static critical behaviors in random-spin systems with short-range interaction

Static critical behaviors in random-spin systems with short-range interaction

YOSHITAKE YAMAZAKI Depcr~.f/l ie/if c?f'Applictl Physics , Tol1oX11 U/lil 'er.sit j~, Sr~~clc i i , .loptiti

Received Aptil28, 1977

Critical behaviors in quenched random-spin systems with N-spin coniponent are studied in the limit M -, 0 of the non-random MN-component models by means of the renormalization group theory. As the static critical phenomena the stability of the fixed points is investigated and the critical exponents q[- O ( E ~ ) ; E = 4 - dl, y , a , and crossover index +[r O(E~) ] and the equation of state [- O(E)] are obtained. Within the approximation up to the order c2, even the random-spin systems with N = 2 or 3 are unstable in the three dimensions and the pure systems are stable there.

Les comportements critiques de systemes reduits de spins aleatoires de N coniposantes sont ttudits dans la limite M + 0 des modeles non aleatoires de composantes M N au moyen de la thCorie du groupe de renormalisation. Comme phenomene critique, la stabilite des points fixes est considCrCe et les exposants critiques q[- O ( E ~ ) ; E = 4 - dl, y, a et I'index de croisement + [=O(E~) ] ainsi que ]'equation d'Ctat [-O(E)] sont obtenus. Dans I'approxiniation jusqu'a I'ordre E', mCme les systemes de spins aleatoires avec N = 2 ou 3 sont instables dans les trois dimensions, alors que les systtmes purs sont stables.

Can. J . Phys., 56, 139 (1978) [Traduit par le journal]

1. Introduction Theories on the second order phase transitions in

pure systems have been greatly advanced since Wilson's theory, and were reviewed by Wilson and Kogut (I), M a (2), and Fisher (3). Recently critical phenomena in random-impurity systems have started to be studied by Grinstein (4), Luther and Grinstein (5), Aharony (6), Emery (7), Lubensky (8), Aharony et al. (9). In general those systems are classified into annealed systems and quenched systems. In annealed systems an average diffusion time for impurities .rd is very much smaller than the typical time .re necessary for measurements, .rd << .re, and these impurities are random in thermodynamic equili- brium. Any distinction between variables, which describe the randomness and the usual dynamic variables of these systems, disappears. For sufficiently high concentration of impurities, the phase separation occurs. Quenched systems are character- ized by .re << .rd, and impurities are frozen in fixed positions because they are interrupted by potential barriers and cannot diffuse freely. The state of these systems is described by a single constant set of values of the random variables for the duration of any experiment. For sufficiently high concentrations of impurities, percolation phenomena occur.

In this article critical behaviors of quenched random systems are studied near the second order phase transition point. The critical phenomena in a quenched random N-component continuous spin system were recently studied by the following authors using the renormalization-group theories near four dimensions. Lubensky (8) derived recur- sion relations for the second cumulant of the distri-

bution function of the disorder, and obtained the static critical exponents to order E' (E -- 4 - 0; cl, the dimensionality of the space). Luther and Grin- stein (5), Emery (7), and Aharony (6), derived the effective Hamiltonian by exploiting the equivalence of the quenched random system with an MN- component non-random (pure) system in the limit M --+ 0. Recently critical behaviors of amorphous magnets were studied by Aharony (10).

The author investigates the critical behaviors of the MN-component spin system by means of the Callan-Symanzik equations. The notations used are rewritten so as to be conveniently compared with Lubensky's expressions, since his results were published after the calculations in this article had been finished. In Sect. 2 an effective Ha~niltonian for quenched systems with N-spin con~ponent is summarized. Static critical behaviors of these systems without an external field are studied in Sect. 3. Those of the systems with an external field are investigated in Sect. 4. In Sect. 5 concluding remarks are summarized. The new results obtained are of the c2 corrections to the eigenvalues for the random- spin system, of the stable regions for the fixed point t o order E ~ , of the c3 correction to q, and of the equation of state.

2. Effective Hamiltonian Let us start to define a primitive quenched random

system with N-spin component in a cl-dimensional space. In a regular Bravais lattice consisting of fi lattice sites in d-dimensional space, each site may be occupied by a classical N-component spin or may be empty, and the probability of site occupied with

Can

. J. P

hys.

Dow

nloa

ded

from

ww

w.n

rcre

sear

chpr

ess.

com

by

UN

IVE

RSI

TY

OF

NE

W M

EX

ICO

on

11/2

8/14

For

pers

onal

use

onl

y.

