A two-dimensional Ising model with non-universal critical behaviour

Physica A 195 (1993) 457-473 North-Holland

IWIIII

SDI: 0378-4371(92)00301-H

A two-dimensional Ising model with non-universal critical behaviour Kazuhiko Minami and Masuo Suzuki Department of Physics, Faculty of Science, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113, Japan Received 28 October 1992 It is shown that the two-dimensional square lattice Ising model with the nearest-neighbour ferromagnetic interaction J, the next-nearest-neighbour antiferromagnetic interaction J' < 0 and the four-body interaction J~ntshows non-universal critical behaviour when J is small. The estimations of critical temperature Tc and the critical exponent of susceptibility 3' are performed within errors of 0.003% and 1.2%, respectively.

1. Introduction It has been generally believed that the critical exponents of the statistical systems depend only on fundamental parameters such as the dimensionality, the symmetry and the interaction range of Hamiltonians. This hypothesis is called the universality hypothesis. T h e r e exist some exceptions when the Hamiltonians include local multi-body interactions. The eight-vertex and the Ashkin-Teller models have been shown exactly to contradict the universality hypothesis. The Ashkin-Teller model on the two-dimensional square lattice is equivalent to the staggered eight-vertex model. The critical isotropic Ashkin-Teller model is, however, equivalent to the isotropic zero-field eight-vertex model which has been solved exactly [1] and whose exponents vary continuously as functions of interaction energies. Both models can be mapped onto two-layer square-lattice Ising models of s = ½ interacting with each other through a four-body interaction. The difference between these two models comes from the four-body interaction. Both are invariant for spin flops of each Ising model on each layer. T h e r e exists a model which includes only local two-body interactions but show non-universal critical behaviour. The square lattice Ising model of s = ½ with nearest-neighbour interaction J and next-nearest-neighbour antiferromagnetic interaction J ' < 0 contradicts the universality hypothesis in the inter0378-4371/93/$06.00 © 1993- Elsevier Science Publishers B.V. All rights reserved

458

K. Minami, M. Suzuki / lsing model with non-universal critical behaviour

action region IJ/J'l <2. (Outside of this interaction region, this model is believed to belong to the universality class of the rectangular Ising model.) This model can also be seen as the two-layer Ising models of the interaction J', interacting with each other through the weak interaction J. The ground state of the above model is ordered as the N~el state in each layer (see fig. 1) in the interaction range IJ/J'l < 2. It is called as the super-anti-ferromagnetic (SAF) state and the above so-called SAF Ising model has been studied by many authors [2-17]. These models have some common aspects. The energy of the eight-vertex and the Ashkin-Teller models are invariant for spin reversals of all spins on a layer at all temperatures T. The SAF Ising model has the same symmetry as that of the eight-vertex and Ashkin-Teller models only at the ground state T = 0 in the range IJ/J'l < 2. This high degeneracy of the ground state energy has already been pointed out by Jiingling [18] and Krinsky and Mukamel [19]. There also exists, of course, a free parameter (i.e. interaction ratio) and critical exponents vary continuously as functions of this parameter, provided the ground state energy is degenerate. The ground state of the SAF Ising model is degenerate when the parameter J/J' changes in the region IJ/J'l < 2 and it is in this range that the model violates the universality hypothesis. These facts indicate the existence of a class of Hamiltonians which show non-universal critical behaviour. In the present paper, we study a model which possesses these properties. Let us consider the Hamiltonian ff~" : - J

~

n.n.

sisj - J '

~

n.n.n.

sisj - Jint ~

SiS]SkSI'

(1.1)

where the first and the second sum are taken over the nearest- and nextnearest-neighbour spin pairs, respectively, and the last sum is taken over the spins which lie on a unit cell (see fig. 2). The ground state of the model is the

÷ ÷

÷ q-

t ÷

÷ ÷

Fig. 1. One of the super-antiferromagnetic (SAF) states.

