Coherent anomalies and critical singularities of the generalized Ising model

Physica A 199 (1993) 154-164 North-Holland SDI: 0378-4371(93)E0188-K

Coherent anomalies and critical singularities of the generalized Ising model Shiladitya Sardar and K.G. Chakraborty Department of Physics, Basirhat College, 24 Parganas (North), West Bengal 743412, India

Received 3 February 1993 The critical behaviour of the generalized Ising model is studied using the power series coherent anomaly method. We work with the eleven-terms susceptibility series of Oitmaa and Enting. Taking the inverse of this series and studying the coherent anomalies we arrived at the results for Curie temperature Tc and the susceptibility exponent 3'. The variation of Tc against r/has been found to be continuous, while the variation of 3' against 77 has been found to have the discontinuity at "q= 0 in agreement with the prediction of earlier authors but in contradiction with the result of apparent continuous variation as obtained from the Pad6 approximant analysis.

I. Introduction T h e cooperative p h e n o m e n a in Ising m o d e l has b e c o m e a subject o f extensive studies during the past few decades and these studies led n o t only to a lot of interesting and i m p o r t a n t results but also to a wide variety of p o w e r f u l and effective theoretical a p p r o a c h e s (see for e x a m p l e ref. [1]). A m o n g these, p e r h a p s the most successful s c h e m e o f calculation of critical quantities is based on exact p o w e r series expansions for the partition function and the free energy. This m e t h o d , invented by D o m b [2] was subsequently e l a b o r a t e d by D o m b , Sykes, Fisher and collaborators [3] and by B a k e r , R u s h b r o o k e [4] and others [5]. This m e t h o d is n o w well k n o w n and has b e e n applied by m a n y a u t h o r s to several different physical models. O u r intention of the present analysis is to bring attention to the w o r k of O i t m a a and E n t i n g [6] w h o e m p l o y e d the m e t h o d of exact series expansion to study the critical b e h a v i o u r of the generalized Ising model. T h e m o d e l they considered consists of Ising spins a r r a n g e d over a simple cubic lattice with n e a r e s t - n e i g h b o u r f e r r o m a g n e t i c exchange interaction of strength J in the x and y directions and a variable interaction of strength ~TJ in the z direction, ~7 being a p a r a m e t e r r e p r e s e n t i n g the dimensionality of the lattice and is restricted in the range 0 ~< 77 ~< 1. This 0378-4371/93/$06.00 © 1993- Elsevier Science Publishers B.V. All rights reserved

S, Sardar, K.G. Chakraborty / Coherent anomalies o f generalized lsing model

155

model implies that one can pass continuously from an isotropic simple cubic lattice (3D) to an isotropic square lattice (2D). Two limiting cases (3D for ~/= 1) and (2D for lq = 0) are well known. For example, the critical exponents for susceptibility y = 1.75 and y = 1.25 represent the values respectively for the 3D and 2D lattices. Other result states that as the anisotropy parameter ~/ is varied from 0 to 1, the Curie temperature varies continuously. This result of continuous variation of T c against 77 is well-established and is now considered as unanimous. However, the nature of variation of y against 7/ is still controversial. The results obtained by Griffiths [7] and Abe [8] state that y retains actually the three-dimensional values for all ~ except at ~ = 0 where it jumps discontinuously from 1.25 to 1.75. On the other hand, such a discontinuity in y has not been obtained by Oitmaa and Enting who, in fact, arrived at the result of continuous variation of y from T/= 0 to ~/= 1. T h e y concluded that such a discontinuity in y cannot be readily achieved from a finite susceptibility series. It is worthwhile to examine the entire situation by means of power series coherent anomaly method ( C A M ) introduced and developed by Suzuki [9]. This method has been recently employed by one of the authors [10] to study the scaling and confluent singularities of the B l u m e - C a p e l model. The purpose of the present paper is to study the eleven terms of the static susceptibility series derived by Oitmaa and Enting for a generalized Ising model. The power series CAM approach which we shall adopt below consists in finding the successive poles of the inverse susceptibility series and to study the coherent anomalies for obtaining the critical properties of the model. The chief motivation is to investigate the nature of variation of the Curie temperature T c and the critical exponent y with respect to the dimensionality (or anisotropy) parameter ~. The plan of the paper is as follows: in section 2, we shall discuss about the Hamiltonian of the generalized Ising model and also we shall compute the coefficients of the direct and inverse susceptibility series. In section 3 the successive poles of the inverse series are found out and the results for Curie temperature have been computed. In section 4, susceptibility exponent y has been computed and the results are discussed. Section 5 contains some concluding remarks.