140 C A N . S . PHYS. VOL. 56. 1978

spin is assumed to be 9 (0 5 9 5 1) independent of the state of occupation of all pther sites on the lattice. In this system there are 2N distinct states in addition to the states due to the internal degrees of freedom of spin. The occupation of the j th site can be specified by the variable $ j , which has the value 1 if the site is occupied with spin and 0 if the site is vacant. The state of the entire lattice can be uniquely described by this set of f? values { I ) ~ ) . The probability p($) of finding the system in any state { I ) ~ ) , which has 177 sites occupied with spin and (m - m) vacant sites, is defined as

The Hamiltonian of the system for this state { I ) ~ ) in a magnetic field 11 may be written as

where Ji j is trai~slationally invariant ferromagnetic interaction and S j is the N-component spin on the j th site. The partition function z($), free energy F($), and spin-spin correlation function gij($) are defined, respectively, as

~ ( $ 1 = 1 exp ( - Pa) i s )

function cij(f?) are given by the ensemble averaging as

F(R) = m- l 1 P($)F($)

C2.41 (ur )

Gij(fJ> = f J - 1 g($>gij($> (ur)

The magnetization in the thermodynamic limit is defined by

~2.51 M(h) = -1im (aF(&)/ah), fi-+ w

In the following let us use the conventional unit (PJ = 1).

Let us first derive the continuous spin Hamiltonian equivalent to the quenched random spin system [2.2]. We assume that the magnitude of spin occupying any i-site is constant, i.e., without loss of generality, Si2 = 1 , which corresponds to the relation Si2$i2 = $ i 2 , in the quenched random spin system. The transformation of [2.3] from the discrete spin variables to the continuous ones is performed as : (1) t o multiply the summand by a set of the Delta functions 6(Si2$i2 - $ i 2 ) , which is expressed by the identity

C2.31 F($) = - P-' In z($) 6(x) = lim (w/n)'I2 exp ( - wx2) >v-+ w

gij($) = .($I-' 1 $isi$jsj exP ( - ~ a ( $ ) ) (2) to replace the sum over a set of discrete spin i s ) variables by the integral with respect to a set of

where p = (k,T)-'. The corresponding observed continuous spin variables. For example, the partition free energy ~ ( f ? ) of the system and the correlation function z($) is expressed as

C2.61 ~ ( $ 1 = 1 d ' ~ exp 2-' 1 P j i j ~ i $ i S j $ j - C w(s?$? - + 1 P / I s ~ $ ~ L i i i i J where terms reduced to a constant are neglected. The second relation is obtained by substituting the integral variables S i with and by taking into account the relations $ i 2 = after the integral over the spin variables has been performed. The free energy F($), excluding the constant terms, is derived from the logarithm of the spin-integrals in [2.6] without the term [17$i]-'. Therefore, with suitable replacement of notations, the random N-component spin system is described by the continuous spin Hamiltonian

12.71 H(S, $1 = H,(s) + H,(s, $1 where Ho and H , are the usual non-random N-component spin Hamiltonian in a magnetic field 17 such as

and the interaction Hamiltonian between the local energy density S ( X ) ~ and the fluctuations of the iocal critical temperature $(x) as

Can

. J. P

hys.

Dow

nloa

ded

from

ww

w.n

rcre

sear

chpr

ess.

com

by

UN

IVE

RSI

TY

OF

NE

W M

EX

ICO

on

11/2

8/14

For

pers

onal

use

onl

y.

in which S = {S,, ..., S,), and $(x) is a random continuous variable whose larger value represents a lower value of the mean field critical temperature. For a given set of configuration {$(x)), the corresponding free energy FN($) is

[2. 101 ~ ~ ( $ 1 = -1n 1 d G N s exp (-H(S, $))

where f? is the number of sites in the system under consideration. The observed free energy F is given by the configurational average of it over the joint probability distribution of the f? variables ($1, P($), as

[2.11] F = 1 ~'$P($)F~($) = - 1 dG$p($) ~n 1 d a N s exp (-H(S, $))

Let us choose the volume of a unit cell to be equal to unity and assume that the joint probability is expressed by the product of a probability distribution 9($j) for $j at the jth spin, as

a ~2.121 P($> = 17 9($j>

j= 1

Let us secondly derive the equivalence of the limit M + 0 of the non-random (pure) MN-component spin model to this quenched random N-component continuous spin system according to the method by Luther and Grinstein (5), Emery (7), and in particular, by Aharony et a/. (9).

One defines the function Y(y) by

C2.131 exp ( - Y(y)) = exp ( - J ddx~[y(x)]) - [ 17 ' . j Sm d$j P(+~))] exp ( - J ddxy(x)$(x)) j = 1 \ - m

That is, the function Y(y) is defined by the second and the third expressions, is equal to the fourth due to the normalization of 9($j), and is abbreviated in the last form. Here 9 is assumed to vanish more rapidly than exponentially for large negative values of $ so that the value of Y may be finite. Then the MN-component spin system can be written as

M

where Y - {S,, ..., SM) and M M N

Y 2 = C S j 2 = C C sja2 j j a

The free energy per spin-component of this system in the limit M + 0 is equal to that obtained in [2.11], i.e.,

C2.151 lim F,vN/(MN) = -lim (MN)-' In J daMNs exp (-HA,,) M-o M-o

= - in^ (MN)- 111 [J dPj$p($)~ dbNs exp ( - H(S, $))lM M+O

= FIN

Similar relations hold for the correlation functions. That is, all the features of the randomness are described through the function Y(y) (which is Y(0) = 0 from the norinalization of 9($) and

where the distribution function

and (0),(+, denotes the average of 0 over d($)). Generally, the ith derivative of Y(y) is

[2.16] Y ( i ) ( ~ ) = (- 1li-l < $ i ) c , d ( q )

Can

. J. P

hys.

