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On the validity of recent phenomenological theories of the Ising model and superfluidity
Volume 28A. number 5
P H Y SIC S L E T T E R S
T h e a s s i s t a n c e of the T e x a s C h r i s t i a n U n i v e r s i t y R e s e a r c h F o u n d a t i o n and of the W r i g h t - P a t terson Air Force Base is appreciated.
ON
THE
VALIDITY OF THE
16 December 1968
1. M. Synek and F. Schmitz, Phys. Letters 27A (1968) 349. 2. M. Synek, A . E . Rainis and E. A. Peterson, J. Chem. Phys. 46 (1967) 2039.
OF RECENT PHENOMENOLOGICAL ISING MODEL AND SUPERFLUIDITY
THEORIES
R. H S I A N G - T A O YEH
Department of Physics, State University of New York at Buffalo, Buffalo, New York, USA Received 2 November 1968
We show explicitly that, the recent phenomenological theories of Wong (for the Ising model) and Mamaladze (for superfluid), are in conflict with exact theory or with experiment.
T h e c r i t i c a l i n d i c e s , a s p r e d i c t e d by the L a n dau t h e o r y of the s e c o n d o r d e r p h a s e t r a n s i t i o n , a r e in c o n f l i c t with e x p e r i m e n t a l r e s u l t s in s u p e r fluid, o r e x a c t c a l c u l a t i o n of the I s i n g - M o d e l [1]. L a n d a u ' s t h e o r y m a k e s u s e of p o w e r s e r i e s e x p a n s i o n of the f r e e e n e r g y in the o r d e r p a r a m e t e r . R e c e n t l y , s e v e r a l a t t e m p t s [2-5] h a v e b e e n m a d e to d e r i v e the c o r r e c t c r i t i c a l i n d i c e s f r o m L a n d a u ' s t h e o r y , by c h o o s i n g a p p r o p r i a t e t e m p e r a t u r e d e p e n d e n c e of the c o e f f i c i e n t in the f r e e e n e r g y e x p a n s i o n . H o w e v e r , all t h e s e t h e o r i e s m a d e u s e of the p r i n c i p l e of m i n i m i z a t i o n of the f r e e e n e r g y , and e q u a t e the m o s t p r o b a b l e v a l u e of the o r d e r p a r a m e t e r to the a v e r a g e v a l u e . G e n e r a l a r g u m e n t s [1] h a v e a l r e a d y shown that t h i s p r o c e d u r e n e g l e c t s the e f f e c t of the f l u c t u a t i o n in the o r d e r p a r a m e t e r , which i s i m p o r t a n t n e a r the t r a n s i t i o n point, e s p e c i a l l y f o r s y s t e m s with s h o r t r a n g e i n t e r a c t i o n s . H e n c e t h i s a p p r o a c h i s not a p p r o p r i a t e f o r s u p e r f l u i d o r I s i n g - M o d e l . It i s o u r p u r p o s e to show t h i s e x p l i c i t l y . F o r the t w o - d i m e n s i o n a l I s i n g - M o d e l , Wong [2] s u g g e s t e d the f o l l o w i n g e x p r e s s i o n f o r the f r e e e n e r g y d e n s i t y d i f f e r e n c e f - f o , in t e r m s of the reduced temperature t = T/Tc:
+ C4(1-t)]~P) 4 + C6(1-t)¼(p)6 +
+ C 1 2 ( 1 - t ) ½ ( p ) 12 + C 1 4 ( 1 - t ) ¼ ( p ) 14 + + c 1 6 ( p ) 16 - h i p ) .
