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Non-universal critical behavior in the planar XY-model with fourth order anisotropy
Journal of Magnetism and Magnehc Matermls 31-34 (1983) I 115-1116 NON-UNIVERSAL CRITICAL BEHAVIOR ORDER ANISOTROPY
1115
IN THE PLANAR XY-MODEL
WITH FOURTH
D.P. L A N D A U
Department of Phystc~ and Astronomy, Umversttv of Georgta. Athens, GA 30602, USA
We have used a Monte Carlo block d~stnbuhon method to study the two-dimensional classical XY-model with a fourth-order amsotropy field h 4 We fred that T~ vanes only slowly wrothh 4 but that v depends strongly o n h 4
1. Introduction
point) and at very low temperature it should approach 2 / 3 ( T = 0 fixed point) An estimate for the critical temperature T~ is obtained by comparing the values of U for two different size lattices L ' and L and using the condmon
It has been known for some time [1] that the two-dimensional XY-model cannot show long-range order at any finite temperature It is now believed [2] that th~s model undergoes a specml kind of phase t r a n s m o n to a state of topological order with a susceptabihty which d~verges exponentmlly as T~ is approached from above. The effects due to the application of symmetry breaking fields to this model are also expected to be interesting. Jos6 et al. [3] suggested that for Hamlltomans of the general form ".3(=
-J~(s,.s).+s,~s/~)+hp~ cos(pO,), tltt
(UL./UL)T
OUbL/a~:LIu. = b ~/~,
We have used a standard importance sampling Monte Carlo method [5] to study the behavior of classical spin vectors [s,[ = 1 (spin dlmenslonallty = 3) on L × L square lathces with periodic boundary conditions Following Binder [4] we calculate higher order moments of the order parameter m c for each size lattice
to obtain v,. = 1 - ( m 4 ) ~ / ( 3 ( m 2 ) ~ )
(3)
At hagh temperatures UL should go to zero ( T = m fixed 0304-8853/83/0000-0000/$03.00
(5)
where b = L'/L. This procedure has been shown to be very effective for Ising systems [4]. To our knowledge this as the first time it has been applied to a continuous spin system. Since the UL must be calculated rather accurately we have restricted our study to lattices of modest size, L < 20, and have used 20000 MCS (Monte Carlo s t e p s / s p i n ) for averaging after first discarding 2000 MCS. In most cases the same data point was then repeated at least once using a different starting configuration so that a total of 40 000-60 000 MCS were kept in all Values of h 4 a s large as 20J were studied.
l
2. Method
(4)
The slope of the curve of Uc, vs Ut yields the critical exponent u through the expresslon
(1)
where 0, is the angle which the classical spin makes with the x-axis, simple critical b e h a w o r was to be expected only for p < 4 For p = 4 they predicted non-universal critical behavior with exponents which d e p e n d upon h4 and for p = 6 they predict that the ordered phase IS separated from the disordered phase by an XY-hke region which exists only between some upper (Tj) and lower (T2) crlUcal temperatures. It should be noted, however, that their calculations using a Migdal transformation did not confirm their predictions. In this paper we shall present results obtained using a M o n t e Carlo block distribution method [4] for the XY-model with fourth order amsotropy l e eq. (1) with p = 4
r. = I
3. Results In fig 1 we show a plot of U20 vs U8 for four different values of h 4 i n c l u d i n g h 4 = 0 The results for h 4 = 0 (the simple XY-model) are particularly Interestms. The curve of U20 vs Us approaches the hne U2o = U8 as the temperature is lowered and at kTJJ = 0.74 _+ 0 03 touches the hne For lower temperatures the data points remain on the line and simply move along the hne towards the zero temperature fixed point We Interpret this behavior as indicating that below T~ rather than an ordered phase there is a line of fixed points consistent with the description of Kosterhtz and Thouless [2]. The fixed point value U * varies between 0 652 _-4-0 006 and 2 / 3 . Our estimate for T~ agrees well with that determined from an analysis of high temperature susceptl b l h t y data obtained by a standard Monte Carlo study [6] When a non-zero h 4 is included, the behavior of the m o m e n t s changes the data show a " b u l g e " above the U20 = Us hne at low temperatures and only cross the
