Exact results of AT model on a special kind of Sierpinski carpets

PHYSlCA ELSEVIER Physica A 227 (1996) 34-42

Exact results of AT model on a special kind of Sierpinski carpets Shu Chen b, Z.R. Yang a,b

a CCAST (World Laboratory), PO.Box 8730, Beijing 100080, China b Department of Physics, and Institute of Theoretical Physics, Beijing Normal University,

Beijing 100875, China

Received 3 November 1995

Abstract

Exact results are obtained for the Ashkin-Teller model on a special Sierpinski carpet (SC). We calculate the partition function and correlation function with the transformation matrix method. Thus the asymptotic behaviour of the correlation function is investigated in the limit of long distance. The results show that no phase transition occurs at any finite temperature for the finite R (the order of ramification) fractal, which is consistent with the conclusion of the Ising and Potts model.

PACS: 05.50.+q; 75.10.Hk

1. Introduct ion

The fractal has always been an intriguing problem in physics since Gefen et al. [ 1-3 ]

started the research work about the phase transition of the Ising and Potts model on

fractals. An important work in fractal physics is researching the phase transition of some

models on a fractal lattice. In previous times, most of the results are obtained with the

renormalization group method; one of the important results is that no phase transition

takes place at any finite temperature for the fractals with finite ramification.

We try to exactly solve some models on fractal lattices and investigate the asymptotic

behaviour of the correlation function in the l imit of long distance, but a rigorous result

is difficult to get on the fractal lattice just as on the general lattice; only those fractal

lattices with finite ramification and no loop structure are exactly solvable. One of the

authors has ever obtained the exact partition function and correlation function of the

Ising model on a special Sierpinski carpet (SC) with graph theory [4,5].

0378-4371/96/$15.00 Q 1996 Elsevier Science B.V. All rights reserved SSD10378-437 I (95)00419-X

S. Chen, ZR. Yang/Physica A 227 (1996) 34-42 35

The Ashkin-Teller (AT) model is also of interest in its own right, for it is related

to other problems of physical interest such as the Ising model, Potts model as well as

the eight-vertex model. It is well known that the general Ashink-Teller model remains

unsolved, the properties are considered through its relation with the staggered eight-

vertex model [6] . In this paper we adopt the transformation matrix method and exactly

calculate the partition function and correlation function of the AT model on SC, thus its

phase transition is discussed.

2. The calculation of the partition function

The Ashkin-Teller model (AT) was introduced by Ashkin and Teller in 1943 [7].

Each site of a lattice £ is occupied by one of four kinds of atoms: A, B, C, or D.

Two neighbouring atoms interact with an energy: e0 for AA, BB, CC, DD; el for AB,

CD; e2 for AC, BD; and e3 for AD, BC. The interaction energy for edge ( i , j ) can be

expressed as

e0, if AA, BB, CC, DD,

8(O-i, O'j ) = 81, if AB, CD, ( I )

s2, if AC, BD,

~3, if AD, BC,

where o'i denotes the state of lattice site i, which can take four different states A, B, C,

D. Thus the Hamiltonian of AT can be written as

7-t = ~--~ e(o-i, o~j), (2)

/i,i)

where (i, J/ denotes summation over all nearest-neighbour atoms on the lattice. It is easy

to see that the AT model identifies with a four-component Potts model with s~, s2, s3 = 0.

Now we will study the AT model on a special SC lattice with finite ramification. The

SC is constructed via an iteration procedure. At first, a square of unit area is divided

into b x b sub-squares, then I x 1 sub-squares are removed in a special way as shown in Fig. 1. The procedure is then repeated again and again for the smaller squares. There

we choose b = 3 and I = 1. Therefore the partition function of the AT model on the SC

lattice is given by

= ~ e x p -,8~_~(o'i,o',)]. Z (3)

,~ (i,j) J

With the same method of our previous paper [ 8 ], we can prove that the partition function

of the AT model on a lattice which can be divided into two parts with only a common

point (Fig. 2) can be expressed as

Z = IZIZII; (4) 4

36 S. Chen, ZR. Yang/Physica A 227 (1996) 34-42

n = !

I

n = 0

n = 2

Fig. 1. The iteration process of generating the Sierpinski Carpet embedded in two Euclidean space.

