Scaling function for two-point correlations in disordered systems

M. CRISAN A N D D. DADARLAT Department of Physics, University of Cllrj, Cllrj, Romania

Received June 9, 1977'

We consider the two-point correlation function for the random spins system near the critical temperature expressed as G(r,q) = Cr-YD(c2q2), r = a(T - T,)/T,. The universal function D(q2c2) is calculated for the random system using the method applied by Fisher and Aharony for pure systems. The deviation from the Orstein-Zernike behaviour of the universal function is found to be small because of the coefficient of the expansion. The expression of the critical index q obtained by this method has the same value as the value obtained by other authors.

Nous considerons la fonction de correlation B deux points pour le systeme de spins au hasard au voisinage de la temperature critique expr~mee sous la forme G(r,q) = Cr-YD(cZqZ), r = u(T - T,)/T,. La fonction universelle D(q2cZ) est calculee pour le systeme de spins au hasard en utilisant la methode appliquee par Fisher et Aharony dans le cas de systemes purs. On trouve que I'ecart entre la fonction universelle et le comportement Orstein-Zernike est faible, a cause du coefficient du developpement. L'expression de I'indice critique1 obtenue par cette methode est la mkme que celle h laquelle sont arriveesd'autres auteurs.

[Traduit par le journal] Can. J Phys ,56,989(1978)

Introduction tibility The scattering intensity near the critical point and

the Fourier transform of the two-point (spin-spin) correlation function

C11 G(q, T I - (S(O) . S(R)) exp (iq . R) R

are directly connected. Recently many theoretical studies were devoted to this problem in various models. The problem was treated by Aharony (I) and Fisher and Aharony (2) for pure systems. Grunstein and Luther (3) pointed out the equivalence between the Lubensky (5) treatment of disordered system and Khmel'nitski's (4) method applied for a pure system containing impurities. The most important idea of the phase transitions in disordered systems, is to transform the randon1 problem into an invariant one.

We consider that this approach may be performed and, we assume that in the critical region, for the disordered system, the correlation function [ I ] has the asymptotic form

[21 C(q, T) = Cr-yD(q2k12)

where r. = a ( T - Tc)/Tc and the second-moment correlation length 5, is given by

[3 I kl(r> = f i r - .

The functions C and f, are obtained from the nor- malized conditions

which give for C the amplitude of the static suscep-

'Revision received March 1. 1978.

[5 1 xo(T) = G(0, T) z C/rY

As was pointed out by Fisher and Aharony (2), C and fl are not universal parameters and as for the pure systems we consider that for the random sys- tems these functions depend on the interaction. In fact we are interested in the calculation of the func- tion D(x2), which is expected to be universal as it is for the pure systems. The method that will be used is given by Fisher and Aharony (2) for the pure sys- tems.

The Scaling Function D

In order t o calculate the universal function D for the disordered systems, we consider from the begin- ning that the mean value in [ I ] contains also the mean value for the impurities. By this method we have transformed the problem of phase transition in disordered system into a translationally invariant one. In this approxin~ation the general equation for C(q, r) is

where the self-energy C is given by the diagrams from Fig. 1. The value of Go(q, r) is given by the Ornstein- Zernike expression Go(q, r ) = (q2 + r)- ' and the vertex functions u and A are given by Lubensky (5), using E - expansion, as

but the case n = 1 will be not considered.

Can

. J. P

hys.

Dow

nloa

ded

from

ww

w.n

rcre

sear

chpr

ess.

com

by

99.2

51.2

51.1

53 o

n 11

/17/

14Fo

r pe

rson

al u

se o

nly.

990 C A N . J . PHYS.

FIG. 1. The diagrams for the self-energy Z.

Following Fisher and Aharony (2) we obtain from [6] G(q, T ) expressed as

[8] G(q, T ) = G,(q, r.) + 32Rd2~r"(n + 2)

x GO2(q, r.)q2K(q, r.1 + K,I2A2Go2(q, 1.1 x q2K(q, r . ) - 8K",,2(~~ + 2)1/AGo2(q, r . )

x q2K(q, r') + 0 ( 1 1 3 , A3, 112A, 11A2)

