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Padé approximant results for the spherical and classical Heisenberg models
Volume28A, number 6
PADF.
APPROXIMANT
PHYSICS
LETTERS
RESULTS FOR HEISENBERG
30 D e c e m b e r 1968
THE SPHERICAL MODELS
AND
CLASSICAL
N. W. D A L T O N
Atomic Energy Research Establishment, Harwell. Berkshire. England and D. W. W O O D
Department of Mathematics. University of Nottingham. Noltinghma. England Received 15 N o v e m b e r 1968
A c o m p u t e r has been used to derive susceptibility s e r i e s for the s p h e r i c a l model. By comparing the Pad~ r e s u l t s for these s e r i e s with those for the two-dimensional c l a s s i c a l H e i s e n b e r g model we conclude that the c r i t i c a l t e m p e r a t u r e of the l a t t e r is finite.
It has long been argued (ref. I) that the spontaneous magnetisation of the Heisenberg model is zero in two-dimensions (i.e. [2]). This has been established rigorously by M e r m i n and Wagner (ref. 2). However work by Stanley and Kaplan (ref. 3) based oh numerical extrapolations of the series expansions for the Heisenberg model indicates that the transition temperature (Tc) of the [2] Heisenberg model is not zero, as might be expected, but finite. The purpose of this letter, is to investigate the validity of their extrapolations by using a computer to extend previous work (refs. 4-7) on the.susceptibility series of the spherical model (ref. 8). The method We have used to derive the series Table i. d-log Pad6 results (spherical model) (simple cubic lattice21 terms) Pad6 [2.2]
}Xq. Kc 1.4415
y 1.4593
[3,3] [4,4]
1.5065
1.8507
[5,5]
1.5144
1.9590
[6,6]
1.5158
1.9858
[7,7]
1.5163
1.9985
[8.8]
1.5163
1.9974
[9,9]
1.5163
1.9974
[10,10]
1.5164
1.9998
Exact
1.516386...
2.0000...
i s to c o m p u t e r i s e t h e c a l c u l a t i o n s o u t l i n e d in r e f . 5. In p a r t i c u l a r w e h a v e c a l c u l a t e d 40 t e r m s in the series for the simple cubic, body-centred cubic, simple quadratic and triangular lattices. We find that whereas the terms for the threed i m e n s i o n a l ([3]) l a t t i c e s a r e a l l p o s i t i v e t h o s e f o r t h e [ 2 J - l a t t i c e s o s c i l l a t e i r r e g u l a r l y in s i g n . f i r s t b e c o m i n g n e g a t i v e a t t h e 18th t e r m f o r t h e s i m p l e q u a d r a t i c l a t t i c e a n d a t t h e 13th t e r m f o r the triangular lattice. [N.B. Previously we reported (Banff conference on Critical Phenomena (1968)) t h a t t h e t e r m s f o r t h e [3] - l a t t i c e s e v e n t u a l l y b e c o m e n e g a t i v e . H o w e v e r it w a s p o i n t e d Table 2. d-log Pad~ r e s u l t s (spherical model) (simple quadratic lattice 9 t e r m s ) Pad~
~Xq.K c
),
[2,2]
-
-
[2.3]
-
-
[3.2]
3.2096
4.0730
[3,3]
-
-
[4,2]
-
-
[2,4]
1.3668
- 0.1047
0.8143
0.0003
[3,4] [4,3] [4,41 [5,3] [3.51
Exact
oo 417
PHYSICS LETTERS
Volume 28A. number 6
Table 3. d-log Pad6 results (classical Heisenberg model) (s.q. l a t t i c e 9 terms} Pad6
Kc
9/
12.2]
0.8607
3.4317
[2.3l
0.8526
3.2967
[3.2]
0.8533
3.3100
[3.3]
0.8599
3.4216
[4.2]
0.9214
5.1311
[2,41
-
-
[3.41
0.9357
5.2349
14.3| [4.4]
0.8068
3.0435
[5,31
0.8716
3.8440
[3.51
1.1239
21.7121
Ratio* method
~- 0.83
~-2.5