K. Minami, M. Suzuki / /sing model with non-universal critical behaviour

J

0

0

0

0

459

Jint

Fig. 2. InteractionsJ, J' and Jint. SAF state and four-fold degenerate for IJ/J'l < 2 and for all Jint/J '. J / J ' and Jint/J' are two free parameters which do not destroy the high degeneracy of the ground state. This Hamiltonian includes the eight-vertex and the SAF Ising models as special cases of it. It is equivalent to the general eight-vertex model with non-zero external electric field and hence has not been solved exactly. This model has been studied by Nightingale [7] for Jint < 0 using the phenomenological renormalization group theory. In the present paper, we estimate the critical temperature Tc and the exponent y of this Hamiltonian mainly for Jint > 0, using the coherent-anomaly method (CAM) [20-22] and the following sophisticated effective field theories. The multi-effective-field theory [20,23-25] is applied to a cluster of 16 spins. The critical temperature Tc and the critical exponent 3' can be estimated simultaneously through the CAM and the multi-effective-field theory. We make, however, use of the method proposed by Lipowski and Suzuki [16] to estimate more accurate values of T~. This enables us to estimate more accurate values of the critical exponent 7. The T¢ and 3' thus obtained agree with their exact values for J / J ' = 0 and for each Jint/J ' within errors of 0.003% and 1.2%, respectively. The basic idea of the CAM is reviewed in section 2. The multi-effective-field theory is explained with its important construction rule in section 3. The method of Lipowski and Suzuki [26] is explained and the "exact" critical point Tc(~) is estimated in section 4. The multi-effective-field theory and the CAM are applied to the model (1.1) and the critical exponent 7 is estimated in section 5 with the use of the "exact" critical point Tc(~) estimated in section 4.

2. The coherent-anomaly method Let us consider a series of mean field type approximations, each of which shows the classical divergence near the approximate critical point. Susceptibilities, for example, behave as

460

K. Minami, M. Suzuki / Ising model with non-universal critical behaviour

T~n) X,,(T) = .,~,, T - T~") '

(2.1)

where x , ( T ) and )~, denote the approximate susceptibility and its coefficient, respectively, of the nth approximation and T(~) denotes the approximate critical temperature. Critical coefficients diverge [21] as n increases, provided that the series is constructed to introduce the degree of approximation systematically, )(n

1 (T(~) _ T*) ~ ,

(2.2)

where T* is the exact critical temperature of the system. This divergence is called the coherent anomaly and the series which show the coherent anomaly are called canonical series. Owing to the CAM scaling theories [21] the asymptotic behavior of thermodynamic functions are obtained, for example, as x(T)

(t-

1 ~,,v , ic)

(2.3)

from a systematic series of mean-field approximations, where 3' : 1 + ~ .

(2.4)

The C A M has been applied #1 to various systems to estimate non-classical critical exponents. There remain, of course, various possibilities to choose the mean-field type approximations and to construct the canonical series. We have formulated the multi-effective-field theory and obtained [23], for example, y = 1.75159 for the two-dimensional square-lattice Ising model. The same construction rule has been applied to the eight-vertex model and continuously varying 3' with errors of maximal 1.2% have been estimated [24]. This fact strongly indicates that the rule is reliable.

3. The multi-effective-field theory

Let us consider a cluster of spins on an infinite lattice. When the spins outside of the relevant cluster are eliminated, there appear various effective fields in the cluster, exp(-- flfftaeff)= Tr' exp(-fli~t~), ~'~For a general review, see for example [22].

/3 = 1 / k a T ,

(3.1)

K. Minami, M. Suzuki / Ising model with non-universal critical behaviour

Yg~ff= ~ a + ~ a a ,

461

(3.2)

where ~g is the original Hamiltonian, ~ a denotes the contributions from the original interactions and from the external fields inside and at the boundary of the cluster, and ~oa consists of the intrinsic effective interactions from the outside of the cluster. Here, Tr' is the trace outside of the cluster I2. For the classical systems, ~oa consists of multi-spin couplings on the boundary of the cluster O as follows: -fl~oa =

~

K q ..... ikSk

(3.3)

" " " Sik .