I I

• . . ° • , ° ° ,

, o , . , °

i

i I ,"

• ~ ° o

Fig. 2. The lattice is divided into two independent parts I and II which are only connected with a joint point 0.

here Z, ZI, Zlt denote the partition functions of the entire lattice, part I and part II

respectively. Employing (4) we conclude that the partition function of the AT model on the nth SC lattice can be written as a function of the partition function of the sub-square,

1 M z . = ~-~Z~ , (5)

where M is the number o f sub-squares on the nth SC latt ice and M ~ is the number o f

j o in t points o f the M parts. The nth SC can be d iv ided into 5 M sub-squares and each

two nearest sub-squares are connected through a jo in t point, therefore i t ' s easy to get

M = 5" and the jo in t point number M t = ~--~."=l 4 x 5 " - j = 5n - 1, namely M p = M - 1.

So far the part i t ion funct ion o f the nth SC latt ice can be got through the calculat ion o f

the part i t ion funct ion o f a sub-square.


For a sub-square we can write the partition function of the AT model on it as

Z, = Z A(o ' I , 0"2)A (0"2, 0"3) A (0"3, O'4)A (o'4, o'1 ) o"

= Tr A 4 , (6)

in which A(o'i, o' j ) = exp [ - - /~ (O ' i , O' j ) ] is an element of matrix A, therefore the matrix

A can be expressed as

0)0 0)1 0)2 0)3 /

0)1 0)0 0)3 0)2 A = , (7)

0)2 0)3 0)0 0)1

0)3 0)2 0)1 0)0

where 0)0 = e-/3~°, 0)1 = e -/3e~, 0)2 = e -~3c2, 0)3 = e-/3e3. For a symmetric matrix which

possesses a structure such as matrix A, one can make it diagonal through a similarity

transformation:

/ ) T A T _ j = A2 (8)

3.3

3.4

where

A1 =0)0+0)1 +0)2-t-0)3,

3.3 = 0)0 -t-- 0)1 -- 0)2 -- 0-)3,

3.2 = 0)0 -- 0)1 q- O92 -- 0)3,

3.4 = 0)0 -- 0)1 -- 0)2 -~- 0)3 (9)

are the eigenvalues of matrix A; the transformation matrix T and its inverse matrix T - i

are

1 - i 1 - 1 1 T - 1 1 1 1 - 1 - 1 ( 1 0 )

T = ~ - 1 - 1 1 1 ' = 2 1 - 1 1 - 1 "

1 - l - 1 1 l 1 1 1

Therefore we get the partit ion function of a sub-square via (8 ) ,

Z~ = Tr A 4

= 3.14 _}_ 3.4 + 3.4 3 +3-4. ( l l )

Substituting (10) into (5 ) , we obtain


Iii t I Iol, "',.

O-ko

Fig. 3. The n = 2 stage of the construction of SC. A sketch of a linear chain part connecting o'k and ~r k, is shown. The heavy lines indicate the linear chain part and the broken lines separate the linear chain part from the other X parts. There are two types of sub-squares, a-type and B-type, in the linear chain.

1 _ _ Z M

Z , = 4M_l .~

1 4 - 4 M-' (,~4 + 12 4 + A3 + A4)M. (12)

3. The correlation function

The correlation function between atom o-~ and o-k, is defined as

1 ~-'~ (o'k~r,,) exp -/3 ~-'~ e(o-i, o~/)], (oko',,) = ~ ~,,} ~i, , (13)

where y]{,q is over all possible atom configurations and 8(O'i, O'j) is given by ( 1 ). The nth SC lattice can always be divided into a linear chain part connecting ark with o'k, by a series of sub-squares and the other X parts which only connect the linear chain part with a joint point [8]. We display Fig. 3 to explain the procedure, the linear chain part connecting o-k and ark, is displayed in terms of heavy lines, obviously the joint point number is X. Employing (4) the partition function Z, can be written as

Z, = ---~ Zc ZI Z2 . . . Zx; (14)

here Zc is the partition function of the linear chain part and Zi, Z2 . . . . . Z x represent the partition function of the other parts respectively. Likewise, y'] {,,} (o-ko-~,) exp[ - f l ~( i , . j ) ,

e(o-i , 0 9) ] has the following expression:


0" 1 0".3

1 0" 0"

4

0"]

2 0"4

0.2

1 0.3

O~ - - t y p e f l - t y p e

Fig. 4. The a-type and fl-type sub-squares.