* ref. 12. out by D r . H:.E: Stanley that this w a s p r o b a b l y due to r o u ~ I h g ~ r ~ o r s s i n c e he had independently c a l culated {~ $)~about 100 t e r m s f o r a ll the c o m mon l a t t i c c ~ a n ~ f~und the [3] s e r i e s to contain a ll p o s i t i v e t e r m s . Having checked this point u s i n g a l a r g e r c o m p u t e r we~now a g r e e with D r . H . E . Stanley and r e f e r to, a~ f u r t h ~ c o m i n g publication by Stanley f o r a listlng:uf-~tl~ t e r m s ]. T h e s p h e r i c a l m o d e l s e r i e s : m a y be u s e d to t e s t the r e l i a b i l i t y of the n u m e r i c a l p r o c e d u r e s f or i n v e s t i g a t i n g c r i t i c a l p h e n o m e n a , s i n c e the c r i t i c a l b e h a v i o u r of the s p h e r i c a l m o d e l is known e x a c t l y . In t a b l e 1 we e x a m i n e the Pad6 a p p r o x i m a n t (ref. 10) m e t h o d by l i s t i n g s o m e of the d log Pad6 a p p r o x i m a n t s to the s p h e r i c a l m o d e l s e r i e s f o r t h e n e a r e s t - n e i g h b o u r s i m p l e cubic lattice. The exact critical t e m p e r a t u r e (notation a s in r e f . 5) and c r i t i c a l exponent (7) a r e c l e a r l y i n d i c a t e d this j u s t i f y i n g the u s e of the Pad6 a p p r o x i m a n t method. S i m i l a r r e s u l t s h a v e a l s o been obtained f o r the b o d y - c e n t e r e d cubic l a t t i c e . T o study the c r i t i c a l b e h a v i o u r of the [2] c l a s -
418
30 December 1968
s i c a l H e i s e n b e r g m o d el we have l i s t e d the d - l o g Padd a p p r o x i m a n t s f o r the s i m p l e q u a d r a t i c l a t t i c e f o r both the s p h e r i c a l m o d el and c l a s s i c a l H e i s e n b e r g m o d el in t a b l e s 2 and 3. Since it is known that the t r a n s i t i o n t e m p e r a t u r e (T c) of the [2] s p h e r i c a l m o d e l i s z e r o and that the s u s c e p t i bility does not obey a power law in the c r i t i c a l r e g i o n (ref. 11) we may a r g u e as follows; if the Padd a p p r o x i m a n t s to the c l a s s i c a l H e i s e n b e r g m o d e l and s p h e r i c a l m o d el s e r i e s a r e s i m i l a r and i n d i c a t e the known b e h a v i o u r of the [2] s p h e r i c a l m o d e l , then we would conclude that T c f o r the c l a s s i c a l H e i s e n b e r g m o d el is z e r o . On the o t h e r hand if t h e Pad6 a p p r o x i m a n t s i n d i c a t e the c o r r e c t r e s u l t s f o r the s p h e r i c a l m o d el and a l s o a finite T c f o r the c l a s s i c a l H e i s e n b e r g m o d e l , in a p p r o x i m a t e a g r e e m e n t with that p r e dicted by the r a t i o method (ref. 12), then we would r e g a r d this as s t r o n g e v i d e n c e f o r T c b e ing finite f o r the [2] c l a s s i c a l H e i s e n b e r g model. F r o m t ab l e 2 and 3 (and s i m i l a r t a b l e s f o r the t r i a n g u l a r lattice) we t h e r e f o r e conclude that T c is finite f o r the [2] c l a s s i c a l H e i s e n b e r g model. We would like to thank C. J. E l l i o t for p r o g r a m m i n g a s s i s t a n c e and D r . H. E. Stanley for c o m m u n i c a t i n g his r e s u l t s to us b e f o r e publication and f o r pointing out rounding e r r o r s in our e a r l i e r work.
References 1. F.Bloch. Z. Physik 74 (1932} 295. 2. N. D. Mermin and H. Wagner, Phys. Rev. Letters 17 (1966) 1133. 3. H. E. Stanley and T. A. Kaplan, Phys. Rev. Letters 17 (1966) 913. 4. N.W. Dalton. Ph.D. Thesis. University of London (1965). 5. N, W. Dalton and C. Domb, Proc. Phys. Soc. 89 (1966) 873. 6. N.W. Dalton. A E R E - P R / T P 13 (1966) p. 19. 7. H. E. Stanley. Phys. Rev. Letters 20 (1968) 589. 8. T.H. Berlin and M. Kac, Phys. Rev. 86 (1952) 821. 9. H.E. Stanley, Phys. Rev. (1968}, to be published. 10. G.A. Baker, Adv. Theor. Phys. 1 (1965} i. 11. N.W. Dalton and D. W. Wood, Proc. Phys. Soc. 90 (1967) 459. 12. H. E. Stanley. Phys. Rev. 158 (1967) 546.