2. The Hamiltonian and the susceptibility series The generalized Ising model may be described by the following Hamiltonian: H :- - Z

Ji]SiSj - (DO Z

Si , i

(1)

S. Sardar, K.G. Chakraborty / Coherent anomalies of generalized Ising model

156

where o)0 = / z H a , t~ being the magnetic moment per site, H a being the applied magnetic field. S i, being a spin variable, is assumed to have the eigenvalues +-1. Jq represents the exchange integral. The summation extends over nearest neighbour pairs. Considering the nearest neighbour exchange interaction only Jij is assumed to take the value J in x or y directions and ~J in z direction in space, rt is assumed to take the values in the range 0 ~
X = I + ~ anxn,

(2)

n=l

where x = tanh(J/kT), k being the Boltzmann constant. The coefficients for a n for different ~7(0~
a2 a7

a3 a8

a4 a9

36.0 5172.0

as alo

100.0 13492,0

276.0 34870,0

a6 all

740.0 89764.0

0

4,0

12.0 1972.0

0.1

4.2

13.62 3817.8068

44.3880 11408.4861

136.5894 34039.1505

419.7309 101117.9509

1267.0221 300052.2301

0.2

4.4

15.2800 6418.2182

53.4240 20916.7906

178.26241 68096.3499

594.7084 220709.8297

1954.8074 714807.8504

0.3

4.6

16.98 9944.1730

63.1160 34693.2233

225.2774 120935.6199

804.0063 419597.64185

2828.9431 1454966.4525

0.4

4.8

18,72 14587.1608

73.4720 53908.33675

277.8944 199082.2348

1050.8017 731682.3664

3916.8477 2687789.51114

0.6

5.0

20.50 20560.3130

84.50 79923.2757

336,3750 310490.0332

1338.3750 1200355.4444

5247.81261 4638495.8625

0.6

5.2

22.32 28099.5104

96,2080 114306.7751

400.98241 464729.6797

1670.1104 1880208.4211

6853.0425 7603744.27024

0.7

5.4

24.18 37464.5073

108.6040 158852.8151

471.9814 673190.2651

2049.4958 2838926.9865

8765.6972 11967139.3776

0.8

5.6

26,08 48940.0717

121.6960 215598.9386

549.63841 949294,6345

2480.1237 4159379.88661

11020.93285 18216941.4483

0.9

5.8

28.02 62837.1425

135.4920 286845.2372

634,2214 1308728,8353

2965.6911 5941913.4119

13655.9430 26966168.5325

1.0

6.0

30.00 79494.0027

150.0 375174.0146

726,0 1769686,0808

3510.0 8306862.4026

16710.0004 38975288.0764

S. Sardar, K . G . C h a k r a b o r t y

/

Coherent anomalies of generalized

Ising m o d e l

157

From eq. (2) the inverse susceptibility series has been derived. It may be expressed as 11

-1 = 1 + ~

(3)

b~x'.

n=l

The coefficients b. have been computed for various values of the anisotropy parameter rl and are shown in table II. This table will be used in the next T a b l e II

