Finite-size crossover in systems with slab geometry

PHYSICA ELSEVIER

Physica A 216 (1995) 489-510

Finite-size crossover in systems with slab geometry D i m o I. U z u n o v a'l, M a s u o S u z u k i b alnstitute of Physical and Chemical Research (RIKEN), Wako, Saitama 351-01, Japan bDepartment of Physics, University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113, Japan

Received 15 November 1994

Abstract A renormalization group study of the finite-size (dimensional) crossover is carried out with the help of e = 4 - d and eo = 3 - d expansion techniques. The finite-size crossover and the invariance relation for the length scale transformation are proven up to the two-loop approximation. The formal equivalence between the finite-size crossover in classical systems and the quantum-to-classical dimensional crossover in certain quantum statistical models is emphasized and exploited. The finite-size corrections to the fluctuation shift of the critical temperature and the width of the critical region are investigated. It is shown that the shift exponent 2 describing the fractional rounding of the critical temperature obeys the relation 2 = D - 2, where D is the dimensionality of the system.

1. Introduction The fluctuation theory of phase transitions and, in particular, the renormalization g r o u p (RG) m e t h o d s are powerful tools in the study of the finite-size scaling (FSS) [ 1 - 5 ] . A central problem within the FSS theory is the FS (dimensional) crossover (FSC) [ 5 - 1 4 ] . As this paper is intended to present new results for the FSC, we shall briefly define the problem and review some of the previous studies. A D-dimensional system can undergo a standard sharp phase transition of second order and, hence, it can exhibit a typical D-dimensional asymptotic (T ~ To) critical behaviour in a close vicinity of the critical point Tc provided its dimensions Li (i = 1 . . . . . D) are macroscopically large and the spatial dimensionality D is not less than DL - the lower borderline dimensionality [15]. N o t e that no sharp phase transition is possible for finite critical temperatures ( T o : / : 0 ) for spatial dimensionalities D ~< DL.

1Permanent address: G. Nadjakov Institute of Solid State Physics, Bulgarian Academy of Sciences, 1784 Sofia, Bulgaria. 0378-4371/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSD1 0 3 7 8 - 4 3 7 1 ( 9 4 ) 0 0 3 0 5 - X

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Under the term "sharp transition" we understand a transition described by the statistical expressions in the thermodynamic limit Li ~ oo. If the limit Li -~ oo in the statistical expressions for the thermodynamic quantities is not taken in its strict mathematical sense, none of the typical nonanalytic properties of the corresponding thermodynamic quantities near the phase transition point Tc can be obtained. So, in a strict mathematical sense for a system to possess a sharp (usual) phase transition, it has to be "infinite". The "infinity" of the system is needed to justify the limit Li -~ 0o from a pure mathematical point of view, and at the same time this limit describes with a great accuracy the properties of macroscopically large but finite bodies. So, in the sense of theoretical physics the limit Li ~ oo stands for a macroscopically large size L ( ~ L~) of the system, which means that the size L should be much larger than all the intrinsic lengths of the system such as the (mean) interparticle distance a, the correlation length ( ( T ) of the thermal fluctuations of the order parameter, or other lengths related with some restrictions on the validity of the model considered or the method of calculation. This general theoretical point of view does not appear to be directly applicable to the asymptotic critical behaviour (T -~ To), where ~(T) -~ oo. There always exists an infinitesimally small (asymptotic) vicinity of Tc where 4 is much larger than any finite L. The typical critical singularities in the strict mathematical sense can be ascribed only to infinite systems. In the narrow temperature region of the finite systems where 4 > L, rounded peaks rather than divergences of the thermodynamic susceptibilities occur. However, the inequality 4 > L for a macroscopic body means, for example, 4 >~ 4' '~ 1022a, which corresponds to an experimentally inaccessible small vicinity of T¢. Rather, any macroscopic scale L ~ 4' ensures, in principle, experimental and theoretical studies of phenomena corresponding to very large but finite correlation lengths 4 < 4' or a ~ 4 ~ ~'. Such phenomena can be considered, to an enormous accuracy, as critical phenomena. The rounding of the typical (for infinite systems) critical divergences or other mathematical singularities for finite macroscopic sizes (L "~ 1022a) is negligibly small. So, in contrast to the pure mathematical understanding of the problem, we can always interpret the thermodynamic limit practically as describing the critical properties of finite macroscopic bodies. The same universal sense is attributed to the thermodynamic limit in other problems of the theoretical physics, where no divergent intrinsic length appears. The above remarks are essential only for the asymptotic critical behaviour described by 4(T) ~ oo. When the nonasymptotic properties of the system [15] are concerned, the correlation length 4 is always finite and no misunderstanding can arise. So we can interpret the limit L -+ oo as the limiting case of macroscopic lengths L > ~, where 4 never reaches infinity. When L ~ 4 or, even, when L --~ ( for any finite 4, the rounding of the critical singularities is considerable and is of real physical interest. Moreover, in the extreme case when a given dimension Lz of the system, say, along the z-axis is very small, L z = m a (m = l, 2 . . . . ), the thermodynamic limit along this direction cannot be taken and one has to use other