i kBT(1 - t)~ g(r,r')-
8A °
~I Jo(ir,~o
I-fiR)
for T > T c
(2) R = I r - r ' I, J o i s the z e r o t h o r d e r B e s s e l f u n c tion. F o r T < T c, g(Jr, r' ) has the s a m e f o r m , e x c e p t that B o i s r e p l a c e d by a d i f f e r e n t c o n s t a n t . So, in the n o t a t i o n of r e f . 1, g(R) ~ R ° a s ]l-tiRe0, a n d g ( R ) ~ R-~ f o r ] l - t i R >>1. In t e r m s of the c r i t i c a l i n d e x , t h i s i m p l i e s that 77 = 0. Eq. (1) a l s o i m p l i e s that 7 = ~ • S i n c e the d i m e n s i o n of the s y s t e m d = 2, t h i s m o d e l d o e s not a g r e e with one of the c o n s e q u e n c e s of s c a l i n g l a w s [1], n a m e l y , d 7 / ( 2 - 7 / ) = 2. M o r e o v e r , the e x a c t c a l c u l a t i o n of the I s i n g m o d e l [1] y i e l d s f o r I I - t l R >>1,
l exp[-Ii-tln]
( ( t - 1)R) ~ J
r rc
I (i:t) ll J 2
f - f o = Ao(1 - t ) - " [ V(P)] 2 + S o ( 1 - t)¼(P )2 +
+ C 8 ( 1 - t ) i p ) 8 + C l O ( 1 - t ) ~ i p ) 10 +
W h e r e Ao, Bo, Cn'S a r e c o n s t a n t s , (P) is the o r d e r p a r a m e t e r and h is the t h e r m a l d y n a m i c c o n j u g a t e of (p). F o l l o w i n g the m e t h o d of r e f . 1, the 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 g ( r , r ' ) i s obt a i n e d as:
(1)
and f o r (1 - t)R << 1
g ( n ) ~ R-¼
T < Tc .
(3)
So the e x a c t s o l u t i o n of g ( R ) i s q u i t e d i f f e r e n t f r o m that g i v e n by W o n g ' s m o d e l . N o t e that 7/ = ¼ 345
PHYSICS
Volume 28A, n u m b e r 5
f o r t h e e x a c t s o l u t i o n , a n d t h e f o r m of g ( R ) i s q u i t e d i f f e r e n t f o r T > T c a n d T < T c. T h e p h e n o menological model always gives similar depend e n c e of g ( R ) f o r b o t h c a s e s . F o r s u p e r f l u i d i t y , e x p e r i m e n t s [6] a l s o s u g g e s t t h a t , t h e r a d i a l d i s tribution function is different for T > T~ and T < Tk cases. For specific heat, all the above phenomenological theories gave finite discontinuity at the transition point, but without anomaly. This is in conf l i c t w i t h l o g a r i t h m i c s i n g u l a r i t y of t h e I s i n g m o d el, and the k-shape singularity in superfluid. Precisely because free energy is not an analytic funct i o n a t T c , e x p a n s i o n of t h e f r e e e n e r g y i n t h e f o r m of eq. (1) i s n o t c o r r e c t . W e c o n c l u d e t h a t , the Landau's theory approach is not appropriate
APPROXIMATE
ANALYTIC QUANTUM
LETTERS
16 D e c e m b e r
1968
for systems with short range interactions. F i n a l l y , i t i s m y p l e a s u r e to t h a n k D r . I s i h a r a , Dr. Lee, Dr. Lin and Dr. Mechetti for useful discussions.
References 1. Kadanoff et al., Rev. Mod. Phys. 39 (1967) 395. 2. V.K.Wong, Phys. L e t t e r s 27A (1968) 358. 3. Yu.G. Mamaladze, Soviet P h y s i c s J E T P 25 (1967) 479. Also P h y s i c s L e t t e r s 27A (1968) 322. 4. V.K. Wong, P h y s i c s L e t t e r s 27A (1968) 269. 5. V . L . G i n z b u r g and L. P. Pitaevskii, Soviet P h y s i c s J E T P 7 (1958) 858. 6. D.G.Henshaw, Phys. Rev. 119 (1960) 9.