The values of the coefficients b . o f the inverse series. rl

bI

b2

b6 btl

b7

b3 b8

b4 b9

b5 bl0

0

-4.0 44.0 -1604.0

4.0 -84.0

-4.0 188.0

12.0 -372.0

-20.0 788.0

0.1

-4.2 46.4399 -2069.4462

4.02 -88.0269

-4.068 212.1902

12.1734 -418.5715

-20.2173 1009.1891

0.2

-4.4 53.7259 -3589.0042

4.08 - 100.9223

-4.144 285.7069

12.6944 -568.2249

-20.8588 1696.9388

0.3

-4.6 66.1769 -6458.5767

4.18 - 124.0670

-4.236 412.1681

13.5654 - 840.2783

-22.0286 2936.6044

0.4

-4.8 83.7643 -11154.1410

4.32 - 158.9710

-4.352 598.1555

14.7904 - 1261.1496

-23.8321 4881.0398

0.5

-5.0 106.8125 - 18376.4970

4.50 - 207.3125

-4.50 853.2735

16.3750 - 1866.0235

-26.3750 7754.9652

0.6

-5.2 135.6489 -29103.0561

4.72 -270.9766

-4.6888 1190.2318

18.3264 -2700.5706

-29.7647 11861.5109

0.7

-5.4 170.6744 -44648.3436

4.98 -352.0947

-4.9240 1624.9529

20.6534 -3822.7291

-34.1103 17590.9673

0.8

-5.6 212.3632 -66734.9038

5.28 -453.0833

-5.2160 2176.7013

23.3664 -5304.5655

-39.5220 25431.7777

0.9

-5.8 261.2640 -97576.3313

5.62 -576.6834

-5.5720 2868.2385

26.4774 -7234.2277

-46.1135 35983.8219

6.0 -726.0

-6.0 3726.0

30.0 -9718.0

-54.0 49974.0447

1.0

-6.0 318.0 - 139974.1767

158

S. Sardar, K.G. Chakraborty / Coherent anomalies of generalized lsing model

section to carry out the p o w e r series coherent anomaly method. We leave this section with the r e m a r k that the model described by eq. (1) with the associated definition of Jij has been called generalized, since it introduces a p a r a m e t e r 7 which signifies the dimensionality of the system. H o w e v e r , it excludes the one-dimensional case.

3. The poles of inverse series and Curie temperature 1

The essence of the power series C A M approach is to find the poles of X (by cutting off the inverse susceptibility series successively) and to examine their coherent anomalies systematically. These poles are represented by x~n) and are shown in table III. Table III shows that X¢cn) increases as n increases. But as n tends to infinity x~") does not tend to infinity, rather it converges to a finite value, which we call c the fixed point x *c. A least-squares fit for z. c~n) against 1/n for 7 = 0 is shown in fig. 1, as a representative one. The value of "~c - ~n) for n ~ ~ gives x *c for 7/ = 0. This procedure is carried out for all values of 7 and in each case x* is found out. The values of x c are shown in table IV. The values of kTc/J are found out from table IV and these are plotted against 77 in fig. 2, which shows a continuous variation of T c against 77. This result is in qualitative agreement with the result obtained from the equation for Curie t e m p e r a t u r e using the molecular-field approximation

kTc/J = 2(2 + 7) and by B e t h e - P e i e r l s - W e i s s approximation. The continuous curve was also obtained by Oitmaa and Enting. The results for Curie t e m p e r a t u r e have been examined in detail and it has been found that they approximately obey the following empirical relationship Tc(7) = Tc(O ) + A71/~ , where A = 2 J / k and ~- lies in the range ~-= 1.5 +--0.25, ( + ) sign referring to 7 = 0 and ( - ) sign to 7 = 1. This result is in contradiction with both the scaling law prediction of ~- = 1.75 and the prediction of ~- = 1.25 by O i t m a a and Enting. This implies that in the present treatment ~- assumes different values for different lattices. This result seems to be m o r e reasonable than the prediction of a fixed ~- for all lattices as predicted by previous theories. F u r t h e r m o r e , it may be noted that in fig. 2 the curve obtained by the present approach (which is shown by the thick curve) almost coincides with that

S. Sardar, K . G . C h a k r a b o r t y /

Coherent anomalies of generalized lsing model

159

Table III Successive poles x c(") of different order polynomials occurring in the inverse susceptibility series. The solutions which form a coherent anomaly set are only shown. 17

X (1) c

g(c 3 )

X(c 5 )

X(c 7 )

X(c9 )

Xc(11)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.250 0.239 0.228 0.218 0.209 0.200 0.193 0.186 0.179 0.173 0.167