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ways of description [17]. The system sizes (Li ~ L) which we shall consider here are either macroscopically large ("infinite"), when L >> 3, or small ("finite'j, i.e. L~<~. If one or more (m = 1, 2, ... ) of the system sizes Li are small enough (or "finite"), the system will undergo a sharp transition and, hence, it will exhibit a d = (D - m)-dimensional asymptotic critical behaviour if only d > DL. In a quite restricted aspect, the FSC is the crossover from a d = ( D - m)-dimensional to a D-dimensional asymptotic (T ~ To) critical behaviour as the m system sizes are enlarged to infinity. This problem is usually treated by RG methods. In a more universal sense the FSC phenomena include not only the asymptotic but the entire (asymptotic and pre-asymptotic) critical behaviour and, even, the behaviour near discontinuous phase transitions. In such a general sense, the FSC becomes a formidable task which requires the use of various theoretical and experimental methods. The investigation of the FSC for the entire critical behaviour of real models, for example, Ising-like models for second-order phase transitions, is also a difficult task. That is why the FSS and the FSC have been extensively investigated on the example of unreal but exactly solvable statistical systems such as the spherical model and the ideal Bose gas [3, 4, 16]. The basic questions of the FSC theory can be solved by considering the Landau-Ginzburg (LG) q~4-theory of critical phenomena in isotropic systems with short-range interactions or Ising-like models which, at a quasimacroscopic level of description, are in many respects equivalent to the ~p4-theory. Here we shall follow this approach to the problem. Within the lowest-order, i.e. mean-field (MF), approximation, the FSC is too simplified. The only FSC effect which can be observed within this approximation for the Ising model is the M F shift of the critical temperature [17]. Besides, this effect is significant for very thin magnets only and can be neglected in usual magnetic films. The fluctuation phenomena can be taken into account by the perturbation theory and, in particular, by RG methods. Several variants of the RG have been used in the previous investigations of the FSC. The scheme of the Wilson RG has been applied by one of the present authors [5] to systems of both box (m = D) and "slab" (m = 1) geometry. In this way the basic FSS relations and the FSC have been verified by using the general RG scheme rather than by an explicit derivation of the RG equations and their analysis. Moreover, an extension of the FSS ideas to dynamic critical phenomena has been proposed [5]. A profound investigation of the FSC has been performed by Lawrie [6] but it has been overlooked in most of the subsequent works [7 14]. It has been shown for the first time how the field theoretical RG technique should be applied in order to perform an explicit derivation of the RG equations and calculate the FSS functions [6]. Moreover, it has been demonstrated [6] that the FSC problem for a slab geometry of the finite system is, under certain conditions, equivalent to the quantum-to-classical crossover (QCC) [15, 18-24] for the quantum Ising model in a transverse external field. Almost at the same time, Bray and Moore [7] have presented the first RG study

