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Physica A 356 (2005) 589–597

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Critical properties of the S4 model ondiamond-type hierarchial lattices

Ying Lia,b, Xiang-Mu Konga,c,�

aDepartment of Physics, Qufu Normal University, Qufu 273165, ChinabInstitute of Physics and Center of Condensed Matter Physics, Chinese Academy of Sciences,

Beijing 100080, ChinacInterdisciplinary Center for Theoretical Studies, Chinese Academy of Sciences, Beijing 100080, China

Received 17 December 2004

Available online 11 May 2005

Abstract

We study the critical properties of the S4 model on diamond-type hierarchical lattices in the

presence of an external magnetic field. It is assumed that for this type of inhomogenous fractal

lattice, the Gaussian distribution constant, the four-spins interaction parameter and the

external magnetic field in the S4 model depend on the coordination number of the site on the

fractal lattices. By combining the real-space renormalization-group scheme with the cumulative

expansion method, we obtain the critical points and further calculate critical exponents

according to the scaling theory. The results show that on diamond-type hierarchical lattices

with branches m44 of 2 bonds, the critical point of the S4 model is just the Gaussian fixed

point, and therefore critical exponents are in full agreement with those of the Gaussian model,

and that on those with mp4, the Wilson–Fisher fixed point as well as the Gaussian fixed point

is obtained. The Wilson–Fisher fixed point has a decisive influence on the critical behavior of

the system, and critical exponents are related to the fractal dimensionality. We found that the

S4 model on the fractal lattices and that on the translation symmetric lattices show similar

behaviors in the dependence of the critical properties on the dimensionality.

r 2005 Elsevier B.V. All rights reserved.

PACS: 64.60.A; 75.10.H

Keywords: The S4 model; Diamond-type hierarchical lattice; Renormalization group; Critical behavior

see front matter r 2005 Elsevier B.V. All rights reserved.

.physa.2005.03.025

nding author. Department of Physics, Qufu Normal University, Qufu 273165, China.

dresses: yli@aphy.iphy.ac.cn (Y. Li), xmkong@yahoo.com.cn (X.-M. Kong).

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Y. Li, X.-M. Kong / Physica A 356 (2005) 589–597590

1. Introduction

The fractal is a type of geometry with self-similar character and fractionaldimensionality. Fractal lattices are good candidates to investigate the physicalproperties in non-integer dimensions. In the past two decades, phase transitions andcritical phenomena of spin models on fractal lattices have been extensively studiedand great progress has been made [1–14]. The pioneer work of these studies wascarried out by Gefen et al. in the 1980s, who studied the phase transitions of the Isingmodel and Potts model [1,2]. As we know, the Ising model is often used to simulatethe ferromagnetic structure, and it plays a crucial role in the theoretical studies ofmagnetic properties. The S4 model, which is an extension of the Ising model, isallowed to take a continuous spin value instead of the discrete value. The study ofthe S4 model is of great value to understand better the properties of the ferromagnetsfrom either a theoretical or practical viewpoint.The critical property of the S4 model on the translation symmetry lattices has

been studied by the momentum space renormalization group (MSRG) method.The results showed that when the space dimensionality d is more than 4, only theGaussian fixed point is obtained, and when d is less than or equal to 4, theWilson–Fisher fixed point as well as the Gaussian one is obtained [15]. However,there are a few studies of the critical properties of the S4 model on fractal lattices sofar. In this present work we study the critical properties of the S4 model onDiamond-type hierarchical lattices by combining the real-space renormalizationgroup (RG) scheme with the cumulative expansion method.

2. The geometry of diamond-type hierarchical lattices

Consider a family of diamond-type hierarchical lattices—for example, thediamond-type hierarchical lattice with m branches of 2 bonds, represented asðmDHÞ2 for convenience. The ðmDHÞ2 lattice is a typical deterministic fractal, andcan be generated iteratively. First start from a simple lattice with two sites and onebond between them (n ¼ 0 in Fig. 1), and then replace it with a lattice with m

branches of 2 bonds (n ¼ 1 in Fig. 1). If this procedure is repeated down to amicroscopic length scale, finally the ðmDHÞ2 lattice is generated. In Fig. 1 we displaythe diamond-type hierarchical lattices with branches m ¼ 2 and m ¼ 3 of 2 bonds.The fractal dimensionality and the order of ramification of such lattices ared f ¼ 1þ ln m= ln 2 and R ¼ 1, respectively.

