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On dynamic scaling in the Heisenberg ferromagnet
Volume 37A, number 5
ON
DYNAMIC
PHYSICS L E T T E R S
SCALING
IN
THE
HEISENBERG
20 December 1971
FERROMAGNET
S. V. M A L E E V A . F. Ioffe Physico-technical Institute, USSR Academy of Science, Leningrad, USSR Received 26 October 1971
The relation between the critical exponents of the dynamic and static scaling theories is rigorously proved in the case of Heisenberg ferromagnets with purely exchange interactions.
On the b a s e of a s u m r u l e , o b t a i n e d f r o m spin c o m m u t a t i o n r e l a t i o n the f o l l o w i n g e q u a t i o n h a s b e e n d e r i v e d in ref. [1]: z : (~ + 3~)f12v .
(1)
H e r e z is the H a l p e r i n - H o h e n b e r g d y n a m i c s c a l i n g e x p o n e n t , ~ and v a r e the e x p o n e n t s of s u s c e p t i b i l i t y and c o r r e l a t i o n length, r e s p e c t i v e l y . Due to the unknown c o n t r i b u t i o n to the s u m r u l e of the high e n e r g y r e g i o n , w h e r e s c a l i n g i s not v a l i d , the d e d u c t i o n in r e f . [1] is not v e r y r i g o r o u s . It i s shown in t h i s p a p e r , that in the c a s e of p u r e l y e x c h a n g e c o u p l i n g , the r e l a t i o n (1) f o l l o w s f r o m t h e t o t a l spin c o n s e r v a t i o n law. I n d e e d , the t o t a l spin c o n s e r v a t i o n law a l l o w s u s to obtain the f o l l o w i n g e x a c t r e l a t i o n f o r the r e t a r d e d G r e e n f u n c t i o n [3]: G_+ ( k = 0,¢o) = -/(u) - 2 t i l l + i S )
H
H e r e ~- = (Tc - T ) / T c . C o m p a r i n g eqs. (2) and (3) and t a k i n g into a c c o u n t , t h a t at H = 0,
406
G:l+(w) at c o ~ 0 i s l e s s than its s i n g u l a r p a r t : G-l.~(w) ~ A w + B w K
and K
s t r o n g c o u p l i n g r e g i m e [5]). But t h i s a s s u m p t i o n l e a d s to the i n e q u a l i t y z > ( 3 v + y ) / 2 v . A s a r e s u l t , eq.(3) should b e b a s e d in a d i f f e r e n t way.
(2)
On the other hand, the dynamic scaling gives the following g e n e r a l expression for t h i s Green function [2,4]: 1 (P w
~ T( 2 v - y ) / 2 one can o b t a i n r e l a t i o n (1). In a d d i t i o n we h a v e at ~- = 0 the w e l l - k n o w n r e l a t i o n
References [1] D.Capoccia, G.Ciccotti and C.Di Castro, Phys. Letters 32A (1970) 359. [2] B. L. Halperin and P. C. Hohenberg, Phys. Rev. 177 (1969) 952. [ 3] A. I. Akhiezer, V. G. Baryakhtar and S. V. Peletminskii Spinovye volny (Spin waves), (Nauka, 1967). [4] A.M. Polyakov, Zh. Eksp. i Teor. Fiz. 57 (1969) 2144. [5] V.N.Gribov and A.A.Migdal, Zh. Eksp.i Teor. Fiz. 55 (1968) 1498.
ON
DYNAMIC
PHYSICS L E T T E R S
SCALING
IN
THE
HEISENBERG
20 December 1971
FERROMAGNET
S. V. M A L E E V A . F. Ioffe Physico-technical Institute, USSR Academy of Science, Leningrad, USSR Received 26 October 1971
The relation between the critical exponents of the dynamic and static scaling theories is rigorously proved in the case of Heisenberg ferromagnets with purely exchange interactions.
On the b a s e of a s u m r u l e , o b t a i n e d f r o m spin c o m m u t a t i o n r e l a t i o n the f o l l o w i n g e q u a t i o n h a s b e e n d e r i v e d in ref. [1]: z : (~ + 3~)f12v .
(1)
H e r e z is the H a l p e r i n - H o h e n b e r g d y n a m i c s c a l i n g e x p o n e n t , ~ and v a r e the e x p o n e n t s of s u s c e p t i b i l i t y and c o r r e l a t i o n length, r e s p e c t i v e l y . Due to the unknown c o n t r i b u t i o n to the s u m r u l e of the high e n e r g y r e g i o n , w h e r e s c a l i n g i s not v a l i d , the d e d u c t i o n in r e f . [1] is not v e r y r i g o r o u s . It i s shown in t h i s p a p e r , that in the c a s e of p u r e l y e x c h a n g e c o u p l i n g , the r e l a t i o n (1) f o l l o w s f r o m t h e t o t a l spin c o n s e r v a t i o n law. I n d e e d , the t o t a l spin c o n s e r v a t i o n law a l l o w s u s to obtain the f o l l o w i n g e x a c t r e l a t i o n f o r the r e t a r d e d G r e e n f u n c t i o n [3]: G_+ ( k = 0,¢o) = -
H
H e r e ~- = (Tc - T ) / T c . C o m p a r i n g eqs. (2) and (3) and t a k i n g into a c c o u n t , t h a t at H = 0,
406
G:l+(w) at c o ~ 0 i s l e s s than its s i n g u l a r p a r t : G-l.~(w) ~ A w + B w K
and K
s t r o n g c o u p l i n g r e g i m e [5]). But t h i s a s s u m p t i o n l e a d s to the i n e q u a l i t y z > ( 3 v + y ) / 2 v . A s a r e s u l t , eq.(3) should b e b a s e d in a d i f f e r e n t way.
(2)
On the other hand, the dynamic scaling gives the following g e n e r a l expression for t h i s Green function [2,4]: 1 (P w
~ T( 2 v - y ) / 2 one can o b t a i n r e l a t i o n (1). In a d d i t i o n we h a v e at ~- = 0 the w e l l - k n o w n r e l a t i o n
References [1] D.Capoccia, G.Ciccotti and C.Di Castro, Phys. Letters 32A (1970) 359. [2] B. L. Halperin and P. C. Hohenberg, Phys. Rev. 177 (1969) 952. [ 3] A. I. Akhiezer, V. G. Baryakhtar and S. V. Peletminskii Spinovye volny (Spin waves), (Nauka, 1967). [4] A.M. Polyakov, Zh. Eksp. i Teor. Fiz. 57 (1969) 2144. [5] V.N.Gribov and A.A.Migdal, Zh. Eksp.i Teor. Fiz. 55 (1968) 1498.