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based on noncyclic but Dirichlet boundary conditions which make it possible to take into account surface effects. In 1982 the R G studies have been renovated by Brezin [8]. In result, a series of studies in various variants of the field theoretical R G theory together with the e and 1In expansions have been done. The FSC has been proven within the one-loop (first order in e = 4 - d) and in the so-called large-n limit (zero-order approximation for the 1/n-expansion). A field theoretical RG investigation by the e = 4 - d expansion up to the two-loop (e 2) approximation has been also carried out for systems with box geometry 1-13]. In this paper we study the FSC in systems having a (hyper) slab geometry and being described by the L G ~04-theory; q~ = (~o,, ~ = 1 . . . . . n) is the n-vector order parameter. In Section 2 we present a discussion of the validity of the L G q~4-model and its interrelationship to the Ising-like lattice models. The problems concerned in the present section and Section 2 are essential for the interpretation of the results for the Ginzburg criterion and the fluctuation and fractional shifts of Tc obtained in Section 3 as well as for the justification and interpretation of the R G analysis (Sections 4 and 5). In Section 6 we summarize our findings. The slab geometry provides a relatively simple mathematical treatment and, as a consequence, a possibility to demonstrate with simple mathematical formulae the size-dependence of the width of the critical (fluctuation) region near T¢ (Section 3) and the validity of the FSC up to the two-loop approximation for the Wilson RG recursion equations [15] (Sections 4, 5). In Sections 4 and 5 we shall use both e = 4 - d and eo = 3 - d expansions within the large-b variant of the RG; d = D - 1 so that we have e = 5 - D and eo = 4 - D in terms of the physical dimensionality D of the (hyper) slab. Results previously presented for the quantum transverse Ising model [6,18] and a model of displacive structure phase transitions [19,24] will be exploited.

2. M o d e l considerations

We shall investigate the L G fluctuation Hamiltonian 3¢g = H~ T in the form 1 (" o

= 5 J d x [c(V~o) z +

ro~o z +

2Uoq~*],

(2.1)

where ro = ~oto, with to = ( T - T¢o)/Tco and Too the bare critical temperature, c, Uo and So are positive parameters, and ~0(x) is the n-vector fluctuation order parameter. Let the system be a large plate with a thickness Lo perpendicular to the Dth direction in the D-dimensional space (or the z-direction for D = 3). The thickness Lo can vary in a broad interval, a ~ Lo < L; L is a macroscopic length and L ~ L j, where L~ withj = 1. . . . . d are all other dimensions of the hyperslab. Without any loss of generality we set all Lj equal to one another so that the volume of the system will be 11"o = VdLo, where Va = L d.

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In the reciprocal space of the wave vectors ("momenta") q we have to work with the Fourier amplitudes tp,(q) of the field components t&(x), given by 1

o~(q)=x/-V-~D~e-iq~O=(x);

(2.2)

