F , I G L Ó I D . V . KAPOR M. S K R I N J A R J . SÓLYOM
S ERIES EXPANSION STUDY OF
F IRST- AND S E C O N D - O R D E R PHASE T R A N S I T I O N S IN A M O D E L WITH flULTISPIN COUPLING
cH ungarian ‘Academ y o f ‘Sciences
CENTRAL RESEARCH
INSTITUTE FOR PHYSICS
BUDAPEST
KFKI-198 5-52
SERIES EXPANSION STUDY OF
FIRST- AND SECOND-ORDER PHASE TRANSITIONS IN A MODEL WITH MULTISPIN COUPLING
F. IGLÖI*, D.V. KAPOR**, M. SKRINJAR** and J. S6LYOM*&
*Central Research Institute for Physics H-1525 Budapest 114, P.O.B.49, Hungary
* *
Department of Physics, University of Novi Sad, Novi Sad, Yugoslavia
i n s t i t u t e for Theoretical Physics, University of Lausanne, Lausanne, Switzerland
HU ISSN 0368 5330
The type of phase transition in a chain of Ising spins with multispin interaction is studied in a transverse field, using strong- and weak-coupling expansions. The transition is shown to be of first order if more than three spins are coupled. The critical exponents for the three-spin coupling model are estimated.
АННОТАЦИЯ
Исследован род фазового перехода модели спиновой цепочки Иэинга с мно
го-спиновыми связями в присутствии внешнего перпендикулярного магнитного по
ля. Показано, что фазовый переход является переходом первого рода, если коли
чество связанных спинов больше трех. Даются оценки критических экспонентов модели с трех-спиновыми связями.
KIVONAT
Többspin kölcsönhatásu Ising spin-lánc fázisátalakulásának tipusát merő
leges mágneses tér jelenlétében erős- és gyengecsatolási sorfejtéssel tanul
mányozzuk. Megmutatjuk, hogy az átalakulás elsőrendű, ha több, mint három spin között van csatolás. Becslést adunk a három-spin kölcsönhatásu modell kriti
kus exponenseire.
1 . I N T R O D U C T I ON
W h i l e s e c ond-order p h ase t r a n s i t i o n s can be studied c o n v e n i ently by using v a r i o u s formulations o f the r e n o r m a l i z a t i o n - g r o u p transformation, the situation is much less s a t i s f a c t o r y for
firs t - o r d e r phase transitions. A l t h o u g h the c o n c e p t of d i s c o n tinuity fixed p o i n t (Nienhuis and N a u e n b e r g 1975) is useful in d e s c r i b i n g f i r st-order phase transitions, these latter are not always associated w i t h a d i s c o n t i n u i t y fixed point. Similarly, altho u g h detailed studies h a v e been p e r f o r m e d in fini t e - s i z e
scaling for systems with first-order p h a s e t r a nsitions (Imry 1980, Fisher and Berker 1982, B l öte and N i g h t i n g a l e 1982, Iglói and Sólyom 1983, Hamer 1983, C a r d y and N i g h t i n g a l e 1983, P r i v m a n and Fisher 1983, B i n d e r and L a n d a u 1984), in many p r a c t i c a l c a s e s it is not easy to d e c i d e from the data for finite systems, w h e t h e r the transition is of first or second order. This is e.g. the case for the m u l t i s p i n - c o u p l i n g m o d e l s t u d i e d in this paper.
The model w a s i n t roduced by T u r b a n (1982) a n d by P e n s o n et al. (1982). The H a m i l t o n i a n can be w r i t t e n in the form
H = ХЕско i
z
i+1 •■* az
i+m-1 (1.1)
w h e r e a* and a r e Pauli operators o n site i. T h e value of m det e r m i n e s the n u m b e r of n e i g h b o u r i n g spins th a t are coupled.
At zero temp e r a t u r e the s y s t e m has a p h a s e t r a n s i t i o n as the
transverse field increases. Since it seems that there is a single phase transition in the system, the s e l f - d u a l i t y of the m o d e l predicts that it should be at (h/X)* = 1, indepen d e n t l y of the number of coupled spins m. For m=2 the model is the standard
Ising m o d e l in t r a n sverse field w hich has a s e c o n d - o r d e r p hase transition. For m -> °°, however, m e a n - f i e l d t h e o r y should be
e x act and the tra n s i t i o n turns out to be of f i r s t order. There are c o n t r o v e r s i a l p r e d i c t i o n s for the crit i c a l v a l u e of m, a b o v e w h i c h the t r a n s i t i o n s h o u l d be of f i rst order. M e a n - f i e l d the
ory gives m c= 2 , however r e n o r m a l i z a t i o n - g r o u p c a l c u l a t i o n s
(Iglói et al. 1983) give a usual s e c o n d - o r d e r b e h a v i o u r for ш^З.