Therefore the correlation function is given by

(15)

(0); 0)I 0)~ 0)'1 3 ! / !

Aa = 0)1 0 )0 ~ 3 0)~ , ! / I I

0)2 0)3 0 )0 0)I

/ ! I 0)3 0)2 0)1 0 ) ;

! 0);, 0),, 04, 0)' 3 are given by

o)g= (o.)g -F 0)2 --F 0)2 + 0)2) 2 0) '1=4(0)00)| +0)20)'~) 2

I 0)2 = 4 (0)o0)2 + ah 0)3) 2 , 0)~ = 4 (0)o0)3 + °)1(-°2) 2

Similarly, for the / / - type sub-square we define

(ig)

(19)

I ~ (~k~k,) exp [-//~ ~(o-,, 0-/)] (,6) (,qc ( i,.j)~

in which the summation ~--]~{,~}c is only over the linear chain part and the other X parts

are reduced, now we only need to calculate the contribution of the linear chain part. On

the linear chain there are two types of different squares, a-type and//-type [5], shown

in Fig. 4. As shown in Fig. 3, at the site ce there are two ways to cross the sub-square

(a-type sub-square), each way includes two bonds; at the site 19 there are also two

ways to cross the sub-square (//-type sub-square), but one way involves one bond and

the other way three bonds. For the a-type sub-square we define

0-2)-- E E exp [-// (8(0-,, o-3) + tr 3 0" 4

-k- 8(o'2, 0"4) -k- 8(0-4, 0-1 ) )] , (17)

here 0-3, 0-4 have four different states A, B, C, D. Taking A,,(0-j, 0-2) as matrix element,

we get


A/~(tr,, o'2) = Z Z exp [-13 (e(o-,, o-2) + e(o'2, tr3) o- 3 CT 4

+ ,E:(O'3, Or4) "t- ~ ( O ' 4 , O r l ) ) ] .

Taking A#(o-1, o'2) as matrix element, we have

A/~ =

where o)D',

I // t/ t! O)I/ 1 O)O 091 °92 3

II tl II fl O)I o)O (°3 012 " o)" o;o' o)~' ' 092 3 " o)'~' o)~' o)'o' O) 3

o)~', ,o~', o)- 3 are given by

(20)

(21)

, 4 ' = o)0 ~ [o)0 ~ + 3

o 4 = o)~ [o,, ~ + 3

o,1' = ,o22 [,o22 + 3

, , = o)2 [,03 + 3 °)3 3

Comparing (18), (21

w~ +w~ + o)~)] +6wow,w2w3, ,o02 + , 4 + o)3)] + 6o)0o),,o2o)3,

(22) o)0 ~ + o),~ + o)3)] + 6o)0o),o)2o)3,

w 2 +o)~ +w~)] +6o)ow,o)zw3. with (7), one finds that matrices A,~, At~ possess the same

structure as matrix A except for the different matrix elements. It is easy to check that they commute with each other, thus they can be diagonal at the same time. So far the correlation function (16) can be expressed as

1 (o'~o'k,) = ~ c Z (° 'k° 'k ')A/3(trk'°-t)A~(°' t ' tr2) '" "a~(crk ' - l ,° 'k ' ) '

{'~k,'~,,...,'r~, } (23)

We define

B(o'k,,o'k) = (O'kO'k,). (24)

AS we see, with each lattice site i we associate a variable o" i which represents four different states A, B, C, D. As in the Ising and Potts model, let us call o- i a 'spin' which takes four values, say a, b, c, d corresponding to A, B, C, D, therefore matrix B is given by

la2 ab ac adl ba b 2 bc bd

B = . (25) ca cb C 2 cd da db dc d 2

Thus we get

1 (o'ko'k,) = Zc Tr BA#Aa... AI3"" Aa


(7 k

I T V1 o-r:,

Fig. 5. The linear chain part. The x denote the points to be decimated.