the vectors x and q will often be denoted by x and q. Moreover, we shall often use q for ]q ] - the modulus of q. We shall denote q by q = (k, ko), where k = (kfl = (k ~. . . . . ka), and k o - kD. For periodic (cyclic) boundary conditions, the components ki (i -- 0 . . . . . d) are given by ki = 2 n n j L i , where n~ = 0, + 1 . . . . . 4- ~ . If there is an underlying lattice structure and if there are reasons to take it into account, the components k~ of q should be restricted to the first Brillouin zone [25]: n/a < k~ <~ n/a, where a is the lattice constant. At this point we have to make an important remark. The Hamiltonian (2.1) has a very universal applicability in many respects but is a quasimacroscopic model and describes phenomena for macroscopic and quasimacroscopic distances only [26]. So, if we try to extend the M F Landau theory (~o, = const.) to a theory of fluctuation fields ~o,(x) using general phenomenological arguments [26], we should keep in mind that only spatial variations of tp,(x) at quasimacroscopic distances can be correctly taken into account. Therefore we have to introduce a momentum cutoff of the type A( = 7n/a), where 0 < ~ ,~ 1. This cutoff is one of the characteristic lengths of the model (2.1) and it has an important role in the studies of both universal and nonuniversal properties of the phase transitions, especially in the description of the FS effects (Sections 3-5). Obviously, A can be used for both crystal and disordered substances, where the constant a in A merely indicates the magnitude of the lattice spacing in crystals or the mean interparticle distance in noncrystalline bodies. The cutoff A can be extended to ~/a or to infinity only in certain studies of the asymptotic critical behaviour where the extremely small momenta (q ~ 0) give the essential contribution to the physical quantities; A ~ vo means a ~ 0 and nothing else, but it works perfectly for ~ ~ 0 and gives sense to the cutoff-free RG schemes [6, 8-143. In studying FS effects we have to know the important lengths in the system. For infinite systems the intrinsic lengths described by (2.1) are a, A, c. = ~.o~o~ ,1,.2, with C,o = (c/cto) 1/z the zero-temperature correlation length of the system; usually, A-1 > 4o > a or A-1 > ~o > a. After some renormalization [15], the initial (bare) critical temperature Too changes to the true critical temperature Tc so that the real 1 correlation length ~ becomes ¢ = ~ot-", where t = (t - T¢)/T¢ and the value v > comes from the renormalization too. All these lengths are important in the study of the phase transitions in infinite systems, but once the asymptotic critical behaviour is concerned the single relevant length which remains is ~. For FS systems, two lengths, ~ and the finite size L, are important for the asymptotic critical behaviour. Whether the system will exhibit a d = ( D - 1)dimensional or a D-dimensional critical behaviour depends on the ratio L/~.. When we -

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study nonuniversal characteristics of the critical behaviour such as the width of the critical region, all the characteristic lengths of the finite system are important together with their interrelationships. In this case we must have in mind that in the domain of the validity of the model (2.1) near T¢ we have ~ > d - 1 > (o > a; for concrete systems and temperatures T some of the signs " > " can be changed to " >> ". Finally, both L > ( and L < 4 are possible. The limiting case L >> 4 describes an almost-infinite D-dimensional system and the limiting case L ~ 4 describes an almost d = (D -- 1)-dimensional infinite system. Intuitively one expects this d - D crossover to occur when L ~ 4, and this argument yields 0 -- 1/v for the so-called "rounding" exponent 0 [-2,3]. Another important circumstance is that we can neglect the spatial dependence of q~,(x) along the spatial directions, for which the size of the system is relatively small (L ~ 4). This means that the corresponding momentum components ki are equal to zero. We shall illustrate this by considering the correlation function

1 G(x

x')

eiq.tx x'}

~q k20 + k 2 + ro/c"

(2.3)

If, for example Lo 4 4 = (c/ro) 1/2, every term in (2.3) including ko = 2rCno/Lo ~ 0 will be much smaller than the term corresponding to ko = 0. Then we can neglect, at least to a first approximation, all terms with ko ~ 0. For the kZ-terms this cannot be applied to the case of a slab geometry, where the Li (i = 1 .... , d) are always assumed to satisfy the inequality Li >> 4, i.e. to be "infinite" and, hence, all k~ = 27zni/Li to smoothly vary from zero to A. An alternative approach for Lo ~ 4 can be used by neglecting the dependence of tp(x) on the component xo of the vector x. Then (2.1) can be written in the form

1;(

Jf =~

uo)

ddx c(Vdp) 2 + ro¢ 2 + 2 L o o ~b4 '

(2.4)

where ¢ = x/~otp and the interaction constant has been changed from Uo to Vo = uo/Lo. In Section 3-5 we shall be faced with this interesting situation. Note that the physical dimensions of tp and ¢ are different. The quantities in (2.1) have the following dimensions compared with the dimension of length [L]: [~o] = [L] -D/z, [C] =[,L1/2], [ro] = [L °] and [Uo] = [LD]. In contrast, in (2.4), we have ['t~] = [,L -d/z] and [,Uo] = [La-1]; ro and c keep their old dimensions. The model (2.4) is a reduced version of (2.1) and describes a system with a physical dimensionality D. However, this system behaves effectively as a d = ( D - 1)-dimensional system. Further, it is not identical with the true d- dimensional system which is described by (2.4) with Uo rather than Vo = uo/Lo. The formal limit Lo -~ 0 which is often used in some studies [6, 11, 14] corresponds to this system. These arguments show that for very small L o we cannot use the continuum (thermodynamic) limit along the Dth spatial direction and the field theory (2.1) fails to