© 1983 N o r t h - H o l l a n d
1116
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d e r s t o o d by e x a m i n i n g the f o r m of the a m s o t r o p 3 T h e a n t s o t r o p y e n e r g y IS e s s e n t i a l l y p a r a b o h c for a n g l e s close to the f o u r m i n i m a axes (see e q n (I)) a n d s m a l l a n g l e e x c i t a t i o n s are h e n c e p o s s i b l e e x c e p t for ext r e m e l y large v a l u e s of h 4 t e h 4 >> [1 cos(40, )] i Estu-nates for v h a v e b e e n o b t a i n e d f r o m the q o p e ~ of the L~0 vs Us c u r v e s In o r d e r to obtali1 r e h a b l c e s t i m a t e s for v we s h o u l d e x t r a p o l a t e x a l u e s o b t , n n e d f r o m c u r x e s o f b~,z vs b'z a s h ~ ~c W e d o n o t y e t h a ~ e s u f f l c m n t a c c u r a t e d a t a for o t h e r lattice s~zes to do th~s, b u t we n o t e that for t h e Islng m o d e l results shov, n m fig 1 v i = 1 2 ± 0 1 as c o m p a r e d with t h e k n o w n cxact result ol v t 1 W e o b t a i n o f c o u r s e v ~ = 0 for h 4 = 0 a n d v a l u e s of v i w h i c h slowly i n c r e a s e Irom v L = 0 6 1 ± 0 0 5 to v i = 1 3 0 ± 0 15 for h 4 = 2 0 T h e v a r i a t i o n ol v with h 4 is c o n s i s t e n t w~th the c h a n g e m /J * m I n d i c a t i n g n o n - u n i v e r s a l critical b e h a v i o r F m ther c a l c u l a t i o n s are in p r o g r e s s
U8 Fig 1 U2o plotted vs Ux for the XY-model with various values of fourth order an,sotropy h 4 The solid curve is obtained for the lsmg square lattice with peno&c boundary con&hons
line at a single t e m p e r a t u r e T h i s b e h a v i o r i n d i c a t e s the p r e s e n c e o f a n o r d e r e d low t e m p e r a t u r e s t a t e N o t e t h a t t h e v a l u e of U * d e p e n d s e x p h c l t l y o n h4 F o r c o m p a r i s o n we h a v e also c a l c u l a t e d t h e c u r v e for a s i m p l e I s l n g m o d e l with i n t e r a c t i o n ( J / 2 ) a n d p e n odlc b o u n d a r y c o n d i t i o n s w h i c h m u s t be t h e h m i t l n g c a s e of o u r m o d e l as h a --+ m [7] T h e result Is s h o w n in fig 1 W e see t h a t t h e low t e m p e r a t u r e d a t a for the XY-model with fourth order anisotropy rather quickly a p p r o a c h t h e I s m g m o d e l c u r v e as h 4 is i n c r e a s e d At h i g h e r t e m p e r a t u r e s the c u r v e s a p p r o a c h t h e l s l n g c u r v e m o r e slowly a n d e v e n for h4 = 2 0 J s u b q a n t m l diff e r e n c e s are still o b v i o u s N o t e t h a t t h e v a l u e s o f the fixed p o i n t U * v a r y c o n t i n u o u s l y a n d m o v e toward~ t h e I s l n g fixed p o i n t U * I s m g as h 4 i n c r e a s e s W e f m d t h a t T~ i n c r e a s e s q m t e slowly with i n c r e a s i n g h 4 a n d for h 4 = 2 0 J is still o n l y k T J J = 0 86 +_ 0 02 w h e r e a s the h m l t m g I s m g v a l u e is 1 / l n ( ~ 2 + 1 ) = 1 13J T h i s slow a p p r o a c h to the a s y m p t o t i c I s m g b e h a v i o r c a n be u n -
4. Conclusions M o n t e C a r l o block d l s m b u t l o n c a l c u l a t i o n s h a v e b e e n s u c c e s s f u l l y carried o u t for t h e classical X Y - m o d e l w , t h f o u r t h o r d e r a n i s o t r o p y W e find that the h x e d p o i n t v a l u e s U * a n d t h e e x p o n e n t v vary with a m s o t rop> h 4 W e wish to t h a n k D r H 1 H e r r r n a n n tor h e l p f u l c o m m e n t s a n d s u g g e s t i o n s T h i s r e s e a r c h w'a,, s u p p o r t e d m p a r t by t h e N a t i o n a l S c m n c e F o u n d a t i o n
References [I] N D Mermln and H Wagner, Ph~s Re', Lett 17 (1966) 1133 [2] J M K o s t e r h t z a n d D J Thouless, J Phys ( 6 ( 1 9 7 3 ) 1IS1 [3] J V Jose, L P Kadanoff, S K~rkpatnck and D R Nelson, Phys Rev B16(1977)1217 [4] K Bmder, Z Phvs B43(1981) 119 [5] See e g K Binder, m Monte Carlo Methods m St,it~stlcal Physws, ed K Binder (Springer Verlag, Berlin, 1979) [6] D P Landau and K Bmder, Phvs Re'~ B24(1981) 1391 [7] M N Barber (private c o m m u m c a n o n ) I) D Betts, Can 1 Ph'~s 42 (1964) 1564 M Suzuki, Progr Theoret Phv,, 37 (1967) 770
1115