Dow

nloa

ded

from

ww

w.n

rcre

sear

chpr

ess.

com

by

UN

IVE

RSI

TY

OF

NE

W M

EX

ICO

on

11/2

8/14

For

pers

onal

use

onl

y.

142 C A N . I . PHYS. VOL.. -76. 1978

where ( $ i ) c , , ( + , stands for the ith cumulant over the d is t r ib~~t ion function d($). Expanding Y(y) near y = 0 (i.e., S2 = O), the following relation is obtained:

where gZi - (- 1)'-'/i! ($'),,, and Y(2)(y) 0 for al ly. In the result the MN-component spin Hamiltonian is rewritten as

with ,no2 - ihO2 + 2($),. In the following sections the static critical behaviors in the quenched random systems will be studied in

the vicinity of the second order phase transition point. As the terms with i 2 3 are irrelevant for that phase transition near fo~r r dimensions, those terms will be omitted in the following.

Static critical behaviors in the q~renched random systems may be investigated by the same renormalization procedure for t.egular spin systems with MN-spin cotnponent as that in ref. 11 . In the renormalized pertur- bation expansion, it is sufficient to note the following three points: (i) the number of the spin-component is equal to MN, i.e.,

(ii) The symmetry of the S4-interactions is of

and of

(iii) The limit M -+ 0 is taken in the Feynman-expanded expressions. The main differences from regular systems, for example, anisotropic c ~ ~ b i c systems by the author (12), are only of different N-dependences in the final expressions, which are associated with the stability of our systems. Let us consider the case with or without an external field separately.

3. Static Critical Behaviors in the Absence of an External Field The effective Hamiltonian in the case can be rewritten by using the renormalized quantities (fields S ,

coupling constants g, < 0, g, > 0, and mass n12) and the renormalization constants (Z,,, Z,,, and Z , for coupling constants g,, g,, and field), as

The last three terms on the right-hand side are the counter terms, which are determined from the following normalization conditions for the two- and four-point one-particle irreducible vertex functions:

By expanding the two- and four-point vertex functions in a power series of g, and g, according to the Feyn- man r ~ ~ l e and taking the norn~alization conditions [3.2] into consideration, the renormalization constants Z,, (w = s, c) and Z, in the limit M -+ 0 are obtained in the Appendix, where the interactions g,, g, are

Can

. J. P

hys.

Dow

nloa

ded

from

ww

w.n

rcre

sear

chpr

ess.

com

by

UN

IVE

RSI

TY

OF

NE

W M

EX

ICO

on

11/2

8/14

For

pers

onal

use

onl

y.

replaced as 11, - -8gs and u, = g, in that limit so as to be convenient to conipare with the results by Luben- sky.

The Callan-Symanzik equations for the amputated one-particle irreducible vertex fi~nctions r("'({p), us, u,, p) and r(L"r)(q; { p ) ; us, L/,, p) in the Hamiltonia~i system [3.1] agree with those for the anistropic cubic systems formally and are given as

where the coefficient f~~nc t ions p, = - ~ [ a In ( L / ~ Z , ~ / Z , ~ ) / ~ L / ~ ] - ~ and p, = - ~ [ a In ( L / , Z , , / Z , ~ ) ~ L / , ] - ~ can be determined to order uSPucq~" with p + q + r I 3, in the Appendix, Sect. 11. The yj ( j = 3, 4 ) functions may be calculated by the relations

L3.41 yj(llS, I/ , ) = P,(U,, L I , ) ~ I l l z j ( l i s , L / , ) / ~ L / , for j = 3, 4 \v=s.c

where Z , is giveii in [A2] . The renormalization constant Z , associated with the vertex fi~nctions with inser- tion of a coniposite operator S 2 is obtained by the Feynnian rule and the normalization condition [3.2] for r(lp2) in the Appendix, Sect. 11. The N-dependences in these coefficient functions P , (w = s, c) and y j ( j = 3, 4 ) are different froin those in the anisotropic cubic systems.

The zeros of the fi~nctions P,(L/,, 11,) = 0 for w = s, c give the fixed point interactions L/,, and u,,, which are determined in the Appendix, Sect. 111.

The eigenvalues for the matrix defined by B,,. - a~,/auwrI, ,=,, , are obtained by using the linearized relations [A41 about their fixed points as

( i ) Gaussian: hsG = kcG = - E

( i i ) Heisenberg: h," = E(N - 4) / (N + 8) + E ~ ( N + 2)(13N + 44)(N + 8) - , + o(E,) 13.51 hcH = E - E ~ ( ~ N + 42)(N + 8) -2 + O(E,)

(iii) uiiphysical: IsU = E - c221/32 + O ( E ~ ) , kcU = ~ / 2 - ~ ~ 1 9 1 6 4 + O(E,)

(iu) random: h I R = E + ~ ~ ( 3 6 5 ~ ~ - 1 4 8 8 ~ ~ + 2112N - 256)/[128(8 - 5 N ) ( N - + 0 ( E 3 )

These results ( h + -1) in case of ( i ) and ( i i ) coincide with the results obtained by other authors. The stable fixed point (u,,, u,,) is for the positive eigenvalues.