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describe a true d-dimensional system. For a geometry of a small box, when all Li are small, only one important mode remains in (2.1) - the ~o(q = 0)-component of ~0. Then the system is zero-dimensional and no phase transition is possible at all because of the very strong fluctuations of the mode ~0(q = 0). In order to describe slabs of a finite thickness Lo, the summation over q in expressions like (2.3) can be represented in the form (2.5t F o k,,

~ k

,~o ko J ( 2 ~ t ~'

that is, the continuum limit for the vector k can be taken, whereas the summation over ko must remain up to the stage at which we choose the limiting case of interest (Lo >> or Lo ~ ~). In many cases it is convenient to perform the summation over ko and to obtain formulae which are valid for any ratio (Lo/¢). It is now easy to see the analogy between the Matsubara frequency o91 = 2 n i T (1 = 0, + 1 . . . . . _+ or) in quantum systems and the momentum component ko and, hence, the correspondence [6] Lo.__~ T -1 Many of the above arguments can be confirmed with the help of microscopic models. We shall briefly discuss the interrelationship between the field model (2.1) and Ising-like lattice models with short-range interactions. The most consistent way to derive field theories from lattice models is to use a well-known Gaussian transformation [6, 15, 18, 27]. The conditions under which such derivations can be done are almost the same for all lattice microscopic models. So we shall focus our attention on the Ising model with nearest-neighbour (NN) interactions defined on a hypercubic D-dimensional lattice. At a certain stage of the derivation [15, 27] the periodicity of the lattice must be taken into account and, hence, the momentum components become confined to the first Brillouin zone as discussed above. Then, the cutoff A = n/a will be very convenient to describe the system if we are able to develop a field theory with a free correlation function G given by 1

G - l ( q ) = j(q) _ T j2(q),

(2.6)

where D

J(q) = 2J0 ~ cos(aki)

(2.7)

i=1

are the Fourier amplitudes of the exchange matrix (Ji~). Neither is the development of a field theory with such a complex propagator easy nor will it correspond to the Ising model. The last circumstance is due to the fact that the amplitudes J(q) are the eigenvalues of the matrix Jij for periodic boundary conditions describing the bulk properties of the system and in the case considered here. The Gaussian transformation quoted above is valid for positive definite matrices