3. Model

The S4 model is derived from the extension of the Ising model. The extensionprocess is described as follows [16]. Consider a cubic lattice in the external field h,and let si denote the spin variable at the lattice site i. If we introduce a probabilitydistribution function W ðs1; s2; . . . ; sN Þ ¼

QNi¼1 dðs

2i � 1Þ into the Ising model, the

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(a)

(b)

Fig. 1. The construction procedure of ðmDHÞ2: (a) two branches of 2 bonds, (b) three branches of

2 bonds.

Y. Li, X.-M. Kong / Physica A 356 (2005) 589–597 591

Ising spin variable si can take any values between �1 and þ1 instead of originaldiscrete value, and accordingly the summation sign in the partition function becomesintegral sign. Thus the partition function of the Ising system can be given by

Z ¼

Z þ1

�1

Ymi¼1

dsi

!W ðs1; s2; . . . ; sN Þe

HðsiÞ (1)

with the effective Hamiltonian

HðsiÞ ¼ KXhiji

sisj þ hX

i

si; �1osi;joþ1 . (2)

If the probability distribution function above is taken as

W ðs1; s2; . . . ; sNÞ ¼ e�ðb=2ÞP

is2i �uP

is4i ; u40 , (3)

the S4 model is obtained. Then the effective Hamiltonian of the S4 model in theexternal field h is given by

H ¼ KXhiji

sisj �X

i

us4i þb

2s2i � hsi

� , (4)

in which K ¼ J=kBT represents the reduced nearest neighbor interaction strength,b is the Gaussian distribution constant, u is the four-spins interaction constant, andh is the external field.

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Y. Li, X.-M. Kong / Physica A 356 (2005) 589–597592

For inhomogenous fractal lattices, let us assume that the Gaussian distributionconstant, the four-spins interaction parameter and the external magnetic field in theS4 model depend on the coordination number qi (or qj) of the site on the lattice, andthe following relations

bqi=bqj

¼ uqi=uqj

¼ hqi=hqj

¼ qi=qj (5)

are satisfied. In this case, the effective Hamiltonian of the S4 model can berewritten as

H ¼ KXhiji

sisj �X

i

uqis4i þ

bqi

2s2i � hqi

si

� , (6)

where bqi, uqi

and hqidenote the Gaussian distribution constant, four-spins

interaction parameter and the external field of the site i on the lattice, respectively.

4. Renormalization group transformation method

For simplicity we study the generator of the ðmDHÞ2 lattice. Generally speaking,the coordination number of a site on a ðmDHÞ2 lattice relies on the position of thesite and the construction stage of the lattice. Here we consider a generator of theconstruction stage. If we note that qa ¼ mn, qb ¼ 2m, qi ¼ 2 ði ¼ 1; 2; . . . ;mÞ inFig. 1(b), in terms of Eq. (6), the effective Hamiltonian of the S4 model on thegenerator is given by

H ¼ H0 þ V (7)

with

H0 ¼Xm

i¼1

Kðsa þ sbÞsi �b2

2s2i þ h2si

��

bmn

2

s2amn�1

�b2m

2

s2b2

� umn

s4amn�1

� u2m

s4b2þ hmn

sa

mn�1þ h2m

sb

2ð8Þ

and

V ¼ �u2Xm

i¼1

s4i . (9)

The partition function of the system is given by

Z ¼

Z þ1

�1

dsadsb

Z þ1

�1

Ymi¼1

dsi

!eH0þV . (10)

In the following, we define the partial trace (PT) as

ðPTÞ ¼

Z þ1

�1

Ymi¼1

dsi

!eH0þV . (11)

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Fig. 2. The renormalization group process of the ðmDHÞ2 lattices.