The analysis of finite-size scaling results lead Penson et al.
(1982) to c o n c l u d e that m c =4. From a c o n j e c t u r e d c r i t e r i o n for d i s t i n g u i s h i n g between con t i n u o u s and di s c o n t i n u o u s transitions
(Livi et a l . 1983), M a r i t a n et al. (1983) p r e d i c t e d that for m=4 the t r a n s i t i o n is a l r e a d y of f i r s t order. Since finite-size
s caling is not ver y s e n s i t i v e to d e c i d e when the c h a r a c t e r of the t r a n s i t i o n changes, o t h e r m e t h o d s should b e used.
In this p a p e r we use the s e r i e s - e x p a n s i o n m e t h o d to d e t e r min e the c r i t i c a l value of m c - The p a p e r is o r g a n i z e d as follows.
The series o b t a i n e d for the g r o u n d - s t a t e e n e r g y in the w e a k - and s t r o n g - c o u p l i n g limits are p r e s e n t e d in Sec. 2. The s e r i e s are a n a l y s e d in Sec. 3, w h e r e we find th a t in fact m (_,=3, and for this case, w h ere the t r a n s i t i o n is still of second order, the critical e x p o n e n t s are also determined. The results are d i s c u s s e d in
S e c . 4 .
2. S E R I E S E X P A N S I O N
The w e a k - and s t r o n g - c o u p l i n g series e x p a n s i o n s for quantum spin systems, w h i c h are analo g o u s to the high- and l o w - t e m p e r a ture e x p a n s i o n s in c l a s s i c a l s t a t i s t i c a l m e c h anics, and the a n aly
sis of the s e ries by u s i n g d i f f e r e n t methods to d e t e r m i n e the critical b e h a viour, have b e e n p r o v e d t o 2*4be v e r y useful in the study of m a n y systems (Hamer et al. 1979, E l i t z u r et al. 1979, Hamer and K o g u t 1980, M a r l a n d 1981) . We will a p p l y this procedure to the m u l t i s p i n - c o u p l i n g model.
The H a m i l t o n i a n of eq. (1.1) c a n be split in two ways. If the m u l t i s p i n coupling X is stron g e r than the t r a n s v e r s e field h, this latter can be t r e a t e d in p e r t u r b a t i o n and an e x p a n s i o n
3
in powers of h/A can be generated. This s t r o n g - c o u p l i n g e x p a n s i o n for the g r o u n d state energy per site will have the form
§ = - AZan (|)n , for A > h . (2.1) n
On the o t h e r hand, if the m u l t i s p i n c o u p l i n g is w e a k e r than the tra n s v e r s e field, a we a k coup l i n g e x p a n s i o n in powers of А/h can be generated. Due to the self-d u a l i t y of the model, the series e x p a nsion c o e f f i c i e n t s will be the same in the two cases, i.e.
§ = - h E a n (£)n , for A < h , (2.2) n
and at A=h the two expressions match. This is valid e v e n if on l y a few finite - o r d e r terms are c a l c u l a t e d in the expansion.
If the tr a n s i t i o n is of s e c o n d order, than the two e x p r e s s ions give n o t only the same e n e r g y at A = h , but the left and r i g h t derivatives, c a l c u l a t e d from the two e x p ressions in their regions of validity, respectively, are als o identical at this point. O n l y the second d e r i v a t i v e s will differ. On the other hand, if the t r a n sition is of first order, the weak- and s t r o n g - c o u p l i n g e x pansions gi v e d i f f e r e n t first d e r i v a t i v e s on the two sides of the tr a n s i t i o n point, indicating a finite latent heat.
If the expansion coeffi c i e n t s are cal c u l a t e d u p to a finite order, the two expansions always give d i f f e r r i n g left and right deri v a t i v e s at the transition point, a l t h o u g h after e x t r a p o l a t i n g to n+°° the d i f f e r e n c e m a y disappear, indicating a s e c o n d order phase transition. If, however, the t r a n s i t i o n is of first order, the dif f e r e n c e be t w e e n the two d e r i v a t i v e s should r e m a i n finite even when n-*-°°.