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J~j, but the cosine in (2.7) takes negative values for sufficiently large q. Therefore two requirements, namely: (1) the necessity to derive the quadratic ( ,,, q2) spectrum of the fluctuations, as in the field theories with short-range interactions, and (2) the condition for the validity of the Gaussian transformation, force us to expand the cosine in (2.7) and to keep small aki, ki < A < yn/a, where ~ ,~ 1. Thus we believe that, at least at a (quasi) macroscopic level, the Hamiltonian (2.1) describes Ising systems. When one (or more) of the sizes L, of the system are small enough, the corresponding term(s) in (2.7) with k~ ~ 0 cannot be expanded for small ak~ because none of the ak, ~ 0 is sufficiently small; in our case v = 0. That is why the model (2.1) fails to describe very thin (Lo = ma, m = 1, 2 . . . . ) films [17]. In such a case the only possibility is to reduce (2.1) to (2.4) and to consider any very thin D-dimensional system with Lo < A - 1 as d = (D - 1)-dimensional with respect to the fluctuation properties. It is the inequality Lo < A-1 which gives sense to the formal limit Lo --* 0. This inequality is possible because of A - 1 >> a, i.e. because of 7 ~ 1 (there is no physical reason to consider values Lo < a and, hence, Lo cannot tend to zero). All these arguments are valid for any microscopic interaction J~j. For long-range interactions [15] Jij, the quadratic spectrum (q2) of the fluctuations must be changed to q'; 0 < a < 2. It is not essential whether we discuss the simple form (2.6) or some special function [15] J(q) coming from more complicated exchange interactions Jij. These arguments as well as other details in the derivation of field models from lattice ones completely support our initial discussion of the Hamiltonian (2.1). The study of the entire, short-range (q ~ A) and long-range (q <~ A), behaviour of lattice models by their field counterparts is a very difficult task. Moreover, it seems that such a task is not very promising. So the thorough problem for the FSS from very thin (L0 ~ a) to very thick (Lo > () slabs cannot be solved for real (field and lattice) models within a single scheme. What we can do with the model (2.1) is to describe the FSC from a thick slab (Lo > A-1) to a very thick one (Lo > ~). In the subsequent sections we shall keep the k = Ikl between 0 and A (an approximation of a spherical Brillouin zone in the d-dimensional subspace of vectors q) but we shall allow ko = 27Zno/Lo to vary without any restriction: no -- 0, _+ 1 . . . . , _+ ~ . The reason is that the large wave numbers ko are of no importance in our investigation which describes the temperature region ~ > A 1 near To. This assumption has been implicitly made in all previous works of FS effects on phase transitions [6-14]. However, this approximation can essentially affect the precise form of the FS corrections to the bulk values of physical quantities such as the specific heat or the magnetic susceptibility in systems with periodic structures [8 14]. This point is clear from our present discussion. The FS corrections in systems with periodic structures, where ko is naturally confined by Jkl < A, should be different from the FS corrections in disordered (noncrystalline) bodies. Of course, they must coincide in the limiting case ~ ~ ~ or for amorphous or very dense (a -~ 0) substances. The last possibility seems unique in condensed matter systems.

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3. Ginzburg criterion and shift exponent 3.1. General considerations The Ginzburg(-Levanyuk) criterion for the validity of the M F approximation can be derived from the equation for the susceptibility )~(T) in a zero external field h. Within the first-order perturbation expansion, which is absolutely enough for our aim here, the inverse susceptibility Z 1 is given by Z-1 = r and the equation [15] r = ro + 4(n + 2)UoAl(r),

(3.1)

where Al(r) =

q2 + r"

Performing the summation over ko = 2r~no/Lo with the help of the formula

__1

,=_ ~, 1 + ~2n2

coth(:)

,33,

and taking the continuum limit over the wave vector k, we obtain

1 f d"k coth[- Lo,/ Al(r) = 2c j ( - ~ a

~

+ r/c3

+ r/c

;

k -Ikl

< a.

(3.4)

The critical temperature Tc is defined by the equation r ( T c ) = 0. F r o m this equation and (3.1) we obtain the fluctuation shift Ate = (Tc - T~o)/Tco of the bare critical temperature T¢o: Ate

Tc - Too _ Tc

4(n + 2) u° AI(0). C~o

(3.5)

Note that T~o in the denominator of A t c = (T~ - Tco)/Tco can be safely replaced by To; see (3.5). The reason is that T~ - Too --~ Uo and, hence, this difference gives a small O(ug) contribution to the R H S of the second equation in (3.5). Eq. (3.1) can be written in the form r = rco + 4(n + 2)uoAAl(r),

(3.6)

AA1(r) = Al(r) - AI(O)

(3.7)

where

and r~o = ~ot, with t = ( T - T~)/T¢, whereas T~ is self-consistently determined by (3.5). The condition r ~ r~o leads to the Ginzburg inequality: r >> 4(n + 2)Uo IAAI(r)I.

(3.8)

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The integral (3.4) can be investigated in the two limiting cases of large a n d small lengths Lo. We have mentioned in Section 2 that these limiting cases can be considered by the formal limits Lo ~ ~ and Lo --, 0. It has been emphasized why the proper limit Lo --* ~ is never needed to describe the bulk properties of a macroscopic body and why the limit Lo ~ 0 in its mathematical sense cannot be taken seriously. For r ,~ A 2, the main contribution to the value of the integral A~(r) comes from the small (k ~ 0) wave numbers k, and, hence, the limit Lo ~ 0 merely means Lo ,~ whereas Lo ~ ~ means Lo >> 4; ~ = (c/r)1/2 is the renormalized correlation length which tends to infinity at the true critical temperature T¢.