{il}EO,O

The effective fields Kq, ~k in (3.3) are classified according to their symmetry; even effective fields behave like the original interactions and odd effective fields decrease the symmetry of the system. The even effective fields are generally non-vanishing both above and below the critical point but the odd effective fields appear only below the critical point. Here, ~0a is written as -[3~(oa = Z [3JtQt re" + ~ flHiQ~ dd , 1

(3.4)

l

where Jt (Ht) is the even (odd) effective field (of the number l) conjugate to the spin variable Q~Ven (Q~,dd) which is the sum of some spin products. The fields are, in the multi-effective-field theory, determined by the following self-consistency conditions: (sq

. . . sik)

= (sj

. . . sjk)

in which ( Q ) denotes the expectation value of Q and ( s j l . . . s j k ) product inside the cluster obtained by some translations of (3.5) is written as (AQT) = 0 , AQ7

= (sl, "

(3.5)

,

denotes a spin Eq.

(sq...si~).

(3.6) " si k) -

(Sh " " " Sjk) ,

(3.7)

where a = "even" or "odd". Near the critical point, the symmetry breaking odd effective fields are small. When we consider the case of a small external field, the expectation value ( Q ) can be expanded with respect to the odd effective fields H~ and the external field H as

462

K. Minami, M. Suzuki / Ising model with non-universal critical behaviour

(Q) =

Tr Q exp(-/3~eff) Tr exp(- fl~ff) Tr Q(1 +/3 E 2Ht~lt-'°dd+ / 3 H Ej st) exp(-/3Y(*) O°dd t H l~t + flHEjsj)exp(-fl~(*)

Tr(l+/3

~ ( Q ) * + fl ~'~H l ( ( Q Q t oad ) , _ ( Q ) * ( Q l d d ) *) -1

(3.8) where ( )* is the expectation value obtained by the Hamiltonian Y(*= - t\ - - E l H I~, t3°da - H Egsa si). Consequently, the self-consistency conditions l (3.6) for each 1 and 1' are reduced [23-25] to _ ~ , H k ( A Q l d O Qko o ,), = H ( dAQ~ d k

2 s t }*

(3.9)

1

for the odd fields and are reduced to (AQ/,~e")* = 0

(3.10)

for the even fields. Eq. (3.9) is rewritten in a matrix form as Mh = Hq ,

(3.11)

where (M)~ k = ( A Qtodd Qkdd) *' (h)k = Hk and (q)t (AQ/da Ejsj)*. The approximate critical point T~ is obtained [23-25] as the point at which the odd effective fields become non-vanishing under the condition H = 0, that is, it is obtained from det Mlr=rc = 0.

(3.12)

As a result, we can obtain the values of approximate critical temperature Tc and those of even effective field Jr at the critical point from eqs. (3.12) and (3.10). Using these results, the approximate susceptibility X near the critical point is obtained [23-25] by definition as

Tc

X=iT_ L and

(3.13)

K. Minami, M. Suzuki / lsing model with non-universal critical behaviour ~=~

IQik~oSkI A

odd

*~Iij(QoQ odd i )*

~

(detM)lr=rc

,

463 (3.14)

where A~tijis the cofactor of the matrix M and

1 ao = -~ 2., Si i

(3.15)

is the arithmetic mean of the N spins inside of the cluster. The conditions (3.5) increase the degree of self-consistency and hence the degree of approximation. Series of approximations are obtained by changing the combination of the fields systematically, i.e., by changing the degree of approximation systematically. The following construction rule has been proposed through the applications in refs. [23] and [24]: 1. introduce effective fields in order of their dominance; 2. introduce dominant odd effective fields and fix a combination of them; 3. construct a series by introducing even effective fields in order of their dominance at each approximate critical point. Here the dominance of the fields is measured by the strength of the fields, i.e. the values of the even fields and the ratios of the odd fields. It has been found through the previous applications [23,24] that the most dominant even fields should be introduced in all the approximations in the series and that various combinations can be included under condition 1 provided that the approximants show the coherent anomaly. In the present paper, an explicit construction and an estimation of 3' through (2.4) will be performed in section 5.