We h a v e p e r f ormed the w e a k - and s t r o n g - c o u p l i n g series e x pansions for the ground-state e n e r g y of the model g i v e n in eq.
(1.1) up to 10th order for m=2,3 and 4, w h i l e for m = 5 , 6 and 7 up to 8th order. The s e r i e s - e x p a n s i o n coefficients are g i v e n in Table I.
3 . A N A L Y S I S O F T H E S E R I E S
As d i s c u s s e d in the p r e v i o u s section, finite-order p e r t u r b a tion t h e o r y always gives a f i n i t e latent heat, a finite d i f f e r ence b e t w e e n the d e r i v a t i v e s of the g r o u n d - s t a t e e n e r g y at A=h, w h e n cal c u l a t e d fr o m the w e a k - or s t r o n g - c o u p l i n g expansions.
This nth order l a t e n t heat L is d e f i n e d as n
,
ГэЕ®(Ь,Х)
E™(h,A)L„ = » { - 5 Н - Ц - -ТЯГ- U j • ,3'1)
Here E S and Ew are the g r o u n d - s t a t e e n e r g i e s c a l culated in the
n n
strong- and w e a k - c o u p l i n g expansion, respectively, k eeping terms up to n t h order. T h e values o b t a i n e d u s i n g the results of the previous section are given in Table II.
In the e x t r a p o l a t i o n to n-*-«5 the e x a c t solution of the m = 2 case (Pfeuty 1970) can be use d as a guide. It is e a s i l y seen, that for the Ising case the e x p a n s i o n c o e f f i c i e n t s can be w r i t t e n in the form
a2n
n П ( i=l
2i-3 v
2i ’ (3.2)
After summing up the series w i t h these c o e f f i c i e n t s one reco v e r s the e x act result of Pfeuty (1970). The latent heat in nth o r der can be a p p r o x i m a t e d by
Ln I(n+1/2) (3.3)
One can see that the latent h e a t goes to zero rou g h l y as 1/n.
This e x p r e s s i o n is the special case of the general scaling for m valid for s e c o n d - o r d e r t r a n s i t i o n s (Iglói 1985)
(3.4)
5
Here a is the s p e c i f i c - h e a t exponent. This k i n d of s c a l i n g b e ha v i o u r has b e e n used by one of us to d e t e r m i n e the c r i t i c a l exponents of v a r i o u s physi c a l q u a n t i t i e s from series expansions.
According to e q s . (3.3) and (3.4), a p l o t of log L n versus log(n+l/2) s h o u l d give a straight line for s e c o n d - o r d e r p hase transitions, w i t h a slope - ( l -а). Thi s plot is shown in Fig. 1 for different v a l u e s of the number of cou p l e d spins. T h e points lie ve r y well on a straight line for m=2 and 3, while for m>4 there are c o n s i d e r a b l e deviations. T h e slope of the line for m=3 is a p p r o x i m a t e l y 1/2, therefore w e p l o t in Fig. 2 the v a l u e of
- 1/2
L n as a function of (n+1/2) ' . As is seen, the values are on a straight line not only for m = 3 , b u t for l arger m v alues as well.
For m=3 the e x t r a p o l a t e d latent h e a t v a n i s h e s , thus the t r a n s i tion is of s e c o n d order. The error in the e x t r a p o l a t i o n of the latent heat is smaller than 0.005. For m >4 , however, the latent heat differs s i g n i f i c a n t l y from zero. In these cases the t r a n s i tion is of first order. The a c c u r a c y of the e x t r a p o l a t i o n is r ather good. This is due to the fact, that the ratio of the c o efficients an (m)/an (m+1) varies o n l y slowly w i t h n, as can be read off from T a ble I. This q u a n t i t y is sma l l e r than o n e for m=2, but it is l arger than u n i t y for rn>3. This r a t i o is e x t r e m e l y stable for m=3. Suppo s i n g that this ratio is the same in higher orders of the e x p a n s i o n as weel, i.e. an (3)/ a n (4)s i .227 , in d e p e n d e n t l y of n, w e esti m a t e the l a t e n t heat for the ca s e m=4 to be
a (4)
L(m=4) = ^lo (m~4) - L l Q (m=3) • &n— jy = 0.218 . (3.5) n
This value is in good agreement w i t h the e s t i m a t e from Fig. 2.