IN THE PLANAR XY-MODEL
WITH FOURTH
D.P. L A N D A U
Department of Phystc~ and Astronomy, Umversttv of Georgta. Athens, GA 30602, USA
We have used a Monte Carlo block d~stnbuhon method to study the two-dimensional classical XY-model with a fourth-order amsotropy field h 4 We fred that T~ vanes only slowly wrothh 4 but that v depends strongly o n h 4
1. Introduction
point) and at very low temperature it should approach 2 / 3 ( T = 0 fixed point) An estimate for the critical temperature T~ is obtained by comparing the values of U for two different size lattices L ' and L and using the condmon
It has been known for some time [1] that the two-dimensional XY-model cannot show long-range order at any finite temperature It is now believed [2] that th~s model undergoes a specml kind of phase t r a n s m o n to a state of topological order with a susceptabihty which d~verges exponentmlly as T~ is approached from above. The effects due to the application of symmetry breaking fields to this model are also expected to be interesting. Jos6 et al. [3] suggested that for Hamlltomans of the general form ".3(=
-J~(s,.s).+s,~s/~)+hp~ cos(pO,), tltt
(UL./UL)T
OUbL/a~:LIu. = b ~/~,
We have used a standard importance sampling Monte Carlo method [5] to study the behavior of classical spin vectors [s,[ = 1 (spin dlmenslonallty = 3) on L × L square lathces with periodic boundary conditions Following Binder [4] we calculate higher order moments of the order parameter m c for each size lattice
to obtain v,. = 1 - ( m 4 ) ~ / ( 3 ( m 2 ) ~ )
(3)
At hagh temperatures UL should go to zero ( T = m fixed 0304-8853/83/0000-0000/$03.00
(5)
where b = L'/L. This procedure has been shown to be very effective for Ising systems [4]. To our knowledge this as the first time it has been applied to a continuous spin system. Since the UL must be calculated rather accurately we have restricted our study to lattices of modest size, L < 20, and have used 20000 MCS (Monte Carlo s t e p s / s p i n ) for averaging after first discarding 2000 MCS. In most cases the same data point was then repeated at least once using a different starting configuration so that a total of 40 000-60 000 MCS were kept in all Values of h 4 a s large as 20J were studied.
l
2. Method
(4)
The slope of the curve of Uc, vs Ut yields the critical exponent u through the expresslon
(1)
where 0, is the angle which the classical spin makes with the x-axis, simple critical b e h a w o r was to be expected only for p < 4 For p = 4 they predicted non-universal critical behavior with exponents which d e p e n d upon h4 and for p = 6 they predict that the ordered phase IS separated from the disordered phase by an XY-hke region which exists only between some upper (Tj) and lower (T2) crlUcal temperatures. It should be noted, however, that their calculations using a Migdal transformation did not confirm their predictions. In this paper we shall present results obtained using a M o n t e Carlo block distribution method [4] for the XY-model with fourth order amsotropy l e eq. (1) with p = 4
r. = I
3. Results In fig 1 we show a plot of U20 vs U8 for four different values of h 4 i n c l u d i n g h 4 = 0 The results for h 4 = 0 (the simple XY-model) are particularly Interestms. The curve of U20 vs Us approaches the hne U2o = U8 as the temperature is lowered and at kTJJ = 0.74 _+ 0 03 touches the hne For lower temperatures the data points remain on the line and simply move along the hne towards the zero temperature fixed point We Interpret this behavior as indicating that below T~ rather than an ordered phase there is a line of fixed points consistent with the description of Kosterhtz and Thouless [2]. The fixed point value U * varies between 0 652 _-4-0 006 and 2 / 3 . Our estimate for T~ agrees well with that determined from an analysis of high temperature susceptl b l h t y data obtained by a standard Monte Carlo study [6] When a non-zero h 4 is included, the behavior of the m o m e n t s changes the data show a " b u l g e " above the U20 = Us hne at low temperatures and only cross the
© 1983 N o r t h - H o l l a n d
1116
1) P L a n d a u / Plana; ,k Y- m o d e l i, tilt fi)ut tit o!der antsotrotn
07
i?
T=O fixed pore,\ /
/
/
/L,Tai' /
/A/ t
/Y /://4: I11
/ 0 5 •
!