SOLUTION MECHANICAL
OF
THE EQUATIONS PROBLEMS
IN T H E
I. V. A M I R K H A N O V a n d V. S. G U R I A N O V
Joint Institute f o r Nuclear Research, Dubna, USSR Received 2 November 1968
The SchrSdinger equation has an analytic solution f o r a s m a l l n u m b e r of simple potentials. In the general case the equation m u s t be solved n u m e r i c a l l y . In the p r e s e n t paper, approximate solutions of the Schrt~dinger equation with a r b i t r a r y local (or non-local) potential, are obtained in analytic f o r m . The same method can be applied to a wide c l a s s of i n t e g r o - d i f f e r e n t i a l equations and of s y s t e m s of such equations.
If t h e p o t e n t i a l V(r) d e c r e a s e s f a s t f o r r ~ ~o, t h e n w e c a n e x p a n d t h e f u n c t i o n V(r)Co(r) i n t h e Fourier series
P o t e n t i a l s c a t t e r i n g of two p a r t i c l e s . L e t u s c o n s i d e r f o r s i m p l i c i t y t h e c a s e l = 0 (the c a s e l ¢ 0, h a s b e e n c o n s i d e r e d i n r e f . 1). a) T h e S c h r 6 d i n g e r e q u a t i o n w i t h l o c a l p o t e n tial has the form dr2
] q~°(r) = V ( r ) c ° ( r ) "
V(r)~o(r) = ~ Bn~Pn(r) n
(1)
where
L e t u s w r i t e eq. (1) i n t h e i n t e g r a l f o r m
B n = fV(r)~oo(r)Ckn(r)dr .
r q~o(r) = s i n k r - # { e x p i k r
f sinkr'
o
~Oo(r) = s i n k r + ~ B n X n ( r ) n
(2)
f e x p i k r ' } V(r')q)o(r')d~/ r
346
(4)
S u b s t i t u t i n g eq. (3) i n t o (2) w e o b t a i n
+
oO
+ sinkr
(3)
.
where
(5)
P H Y SIC S L E T T E R S
T h e a s s i s t a n c e of the T e x a s C h r i s t i a n U n i v e r s i t y R e s e a r c h F o u n d a t i o n and of the W r i g h t - P a t terson Air Force Base is appreciated.
ON
THE
VALIDITY OF THE
16 December 1968
1. M. Synek and F. Schmitz, Phys. Letters 27A (1968) 349. 2. M. Synek, A . E . Rainis and E. A. Peterson, J. Chem. Phys. 46 (1967) 2039.
OF RECENT PHENOMENOLOGICAL ISING MODEL AND SUPERFLUIDITY
THEORIES
R. H S I A N G - T A O YEH
Department of Physics, State University of New York at Buffalo, Buffalo, New York, USA Received 2 November 1968
We show explicitly that, the recent phenomenological theories of Wong (for the Ising model) and Mamaladze (for superfluid), are in conflict with exact theory or with experiment.