Y. Li, X.-M. Kong / Physica A 356 (2005) 589–597 593

Under the RG technique, that is, removing the sites 1; 2; 3; . . . , m after one stepiteration, as shown in Fig. 2, the partition function of the system remains unchanged.Thus the effective Hamiltonian of the system after the RG transformation can bewritten as

H 0 ¼ lnðPTÞ . (12)

It follows from Eq. (11) that we have

ðPTÞ ¼

Z þ1

�1

Ymi¼1

dsi

!eH0

Rþ1

�1ðQm

i¼1 dsiÞeH0þVRþ1

�1ðQm

i¼1 dsiÞeH0

¼

Z þ1

�1

Ymi¼1

dsi

!eH0heV i0 ,

(13)

where

h i0 ¼

Rþ1

�1ðQm

i¼1 dsiÞð ÞeH0Rþ1

�1ðQm

i¼1 dsiÞeH0

is called the cumulative expansion average. On the supposition that V is a smallvariable, we expand eV

heV i0 ¼ ehVi0þð1=2Þ½hV2i0�hVi20�þ . (14)

Thus Eq. (13) can be expressed as

ðPTÞ ¼

Z þ1

�1

Ymi¼1

dsi

!eH0ehVi0þð1=2Þ½hV2i0�hVi20�þ . (15)

From Eq. (8) one hasZ þ1

�1

Ymi¼1

dsi

!eH0 ¼ eH 0

0 (16)

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Y. Li, X.-M. Kong / Physica A 356 (2005) 589–597594

with

H 00 ¼ �

m

2

b2

2ðs2a þ s2bÞ � m

u2

2ðs4a þ s4bÞ þ m

h2

2ðsa þ sbÞ

þm

2b2ðKsa þ Ksb þ h2Þ

2 .

As a result, by means of the cumulative expansion approximation, from Eqs. (5),(12), (15) and (16) one obtains

H 0 ¼ �m

2

b2

2ðs2a þ s2bÞ � m

u2

2ðs4a þ s4bÞ þ m

h2

2ðsa þ sbÞ

þ mðKsa þ Ksb þ h2Þ

2

2b2þ hVi0 þ

1

2ðhV 2i0 � hVi20Þ . ð17Þ

In the following we focus on calculating the cumulative expansion terms in Eq. (17).The first-order expansion is

hVi0 ¼

Rþ1

�1ðQm

i¼1 dsiÞVeH0Rþ1

�1ðQm

i¼1 dsiÞeH0

¼

Rþ1

�1ðQm

i¼1 dsiÞ½�u2ðs41 þ s42 þ s43 þ þ s4mÞ�e

H0Rþ1

�1ðQm

i¼1 dsiÞeH0

.

(18)

Noticing that the position of sites 1; 2; . . . ;m is quite equal, therefore, we can simplifyEq. (18) as

hVi0 ¼ � mu2

Rþ1

�1dsis

4i e

H0Rþ1

�1dsieH0

¼ � mu26ðKsa þ Ksb þ h2Þ

2

b32� mu2

ðKsa þ Ksb þ h2Þ4

b42. ð19Þ

In the same way, the second-order term is approximately obtained

hV 2i0 ¼

Rþ1

�1ðQm

i¼1 dsiÞV2eH0Rþ1

�1ðQm

i¼1 dsiÞeH0

¼ mu22420ðKsa þ Ksb þ h2Þ

2

b52þ mu22

174ðKsa þ Ksb þ h2Þ4

b62

þ m2u2236ðKsa þ Ksb þ h2Þ

4

b62, ð20Þ

hVi20 ¼ m2u2236ðKsa þ Ksb þ h2Þ

4

b62(21)

and

1

2ðhV 2i0 � hVi20Þ ¼ mu22

210ðKsa þ Ksb þ h2Þ2

b52þ mu22

87ðKsa þ Ksb þ h2Þ4

b62.