PADF.
APPROXIMANT
PHYSICS
LETTERS
RESULTS FOR HEISENBERG
30 D e c e m b e r 1968
THE SPHERICAL MODELS
AND
CLASSICAL
N. W. D A L T O N
Atomic Energy Research Establishment, Harwell. Berkshire. England and D. W. W O O D
Department of Mathematics. University of Nottingham. Noltinghma. England Received 15 N o v e m b e r 1968
A c o m p u t e r has been used to derive susceptibility s e r i e s for the s p h e r i c a l model. By comparing the Pad~ r e s u l t s for these s e r i e s with those for the two-dimensional c l a s s i c a l H e i s e n b e r g model we conclude that the c r i t i c a l t e m p e r a t u r e of the l a t t e r is finite.
It has long been argued (ref. I) that the spontaneous magnetisation of the Heisenberg model is zero in two-dimensions (i.e. [2]). This has been established rigorously by M e r m i n and Wagner (ref. 2). However work by Stanley and Kaplan (ref. 3) based oh numerical extrapolations of the series expansions for the Heisenberg model indicates that the transition temperature (Tc) of the [2] Heisenberg model is not zero, as might be expected, but finite. The purpose of this letter, is to investigate the validity of their extrapolations by using a computer to extend previous work (refs. 4-7) on the.susceptibility series of the spherical model (ref. 8). The method We have used to derive the series Table i. d-log Pad6 results (spherical model) (simple cubic lattice21 terms) Pad6 [2.2]
}Xq. Kc 1.4415
y 1.4593
[3,3] [4,4]
1.5065
1.8507
[5,5]
1.5144
1.9590
[6,6]
1.5158
1.9858
[7,7]
1.5163
1.9985
[8.8]
1.5163
1.9974
[9,9]
1.5163
1.9974
[10,10]
1.5164
1.9998
Exact
1.516386...
2.0000...
i s to c o m p u t e r i s e t h e c a l c u l a t i o n s o u t l i n e d in r e f . 5. In p a r t i c u l a r w e h a v e c a l c u l a t e d 40 t e r m s in the series for the simple cubic, body-centred cubic, simple quadratic and triangular lattices. We find that whereas the terms for the threed i m e n s i o n a l ([3]) l a t t i c e s a r e a l l p o s i t i v e t h o s e f o r t h e [ 2 J - l a t t i c e s o s c i l l a t e i r r e g u l a r l y in s i g n . f i r s t b e c o m i n g n e g a t i v e a t t h e 18th t e r m f o r t h e s i m p l e q u a d r a t i c l a t t i c e a n d a t t h e 13th t e r m f o r the triangular lattice. [N.B. Previously we reported (Banff conference on Critical Phenomena (1968)) t h a t t h e t e r m s f o r t h e [3] - l a t t i c e s e v e n t u a l l y b e c o m e n e g a t i v e . H o w e v e r it w a s p o i n t e d Table 2. d-log Pad~ r e s u l t s (spherical model) (simple quadratic lattice 9 t e r m s ) Pad~
~Xq.K c
),
[2,2]
-
-
[2.3]
-
-
[3.2]
3.2096
4.0730
[3,3]
-
-
[4,2]
-
-
[2,4]
1.3668
- 0.1047
0.8143
0.0003
[3,4] [4,3] [4,41 [5,3] [3.51
Exact
oo 417
PHYSICS LETTERS
Volume 28A. number 6
Table 3. d-log Pad6 results (classical Heisenberg model) (s.q. l a t t i c e 9 terms} Pad6
Kc
9/
12.2]
0.8607
3.4317
[2.3l
0.8526
3.2967
[3.2]
0.8533
3.3100
[3.3]
0.8599
3.4216
[4.2]
0.9214
5.1311
[2,41
-
-
[3.41
0.9357
5.2349
14.3| [4.4]
0.8068
3.0435
[5,31
0.8716
3.8440
[3.51
1.1239
21.7121
Ratio* method
~- 0.83
~-2.5