T h e c r i t i c a l i n d i c e s , a s p r e d i c t e d by the L a n dau t h e o r y of the s e c o n d o r d e r p h a s e t r a n s i t i o n , a r e in c o n f l i c t with e x p e r i m e n t a l r e s u l t s in s u p e r fluid, o r e x a c t c a l c u l a t i o n of the I s i n g - M o d e l [1]. L a n d a u ' s t h e o r y m a k e s u s e of p o w e r s e r i e s e x p a n s i o n of the f r e e e n e r g y in the o r d e r p a r a m e t e r . R e c e n t l y , s e v e r a l a t t e m p t s [2-5] h a v e b e e n m a d e to d e r i v e the c o r r e c t c r i t i c a l i n d i c e s f r o m L a n d a u ' s t h e o r y , by c h o o s i n g a p p r o p r i a t e t e m p e r a t u r e d e p e n d e n c e of the c o e f f i c i e n t in the f r e e e n e r g y e x p a n s i o n . H o w e v e r , all t h e s e t h e o r i e s m a d e u s e of the p r i n c i p l e of m i n i m i z a t i o n of the f r e e e n e r g y , and e q u a t e the m o s t p r o b a b l e v a l u e of the o r d e r p a r a m e t e r to the a v e r a g e v a l u e . G e n e r a l a r g u m e n t s [1] h a v e a l r e a d y shown that t h i s p r o c e d u r e n e g l e c t s the e f f e c t of the f l u c t u a t i o n in the o r d e r p a r a m e t e r , which i s i m p o r t a n t n e a r the t r a n s i t i o n point, e s p e c i a l l y f o r s y s t e m s with s h o r t r a n g e i n t e r a c t i o n s . H e n c e t h i s a p p r o a c h i s not a p p r o p r i a t e f o r s u p e r f l u i d o r I s i n g - M o d e l . It i s o u r p u r p o s e to show t h i s e x p l i c i t l y . F o r the t w o - d i m e n s i o n a l I s i n g - M o d e l , Wong [2] s u g g e s t e d the f o l l o w i n g e x p r e s s i o n f o r the f r e e e n e r g y d e n s i t y d i f f e r e n c e f - f o , in t e r m s of the reduced temperature t = T/Tc:
+ C4(1-t)]~P) 4 + C6(1-t)¼(p)6 +
+ C 1 2 ( 1 - t ) ½ ( p ) 12 + C 1 4 ( 1 - t ) ¼ ( p ) 14 + + c 1 6 ( p ) 16 - h i p ) .
i kBT(1 - t)~ g(r,r')-
8A °
~I Jo(ir,~o
I-fiR)
for T > T c
(2) R = I r - r ' I, J o i s the z e r o t h o r d e r B e s s e l f u n c tion. F o r T < T c, g(Jr, r' ) has the s a m e f o r m , e x c e p t that B o i s r e p l a c e d by a d i f f e r e n t c o n s t a n t . So, in the n o t a t i o n of r e f . 1, g(R) ~ R ° a s ]l-tiRe0, a n d g ( R ) ~ R-~ f o r ] l - t i R >>1. In t e r m s of the c r i t i c a l i n d e x , t h i s i m p l i e s that 77 = 0. Eq. (1) a l s o i m p l i e s that 7 = ~ • S i n c e the d i m e n s i o n of the s y s t e m d = 2, t h i s m o d e l d o e s not a g r e e with one of the c o n s e q u e n c e s of s c a l i n g l a w s [1], n a m e l y , d 7 / ( 2 - 7 / ) = 2. M o r e o v e r , the e x a c t c a l c u l a t i o n of the I s i n g m o d e l [1] y i e l d s f o r I I - t l R >>1,
l exp[-Ii-tln]
( ( t - 1)R) ~ J
r rc
I (i:t) ll J 2
f - f o = Ao(1 - t ) - " [ V(P)] 2 + S o ( 1 - t)¼(P )2 +
+ C 8 ( 1 - t ) i p ) 8 + C l O ( 1 - t ) ~ i p ) 10 +
W h e r e Ao, Bo, Cn'S a r e c o n s t a n t s , (P) is the o r d e r p a r a m e t e r and h is the t h e r m a l d y n a m i c c o n j u g a t e of (p). F o l l o w i n g the m e t h o d of r e f . 1, the 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 g ( r , r ' ) i s obt a i n e d as:
(1)
and f o r (1 - t)R << 1
g ( n ) ~ R-¼
T < Tc .