3.2. Lo ~ Expanding the integral

A l(r) for small kLo, we obtain from (3.5) the fluctuation shift

Atc(Lo) = 4(n+2)Ka d- 2

v°Aa (

~o2(~oA)z 1 +

~]~-

(LoA) 2 +

O(L4A 4) ) ,

(3.9)

where d > 2, Kd = 21-a~t-d/2F-X(½d) and the parameter Vo has been introduced in Section 2. The expansion parameter LoA is small for a < Lo ~ a/n7 as it can be expected from the analysis in Section 2. The first term on the RHS of Eq. (3.9) is precisely the fluctuation shift Ate(0) of a d-dimensional system with a fluctuation interaction Vo. According to our discussion in Section 2, this is, in fact, the original D-dimensional system, where the spatial fluctuations of the order parameter along the Dth axis are absent because of the small thickness Lo, and the fluctuation interaction constant Uo describes the fluctuations along the remaining d = D - 1 spatial directions. This is effectively a d-dimensional behaviour of the system. The second term on the RHS of(3.9) gives the first correction to such a nearly d-dimensional behaviour. The correction is positive because it indicates fluctuation effects along the Dth direction of the space. The second term in (3.9) is part of a series with alternating signs. So it gives a slightly overestimation of the real enhancement of At~(Lo). The results (3.9) can be used to compare two systems with different thicknesses, Lo and/0; both of them must be much larger than a. Obviously the result will be At¢(Lo) ,~_ lo , Ate(/o) Lo

(3.10)

by neglecting a numerical factor and terms of type O[A2(Lo - lo)2]. Eq. (3.9) can also be written for the physical dimensionality D = d + I. From such an expression for At¢(Lo) and from the following fluctuation shift of Tc for a D-dimensional infinite system: At o,.~ 4(n + 2)Ko uoA ° D - 2 ~2(~oA)2'

(3.11)

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we obtain the relation At~(Lo) 1 - At ° - LoA

>

1.

(3.12)

This result confirms that the fluctuation effects becomes stronger when the spatial dimensionality of the system is virtually lowered. The integral difference AA~(r) in (3.8) is calculated by expanding the integral (3.4) for small Lo x / ~ + r/c. The Ginzburg criterion (3.8) can be written in the form 12 d/2 >~

+ 2) CdVo . 1 +~--z~-~-~ - -

(A¢)n ,

(3.13)

where Ca=~KaF(½d-

1) F ( 2 - ~ d ) ;

2
(3.14)

The discussion of Eq. (3.9) is valid for the inequality (3.13) as well. The second term on the RHS of(3.13) is a correction to the Ginzburg criterion for a D-dimensional system with spatial fluctuations along d = D - 1 dimensions of the space. Note that the correction term in (3.13) is small mainly because of the small numerical coefficient; in our present approximation Lo ,~ ~ but A~ ~ 1 and, especially for d ~ 4, (Lo/~) ~ (A~) d may even become unity. Let us mention that the next correction to (3.13) is of the order of (Lo/() 6 (A~) d ,~ 1 (for 2 < d < 4). The inequality (3.13) can be compared with the Ginzburg inequality for a D-dimensional infinite system with a fluctuation interaction constant Uo: t 2 o/2>>

4(n + 2)Couo 2 D ~0~0

" 2
(3.15)

