finite-size scaling of o(n) models in higher dimensions

Volume 118, number 3 PHYSICS LETTERS A 6 October 1986

FINITE-SIZE SCALING OF O(n) MODELS IN HIGHER DIMENSIONS

Surjit S I N G H and R.K. P A T H R I A

Department of Physics, University of Waterloo, Waterloo, Ontario, Canada N2L 3GI

Received 11 July 1986; accepted for publication 5 August 1986

We propose a finite-size scaling hypothesis for O(n) models, with n >~ 2, in geometry L d - d ' X o0 a', with d > 4 and d ' ~< 2, subject to periodic boundary conditions. Several predictions, for T< ~ as well as T= T~, are made and are verified analytically for the special case of the spherical model (n = ~).

Finite-size scaling theory for the critical region, first introduced by Fisher [1] in 1970 and subse- quently elaborated by a number of authors (for a detailed review see ref. [2]), has been based on the fact that the physical behavior of a finite-sized system in d dimensions, where d is less than the upper critical dimensionali ty d > , in the vicinity of the bulk critical temperature T¢ is governed by a single scaled length L / ~ ( T ) , where L is a length characterizing the size of the system while ~(T) is the correlation length. For d > d > , however, this theory needs modificat ion - primarily because of the breakdown of hyperscaling which, in turn, is due to the presence of a dangerous irrelevant variable (DIV) in the problem [3,4]. With this in view, several authors in recent years have put forward modified hypotheses for application to d > d> and have explored their consequences for different geometries such as the block and the cylinder [4-9].

In this letter we present a finite-size scaling hypothesis for O(n) models, with n >i 2, in a gen- eral geometry L d d ' x O0 d', with d > 4 (which is the upper critical dimensionali ty in this case) and d ' ~< 2 (which is the lower critical dimensionality); the boundary condit ions imposed on the system are assumed to be periodic. In the region of the first-order phase transition ( T < T~), the new hypothesis yields results which are qualitatively the same as the ones pertaining to the case 2 < d < 4. In the region of the second-order phase transition

(T- -T¢) , however, the results following from the new hypothesis turn out to be strikingly different f rom the ones for 2 ~< d < 4. Notwiths tanding this, we find that in both these regions, T < T~ as well as T- -T¢ , the system in zero field can be de- scribed by a single, uni fy ing variable z, as given by eq. (21) in conjunct ion with eqs. (8). The emergence of the variable z enables us to make several new predictions in the region T - - - T c - most notably, eq (22) for the zero-field susceptibility, X0, and eq. (26) for the correlation length, ~, of the given (finite) system at T = T~. It is gratify- ing that these, and all other, predictions emanat- ing from the new hypothesis are analytically verified for the special case of the spherical model (n = oo). For the latter model, some results pertaining to the borderline case d = 4 are also in- cluded.

In earlier work [10,11], it was postulated that, for 2 < d < 4, the singular part of the free-energy density of a system with O(n) symmetry (n >~ 2), confined to geometry L d - d ' > ( O0 d' and subject to periodic boundary conditions, may be expressed in the form

f"'(T, H; L)= TL-"Y(x,, x2), (1) where

x , = r i l L ' / " i , x 2 = C : L ~ / " I 4 / r , (2)

(~1 and (~2 are nonuniversal, system-dependent

0375-9601/86 /$03 .50 © Elsevier Science Publishers B.V. (Nor th-Hol land Physics Publishing Division)

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Volume It8, number 3 PHYSICS LETTERS A 6 October 1986

scale factors, /" is an appropriate temperature variable (such that, for T-= T~, 7~ t = ( T - T~)/T~) while other symbols have their usual meanings; the corresponding hypothesis for the correlation function is [12]

G ( R , T, /-/; 2 "

(3)

where a is a microscopic length (such as the lattice constant) while ~5 denotes the correlation length (of the finite-sized system) which obeys the sub- sidiary hypothesis

~x(T, H; L ) = L S ( x , , x2). (4)

For d > 4, on the other hand, we propose a new hypothesis, viz.