The latent he a t for larger values of m can be e x t r a p o l a t e d in the same way, however, the accu r a c y is s o m e w h a t smaller. The calculated l a t e n t heats are shown in Fig. 3, together w i t h the s e r i e s - e x p a n s i o n results. The l atent heat for large v a l u e s of m behaves as L = l - 3 /2m. At m=3 the l a t e n t heat b ecomes zero and remains i d e ntically zero for sm a l l e r values of m.
Next we e s t i m a t e the critical pr o p e r t i e s of the m o d e l for m=3. Since we h a v e c a l c u l a t e d the g r o u n d - s t a t e e nergy only, the s p ecific-heat e x p o n e n t a c a n be o b t a i n e d from the second d e r i v a tive. The series is rather short, t h e r e f o r e d i f f e r e n t m e t h o d s have b e e n used to get a b e s t estimate. The r e s u l t of the ratio m e t h o d (Gaunt a n d Guttmann 1974) is a = 0 .53+0.03. By u s ing the scaling relat i o n (3.4) we o b t a i n a = 0 . 54+0.02. The best r e s u l t is achie v e d by P á d é analysis (Gaunt a n d Guttmann 1974) of the series. A c c o r d i n g to the P á d é a p p r o x i m a n t s (Table III) w e obtain a = 0 . 554+0.001. Thu s all these e s t i m a t e s are c o n s i s t e n t w i t h the pre d i c t i o n
a = 0.55 + 0.01 (3.6)
The critical e x p o n e n t of the c o r r e l a t i o n length can be c a l c u lated from the h y p e r s c a l i n g rela t i o n dv=2-a, and we get v =
= 0.73+0.01. This value is somewhat smaller t h a n the r e s u l t o b tained by Iglói et al. (1983) from the r e n o r m a l i z a t i o n - g r o u p calcu l a t i o n and is close to the v a l u e det e r m i n e d by P e n s o n et al. (1982) from f i n ite-size scaling.
k. D I S C U S S I O N
In the p r e s e n t paper the phase t r a nsition in a c h a i n of Ising spins c o u p l e d by a m u l t i s p i n i n t eraction and s u b m i t t e d to a t r a n sverse f i e l d has b e e n studied. T h e weak- and s t r o n g - c o u p ling series e x p a n s i o n s for the g r o u n d - s t a t e e n e r g y have b e e n p e r f o r m e d up to 10th order in the p e rturbation. It has b e e n shown that in the cases w h e n more t h a n 3 neighb o u r i n g s p ins are coupled, the t r a n s i t i o n is of first order. This m e t h o d is thus mor e sensitive t h a n f i n ite-size s c a l i n g to d e t e r m i n e the order of transition. In this l a t t e r m e t h o d the m=4 c a s e still seemed to b e h a v e like h a v i n g s e c o n d - o r d e r p h a s e tr a n s i t i o n (Penson et al. 1982). As s h o w n by Iglói and S ó l y o m (1983) and H a m e r (1983) the finite l atent heat in a f i r s t - o r d e r t r a n sition can be de-
7
termined from finite-size sca l i n g calcul a t i o n s as well but the limits L->°°, w h e r e L is the length of the system, and A-»-A* cannot be interchanged.
The analysis of the series all o w e d us to e s t i m a t e the c r i t i cal exponents a and v for the case m=3. The values o b t a i n e d d i f fer from the v a l u e s known for the 4-state Potts m o d e l indicating once more that these models do not b e l o n g to the same u n i v e r s a l ity class (Iglói et al. 1983), a l t h o u g h in both cases a fourfold deg e n e r a c y is l i fted at the transition.
A C K N O W L E D G E M E N T S
The authors are indebted to Prof. R. Dekeyser (Leuven) for his hel p in the Pade analysis of the series. One of us (J.S.) is grateful to Prof. P. Erdős for his h o s p i t a l i t y at the U n i v e r s i t y of Lausanne and to the Swiss Nati o n a l S cience Fo u n d a t i o n for the financial support. The n u m e r i c a l calcul a t i o n s have bee n performed at the Central Research I n s t itute for Physics and at the U n i v e r sity of L a u s e n n e . We ackn o w l e d g e the hel p of Dr. A. Simonits at the early stages of the n u m e r i c a l calculations.