The Green function [8] was calculated in fact using the corrections given in Fig. 1 to the free propagator. In order to obtain this expression we may use the R N G method or the 'parquet' equations. Impurity scattering effect is calculated in the second order approximation. The equivalence between these net hods was pointed out by Grinstein and Luther (3). If the relations [7] and [9] are used in the expression [8] we get for the corre la t io~~ function the E-expansion

where KO is the zero order of E-expansion for K. From [2] and [4] we obtain for the scaling func-

tion D the expression

In 4-E dimension we write down the E-expansion for [ l l ] as

VOL. 56. 1978

[12] as

Jo = rG,(q, r.) = [I + q 2 / ~ ' ] - 1 ; J , = 0

The scaling function D(x2, E) will be written as

[14] D(x2, E) = DO(xZ) + E D ~ ( x ~ ) + E ' D ~ ( x ~ ) + ... where

[151 k 1 2 q 2 = cq2r.-2vI'

with the constant C and 2vly expanded as

With these results we may obtain Do, D l , D,, ... from [14]-[17]. The first term gives

and using [4] we get Co = 1 . In this approximation we get for D , ( S ~ ) the

Orstein-Zernike expression :

~191 D ~ ( X . ~ ) = [ I + . X ~ I - ~

The second term gives the relation

and using [4] we get

Dl(0) = D2(0) = ... = 0 [211 D1'(O) = D2'(0) = ... = 0

and

which is satisfied for y 4 0 only if C , = 0 and i j l =

0 , and then

The next tern1 that cotltains the E~ contribution will give the expression for D2(x2) . Indeed, from [12] and [14], D , ( x ~ ) is

.x2 [24] D ~ ( x ~ ) = [C2 - $ij2 111 r] ( 1 + x ~ ) ~

where J,, J,, and J2 are obtained from [ l o ] and with the condition

Can

. J. P

hys.

Dow

nloa

ded

from

ww

w.n

rcre

sear

chpr

ess.

com

by

99.2

51.2

51.1

53 o

n 11

/17/

14Fo

r pe

rson

al u

se o

nly.

CRlSAN A N D DADARLAT 99 1

[25 1 DZ1(0) = 0 If the scaling laws are respected for the disordered systems, then which gives

17(5/2 - 8) [26] C2 - 9 p 2 In r + Ko(O, 1.) = 0 [331

64(n - and The value KO has been calculated by Fisher and 2v Aharony (2 ) as [34] --E I +g= 1 + i p , ~ 2 + ..., F 2 = p2 Y 2 [27] Ko(O, 1.) = In r. - -- L In h + I< + O(r1I2)

Using [28], [34] the critical index q is obtained as where h and lc are constants.

From [26] and [27] we get C351 n (512-8 )

11 = 2 j ~ ( ~ - 112 + 0'~')

[28] p2 = , C2 = P 2 ( h h - 4k) for 17 # 1, obtained also by other methods by Khmel'nitsky (4) and Lubensky (5).

and S

2

[29] D2(x2) = 4p2 ( I + ~ 2 ) ~

[Ko(q, r ) - KO(0, ).)I or if we denote

Ko(q, r . ) - Ko(O, r . ) = x2Q(s2)

the expression [29] becomes

With these results for the coefficients D,, D,, and D2 the expression [L4] is given as

+ 0 ( E 3 )

or, if Q(x2) is calculated as

~ ( x * ) = +(0, - b2x2) ; 0 , = 0.00752; b2 = 0.00019 L

Conclusions The expression [32] obtained for D - ' ( x ~ ) shows

that, for the disordered systems, in the limit x -t 0 , the deviation from the Ornstein-Zernike behaviour is small because the coefficients 0 , and 0 , are quite small.

The scaling laws are respected in the critical re- gion of the disordered systems, as it is proved by the expression [35], which is in agreement with the value obtained by Lubensky (5) and Khmel'nitsky (4). From [35] and [31] we get for the scaling function D - ' ( x 2 ) the expression

which has the same form as for the pure systems, the difference being the critical index 11 given by [35].

[31] becomes 1. A. AHARONY. Phys. Rev. B, 12, 1038 (1975). 2. M . Frs~~Rand A. AHARONY. Phys. Rev. B, 10,2818(1974). t7(5n - 8)

[32] D - ' ( x ~ ) = 1 + x 2 - & 2 3. G. GRINSTEIN and A. LUTHER. Phys. Rev. B, 13, 1329 128(n - 1)' (1976).

2 4. D. E. KHMEL'NITSKY. Zh. Eksp. Teor. Fiz. 68,1960(1975).

x ~ " ( 6 1 - 62-'i ) 5. T. C. LUBENSKY. Phys. Rev. B, 11,3573 (1975).

Can

. J. P

hys.

Dow

nloa

ded

from

ww

w.n

rcre

sear

chpr

ess.

com

by

99.2

51.2

51.1

53 o

n 11

/17/

14Fo

r pe

rson

al u

se o

nly.