f ( ° ( T , H; L ) = TL dy(Zl, z 2, z3), (5)

along with

G(R, T, H; L)"~ aZd(A22/A3)R2-d

X,~( R/l~, L/~, z2, z3) ( 6 )

and

~(T, H; L).~ LS(z , , z2, z3) , (V)

where

z 1 = A1L27, Z 2 = A2L3H/T, z 3 = A3L-'* ; (8)

here, A 1, A 2 and A 3 a r e nonuniversal, system-dependent scale factors, ¢0" ( = d - 4) is the so-called anomalous dimension [3], while use has been made of the mean field exponents ~,= ½, A = 3 2 and

= 0. The amplitudes A, in the vicinity of the bulk critical point can be determined, as usual, from a knowledge of three independent pieces of information on the bulk system, such as the susceptibility, the specific heat and the correlation length, for T>_ T~; for ascertaining full temperature dependence of 7 and A 2 , for T~< T~, one must instead follow a procedure similar to the one laid down in refs. [12] and [13]. Quite expectedly, these amplitudes will depend not only on the universal parameters n and d but also on the nonuniversal aspects of the system studied. The

stage is now set for making predictions for d > 4. For simplicity, we start with the scaling for-

mula for the zero-field susceptibility of the system, which follows readily from hypothesis (5), namely

= L -,~" Xo (A2LZ/A3T)Z(A, L27, A3 ). (9)

Concentrating first on the regime [ < 0, we wonder if the function 2 can be expressed in terms of a single argument in the variables L and [, rather than two as in (9). To see this, we invert eq. (9) to write

]71 =(A~/A1A3TXo).~-(A2L2/A3TXo, A3L '~*) (lo)

and note that, for L ~ ¢¢, since both arguments of the function ~ (z~ , z3) tend to zero, we may invoke a limiting form such as

g ( z ¢ , z ~ ) - z F z ~ (z¢, z3-~O), (11)

where r and s are as yet unknown. Eq. (10) thereby yields

X o - ( A~/A3T)[ A( 'A;L2r-'~*'[[I , ] , /(r+, , .

(12)

Now, making use of the temperature dependence of A 2 and 7, examining the limit T---, 0 and utiliz- ing some known results [5] for geometries d ' = 0 and 1, we conclude that r = - (5 , + 1)/~,, where 5' is the critical exponent for the susceptibility of the d '-dimensional bulk system with d ' < 2, while s = 1. It then follows that

X o - ( AI/A3) ~'( A2/A3T ) L¢ [ [[ +, (13)

where

~ ' = 2 + ( d - Z ) ~ , = Z ( d - d ' ) / ( Z - d ' ) ( d ' < 2 ) , (14)

since ~ , = 2 / ( 2 - d ' ) . Thus, for d > 4 the "approach exponent" ~" in the regime t < 0 turns out to be the same as for 2 < d < 4 [11,13].

Expressing (13) in the form (9), we find that

Xo - (A2LZ/A3T)[(A1/A3)La-217[]~, (15)

which shows that, in the region of the first-order phase transition, the function Z can indeed be

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Volume 118. number 3 PHYSICS LETTERS A 6 October 1986

expressed in terms of a single argument in the variables L and 7, viz. 1 z, l/z3. We also note that in this regime a direct correspondence exists be- tween cases 2 < d < 4 and d > 4, which becomes even more vivid if the final result for x0 is written

in terms of the spontaneous magnetization M,(T) and the helicity modulus ‘l’(T) of the bulk system,

viz.

x0 - [LdM,:(T)/aZdT] [ L+v(T)/T] 9-1.

(16)

It can be shown that, in terms of M,(T) and Z’(T), the corresponding expression for x0, for 2 < d c 4, is precisely the same as (16). In passing, we observe that, for d’ = 2, the approach of x0 toward standard bulk behavior in this regime would be exponential in nature.

If one (naively) assumes that the variable z,/z3 (= (A,/,43)Ld-2r) governs the situation in the

“core” region (r= 0) as well, one is led to infer

that (x0), - L2, which is known to be untrue [4,6,9]. This means that for the “core” region

z,/z3 is not the right variable. To determine a variable that governs the behavior of the system in the region i = 0 as well as the regime I -Z 0, we re-write (9) in the form

X0” (A:L”/A,T)(A,L’lil)hZ(A,L2i, A,L_“I)

(17)

where b is as yet unknown. To recover (15) from (17), we must have:

Z( Zi, zj)- (z,(+-h/z; (Zi-, -cc, z3+O).