R E F E R E N C E S
Binder K. and L a n d a u D.P.: 1984 P h y s . Rev. В _30 1477 Blöte H.W.J. and Nigh t i n g a l e M.P.: 1981 Physica 112A 405 Cardy J.L. and Night i n g a l e P.: 1983 Phys. Rev. В 2J7 4256
Elitzur S. , Pe a r s o n R. and Sh i g e m i t s u J. : Phys. Rev. D 19_ 3698 Fisher M.E. and Berker A.N. : 1982 Phys. Rev. В 26_ 2507
Gaunt D.S. and Guttmann A.J.: 1974, in Phase T r a n s i t i o n s and Critical Phenomena, ed. C. Domb and M.S. Green, Vol. 3.
(New York: Academic)
Hamer C.J.: 1983 J. Phys. A: Math. Gen. 1_£ 3085
Hamer C.J. and Kogut.J.B.: 1980 Phys. Rev. В 22^ 3378
Hamer C.J., Kogut J.B. and Susskind L . : 1979 Phys. Rev. D 19 3091
Iglói F . : (to be published)
Iglói F., Kapor D . V . , Skrinjar M. and S ó l y o m J . : 1983 J. Phys.
A: Math. Gen. lj> 4067 {
Iglói F. and S ólyom J.: 1983 J. Phys. C.: Solid State Phys.
16 2833
Imry Y. : 1980 Phys. Rév. В 2_1 2042
Livi R. , Mar i t a n A., Ruffo S. and S t e l l a A.L.: 1983 Phys. Rév.
Lett. 50 459
Ma r i t a n A., S tella A. and V a n d e r z a n d e C.: 1984 Phys. Rev. В .29 519
M a r l a n d L.G. : 1981 J. Phys. A.: Math. Gen. 14: 2047
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Penson K.A., Jul i i é n R. and Pfeuty P.: 1982 Phys. Rev. В _26 6334 Pfeuty P.: 1970 Ann. Phys., NY 51_ 79
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Í I 1
Table I
Series expansion coefficient a R (m ) for the model with m coupled spins in nth order of perturbation
O r d e r a n ( 2 ) a „ ( 3 ) a n ( 10 a n ( 5 ) a n < 6 ) a ( 7 )
n
2 1/1* 1 / 6 1 / 8 1 / 1 0 1 / 1 2 1/11*
1» 0 . 0 0 1 5 6 2 5 0 0 O . O I 8 5 I 8 5 2 0 . 0 1 1 * 9 7 3 9 6 0 . 0 1 1 8 3 3 3 3 0 . 0 0 9 1 * 9 0 7 1 * О . О О 7 7 6 2 3 9 6 0 . 0 0 3 9 0 6 2 5 О. ОО 5 6 О7 О О 0 . 0 0 1 * 5 7 3 6 8 0 . 0 0 3 1 * 6 6 1 9 0 . 0 0 2 6 2 8 7 1 0 . 0 0 2 0 2 6 7 9 8 0 . 0 0 1 5 2 5 8 8 0 . 0 0 2 5 1 2 6 1 0 . 0 0 2 0 5 1 1 * 3 0 . 0 0 1 1 * 8 1 * 7 6 0 . 0 0 1 0 6 1 1 * 5 0 . 0 0 0 7 6 9 5 3 10 0 . 0 0 0 7 1 * 7 6 8 O . O O I 3 7 9 2 9 0 . 0 0 1 1 21+33
Table I I
The latent heat l_n (m) in nth order of perturbation theory calculated from eq. (3*1) for the model with m coupled spins
Order
} L (2)
n L (3)
n Ln (l*)
1 1
L„<5>
. . . . . . L
L (6)
n Ln (7>
2
L.U . ■ "IIT f
0.25 0.5 0.625 0.7 0.75 0.7857
1* 0.11*06 0.3701* O. 5 2 0 2 О. 6 1 7 2 0 . 6 8 3 6 0.7311*
6 0.0976 0.3087 0.1*699 0.5790 0.651*6 0 . 7 0 9 1
8 0.071*8 O.2 7IO 0.1*391 0.5568 0.6387 0.6975
10 0.0606 0.21*1*8 0.1*178
11
Table I I I
Pádé analysis of the series for the logarithmic derivative of the specific heat in the m=3 model
x
The nth order latent heat Ln versus n+1/2 on a log-log plot for different values of the number of
coupled spins
13
-1 /2 The nth order latent heat Ln versus (n+1/2) ' for
the models with m>3
Csj
CN I
Vf\
CO
GO Ö
(n + 1 /2 )
Fig. 3
The nth order latent heat and the extrapolated value for n->°° plotted for different m values
3
*
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