4. Estimations of the critical temperature

Lipowski and Suzuki [26] have proposed a method which enables us to estimate critical temperatures exactly for the s = ½ two-dimensional rectangular, triangular and honeycomb Ising models. Their method is also useful to estimate critical temperatures with accuracy, even for Ising models with higher spins or with next-nearest-neighbour interactions. Let us introduce a one-body effective field to spins on the Lth line of the strip in fig. 3. This corresponds to the treatment of a half-infinite system in which all spins on the L'th line for all L' larger than L are eliminated. When spins on the last d lines are also eliminated, we obtain the strip of the width L - d with some effective fields applied to spins on the ( L - d)th line. Lipowski and Suzuki [26] omitted

464

K. Minami, M. Suzuki

lsing model with non-universal critical behaviour

L

Fig. 3. The two-dimensionalstrip of width L, in which the interactions in the 0th line are omitted.

multi-effective fields except the one-body field and assumed that the one-body field is exactly the same, at the critical point, as the one before the elimination. T h e y have imposed the self-consistently condition of the form = / s (°)\

(4.1)

where sl j) is the ith spin on the jth line of the strip, ( Q ) L denotes the expectation value of Q taken in the strip of width L. This condition results in the exact critical temperature for the models above mentioned. T h e r e exists, fortunately, a theory which proves this coincidence of T¢ to the exact critical temperature Tc*. Morita [27] has already shown a method, for these models, to obtain the exact critical temperature using the sum of correlation functions. Let us again consider the infinite strip of width L shown in fig. 3. Let f L ( T ) be the sum of two spin correlations, being a function of temperature T, defined as A(T) = ~

_(O)o(L)\] '

("~0 ~i

(4.2)

where the sum is taken over all the spins on the L t h line. If the lattice is divided into some sublattices according to orders which will appear below the critical point, the summation is restricted to spins on the sublattice on which S~o°) lies. Then the correlation length ~ is obtained from the equation 1

lim 1 L-,~ Z lnlfL(T)l "

(4.3)

K. Minami, M. Suzuki / Ising model with non-universal critical behaviour

465

ai /\ can be calculated as the sum of contributions, The correlations (% ^(0)_(L) each of which is expressed as a path from s(0°) to s}L) , provided that the sign - 1 is assigned for each crossing of the path. Morita [27] has shown that (4.2) can be calculated when the system is reflectionally or translationally invariant (and has no cross-bond interactions). In the square lattice, for example, each path with any backward step has its counterpart which gives the same contribution with opposite sign (see fig. 4) and hence they cancel each other. There remains a path which does not go backwards. As a result, the correlation (4.2) is decoupled into the following product: fL(T) = f l ( T ) L .

(4.4)

Hence, from (4.3) we arrive at the formula 1 - -

- -

~___

lnlfl(Z)l

(4.5)

•

The critical temperature T~ is obtained as the point at which ~--->oo, and consequently we have [f~(T~) I = 1.

(4.6)

These two formulae are intrinsically equivalent to each other. Neglecting higher orders of the effective field, the condition (4.1) reduces to Ih(Tc)l

= If~_~(Tc)l

I I

(4.7)

•

I

I

Fig. 4. Two contributions denoted by the solid lines cancel each other because of the minus sign assigned to the crossing of a line.

K. Minami, M. Suzuki / Ising model with non-universal critical behaviour

466

Considering (4.4), one obtains (4.6). T h e same is true for the models which consist only of the cross bond interaction. It is easy to convince that the cancellation occurs when the path goes back to the line before and the factorization formula (4.4) is also valid. In our Hamiltonian (1.1), the cancellation does not occur, because of the existence of the interactions J and Jint" As a result, fL(T) cannot be factorized as (4.4). We can, however, obtain an approximate critical temperature Tc(L ) from the condition (4.7) when J and Ji,t are small. The critical temperature Tc(L) obviously approaches the exact critical temperature T~* as L ~ oo. Hence the true critical temperature T* can be estimated through some extrapolation of the series of Tc(L ) with an increasing L. In the present paper the nearest-neighbour interactions on the 0th line have been omitted as shown in fig. 3. This is a slight modification of the two formulae and this improves the estimations. In our case, our desired infinite system can be reproduced by connecting our lattices iteratively. We treat, as a result, an infinite system, in which the correlation lines in fig. 4 are permitted to go backward only inside each strip with width L. To summarize, we can obtain the approximate critical temperature T~(L) from the condition (4.7). The critical temperature Tc(L ) approaches the exact critical temperature T* as L increases. Thus T* is estimated by some extrapolation of a finite series of Tc(L ). We apply this method to our model (1.1) and estimate T¢(~) for each -J/J'

Tc(L)

Tc(L)

3.07

3.07 0

0 0 ~

0

0

0 0 ot~" ....