(22)

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Substituting Eqs. (19) and (22) into Eq. (17), finally we have

H 0 ¼ H 01 þ H 0

2 þ H 03 þ H 0

4 , (23)

with

H 01 ¼ m

K2

b2�12u2K

2

b32þ420u22K

2

b52

!sasb ,

H 02 ¼ �

b2

2

m

21�

2K2

b22þ24u2K

2

b42�840u22K

2

b62

!ðs2a þ s2bÞ ,

H 03 ¼ �u2

m

21þ

2K4

b42�174u2K

4

b62

!ðs4a þ s4bÞ ,

H 04 ¼ h2

m

21þ

2K

b2�24u2K

b32þ840u22K

b52

!ðsa þ sbÞ .

If we remark the spins, i.e.,

s0a;b ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim 1�

2K2

b22þ24u2K

2

b42�840u22K

2

b62

!vuut sa;b , (24)

then Eq. (23) can be rewritten as follows:

H 0 ¼ K 0s0as0b �bmn�1

2

s0a2

mn�1�

b2

2

s0b2

2� u0

mn�1

s0a4

mn�1

� u02

s0b4

2þ h0

mn�1

s0amn�1

þ h02

s0b2, ð25Þ

with

K 0 ¼1

A

K2

b2�12u2K

2

b32þ420u22K

2

b52

!(26)

u02 ¼

u2

mA21þ

2K4

b42�174u2K

4

b62

!(27)

h02 ¼

m1=2h2

A1=21þ

2K

b2�24u2K

b32þ840u22K

b52

!(28)

A ¼ 1�2K2

b22þ24u2K

2

b42�840u22K

2

b62

!.

Eqs. (26)–(28) are the recursion relations of the RG transformation.

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Y. Li, X.-M. Kong / Physica A 356 (2005) 589–597596

5. Results

From the recursion relations above, one finds that when there are more than 4branches in the ðmDHÞ2 lattice, there exists one fixed point, i.e., the Gaussian fixedpoint, and that when m is less than or equal to 4, the Wilson–Fisher (WF) fixed pointas well as the Gaussian one is obtained. The WF fixed points have a decisiveeffect on the critical properties of the system in the case of mp4, that is, thefractal dimensionality dfp3. In Table 1, we show the WF fixed points for the casem ¼ 2; 3; 4.In terms of the RG transformation and scaling theory, we calculate the critical

exponents for m ¼ 2; 3; 4, which are listed in Table 2. The critical exponents in Table2 a, b, d, g, n and Z describe in the neighborhood of the critical point the variations inheat, magnetization, magnetic field, magnetic susceptibility, correlation length andcorrelation function, respectively.

6. Summary

Using a real-space renormalization-group method together with the cumulativeexpansion technique, we investigate the critical properties of the S4 model ondiamond-type hierarchical lattices with m branches of 2 bonds in the external field,and draw the following conclusions: (1) For the case m44 (or df43Þ, there existsone fixed point, i.e., the Gaussian fixed point Kn ¼ b2=2; un ¼ 0; hn

¼ 0, and thuscritical exponents are in full agreement with those of the Gaussian model. In otherwords, for m44 (or d f43), critical properties of the S4 model are identical withthose of the Gaussian model on diamond-type hierarchical lattices, and they belongto the same universality type. On the other hand, our results also give the same

Table 1

The WF fixed points in the case of m ¼ 2; 3; 4

m Kn un2 hn

2

2 0:218b2 0:080b22 0

3 0:468b2 0:035b22 0

4 0:530b2 0:016b22 0

Table 2

Critical exponents of the S4 model system when m equals 2, 3, and 4

m a b d g n Z

2 1.010 �0.009 �102.523 0.918 0.450 �0.036

3 0.864 0.128 7.842 0.879 0.443 0.00024

4 0.544 0.243 5.000 0.971 0.485 0.002

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Y. Li, X.-M. Kong / Physica A 356 (2005) 589–597 597

critical exponents as those of the mean field theory for m ¼ 8 (or df ¼ 4Þ [17]. (2) Forthe case mp4 (or d fp3), the WF fixed points as well as the Gaussian one areobtained, and in this case, only the former has a decisive effect on the criticalproperties of the system. In addition, here the critical exponents are found to berelatively close to those of the Gaussian model [11], but very different from those ofthe Ising model. We found that for the S4 model, the dependence of the criticalproperties on the dimensionality on the fractal lattices shows qualitative agreementwith that on the translation symmetric lattices [15].

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