0.320 0.298 0.279 0.264 0.251 0.240 0.230 0.221 0.213 0.205 0.199

0.347 0.318 0.296 0.278 0.263 0.250 0.239 0.230 0.223 0.214 0.207

0.362 0.327 0.303 0.283 0.268 0.255 0.244 0.234 0.225 0.217 0.210

0.372 0.333 0.306 0.287 0.271 0.258 0.246 0.236 0.227 0.220 0.212

0.379 0.336 0.309 0.289 0.273 0.259 0.249 0.238 0.229 0.221 0.214

0.4

c~ ~0.3 x

0.2i 0

I 0.5 1In

I 1.0

Fig. 1. Least-squares fit for x(c") versus 1 / n for T/= O.

obtained by Oitmaa and Enting shown by the dotted curve in fig. 2, implying that the two methods yield almost identical Curie temperature except for small r/ where the C A M approach leads to slightly larger curvature than that obtained by Oitmaa and Enting [6]. To conclude this section it is necessary to mention that as in the case of the B l u m e - C a p e l model we have found multiple values for each pole corresponding to every value of r/ we find here the similar situation also. A m o n g these values we have chosen those values which are coherently anomalous. That is, the "distance" between two consecutive poles decreases systematically in almost linear fashion. Only these values are shown in table III, other values being discarded.

160

S. Sardar, K.G. Chakraborty / Coherent anomalies of generalized lsing model

Table IV The values of the fixed point x* for different values of 7/. Values of kTc/J are also shown calculated from x*~. 11

x*

kT¢/J

0 0.1 0.2 0.3 0.4 O.5 0.6 0.7 0.8 0.9 1.0

0.3808 0.3413 0.3144 0.2940 0.2776 0.2639 0.2523 0.2418 0.2331 0.2247 0.2174

2.4938 2.8124 3.0729 3.3010 3.5078 3.6997 3.8780 4.0538 4.2111 4.3744 4.5264

4£

3.5

~

i.

~

3.C

2£

2.0

I 0

0.5

1.0

11. Fig. 2. The variation of kTc/J against ~/. The full curve refers to the results of the present paper and the dotted one has been obtained from the Pad6 approximant results.

4. Critical exponent for susceptibility A c c o r d i n g to Suzuki the susceptibility X can be w r i t t e n in the following f o r m : x ( T ) = [xc/(x c - x)]2~(Xc) ,

w h e r e )~(Xc) is called the critical coefficient d e f i n e d b y £(Xc)

: (X*c - x)

,

b e i n g the non-classical scaling e x p o n e n t , so that y = 1 + ~. N o r m a l l y , the data ,~(xc) a n d x * ~ - x s h o u l d b e c o h e r e n t l y a n o m a l o u s . T h e m e t h o d of

S. Sardar, K.G. Chakraborty / Coherent anomalies of generalized Ising model

161

obtaining the data for the critical coefficient )~(x~) is lengthy and tedious. We adopt here a novel approach which is simple and transparent. This approach is crystallised in the following formula: ( X Xc X txc ) implying that we take the limit x-->X~c~ first and then take the limit "~c ..~n)___~X c, " For example, in the first pole approximation the inverse susceptibility is -1 X1 = 1 - 4x, which gives X1 = (1

