' n , e x t r a p o l a t i o n to 1 'n = 0 will s t i l l g i v e the c o r r e c t T (rl. In c o n t r a s t t h e Pade" a p p r o x i m a n t m e t h o d d o e s not p r o v i d e a m e a n s of e x t r a p o l a t i n g s u c h t r e n d s w h e n t h e s i n g u l a r i t y d o e s not c o r r e s p o n d to the f o r m a s s u m e d . T h e r a t i o e s t i m a t e s a r e 7 (l) = 3.5; y(2) = 5.0 ± 0.1; 7,(3) : 6 5 ± 0.2; 7 ( 0 = 8.0 ± 0.3. It i s not s u r p r i s i n g t h a t t h e r e s u l t f o r 7 (1) agrees with the scaling prediction, because the g r a p h s f o r t h i s s e r i e s h a v e only o n e b o n d in t h e " w e a k " d i r e c t i o n a n d s o do not a d e q u a t e l y s a m p l e t h e t h r e e - d i m e n s i o n a l n a t u r e of t h e l a t t i c e . In *2 holds for all a v a i l a b l e t e r m s . The f a c t Xl* : 4(X~) e s t i m a t e s fo~ 7 (2) . y(3). ? (4) a r e in definite disagree ment with the scaling predictions of 5.25,7.0, 8.75 respectively. We thus conclude that Ilapaports analysis ]41 is inconclusive and that the ratio analysis p r e sented is in d i s a g r e e m e n t with the scaling p r e dictions. We a r e unable to conclude whether or not the gap index tends to a constant but if so, our r e s u l t s indicate this limit is probably l e s s than or equal to 1.5 in c o n t r a s t to the scaling r e s u l t s 1.75.

m=0

of )0") a r e g i v e n by

1 ~ n(t,21J_ )~ - 1)

171 (5

w h e r e ~zn a r e t h e r a t i o s of s u c c e s s i v e c o e f f i c i e n t s . T h e s e e s t i m a t e s a r e e x p e c t e d to a p p r o a c h t h e c o r r e c t e x p o n e n t ?,(r) in t h e l i m i t t z - ' ~ , a n d a r e p l o t t e d in fig. 1. S i n c e t h e s e r i e s f o r x r s t a r t s at z,r t h e r e is s o m e a m b i g u i t y a s to w h e t h e r t h i s f a c t o r s h o u l d b e d i v i d e d o u t o r not. S p e c i f i c h e a t s e r i e s w h i c h s t a r t at K 2 h a v e t h e s a m e d i f f i c u l t y [81. R a p a p o r t ' s a n a l y s i s [4] a n d t h e r a t i o e s t i m a t e s in fig, 1 w e r e o b t a i n e d b y d i v i d i n g out t h i s f a c t o r . T h i s d i f f i c u l t y i s a v o i d e d w i t h t h e Xr s e r i e s . T h e f a c t t h a t t h e e s t i m a t e s of y do not a p p r o a c h the 1 n= 0 axis horizontally indicates the ass u m e d f o r m tz n = ( 1 / ) ' 2 ) {1 + (7}f) - 1) n } h a s ml a d d i t i v e c o r r e c t i o n of o r d e r 1 )e2, i.e. t h e s i n g u l a r i t y i s not e x a c t l y of t h e f o r m (~'2 - r ) - 7 °9 .

108

LETTI~,tlS

One of t h e a u t h o r s ( I . G . E . ) a c k n o w l e d g e s support through a Monash University research s c h o l a r s h i p . T h e o t h e r ( J . O . ) a c k n o w l e d g e s the s u p p o r t of t h e A u s t r a l i a n C o m m o n w e a l t h G o v e r n m e n t t h r o u g h t h e a w a r d of a Q u e e n E l i z a b e t h II F e l l o w s h i p . We a r e g r a t e f u l to D. C. R a p a p o r t f o r s e n d i n g u s a p r e p r i n t of h i s p a p e r .

RcJ-ere~ce s [3] J . O i t m a a and I.G. E n t i n g , P h y s , L e t t . 36A(1971) 91. [2] J. O i t m a a and I.G.~':nting, J. P h y s . C(1972) to be published. [3] R . J . B a x t e r , P h y s . Rev. Lett. 26 (1971) 832. [t] D . C . R a p a p o r t , P h y s . I , e t t e r s 37A (1971) 4(t7. [5] R. Abe, P r o g . T h e o r . Phys.-{4 (1971) 339. [61 A. Coniglio (197l), to be p u b l i s h e d . [7} C. Domb, A d v a n c e s in P h y s i c s 19 (1970) 339. [8] D. L. t t u n t e r , ,l. P h y s . C2 (1969) 9{1.