0

0

000tl~"

0

0

0 000~,....

0

2.67 0

2.67

g glgF J

0

2.27

0

O

2.27

....

0

1.87

1.87

1.47

1.47 1/1

1/2

Fig. 5.

1/3

1/oo

1/L

. 1/1

.

.

.

. 1/2

Fig. 6.

""' 1/3

1/oo

1/L

Fig. 5. The critical temperatures Tc(L ) for L = 1 . . . . . 9 for the interaction ratio - J / J ' = 0.2, -J~,t/J'= 0.4, 0.2, 0.0 and - 0 . 2 , downwards. They are extrapolated by the ~-algorithm. Fig. 6. The critical temperatures Tc(L ) for L = 1 . . . . . 9 for the interaction ratio - J i . t / J ' = 0.2, - J / J ' = 0.0, 0.1, 0 . 2 , . . . and 0.8, downwards. They are extrapolated by the ~-algorithm.

K. Minami, M. Suzuki / Ising model with non-universal critical behaviour

467

TC(OO) 3.069 0

0

0000.4

0

0

0

S1

2.669 0

0

0

d

0.2

0

s2

$3

$4

d,

0

$12 2.269

0

0

0oo

o

0.0 0

0

0

X

0 0

1.869

0

0

0

0

0

-J,.JJ' = -0.2 1.469 .0

I

I

0.2

0.4

1

i

0.6

I

i

0.8

1.0

1.2

slO

-J/J' Fig. 7.

s9

$8

87

Fig. 8.

Fig. 7. The extrapolated critical temperature Tc(o9 for the interaction ratios -Ji,,/J '= -0.2, 0.0, 0.2 and 0.4, downwards. Fig. 8. The 4 × 4-cluster. a n d - J i n t / J '. T h e a p p r o x i m a t e critical t e m p e r a t u r e s T c ( L ) is c a l c u l a t e d u p to t h e w i d t h L = 9 with d = 1. ( T h e critical t e m p e r a t u r e s { T o ( L ) } , for i n s t a n c e f o r t h e i n t e r a c t i o n r a t i o s - J / J ' = 0.2 a n d - J i n t / J ' = 0.2 a r e s h o w n in fig. 5 a n d fig. 6, r e s p e c t i v e l y . ) T h e s e a r e e x t r a p o l a t e d b y t h e e - a l g o r i t h m (see for e x a m p l e [28]) a n d t h e results { Tc(~)} a r e s h o w n in fig. 7 a n d t a b l e I. F o r t h e c a s e Table I Critical temperature Tc(~) obtained by the method of Lipowski and Suzuki and the e-algorithm. -

j/j,

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

-

Ji.t/J

'

-0.2

0.0

0.2

0.4

1.93227 1.93081 1.92640 1.91905 1.90874 1.89548 1.87895 1.85437 1.82755

2.26919 2.26749 2.26237 2.25414 2.24194 2.22630 2.20718 2.18248 2.15240 2.11926 2.07670

2.57731 2.57544 2.56979 2.56030 2.54688 2.52939 2.50768 2.48170 2.45183

2.86441 2.86237 2.85625 2.84602 2.83174 2.81408 2.79710 2.76656

K. Minami, M. Suzuld / Ising model with non-universal critical behaviour

468

- J / J ' = - J i n t / J ' = O, all the Tc(L ) equal the exact T*. Otherwise the series of Tc(L), analyzed by the e-algorithm, shows a rapid convergence to T¢(~), especially for small values o f . - J / J ' and -Jint/J'. T h e cases - J / J ' = 0 corresponds to the eight-vertex model solved by Baxter [1]. We obtain the following values of Tc(o0): T¢(~) = 1 . 9 3 2 2 7 . . . , 2.26918 . . . . 2 . 5 7 7 3 1 . . . , and 2 . 8 6 4 4 0 . . . , for -Jint/J '= - 0 . 2 , 0.0, 0.2 and 0.4, respectively. They agree with the exact critical temperatures: T~* = 1 . 9 3 2 3 0 . . . , 2 . 2 6 9 1 8 . . . , 2.57735 . . . . and 2.86450... exactly or with high accuracy (within errors of 0.003%).

