Critical curves for the Ising model with quenched impurities

Volume 37A, n u m b e r 1

CRITICAL

PHYSICS

CURVES

FOR

THE

ISING

LETTERS

MODEL

25 October 1971

WITH

QUENCHED

IMPURITIES

D. C. R A P A P O R T

Wheatstone Physics Laboratory, King's College, London WC2R 2LS, UK Received 24 September 1971

The effects of two kinds of quenched random i m p u r i t y on the c r i t i c a l t e m p e r a t u r e of the Ising model a r e examined with the aid of s e r i e s methods.

L i t t l e i s k n o w n a b o u t t h e e f f e c t of q u e n c h e d r a n d o m i m p u r i t i e s on t h e c r i t i c a l p r o p e r t i e s of l a t t i c e s p i n s y s t e m s . O n e w a y of i n t r o d u c i n g s u c h i m p u r i t i e s i n t o a s y s t e m i s to r e p l a c e a s e t of s p i n s c h o s e n a t r a n d o m b y n o n - m a g n e t i c e l e m e n t s . In t h e c a s e of t h e I s i n g m o d e l w i t h i m p u r i t i e s of t h i s k i n d , t h e s e r i e s e x p a n s i o n of t h e initial susceptibility per spin can be written in two different forms,

x -- n~o x (p)w~

(1)

x = ~

(2)

n>~O

Yn(w)pn,

where p = 1 - (impurity concentration) and w = = t a n h ( J / k B T ) . If X i s a s s u m e d to d i v e r g e a t a c r i t i c a l t e m p e r a t u r e Tc(P ) f o r p g r e a t e r t h a n a c e r t a i n v a l u e p cS t h e n a n a n a l y s i s of t h e s e s e r i e s s h o u l d y i e l d , a m o n g o t h e r t h i n g s , d e t a i l s of t h e f o r m of t h e c r i t i c a l c u r v e . P r e v i o u s w o r k h a s i n c l u d e d s t u d i e s of s e r i e s (1) a s f a r a s n = 8 [1] a n d (2) t o n = 3 [2] f o r t h e m o d e l o n t h e F C C l a t t i c e . W e h a v e e x t e n d e d b o t h of t h e s e s e r i e s b y t h e a d d i t i o n of two f u r t h e r t e r m s to eq. (1) a n d t h r e e to eq. (2). W e h a v e a l s o i n v e s t i g a t e d a new t y p e of I s i n g system in which non-magnetic impurities are l o c a t e d o n a r a n d o m s e l e c t i o n of l a t t i c e b o n d s w h e r e t h e i r e f f e c t i s to p r e v e n t n e i g h b o u r i n g s p i n s f r o m i n t e r a c t i n g . If X d i v e r g e s a t Tc(P) f o r p g r e a t e r t h a n s o m e v a l u e pBc, t h e c r i t i c a l curve should again be obtainable by analyzing s e r i e s of t y p e s (1) a n d (2). T h e c o e f f i c i e n t s of these series are readily derived - we have so far c o m p u t e d t h e f u n c t i o n s Xn(P ) a s f a r a s n = 10. T h e t y p e (1) s e r i e s f o r b o t h a n d s i t e i m p u r i t y problems were analyzed by ratio and Pad4 techn i q u e s to o b t a i n e s t i m a t e s f o r T c ( a c t u a l l y Wc) a n d t h e c r i t i c a l e x p o n e n t ~, a s f u n c t i o n s of p. W e

discuss the bond impurity results first. The acc u r a c y of t h e T c e s t i m a t e s w a s f o u n d to d e crease as p was lowered, but we have been able to construct the critical curve over the range 0.25 --< p < 1.0 w i t h a r e a s o n a b l e d e g r e e of a c c u r a c y , e v e n n e a r t h e l o w e r l i m i t . In t h e f i g u r e we have plotted the complete critical curve f o r t h e a n n e a l e d v e l s i o n of t h i s m o d e l [3] - t h a t for the quenched model differs from this by an a m o u n t l e s s t h a n t h e t h i c k n e s s of t h e l i n e i t s e l f over the range considered. The temperature s c a l e i s n o r m a l i z e d b y d i v i d i n g by T c (1), t h e c r i t i c a l t e m p e r a t u r e w h e n no i m p u r i t i e s a r e present. Since T c for the annealed model can be shown to be zero when p = 0.115, and for the quenched model, because pB is just the critical concentration in the bond percolation problem, T c = 0 w h e n P= 0.1_19 [4], s o m e s e p a r a t i o n of t h e c u r v e s i s to b e e x p e c t e d a s p d e c r e a s e s . T h e e s t i m a t e s f o r ~, i n c r e a s e s t e a d i l y f r o m 1.25 a s p decreases: this anomalous behaviour was also e n c o u n t e r e d i n t h e a n n e a l e d m o d e l [3], a n d it i s 1.0 Tc,_p,

05

0

0-5

p

I0

Fig. 1. Critical c u r v e s for (from left) m o l e c u l a r field approx., annealed (and quenched) bond impurity model. site i m p u r i t y model. 15

Volume

37A,

number

i

PHYSICS

u n l i k e l y that it r e f l e c t s the t r u e b e h a v i o u r of 7. T h e s e r i e s (1) f o r the s i t e i m p u r i t y p r o b l e m is not s o w e l l b e h a v e d and is u n a b l e to p r o v i d p e s t i m a t e s of T c for p < 0.6" e v e n f o r l a r g e r p, T c s e e m s to be s l i g h t l y u n d e r e s t i m a t e d , y a g a i n i n c r e a s e s w i t h d e c r e a s i n g p. T h e c r i t i c a l c u r v e f o r p >1 0.6 a p p e a r s in the f i g u r e , as d o e s the s m a l l p p a r t w h i c h f o l l o w s by a p p l y i n g the r a t i o m e t h o d to s e r i e s (2). T h i s s e r i e s is v e r y s h o r t and is u n a b l e to p r o v i d e r e s u l t s of the s a m e d e g r e e of r e l i a b i l i t y as (1). W h e n T = 0, eq. (2) r e d u c e s to the m e a n c l u s t e r s i z e e x p a n s i o n of s i t e p e r c o l a t i o n t h e o r y [4] w i t h the a d d i t i o n of an e x t r a t e r m . The c r i t i c a l v a l u e pS is about 0.199. F i n a l l y , the s t r a i g h t l i n e in the ~ g u r e is the r e s u l t of the m o l e c u l a r f i e l d a p p r o x i m a t i o n for t h i s

16

LETTERS

25 October

1971

model. T h e s e and o t h e r r e s u l t s w i l l be d i s c u s s e d in greater detail elsewhere. T h e a u t h o r w i s h e s to thank P r o f e s s o r C. D o m b for helpful c o m m e n t s and to a c k n o w l e d g e the s u p p o r t of a C o m m o n w e a l t h S c h o l a r s h i p .

References [1] G. S. Rushbrooke, Battelle Colloquium on Critical phenomena {1970). [2] G. S. Rusbbrooke and D. J. Morgan, Mol. Phys. 4 (1961) 29. [3] D. C. Rapaport, J. Phys. C., to be published. [4] M. F. Sykes and J. W. Essam, Phys. Rev. 133 (1964) A310.