(3)
So the e x a c t s o l u t i o n of g ( R ) i s q u i t e d i f f e r e n t f r o m that g i v e n by W o n g ' s m o d e l . N o t e that 7/ = ¼ 345
PHYSICS
Volume 28A, n u m b e r 5
f o r t h e e x a c t s o l u t i o n , a n d t h e f o r m of g ( R ) i s q u i t e d i f f e r e n t f o r T > T c a n d T < T c. T h e p h e n o menological model always gives similar depend e n c e of g ( R ) f o r b o t h c a s e s . F o r s u p e r f l u i d i t y , e x p e r i m e n t s [6] a l s o s u g g e s t t h a t , t h e r a d i a l d i s tribution function is different for T > T~ and T < Tk cases. For specific heat, all the above phenomenological theories gave finite discontinuity at the transition point, but without anomaly. This is in conf l i c t w i t h l o g a r i t h m i c s i n g u l a r i t y of t h e I s i n g m o d el, and the k-shape singularity in superfluid. Precisely because free energy is not an analytic funct i o n a t T c , e x p a n s i o n of t h e f r e e e n e r g y i n t h e f o r m of eq. (1) i s n o t c o r r e c t . W e c o n c l u d e t h a t , the Landau's theory approach is not appropriate
APPROXIMATE
ANALYTIC QUANTUM
LETTERS
16 D e c e m b e r
1968
for systems with short range interactions. F i n a l l y , i t i s m y p l e a s u r e to t h a n k D r . I s i h a r a , Dr. Lee, Dr. Lin and Dr. Mechetti for useful discussions.
References 1. Kadanoff et al., Rev. Mod. Phys. 39 (1967) 395. 2. V.K.Wong, Phys. L e t t e r s 27A (1968) 358. 3. Yu.G. Mamaladze, Soviet P h y s i c s J E T P 25 (1967) 479. Also P h y s i c s L e t t e r s 27A (1968) 322. 4. V.K. Wong, P h y s i c s L e t t e r s 27A (1968) 269. 5. V . L . G i n z b u r g and L. P. Pitaevskii, Soviet P h y s i c s J E T P 7 (1958) 858. 6. D.G.Henshaw, Phys. Rev. 119 (1960) 9.
SOLUTION MECHANICAL
OF
THE EQUATIONS PROBLEMS
IN T H E
I. V. A M I R K H A N O V a n d V. S. G U R I A N O V
Joint Institute f o r Nuclear Research, Dubna, USSR Received 2 November 1968
The SchrSdinger equation has an analytic solution f o r a s m a l l n u m b e r of simple potentials. In the general case the equation m u s t be solved n u m e r i c a l l y . In the p r e s e n t paper, approximate solutions of the Schrt~dinger equation with a r b i t r a r y local (or non-local) potential, are obtained in analytic f o r m . The same method can be applied to a wide c l a s s of i n t e g r o - d i f f e r e n t i a l equations and of s y s t e m s of such equations.
If t h e p o t e n t i a l V(r) d e c r e a s e s f a s t f o r r ~ ~o, t h e n w e c a n e x p a n d t h e f u n c t i o n V(r)Co(r) i n t h e Fourier series
P o t e n t i a l s c a t t e r i n g of two p a r t i c l e s . L e t u s c o n s i d e r f o r s i m p l i c i t y t h e c a s e l = 0 (the c a s e l ¢ 0, h a s b e e n c o n s i d e r e d i n r e f . 1). a) T h e S c h r 6 d i n g e r e q u a t i o n w i t h l o c a l p o t e n tial has the form dr2
] q~°(r) = V ( r ) c ° ( r ) "
V(r)~o(r) = ~ Bn~Pn(r) n
(1)
where
L e t u s w r i t e eq. (1) i n t h e i n t e g r a l f o r m
B n = fV(r)~oo(r)Ckn(r)dr .
r q~o(r) = s i n k r - # { e x p i k r
f sinkr'
o
~Oo(r) = s i n k r + ~ B n X n ( r ) n
(2)
f e x p i k r ' } V(r')q)o(r')d~/ r
346
(4)
S u b s t i t u t i n g eq. (3) i n t o (2) w e o b t a i n
+
oO
+ sinkr
(3)
.
where
(5)