5. Estimations of the exponent In this section, we estimate the critical exponent 3' by the multi-effective-field theory and the C A M with the use of the Tc(~ ) obtained in section 4. The cluster g2 is chosen as the 4 × 4-cluster shown in fig. 8. In the interaction range IJ/J'l < 2, the interaction J ' is dominant and the ground states are ordered as the N6el ordering in each sublattices, the so-called super-antiferromagnetic (SAF) state, one of which is shown in fig. 1. The one-body effective field/-/1, for example, is imposed on the spins s,, s 4, s 7 and s10 in fig. 8, thus Q1 dd is defined as Q odd 1

=S1--$4--$7

+SIo ,

(5.1)

where the minus signs come from the expected SAF ordering. The quantity AQ~ dd should be carefully chosen according to the sublattice structure and the SAF ordering as follows: A Q 1 dd = s 1 -

s15 ,

A Q 6 dd = s12sis 2 - $8s15s 6 ,

(5.2)

and so on. Some series of approximations are constructed according to the construction rules in section 3. Corresponding to rules 2 and 3 mentioned there, we choose the combinations of effective fields as follows: 2. We introduce all the one-body effective fields H 1 , n 2 and H 3 and dominant three-body effective fields H 4, H 5 and H 6 for the symmetry breaking fields. Three-body fields are found to be important to obtain the coherent anomaly when the interaction ratios - J / J ' or - J i , t / J ' are large. 3. The symmetry breaking occurs in each sublattice in our model. We should

K. Minami, M. Suzuki / lsing model with non-universal critical behaviour

469

increase the degree of approximation by introducing even effective fields, acting inside each sublattice. The strength of them is found to be tic J1 ---0.25, flcJ2 = - 0 . 1 , [ 3 ¢ J a = - 0 . 0 3 and flJ4 =0.01, where fl~ = 1 / k B T ~ and T¢ is the critical temperature of each approximation. There also exist effective fields which act between the two sublattices. The most dominant effective fields among them should be introduced to obtain the coherent anomaly, especially when ]J/J'l is large. They equal, on the other hand, exactly zero when J = 0. This is an example of so-called "irrelevant fields" which have been classified

odd (J

~j

H1

,

H2

( ) k.) %/

H3

H4

) H5

H6

even (J

-\~

()

\J ()

J1

J2

13

Ja

Jb

Jc

14

Fig. 9. The effective fields for the 4 x 4-cluster. The next-nearest-neighbour interactions are omitted for simplicity.

470

K. Minami, M. Suzuki / lsing model with non-universal critical behaviour

and proved in refs. [23] and [24] for more complicated cases. The strength of them is:/3cJ a = - 0 . 0 2 R , / 3 c J b = - 0 . 0 2 R , and/3¢J 3 = 0.001R, where R = -J/J'. Finally, we construct a series of approximations (see fig. 9) as

odd (H,, H2, Ha,/-/4,/-/5, H6}, even {J~, Jb, 4, 4 ) (4, 4, 4, Jl, J2} (L,A,4,4, 4} (L, A, 4, J,,J2,4} (J~, Jb, 7~, J,, J2, ]4}

(j = 1), (j=2), (j=3), (j=4), (j=5),

(5.3)

where each j denotes a combination of effective fields and hence an approximation. This series has been used in the applications to the square-lattice Ising model and the eight-vertex model in the previous papers [23,24] and found to be most reliable. We estimate ~p for each interaction ratio - J / J ' and - J i n t / J ' by the leastsquares fitting of the data from j = 1 , . . . , 5 by in ~

In

0.8"

0.8

0.4

0.4

0.0

0.0

-0.4

-0.4

-0.8

-0.8

-1.2 -3.2

i -2.4

I -1.6

~n(T(S)_ T0(oo)) Fig. 10.

i

j=5

-1.2

-0.8

I

I

.2

-2.4

-1.6

I

-0.8

ln(Tc(j) -- Tc(oo)) Fig. 11.