-

4x) -1 =

Xc -x xc(1-4x

)) -1 e

,

which can be written as X1 = )~1/e, e = (x c - x ) / x c and )(1 can be obtained from the braketted part in the limit x---~xc, xc---~x*. It is worth noting at this point that in computing the values of the critical coefficients for the B l u m e - C a p e l model one of the authors adopted a different approach. In that problem each value of ,t" c(n) was substituted successively in the complete inverse susceptibility series. These values which are rapidly convergent lead to certain difficulties. We have got negative value of the coefficient at a certain pole. However, using the present approach one will arrive at the successively increasing value of the critical coefficient so that no negative value appears and the values form a coherent anomaly set. In this manner we have estimated the magnitude of the confluent singularities of the B l u m e - C a p e l model. We shall present these calculations in separate publications. Returning to the present problem of the generalized Ising model, we computed the values of the critical coefficients ,~(Xc) for all values of ~1 and ..(n) these data are tabulated in table V. This table shows distinctly that as xc increases ~(Xc) increases. However, the increase in ,~(Xc) between two consecutive poles decreases as one moves on to higher order poles, implying that these data are coherently anomalous. A least-square fit of log )~(Xc) and log (x* - xc) has been carried out for all values of 7/. A representative graph for 7/= 0 is shown in fig. 3. The slope of the straight line gives an estimate of the non-classical critical coefficient qJ. The values of ~O and y calculated in this manner are shown in table VI. The values of 3/ are plotted against ~7 in fig. 4. Table VI and fig. 4 demonstrate clearly that the value of 7(7/) for rt = 0 does not differ appreciably from its three-dimensional value ~/(1). Closer scrutiny does, in fact, imply that one can conceive a range 7(r/) = y(1) --- 0.03 except for 77 = 0, where 7 is comparatively much larger. It naturally leads us to the conclusion that 3" has an appreciable jump at ~7 = 0, 3' being nearly uniform throughout all

162

S. Sardar, K.G. Chakraborty / Coherent anomalies of generalized lsing model

Table V Values of the critical coefficients )~(x(°)) obtained from the new formula as described in the text.

n

~(x(c '))

~(x': ~')

~(x?')

~(x2)

,~(x'c9))

,~(x(?'))

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.6036 0.6976 0.7229 0.7394 0.7505 0.7579 0.7622 0.7659 0.7661 0.7673 0.7666

0.9046 1.0146 1.0290 1.0374 1.0430 1.0475 1.0505 1.0549 1.0565 1.0600 1.0636

1.1425 1.2493 1.2436 1.2368 1.2317 1.2281 1.2263 1.2298 1.2344 1.2342 1.2370

1.3421 1.4358 1.4081 1.3861 1.3772 1.3718 1.3700 1.3707 1.3685 1.3717 1.3755

1.5563 1.6108 1.5435 1.5162 1.4976 1.4908 1.4807 1.4810 1.4781 1.4917 1.4865

1.7706 1.7556 1.6628 1.6212 1.6000 1.5819 1.5944 1.5827 1.5801 1.5846 1.5916

0.3

,5~ x

oo,

-0.3

0

I -0.5

-1.0

-1.5

tog (x c - x c ) Fig. 3. Least-squares fit for log)~(x(cn)) against log (x* - x c(") ). The slope of the curve yields the coherent anomaly exponent 4,.

values of 7. This result is in good a g r e e m e n t with those of Griffiths, A b e but in disagreement with the result obtained by O i t m a a and Enting. The drop in the value of y may be estimated by the following relation Ay = [y(0) - y ( ~ ) ] / y ( 0 ) . Table VI shows the values of Ay in percentage. We note that as y changes f r o m 0 to 0.1, Ay drops by nearly 23%, while as ~ changes from 0.1 to 1, Ay drops by nearly 25%. F u r t h e r m o r e , taking ten values of y from ~ = 0.1 to 1 we have found that their average value is represented by 1.2837. Let us denote by 5~ as the deviation from this average. The values of 5~ are also shown in table VI. These values are definitely very small. A n average curve has been drawn as shown in fig. 4 and it is seen that the real values lie almost close to this line.

S. Sardar, K.G. Chakraborty / Coherent anomalies of generalized Ising model

163

Table VI The estimates of the CAM exponent ~/,, critical exponent % the values of the quantities A3,,~ for various values of the anisotropy parameter "0.

(in %) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.6942 0.3090 0.3013 0.2885 0.2811 0.2822 0.2626 0.2744 0.2853 0.2723 0.2774

1.6942 1.3090 1.3013 1.2885 1.2811 1.2822 1.2626 1.2744 1.2853 1.2723 1.2774

22.736 23.191 23.946 24.383 24.318 25.475 24.602 25.135 24.903 24.602

-0.4105 - 0.0253 -0.076 0.0048 0.0026 0.0015 0.0211 0.0063 -0.0016 0.0114 0.0063

1.7 1.6

1.5

1.3

G

1.2 0

I O. 5

J 1.0

%

Fig. 4. The estimates of 3' are shown against ~. The line refers to the average curve drawn for convenience in estimating the small departure from this line.