'/~/ /
/ I
Iz D// /; //z i el i x
04
I 04
o"
I 06
05
J 07
d e r s t o o d by e x a m i n i n g the f o r m of the a m s o t r o p 3 T h e a n t s o t r o p y e n e r g y IS e s s e n t i a l l y p a r a b o h c for a n g l e s close to the f o u r m i n i m a axes (see e q n (I)) a n d s m a l l a n g l e e x c i t a t i o n s are h e n c e p o s s i b l e e x c e p t for ext r e m e l y large v a l u e s of h 4 t e h 4 >> [1 cos(40, )] i Estu-nates for v h a v e b e e n o b t a i n e d f r o m the q o p e ~ of the L~0 vs Us c u r v e s In o r d e r to obtali1 r e h a b l c e s t i m a t e s for v we s h o u l d e x t r a p o l a t e x a l u e s o b t , n n e d f r o m c u r x e s o f b~,z vs b'z a s h ~ ~c W e d o n o t y e t h a ~ e s u f f l c m n t a c c u r a t e d a t a for o t h e r lattice s~zes to do th~s, b u t we n o t e that for t h e Islng m o d e l results shov, n m fig 1 v i = 1 2 ± 0 1 as c o m p a r e d with t h e k n o w n cxact result ol v t 1 W e o b t a i n o f c o u r s e v ~ = 0 for h 4 = 0 a n d v a l u e s of v i w h i c h slowly i n c r e a s e Irom v L = 0 6 1 ± 0 0 5 to v i = 1 3 0 ± 0 15 for h 4 = 2 0 T h e v a r i a t i o n ol v with h 4 is c o n s i s t e n t w~th the c h a n g e m /J * m I n d i c a t i n g n o n - u n i v e r s a l critical b e h a v i o r F m ther c a l c u l a t i o n s are in p r o g r e s s
U8 Fig 1 U2o plotted vs Ux for the XY-model with various values of fourth order an,sotropy h 4 The solid curve is obtained for the lsmg square lattice with peno&c boundary con&hons
line at a single t e m p e r a t u r e T h i s b e h a v i o r i n d i c a t e s the p r e s e n c e o f a n o r d e r e d low t e m p e r a t u r e s t a t e N o t e t h a t t h e v a l u e of U * d e p e n d s e x p h c l t l y o n h4 F o r c o m p a r i s o n we h a v e also c a l c u l a t e d t h e c u r v e for a s i m p l e I s l n g m o d e l with i n t e r a c t i o n ( J / 2 ) a n d p e n odlc b o u n d a r y c o n d i t i o n s w h i c h m u s t be t h e h m i t l n g c a s e of o u r m o d e l as h a --+ m [7] T h e result Is s h o w n in fig 1 W e see t h a t t h e low t e m p e r a t u r e d a t a for the XY-model with fourth order anisotropy rather quickly a p p r o a c h t h e I s m g m o d e l c u r v e as h 4 is i n c r e a s e d At h i g h e r t e m p e r a t u r e s the c u r v e s a p p r o a c h t h e l s l n g c u r v e m o r e slowly a n d e v e n for h4 = 2 0 J s u b q a n t m l diff e r e n c e s are still o b v i o u s N o t e t h a t t h e v a l u e s o f the fixed p o i n t U * v a r y c o n t i n u o u s l y a n d m o v e toward~ t h e I s l n g fixed p o i n t U * I s m g as h 4 i n c r e a s e s W e f m d t h a t T~ i n c r e a s e s q m t e slowly with i n c r e a s i n g h 4 a n d for h 4 = 2 0 J is still o n l y k T J J = 0 86 +_ 0 02 w h e r e a s the h m l t m g I s m g v a l u e is 1 / l n ( ~ 2 + 1 ) = 1 13J T h i s slow a p p r o a c h to the a s y m p t o t i c I s m g b e h a v i o r c a n be u n -
4. Conclusions M o n t e C a r l o block d l s m b u t l o n c a l c u l a t i o n s h a v e b e e n s u c c e s s f u l l y carried o u t for t h e classical X Y - m o d e l w , t h f o u r t h o r d e r a n i s o t r o p y W e find that the h x e d p o i n t v a l u e s U * a n d t h e e x p o n e n t v vary with a m s o t rop> h 4 W e wish to t h a n k D r H 1 H e r r r n a n n tor h e l p f u l c o m m e n t s a n d s u g g e s t i o n s T h i s r e s e a r c h w'a,, s u p p o r t e d m p a r t by t h e N a t i o n a l S c m n c e F o u n d a t i o n
References [I] N D Mermln and H Wagner, Ph~s Re', Lett 17 (1966) 1133 [2] J M K o s t e r h t z a n d D J Thouless, J Phys ( 6 ( 1 9 7 3 ) 1IS1 [3] J V Jose, L P Kadanoff, S K~rkpatnck and D R Nelson, Phys Rev B16(1977)1217 [4] K Bmder, Z Phvs B43(1981) 119 [5] See e g K Binder, m Monte Carlo Methods m St,it~stlcal Physws, ed K Binder (Springer Verlag, Berlin, 1979) [6] D P Landau and K Bmder, Phvs Re'~ B24(1981) 1391 [7] M N Barber (private c o m m u m c a n o n ) I) D Betts, Can 1 Ph'~s 42 (1964) 1564 M Suzuki, Progr Theoret Phv,, 37 (1967) 770