(18)

This suggests a new variable, z = L,/,z~/(~-~), in terms of which one may write, for i= 0 as well as i<O,

x0 = (4L2/A3T>(4L2 ljfX4. For I il + 0, one would naturally

f(z)- lZl-h> with the result that

( xo)c - (Ai LZ/A,T)cZy(+-?

09)

require that

(20)

Comparing (20) with known results [4,6,9] for d’ = 0 and 1, we infer that b = - 1. Eqs. (19) and

(20) then become

x0 = (-&‘G%~) I iI -‘.fW (21)

( z +zI/,y(++~) _ ~*(d-d')/(4-0~

1

and

( xo)c - (&AJ), [ A,T2+‘d-2’q t’(f+‘! (22)

It follows that

(x0)= a L 2(&d')/(4-d')

(d'<%

a Ld- ‘/ln L (d’= 2). (23)

The corresponding result for the correlation length is

+/t;“*]tl]-“*+(z),

whence

(24)

[(i<O) - L[(AI/A,)L~-2JTl]~‘2,

while

(25)

[( j=O)_ [A;~L*+(d-2)~]'/"~+*'.

(26)

Thus, c(j < 0) a L (d-d’)/(2-d’) for d’ < 2 but

grows exponentially with Ld-* for d’ = 2. On the other hand, [(j= 0) a L’d-d’)/(4-d’) for d’ < 2 and a [Ld-*/ln L] 1/2 for d’ = 2. A closer look at the foregoing results shows that, in fact, f(z) -

[+(z)]~ and, hence, x0 - (A:/A3t)t2. To verify these predictions we have analyzed

the special case of the spherical model in geometry Ldwd’ x cod', with d > 4, under periodic boundary conditions. Following standard procedure [ll-141,

we first determined the three nonuniversal scale factors A,, on the basis of known bulk properties of the system [15] and found them to be

A, = K,a-*w-i, A, = K-‘/24-$,-‘/2,

A = ad-4,,-1 (27) 3 9

where K = J/k,T, K, = 4 W,(O) while w = (dW,(+)/d+ I 9so, W,(+> being the familiar Wat- son integral [16]. Next, we showed that the free energy density of the finite-sized system in zero field does indeed conform to the prop_osed hypothesis (5), while the scaling function Y is given

133


by the parametric equations

z3 ~ [a-'F(½(4- d))

- J l ( ½ d Id* ; y )

- X ( ½ ( d - 2 ) ld*" Y)]

and

(28)

Y~ 2z' [c(~(2 - a)) z 1 = 2y 2 8,Trd,/2

+ 2 J ' U ( ½ ( d - 2 ) ] d * ; y ) ] . (29)

Here, d* = d - d ' while other symbols have the same meanings as in ref. [14]. The variable y(z~, z3), in particular, denotes a scale length parameter of the system, viz. y = L/2(; accord- ingly, the scaling function for the correlation length is simply

S (z , , z3) = 1 / 2 y , (30)

and for the susceptibility, see eq. (9),

ZZ(z,, z3) = 1 / 8 y 2 = ½[S(z, , z3)] 2. (31)

A detailed study of the various regimes now re- veals the following results.

For [ > 0, finite-size corrections to bulk properties turn out to be O(e -*/*(~)) if ?>> (alL)2; for t '= O(a/L) 2, however, corrections are found to be O(L-'~*). Next, over a broad range of tempera- tures (which includes the regime 7< 0 as well as the region [ = 0), eq. (29) reduces to

• V(½(2- d ' ) ) z, = 0, (32) y4 d' 12Zly2 d 16"rr u'/2

where d ' < 2. Explicit solutions of eq, (32) can be written down for d ' - - 0 and 1:

L (e-- +zl) 1/2 Y= 2~ =½ z21+z3

=~-~'z'lsinh{~sinh-'[16,3v~Z3z113/2 ]}

( d ' = l , z l ~ O )-

(33)

More generally

>,= L / 2 ~ = I z , 1 ' /2g( z ) (34)

Z /.,2/(4-d')~ The emergence of a single com- Z ~ 1 / ~ 3 , .

bination of z~ and z 2 confirms eqs. (21) and (24), the form of z also being the same as predicted. At

zl/(4 a'l which verifies [ = 0, eq. (32) gives: y - 3 predictions (22) and (26); for [ < 0 and L-- , ac,

g I / ( 2 d ' ) on the other hand, y - ( ] z 1 l / 3] which, in turn, verifies predictions (15) and (25). In the latter regime, the final results for d > 4 are found to be qualitatively the same as for 2 < d < 4.