* ref. 12. out by D r . H:.E: Stanley that this w a s p r o b a b l y due to r o u ~ I h g ~ r ~ o r s s i n c e he had independently c a l culated {~ $)~about 100 t e r m s f o r a ll the c o m mon l a t t i c c ~ a n ~ f~und the [3] s e r i e s to contain a ll p o s i t i v e t e r m s . Having checked this point u s i n g a l a r g e r c o m p u t e r we~now a g r e e with D r . H . E . Stanley and r e f e r to, a~ f u r t h ~ c o m i n g publication by Stanley f o r a listlng:uf-~tl~ t e r m s ]. T h e s p h e r i c a l m o d e l s e r i e s : m a y be u s e d to t e s t the r e l i a b i l i t y of the n u m e r i c a l p r o c e d u r e s f or i n v e s t i g a t i n g c r i t i c a l p h e n o m e n a , s i n c e the c r i t i c a l b e h a v i o u r of the s p h e r i c a l m o d e l is known e x a c t l y . In t a b l e 1 we e x a m i n e the Pad6 a p p r o x i m a n t (ref. 10) m e t h o d by l i s t i n g s o m e of the d log Pad6 a p p r o x i m a n t s to the s p h e r i c a l m o d e l s e r i e s f o r t h e n e a r e s t - n e i g h b o u r s i m p l e cubic lattice. The exact critical t e m p e r a t u r e (notation a s in r e f . 5) and c r i t i c a l exponent (7) a r e c l e a r l y i n d i c a t e d this j u s t i f y i n g the u s e of the Pad6 a p p r o x i m a n t method. S i m i l a r r e s u l t s h a v e a l s o been obtained f o r the b o d y - c e n t e r e d cubic l a t t i c e . T o study the c r i t i c a l b e h a v i o u r of the [2] c l a s -
418
30 December 1968
s i c a l H e i s e n b e r g m o d el we have l i s t e d the d - l o g Padd a p p r o x i m a n t s f o r the s i m p l e q u a d r a t i c l a t t i c e f o r both the s p h e r i c a l m o d el and c l a s s i c a l H e i s e n b e r g m o d el in t a b l e s 2 and 3. Since it is known that the t r a n s i t i o n t e m p e r a t u r e (T c) of the [2] s p h e r i c a l m o d e l i s z e r o and that the s u s c e p t i bility does not obey a power law in the c r i t i c a l r e g i o n (ref. 11) we may a r g u e as follows; if the Padd a p p r o x i m a n t s to the c l a s s i c a l H e i s e n b e r g m o d e l and s p h e r i c a l m o d el s e r i e s a r e s i m i l a r and i n d i c a t e the known b e h a v i o u r of the [2] s p h e r i c a l m o d e l , then we would conclude that T c f o r the c l a s s i c a l H e i s e n b e r g m o d el is z e r o . On the o t h e r hand if t h e Pad6 a p p r o x i m a n t s i n d i c a t e the c o r r e c t r e s u l t s f o r the s p h e r i c a l m o d el and a l s o a finite T c f o r the c l a s s i c a l H e i s e n b e r g m o d e l , in a p p r o x i m a t e a g r e e m e n t with that p r e dicted by the r a t i o method (ref. 12), then we would r e g a r d this as s t r o n g e v i d e n c e f o r T c b e ing finite f o r the [2] c l a s s i c a l H e i s e n b e r g model. F r o m t ab l e 2 and 3 (and s i m i l a r t a b l e s f o r the t r i a n g u l a r lattice) we t h e r e f o r e conclude that T c is finite f o r the [2] c l a s s i c a l H e i s e n b e r g model. We would like to thank C. J. E l l i o t for p r o g r a m m i n g a s s i s t a n c e and D r . H. E. Stanley for c o m m u n i c a t i n g his r e s u l t s to us b e f o r e publication and f o r pointing out rounding e r r o r s in our e a r l i e r work.
References 1. F.Bloch. Z. Physik 74 (1932} 295. 2. N. D. Mermin and H. Wagner, Phys. Rev. Letters 17 (1966) 1133. 3. H. E. Stanley and T. A. Kaplan, Phys. Rev. Letters 17 (1966) 913. 4. N.W. Dalton. Ph.D. Thesis. University of London (1965). 5. N, W. Dalton and C. Domb, Proc. Phys. Soc. 89 (1966) 873. 6. N.W. Dalton. A E R E - P R / T P 13 (1966) p. 19. 7. H. E. Stanley. Phys. Rev. Letters 20 (1968) 589. 8. T.H. Berlin and M. Kac, Phys. Rev. 86 (1952) 821. 9. H.E. Stanley, Phys. Rev. (1968}, to be published. 10. G.A. Baker, Adv. Theor. Phys. 1 (1965} i. 11. N.W. Dalton and D. W. Wood, Proc. Phys. Soc. 90 (1967) 459. 12. H. E. Stanley. Phys. Rev. 158 (1967) 546.