The other coefficient f~~nc t ions y, and y, in [3.3] are obtained by [3.4] as for (iii) ~~nphysica l :

and for (iv) random:

From these results the main critical exponents can be derived by the relations q = y,(u,,, urn,), y = ( 2 - q ) / [ 2 - q + y4(~irns, 21,,)], and ci = 2 - rlyl(2 - q ) (d = dimension of space) as

(ii) Heisenberg:

q H = E2(N + 2) /[2(N + 8)2] + E,(N + 2 ) ( - N 2 + 56N + 272)/[8(N + + O(E,)

yH = 1 + E(N + 2) /[2(N + 8)] + E ~ ( N + 2) (N2 + 22N + 52)/[4(N + 8),] + O(E,)

(iii) unphysical:

qu = ~ ~ 1 6 4 + E317/1024 + 0(E4)

P.71 yU = 1 + E/8 + ~ ~ 1 3 1 2 5 6 + O ( E ~ )

clU = E/4 + ~ ~ 7 1 1 2 8 + 0(E3)

Can

. J. P

hys.

Dow

nloa

ded

from

ww

w.n

rcre

sear

chpr

ess.

com

by

UN

IVE

RSI

TY

OF

NE

W M

EX

ICO

on

11/2

8/14

For

pers

onal

use

onl

y.

C A N . J . PI-IYS. VOI.. -56. 1978

(io) random

The critical exponents in case ( i i ) were firstly derived by Wilson. In case (iii) the lowest order terms in y3(t/m,, 0), y4(tim,, 0) , q , y, andci coincide with the results obtained by Lubensky (8). In the random system the terms up to order E' coincide with his results and the term of order c3 is the new correction. The crossover exponent 4 t o the random fixed point is = ci" since -hSF' is e q ~ ~ a l to ciH/v" as Lubensky (8) has pointed out. Thus the Heisenberg fixed point is stable as long as h," > 0 and hcll > 0 , i.e., ci" < 0. It becomes unstable at a critical value N, of the spin component N determined by ciF'(Nc) = 0 , which is N, = 4 ill order E

(i.e., the randonl fixed point with N = 2 and 3 in E > 0 is stable). Let us discuss the stability of the fixed points with positive t/,,, u,, of order c2. The Gaussian fixed point is

stable for E < 0 and unstable for E > 0. The Heisenberg fixed point is stable for the dimensions satisfying 0 < E < ( N + Q 2 / ( 9 ~ + 42) for N > 4 or (4 - N ) ( N + 8)2 / [ (N + 2)(13N + 44)] < E < ( N + 8 ) 2 / ( 9 ~ + 42) for 0.8456 < N < 4. For example, 0 < E < 1.943 for N = 5, 0 < E < 2.042 for N = 6, ... . The unphysical fixed point is always stable for the dimension 0 < E < 1.524 independent of N. The random fixed point is stable for ILsR > 0 and kcR > 0. That is, p ~ ~ t t i n g

y = - . - [365N3 - 1488N2 + 2112N - 256]/[128(8 - 5N)(N - and

y = N[35N3 - 1932N2 + 3840N - 512]/[128(8 - 5N)(N - 1)2(4 - N ) ]

the stable regions of the random fixed point are as follows: E > y-' for N,, < N < 1 , 0 < E < y-' for 1 < N < 1.6 or N,, < N < 4, 0 < E < x- ' for 1.6 < N < N,,, and none otherwise, where N,, and N,, are the values satisfying s = y , and 0.1 < N,, < 0.2 and 2.1 < N,? < 2.2. Note that there exists a stable random fixed point for N < 1 t o order z2. For example, 0 < E < 0.274 for N = 2 and 0 < E < 0.220 for N = 3.

Let LIS summarize the stability of the fixed points. There are two stable fixed points, the unphysical and Heisenberg fixed points, for N > N,. For N,, < N < 1 or 1 < N < N, the ~~nphysical and random fixed points are stable. In the region N < N,, there is a stable ~~nphysical fixed point and the effective interactions ti,,, 1/,, for the random system become negative, i.e., unphysical. The results obtained for the random system are applicable to the regions where the stable fixed point exists.

For the typical cases cl = 3 and N = 3 (2) the numerical values are calculated as follows: q R = 0.042 (0.095), yR = 1.160 (1.045), ciR = 0.300 (1.242), AIR = 0.291 (-2.656), = -0.444 (-0.047). On the other hand the Heisenberg system has the following values for their exponents; q H = 0.039 (0.039), yF' = 1.347 (1.300), ci" = -0.100 (-0.020). In these typical cases the random system is unstable but can be conveniently compared with the Heisenberg system because the critical exponents for the random system in the stable region almost coincide with those for the Heisenberg system within the inagni t~~de of the error bar due to experiments. The values of q , y, and ci are slightly different between the quenched random and Heisen- berg systems.