5. Conclusions T h e critical b e h a v i o u r o f t h e g e n e r a l i z e d Ising m o d e l has b e e n s t u d i e d in t h e p r e c e d i n g sections using t h e p o w e r series c o h e r e n t a n o m a l y m e t h o d . R e s u l t s o b t a i n e d are s u m m a r i z e d b e l o w : (i) W e h a v e f o u n d a c o n t i n u o u s v a r i a t i o n o f k T c / J w i t h r e s p e c t to t h e a n i s o t r o p y p a r a m e t e r r/, in close q u a n t i t a t i v e o b t a i n e d f r o m th e e x a c t series e x p a n s i o n .

agreement

with

the

result

(ii) T h e critical e x p o n e n t 3' f o r susceptibility has b e e n e s t i m a t e d f o r all

164

s. Sardar, K.G. Chakraborty / Coherent anomalies of generalized Ising model

v a l u e s o f 77 in t h e r a n g e 0 ~< 77 ~< 1 a n d t h e results s h o w d i s c o n t i n u i t y n e a r r / = 0 a n d ~, t a k e s t h e t h r e e - d i m e n s i o n a l v a l u e t h r o u g h o u t t h e r a n g e . T h e r e a s o n t h a t a d i s c o n t i n u i t y n e a r r / = 0 was n o t o b t a i n e d b y O i t m a a a n d E n t i n g is d u e to t h e fact t h a t t h e y w o r k e d with t h e d i r e c t series w h i c h is finite. B u t in t h e p o w e r series C A M a p p r o a c h , as S u z u k i p o i n t e d o u t , t h e c o n s i d e r a tion o f t h e i n v e r s e series effectively serves to t h e i n c l u s i o n o f s o m e p a r t i a l s u m m a t i o n o f the c o n t r i b u t i o n d u e to infinite n u m b e r o f t e r m s . T h i s m i g h t b e t h e r e a s o n for g e t t i n g t h e r e s u l t o f n e a r l y d i s c o n t i n u o u s j u m p at rl = 0. I n t h e f u t u r e e x p o s 6 s o f t h e p r o b l e m we shall e m p l o y t h e c o n t i n u e d f r a c t i o n a p p r o a c h to r e - e x a m i n e a n d r e - e s t a b l i s h t h e a b o v e results.

References [1] M.E. Fisher, Rept. Prog. Phys. 30 (1967) 615. [2] C. Domb, Proc. Roy. Soc. A 199 (1949) 199; Adv. Phys. 9 (1960) 149. [3] C. Domb and M.F. Sykes, Phys. Rev. 108 (1957) 1415; Phys. Rev. 128 (1962) 168; M.E. Fisher, Physica 25 (1959) 615; Phys. Rev. 162 (1967) 480. C. Domb and D.W. Wood, Phys. Lett. 8 (1964) 20; Proc. Phys. Soc. (London) 86 (1965) 1; M.F. Sykes, J. Math. Phys. 2 (1961) 52. [4] G.S. Rushbrooke, J. Math. Phys. 5 (1964) 1106; G.A. Baker, H.E. Gilbert and G. Rushbrooke, Phys. Lett. 20 (1966) 146; G.S. Rushbrooke and P.J. Wood, Proc. Phys. Soc. A 68 (1955) 1161. [5] D.W. Wood, Phys. Lett. 14 (1965) 191; G.S. Rushbrooke and P.J. Wood, Mol. Phys. 1 (1958) 257. [6] J. Oitmaa and I.G. Enting, J. Phys. C 5 (1972) 231. [7] R.B. Griffiths, Phys. Rev. Lett. 24 (1970) 1479. [8] R. Abe, Prog. Theor. Phys. 44 (1970) 339. [9] M. Suzuki, J. Phys. Soc. Jpn. 55 (1986) 4205; 56 (1987) 4221; Phys. Lett. A 116 (1986) 375; J. Phys. Soc. Jpn. 56 (1987) 4221. [10] K.G. Chakraborty, Physica A 189 (1992) 271.