For d ' = 2, on the other hand, y(z 1 ~ -oc) e x p ( - 4 - ~ I Zl l / z 3 ) whereas y ( z I = O)

V'z3 Iln z3l . It follows, as predicted, that in this case

( ( [ < 0 ) - L exp[4"rrK~(L/a)" 21~11, (35)

whereas

I / 2 ( w ( L / a ) "-4

~(?= 0)- L t lni w--~a~74] (36)

We have also carried out a detailed investiga- tion of the borderline case d = 4. While for T < T, we obtain essentially the same results as for d < 4 or d > 4 , the ones for T = T ~ turn out to be distinct. For instance, at T = T c, we find that

~c~L[ln(L/a)] l / (4 d') ( d ' < 2 ) ,

( l n ( L / a ) ) 1 / 2 - L In ln(L/a) ( d ' = 2), (37)

which generalizes known results [4,9] for d ' = 0 and 1. In addition, we have examined the behavior of the free energy and the specific heat as well; results of those studies will be reported in a subse- quent communication.

We are thankful to the Natural Sciences and Engineering Research Council of Canada for financial support. We also thank Professor V. Privman for valuable discussions and for sending us copies of his work prior to publication.

134


References

[1] M.E. Fisher, in Critical phenomena, Proc. Enrico Fermi Intern. School of Physics, Course LI, ed. M.S. Green (Academic Press, New York, 1977) pp. 73-99; M.E. Fisher and M.N. Barber, Phys. Rev. Lett. 28 (1972) 1516.

[2] M.N. Barber, in: Phase transitions and critical phenomena, Vol. 8, eds. C. Domb and J.L. Lebowitz (Academic Press, New York, 1983) pp. 145-266.

[3] M.E. Fisher, in: Renormalization group in critical phenomena and quantum field theory, eds. J.D. Gunton and M.S. Green (Temple Univ. Press, Philadelphia, 1974) pp. 65-68; in: Collective properties of physical systems, Nobel Symposium No. 24, eds. B. Lundqvist and S. Lundqvist (Academic Press, New York, 1974) pp. 16-37; in Lecture notes in physics, Vol. 186. Critical phenomena, ed. F.J.W. Hahne (Springer, Berlin, 1983) pp. 1-139, appendix D.

[4] E. Br~zin, J. Phys (Paris) 43 (1982) 15. [5] V. Privman and M.E. Fisher, J. Stat. Phys. 33 (1983) 385. [6] M.E. Fisher and V. Privman, Phys. Rev. B 32 (1985) 447. [7] K. Binder, M. Nauenberg, V. Privman and A.P. Young,

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[8] J. Rudnick, G. Gaspari and V. Privman, Phys. Rev. B 32 (1985) 7594; J. Rudnick, H. Guo and D. Jasnow, J. Stat. Phys. 41 (1985) 353.

[9] J.M. Luck, Phys. Rev. B 31 (1985) 3069; E. Br6zin and J. Zinn-Justin, Nucl. Phys. B 257 [FS14] (1985) 867; J. Shapiro and J. Rudnick, J. Stat. Phys., to be published; M.E. Fisher and V. Privman, Commun. Math. Phys. 103 (1986) 527.

[10] V. Privman and M.E. Fisher, Phys. Rev. B 30 (1984) 322. [11] S. Singh and R.K. Pathria, Phys. Rev. Lett. 55 (1985) 347. [12] S. Singh and R.K. Pathria, Phys. Rev. B 33 (1986) 672. [13] J. Shapiro, Phys. Rev. Lett. 56 (1986) 2225;

S. Singh and R.K. Pathria, Phys. Rev. Lett. 56 (1986) 2226.

[14] S. Singh and R.K. Pathria, Phys. Rev. B 31 (1985) 4483. [15] G.S. Joyce, in: Phase transitions and critical phenomena,

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finite-size scaling of o(n) models in higher dimensions

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