In order to establish the stable region for the fixed points the next order corrections would be required from these facts.

4. Static Critical Behaviors in the Presence of an External Field Let us now consider the case in a nlagnetic field 11 with g,, g, > 0 , in a way similar to that used to discuss

the equation of state in the second order phase transition by Brezin et a/. (13), Aharony (14), Wallace ( 1 5 ) , and Yamazaki (I 1) :

where a - m2 - Z,I?I,~, b = Z 3 - 1 , C, = ( 1 - Z , ,)g, for w = S, C. The easy axes in the system are in the

Can

. J. P

hys.

Dow

nloa

ded

from

ww

w.n

rcre

sear

chpr

ess.

com

by

UN

IVE

RSI

TY

OF

NE

W M

EX

ICO

on

11/2

8/14

For

pers

onal

use

onl

y.

directions that minimize the additional terms ,v

where

i.e., S12 = S2' = ... = sM2 3 CM2 (constant), which means that all the spin components S,, (1 I cr I N ) constructing any S, satisfy the relation

and may be continuously symmetric-variable except for the component along the applied field. That is, the easy axes are the diagonals ( I , , ..., in the S,-space (1 I j I M). Let us choose the magnetic field in the direction of an easy axis h = elu = M-'I2 (I1, ..., lM) and assume that the direction of the inagneti- zation fi is along that of the field, M = f ie,u. In order to subtract the spontaneoi~sly broken-symmetric contribution of magnetization in this direction and symmetrize the theory, one must separate the spin components into the longitudinal and transverse components, Sll and S, as S = (SII + iQ)e, + S, where the vector S, consists of the hf - 1 orthonormal unit vectors perpendicular to h, e j ( j = 1, ..., M - 1) in the Sj-space (each of e, is the sum of N orthonorinal unit vectors perpendicular to h, ej, (a = 1, ..., N) in the Sj,-space), i.e.,

M - 1 M

S,= x S,,ej, x e T = O ( j = 1 , ..., M - 1 ) j= 1 v = 1

By introducing these spin coordinates, the Hamiltonian of the system is rewritten as

HAfN, - J ddx[[{rn2 - + [g, - c, + ehf2(gC - cc)]A2/3!)iil - II]S,

+ bC(VS11)2 + (VS1)21/2 + [t7z2 - a - niIl2 + {g, - C, + eM2(gC - ~ , ) ) i i i ~ / 2 ] ~ ~ , ~ / 2 2 + [m2 - fl - HI, + {g, - c, + eM2(gC - ~ , ) ) h ? ~ / 3 ! ] ~ , ~ / 2 + (g, - c ~ ) ~ ~ s ~ ~ ( S ~ ~ ~ + SL2)/3!

M - 1 k M - 1 I

i k p = i i j k , 1 p=r

where

The counter terms are determined by the normalization conditions

That is, the longitudinal and transverse Inass terms, 111~1 and mL2, are defined by the corresponding inverse susceptibilities as

Can

. J. P

hys.

Dow

nloa

ded

from

ww

w.n

rcre

sear

chpr

ess.

com

by

UN

IVE

RSI

TY

OF

NE

W M

EX

ICO

on

11/2

8/14

For

pers

onal

use

onl

y.

146 C A N . J . PHYS. VOL. 56. 1978

and obtained of order unity as

1)711 = 1)12 + (g, + eM2&'=)fi2/2 + O(E) L4.51 mL2 = m 2 + (g , + 3eM2gC) f i2 /3 ! + O(E)

The equation of state can be derived from the relation T ( ' ) = 0 , i.e.,

P.61 ( S I I ( X > > = 0 , (SL(x )> = 0

The second relation holds valid due to the orthonormality of the basis vectors e j . From the first relation the equation of state of order E is obtained to be

[4.7] h l f i = rn2 - a ( ' ) + [g, - c,") + eM2(,yC - c , ( " ) ] f i2 /3 ! + ( 2 ~ ~ - ~ ~ d ~ l r { g , [ 3 G ~ ~ ( / r ) + ( M N - 1 )

x G,(k)]/3! + eM2gC[Gll(lr) + ( M N - l )G,(k)]/2) + O ( E ~ )

where Gll(lc) and GL(/r) stand for the longitudinal and transverse propagators, respectively, and a ( ' ) , c,('), and c,(') the corresponding counter terms n, c,, and c, of order E. From the identical relation rnl12 = (ah/ aa),, the longitudinal mass term is related to the magnetic field as

[4.81 1 7 7 1 1 ~ = / ~ / f i + [g, - c,(') + eM2(gc - eC( ' ) )1 f i2 /3 + O(E)

which is finite at T < T, and / I = 0. On the other hand, the renormalization-group equation for the vertex function T ( " ) is described as