Fig. 10. The values of In )~ plotted versus ln[T(cj ) - T¢(o0)], where the combinations of even effective fields are denoted by j and the lines are obtained by the least-squares fittings for - J / J ' = 0.2, -Ji,=/J'= 0.4, 0.2, 0.0 and - 0 . 2 , downwards. Fig. 11. The values of In ~ plotted versus ln[TCcj ) - Tc(oo)] , where the combinations of even effective fields are denoted by j and the lines are obtained by the least-squares fittings for - J i , t / J ' = 0.2, - J / J ' = 0.0, 0.1, 0.2 . . . . and 0.8, downwards.

K. M i n a m i , M. S u z u k i / Ising m o d e l with non-universal critical behaviour

471

Table II Critical exponent of susceptibility Y, obtained by the multi-effective field theory and the coherent-anomaly method. - j/j,

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

- Ji.t/J '

-0.2

0.0

0.2

0.4

1.9788 1.9755 1.9654 1.9479 1.9227 1.8881 1.8443 1.8091 1.7595

1.7638 1.7607 1.7512 1.7337 1.7113 1.6810 1.6423 1.6031 1.5623 1.5085 1.4699

1.6203 1.6175 1.6092 1.5953 1.5758 1.5507 1.5198 1.4824 1.4351

1.5284 1.5260 1.5189 1.5068 1.4891 1.4620 1.4048 1.3829

In ;f(J) = -q~ In[ T~ j) - T~(oo)] + const. ,

(5.4)

where Tc(oo) is the critical temperature obtained in section 4. The estimation of the critical exponent 7 of the susceptibility is obtained from (2.4). The approximate critical temperatures T~ j) and the critical coefficients ;f(J) for J / J ' = 0.2 and - J i n t / J ' = 0.2 are shown, respectively, in figs. 10 and 11 in which In )~(J) and ln[T~ j) - Tc(oo)] are plotted. The estimated values of y are shown as functions of - J / J ' and -Jint/J in table II and they are plotted in fig. 12. '

2.15

1.95

0

0 o

o

- J,.,/J' = - 0 . 2 o o o

1.75

0

0

0

o o o o

0

1.55

0

0 0

0

o

.

o

0

o 0

0

0.0 0

0

0

0 0

0.4

1.35

1.15

'

0.0

0.2

I

0.4

,

0.2

I

0.6

*

I

0.8

,

I

1.0

,

1.2

. JIJ'

Fig. 12. The estimated Y for the interaction ratios - J i = l ] ' = -0.2, 0.0, 0.2 and 0.4, downwards.

472

K. Minami, M. Suzuld / lsing model with non-universal critical behaviour

6. Condu~on We have analyzed the two-dimensional Ising model with the nearest-neighbour interaction J, the antiferromagnetic next-nearest-neighbour interaction J' and the four-body interaction Jint. We have estimated the critical temperature by the method of Lipowsky and Suzuki [26] and have estimated the critical exponent 3, with the use of the multi-effective-field theory and the coherentanomaly method (CAM). The estimated critical temperature Tc(oo) is reliable at least up to four digits and the critical exponent 3' is reliable up to almost three digits. This accuracy has been convinced through the comparison to the exactly solved eight-vertex model. Our results give a strong evidence of the non-universal critical behaviour of the model (1.1). The case Jint = 0 corresponds to the super-antiferromagnetic (SAF) Ising model which has been studied by almost all the standard techniques. We have already analyzed the SAF Ising model in ref. [25] and obtained the results of the same accuracy. They have been compared to the results of the Monte Carlo renormalization-group method and that of the high temperature expansions.

Acknowledgements This work is supported by Grant-in-Aid for Scientific Research on Priority Areas "Computational Physics as a New Frontier in Condensed Matter Research", from the Ministry of Education, Science and Culture, Japan.

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K. Minami, M. Suzuki / Ising model with non-universal critical behaviour

[16] [17] [18] [19] [20]

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