= 1_~ Tr TBT-1TAt~ T - l TA,~T- I . . . T A , , T - I. ( 26 ) Zc

Let us assume that we have L o~-type and L I /3-type sub-squares along the linear chain (Fig. 5), thus there are L A , ' s and U A f s in (26). Because T A , T -1 and TA/3T - I are

diagonal matrices, they can commute with each other. Thus

T r T B T - 1 ( T A , T - I ) L ( T A # T - 1 )L'. (27)

Similar to (12), Zc is given by

1 (a~ + a~ + a~ + a~)~+~'. Z c - 4L+L,_ I . (28)

Substituting (9) , (18), (21) , (25) and (28) into (27), we get

L I I I L t ~ , I L ~ I I L I - - ~ , I L ~ I I L t I l L . I l L I

(,,~14 q- ,~2 4 --[.- ~3 4 + /~44 )

where a~ a~, a~, a~, and V , a~', a" a~' , 3, are the eigenvalues of matrix A, and A[~

respectively and given by

! ! a', = 0)~ + 0)1 + 0)~ + 0)3,

---- 0)0 + 0)1 -- 0)5 -- 0)3 '

and

I I I I I t 11 Atl t = 0)0 -I- 0)1 -~- 0)2 q- 0)3,

a ~' = 0)'0' + 0)7 0)'~' " - - - - 0 ) 3 '

! ! a; = 0)~ - 0)1 + 0)~ - 0)3,

t t t q._ 0)~, "~'4 = 0)0 - - Wl - - 0)2

(30)

II II I I I I ,~t = 0)0 -- 0)1 + 092 -- 0)3 '

I! II II II .,~r = 0)0 --.0)1 -- 0)2 -~- °)3"

(31)

p l , P 2 , P3, P4 arethe diagonalelements ofmatr ix TBT -1 , which are constants determined

by spin values a, b, c, d:

Pl = ( a + b + c + d ) 2, P 2 = ( a - b + c - d ) 2, (32)

P3 = ( a + b - c - d) 2, P4 = (a - b - c + d) 2.


The expression (29) gives the exact correlation function. When L + L' tends to infinity,

4L+L ' ,~t L ~II L' (o ' kok , ) "~ "'max "',,,,,x (33)

(/~4 + 34 +,~n3 + /~4)4 L + L ' '

i i i where /~max, "~max are the maximum eigenvalues of matrix A,~, Ate.

In general, we chose eo < 0 and eo < e l , e2 ,e3 in expression (1) . For the finite

temperature, namely wo, w j, w2, o93 :/: 0<3, (OrkO'k ,) tends exponentially to zero as L + U

becomes large. L + L p represents the distance between o-k and o'k,. Generally L, L' are

determined by the structure of the fractal and the linear path connecting o'k and o-k,.

The result indicates that the correlation between spins will decrease rapidly with the

increase of distance L + U. Correlation length ( is defined by

(o'ko'k,) ~ e x p [ - ( L + L ' ) / ~ ] . (34)

Because L' is much less than L, we ignore the factor corresponding to L' , thus one gets

Fin + + + a2]-1 L I 4 Amax J

When temperature approaches zero, equivalent to w0 approaching infinity, the correlation length ( --~ oo. It thus indicates the existence of a zero-temperature phase transition. The result rigorously confirms the following conclusion [ 1-3]: no finite-temperature phase transition occurs for fractals with finite ramification.

Acknowledgements

This work was supported in part by the Non-linear Science Project of China and the

Nation Nature Science Foundation of China.

References

[ 1 ] Y. Gefen, A. Aharony and B.B. Mandelbrot, J. Phys. A: Math. Gen. 16 (1983) 1267. 121 Y. Gefen, A. Aharony, Y. Shapir and B.B. Mandelbrot, J. Phys. A: Math. Gen. 17 (1984) 435. 131 Y. Gefen, A. Aharony and B.B. Mandelbrot, J. Phys. A: Math. Gen. 17 (1984) 1277. 141 Z.R. Yang, Phys. Rev. E 49 (1994) 2457. 151 Z.R. Yang, Ic/93/342 (preprint), International Center for Theoretical Physics, Trieste, Italy. 161 R.J. Baxter, Exactly solved models in statistical mechanics (Academic Press, New York, 1982). 171 J. Ashkin and Teller, Phys. Rev. 64 (1943) 178. 181 S. Chen and Z.R. Yang, Z. Phys. B (to be published).

Exact results of AT model on a special kind of Sierpinski carpets

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