[4.91 [ W ~ P + ~,a /ag , + ijCa/agc - ~ , ~ ~ / ~ I ~ ( " ) ( { P } ; g,, g,, P) = 0

At the criticality the interactions g, and g, must be replaced by the effective interactions g,, and g,, satisfying ij, (g,,, g,,) = 0 = PC (g,,, g,,). In the lowest order, g,, and g,, are obtained by using the relations g,, E

bg,, for w = s, c, which are

As one considers the case g,, g, > 0 only, the case 1 < N < 4 in this order will be discussed in the following. In the limit M -+ 0 the following relations hold: eIw2 = 1 and M N = 0 in [4.5]-[4.7]. In the result the equation of state can be described as

[4.111 l7lfi = t + (g,, + g,,)fi2/3! + 4-'(2,, + Em,)[[ + (g,, + g,,)fi2/21 In [t + (g,, + g,,)

x f i2/21 - 12-'(2,, + 32,,>[t + (g,, + 3g,,)fi2/3!1 In [t + (g,, + k , , ) f i 2 /3 !1 + O(E2)

where t E 117' - 1 1 7 , ~ ( 1 1 7 , ~ = critical mass). Adopting the standard normalizations for the equation of state: /7/A% f ( x ) = 1 at the critical isotherm t = 0 and t / f i l ' k x = - 1 a t the coexistence curve h = 0 , T < T,, the following scaling form for the equation of state is obtained:

[4.12] f ( x ) = x + 1 + ~[32(Ar - I ) ] - ' { [3 (8 - hl) (x + 3) In (s + 3) - (16 - N ) ( x + (16 - N)/(8 - N ) )

x In ( X + (16 - N)/(8 - N ) ) ] - [9(8 - N ) In 3 - (16 - N)2/(8 - N ) In ((16 - N) / (8 - N ) ) ]

x ( X + 1 ) + [6(8 - N ) In 2 - 8(16 - N)/(8 - N ) In (8/(8 - N ) ) ] x ) + O ( E ~ )

where F = 3 + E + O ( E ~ ) and p- ' = 2 + E ( ~ N - is finite, i.e., the longitudinal susceptibility is finite a t 8) /[8(N - I ) ] + O ( E ~ ) . AS f ( x ) must be positive, zero field below T,. For the typical cases N = 3 ( 2 ) N must be 1 < N < 4 on the right-hand side of the the numerical values are calculated as follows: equation. This situation is associated with the sta- f (5) = 6 + ~ 0 . 0 4 4 (0.091), f (4 ) = 5 + ~ 0 . 0 2 4 (0.051), bility of the random fixed point. In this region f '(- 1) f (3 ) = 4 + ~0 .010 (0.021), f (2 ) = 3 + ~ 0 . 0 0 2 (0.004),

Can

. J. P

hys.

Dow

nloa

ded

from

ww

w.n

rcre

sear

chpr

ess.

com

by

UN

IVE

RSI

TY

OF

NE

W M

EX

ICO

on

11/2

8/14

For

pers

onal

use

onl

y.

f(1) = 2 - ~0.001 (0.003), f ( 0 ) = 1 + EO ( O ) , a n d f ( - 1) = EO ( O ) , respectively. O n the other hand, fo r the Heisenberg system with N = 3 (2) , f (5) = 6 + 61.494 (l . l06), f (4) = 5 + ~1.098 (0.805), f (3) = 4 + ~0.739 (0.536), f (2) = 3 + ~0.427 (0.305), f(1) = 2 + ~0.172 (0.120), f(0) = 1 + EO (O), and f ( - 1 ) = EO ( O ) , respectively. These results show tha t the stronger field is requlred with the larger N t o keep the same value o f magnetization in the Heisenberg system, bu t in the quenched random system this s i t ~ ~ a t i o n is inverse because the dominant terms for N large d r o p o u t a n d N = 1 becomes s i n g ~ ~ l a r in the limit M + 0.

5. Concluding Remarks Critical behaviors in quenched random N-corn-

ponent spin systems have been studied in the MN- component spin systems by means of the renormalization group theory. T h e stability of the fixed points of order E~ has been studied. F o r N > N, (satisfying a H ( N c ) = 0) the stable fixed points a re of the unphysical a n d Heisenberg fixed points. T h e unphysical and random fixed points a r e stable for Nc, < N < 1 o r 1 < N < N,. F o r N < N c , there is n o stable a n d physical fixed point. I t should b e noted tha t the random-spin systems even with N = 2 o r 3 a r e unstable in the three dimensions contrary t o the Lubensky conclusion (8) u p t o order E . T h e .c3 correction t o the critical exponent q and the equation of state have been derived. These expressions have a singularity for N = 1 .

Finally let us comment a b o u t the suspicion based o n the replica trick, M + 0, which may involve a basically incorrect interchange of limits, i.e., the limit M + 0 a n d the thernlodynanlic limit # + a. T h e replica trick, M + 0, has recently led t o unphysical results in models o f spin glasses (16). O n the contrary, the replica method derives physical results in the present quenched random model. T h e reasons are as follows: (I) we use n o t the steepest descent method, which gives the restriction M 2 2,

b u t the Gaussian integral, which does n o t give M any restriction, since the interaction J i j in [2.2] is positive constant, i.e., since the integral over J i j is no t included; (2) consequently the results obtained i n the present article coincide with those derived by Lubensky (8) without using the replica trick.

Acknowledgements T h e au thor would like t o thank Professors S.

Katsura, M. Suzuki, a n d S. Inawashiro for their useful discussions. T h e au thor is particularly grate- ful t o Professors A. Aharony and G . Grinstein fo r stimulating correspondence o n the earlier version of this work. Particularly thanks a re d u e t o Professors A. Aharony, G. Grinstein, a n d D. J. Wallace, f o r sending their preprints t o the author .

I . K. G. WILSON and J . KOGUT. Phys. Rep. 12C, 75 (1974). 2. S. M A . Rev. Mod. Phys. 15,589 (1973). 3. M. E. F ISHER. Rev. Mod. Phys. 46,597 (1974). 4. G. GRlNSrElN. Ph.D. Thesis. Harvard University, Cam-

bridge, MA. 1974. 5. A . LUTHER and G. GRINS.I.EIN. 1973. AIP Conf. Proc. 24,

313 (1974). 6. A . AHARONY. Phase transitions and critical phenomena 6.

Gli/c,tl/).~ C. Domband M . S. Green. Academic Press, New York, NY. 1976.

7. V. EMERY. Phys. Rev. B. 11,239 (1975). 8. T. C. LUBENSKY. Phys. Rev. B. 11,3573 (1975). 9. A . A H A R O N Y , Y. I IVIRY, and S. M A . Phys. Rev. B, 13,466

(1976). 10. A. AHARONY. Phys. Rev. B, 12, 1038, (1975): 12. 1049

(1975). 1 1 . Y. YAMAZAKI. Prog. Theor. Phys. 12. Y. Y A ~ I A Z A K I . Prog. Theor. Phys. 55, 1093 (1976):55, 1733

(1976). \~ - , ~

13. E. B K E Z I N , D. J . WALL ACE.^^^ K. G. WILSON. Phys. Rev. B, 7.232 (1973).

14. A. AHARONY. Phys Rev. B, 10.3006 (1974). 15. D. J . WALLACE. Phase transitions and critical phenomena

6. Glirctl /)y C. Domb and M. S. Green. Academic Press, New Yol-k. NY. 1976.

16. D. SHERRINCTON and S. KIRKP,\.I-KICK. Phys. Rev. Lett. 35, 1792 (1975).

Appendix : Lists of Expressions [ I ] Renormalizatioi7 Constants Z , , and Z 3 and Co~~tributioiz of Feytzinaiz Diag~ztns A , B, C , and D

T h e renormalization constant Z , , (w = s,c) in the limit M + 0 are obtained as

Z , , = 1 + Z , w ( 2 ) + Z1,v33) + 0 ( u 3 )

[A1 I Z 1 w ( 2 ) - ~ , , ( ~ ) ( 1 , 0)ks + Z , ~ ( ~ ) ( O , I ) &

Z l w ( 3 ) = z1w(3)(2, O)kS2 + Z ~ , ( ~ ) ( I , l)nsnc + Z ~ , ( ~ ) ( O , 2)kc2, fo r w = s or c where

~ ~ , , , ( ~ ) ( 1 , 0) = - A/6, -A14

~ , , ( ~ ) ( 0 , 1 ) = ( N + 2)A/3, ( N + 8)A/6

~ ~ ~ ( ~ ~ ( 2 , 0 ) 3 A2/36 - 1 I B/288, 5A2/96 - 7B/96

~ l , ' ~ ' ( l , 1 ) = - ( N + 2)A2/8 + ( N + 2)B/6, -(5N + 28)A2/48 + ( N + 5)B/6

Can

. J. P

hys.

Dow

nloa

ded

from

ww

w.n

rcre

sear

chpr

ess.

com

by

UN

IVE

RSI

TY

OF

NE

W M

EX

ICO

on

11/2

8/14

For

pers

onal

use

onl

y.

148 C A N . J . I'HYS. V O L . 56. 1978

and Z ~ ~ ( " ( O , 2) = ( N + 2)(N + 4 ) ~ ~ / 1 2 - ( N + 2)B/3, ( N + 8 ) 2 ~ 2 / 3 6 - (5N + 22)B/9

for w = s,c, respectively. For Z , the following relation holds:

Z , = I + z , ( ~ ) + z3(,) + 0 ( u 4 )

[A2 1 Z , ( ' ) = ~ ~ ( ~ ~ ( 2 , O)ir,' + ~ ~ ( ~ ~ ( 1 , l)li,ti, + z , ( ~ ) ( O , 2)lic2

z , ( ~ ) = z3(,)(3, 0)lis3 + z3(,)(2, l)liS2uc + ~ ~ ( ~ ~ ( 1 , 2)li,lic2 + z3(,)(0, 3)irC3 where

~ ~ ' ~ ' ( 2 , 0 ) = C/576, ~ ~ ( ~ ~ ( 1 , 1 ) - - ( N + 2)C/72

z , ( ~ ) ( ~ , O ) = ( N + 2)C/18, ~ , ( ~ ) ( 3 , 0 ) = - 011728

~ ~ ( ~ ' ( 2 , 1 ) = ( N + 2)D/144, ~ ~ ( ~ ~ ( 1 , 2) = - ( N + 2)(N + 8)D/144

~ ~ ( ~ ~ ( 0 , 3) = ( N + 2)(N + 8)D/54 Here

ir, = 511, for w = s, c ( 3 - 27~" /~ / { r (d /2 ) (27~)"} )

The constants A , B, C , and D in [A31 stand for the contribution o f Feynman diagrams expanded in powers o f E ( = d - 4) as

[A31 A E (27~)-"Jd"k(k2 + 1 ) - 2 / 3 = ( I + E / ~ ) / E + O(E)

B = ( 2 7 ~ ) - ~ " J d1'k1 ddk2 { ( k I 2 + ~ ) ~ ( k , ~ + I ) [ ( / ( , + 1 ~ ~ ) ~ + 1 ] ) - 1 / 5 2 - A2/2 = 1 / ( 4 ~ ) + O(1)

C = ( 2 7 ~ ) - ~ " c / / d ~ ' d"k1 d"k2 { ( k , + 1)(1(22 + l)[(p + k 1 + k 2 ) 2 + 1 ] ) - ' 1 , ~ = , , 2 / 3 ~

= - [ l + ~ ( 5 1 4 - 1 ) ] / ( 8 ~ ) + o(1) D = A B - 1 / 2 ( 2 7 ~ ) - ~ " d / d ~ > ~ J d ~ k ~ [ ( p + + l ] - 1 { ~ d " l c 2 ( k 2 2 + I ) - ' [ ( k , + k2)2 + 1 ] - 1 ) 2 1 p ~ = L l ~ / 5 3

= - ( I + ~ 5 / 4 ) / ( 2 4 ~ ~ ) + 0(1)

[ I I j The Coeflcient Filnctions P,.,(w = s, c ) nnd the Renorninlizntion Corislar~l Z ,

[A41 P,"(u,, 11,) = -~l i , { l - [ l i , ~ ~ , ( ~ ) ( l , 0 ) + ~ ? , z ~ , ( ~ ) ( 0 , l ) ] - [ L ? , ~ Z ~ , ~ ( ~ ) ( ~ , 0 ) + irSilC ~ ~ ~ ( ~ ' ( 1 , 1 )

+ nc2 ~~, \ , (3)(0 , 211 + o ( ~ 4 ) where

Z ~ , ( ~ ) ( I , 0 ) - - A/6, - A/4

zp,(')(O, 1 ) -- ( N + 2)A/3, ( N + 8)A/6

~ ~ ~ ( " ( 2 , 0 ) - l lB/144 - C/144, -7B/48 - C/144

~ ~ ~ ( ~ ' ( 1 , 1 ) ( N + 2)B/3 + (Ar + 2)C/18, ( N + 5)B/3 + ( N + 2)C/18

z , , ( ~ ) ( O , 2 ) 3 -2(N + 2)B/3 - 2(N + 2)C/9, -2(5N + 22)B/9 - 2(N + 2)C/9

for w = s and c, respectively.

[A51 Z4-' = 1 + A/24[li, - 4(N + 2)i?,] + ( B - A2/2)/192 [irs2 - 8(N + 2)ii,lic + 32(N + 2)lic2] + O(i13)

[III] The Fixer1 Poinr Inieraclions u,, crrlcj ti,,

The zeros u,, and u,, o f order E o f these functions P,(u,, 11,) = 0 for w = s, c are determined as

( i ) Gaussian: c,,~ = ilmcG = 0

( i i ) Heisenberg: ir,," = 0, li,," = E ~ ( N + 8)-' [ 1 + ~ { 3 ( 3 N + 14)(N + 8)-2 - 1/21] + O(E,)

[A61 (iii) unphysical : ir,," = - ~ 6 [ 1 + ~(21132 - 1/2)] + O ( E ~ ) , l imCU = 0

(iu) random: iimsR = E ~ ( N - 1) - ' [ (4 - N ) - ~ ( ( 4 - N ) / 2 + (105N3 - 364N2

+ 992N - 256)/128(N - + 0(E3)

G m C R = ~ 3 / [ 2 ( N - 1 ) ] [ 1 + & { - 112 + (25N2 - 248N + 64)/128(N - + 0(E3)

Can

. J. P

hys.

Dow

nloa

ded

from

ww

w.n

rcre

sear

chpr

ess.

com

by

UN

IVE

RSI

TY

OF

NE

W M

EX

ICO

on

11/2

8/14

For

pers

onal

use

onl

y.

static critical behaviors in random-spin systems with short-range interaction

Documents