The critical region above T~ is defined by tc = ( T G - Tc)/T~, where T~ is the temperature at which the inequalities (3.13) and (3.15) become equalities. The critical region of infinite systems with D = 4 is exponentially small [-15]. For D > 4 the critical region does virtually not exist. Using the inequalities (3.13) and (3.15) we can determine three different critical regions: tc from (3.13) corresponding to the system of interest, and t~m and t~ ) - the critical regions of D- and d-dimensional infinite systems. The inequality (3.15) is not enough to define the critical region ,G'ca)properly. The constant Uo in (3.15) will have another physical dimension in the d-dimensional case; see the dimensional analysis in Section 2. In order to avoid confusion we must introduce another notation (rio) for the interaction constant of the infinite d-dimensional system. According to the above definitions, the critical regions t G and "G'ta)are applicable to the region 2 < d < 4, and t~m is applicable to the region 2 < D < 4. Therefore, the common domain is 2 < d < 3 or, which is the same, 3 < D < 4. Let us choose d and D = 7, and compare tG, -~ttd) and t~m. Neglecting numerical factors and the

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dimensionless parameter ~o which are not essential in discussions of the size of critical regions, we easily obtain that

,(a)_Vo 4/3 1 + (AO a tG - ,G kt~o/ 540n \ ¢ )

(3.16a)

(¢°'~4/3[1+ tG ~ (t(GD))1/3 \~o//

(3.16b)

and x/~ (_%_q)4 ] (A~) a •

~

The relation (3.16a) shows the enlargement of the critical region tG due to the increase of the thickness Lo. When Lo is sufficiently small so that no fluctuations along the direction perpendicular to the slab are possible, the second term in (3.16a) can be neglected and t~o can be chosen equal to Vo. Then, we have tG ,(a) It is seen from (3.16b) how the critical region tG is related with that of the infinite D-dimensional system. As in usual cases t~°) ~ 1, (t~°)) ~/3 >>t(G°). Therefore, tG >> ~G ,(D), provided Lo is much larger than ¢o.

3.3. Lo >>~ The calculation of the integral Al(r) is carried out for L o x / - ~ + r/c >> 1. It is convenient to represent this integral as a sum of two integrals:

Al(r) = A~°)(r) + AL(r),

(3.17)

where A~°°)(r) is the value of Al(r) for Lo ~ oo and AL(r) = Al(r) -- A~)(r): A

A(l~)(r) = K o - I [ k ° - 2 dk 2--7- . / x / ~ + ~ c

(3.18)

0

and A

AtL)(r) _ K , _ c

1

f k " - 2 dk 3° x / ~ + r / c ( e L ° ~

(3.19)

-- 1)"

In fact, the integral A(~°~)(r)is divergent for D ~< 2 and r ~ 0. The integral AL(r) exhibits the same "infrared" divergence for D ~< 3. The divergence of A(l~)(r) is well known [15]. This is the typical divergence of the perturbation integrals at and below the lower borderline dimensionality DE(----2) of an infinite D-dimensional system described by (2.1). The divergence of AL(r) below D E + 1 dimensions has the same origin but it is not well understood. Obviously, this divergence comes from the fact that a D-dimensional system with a finite size Lo will behave effectively in the asymptotic vicinity of T¢ as a (D - 1)-dimensional system.

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Some analytical and numerical calculations can be performed for a general dimensionality D, as follows. First note that the cutoff A in (3.19) can be approximately set to infinity. For D = 4, we obtain A2 ~(2) A1 (0) = ~ + 2cL---~o'

(3.20a)

and, for D = 5, A3

~(3)

A~ (0) - 48rtz~ + 4rt2cL--~ .

(3.20b)

In general, we have

AI(O) -

Ko-lA °-2 K o - I ~ ( D - 2) F(D -- 2) 2c(D -- 2) + cLOo z ,

D > 2,

(3.21)

where if(2) = gn 1 2 and ~(3) = 1.202 . . . . While in Subsection 3.2 we have investigated the fluctuation shift of T¢, we shall use here the integrals (3.20), (3.21) to obtain the shift exponent 2 describing the fractional shift [2] AtL of T¢ due to FS effects. This shift is again a fluctuation effect but it is smaller than the fluctuation shift Ate; while Ate is proportional to uoA~(0), At L ~ AL(0) < AI(0).

As long as Lo is considered to be finite, the critical temperatures Too and T¢ in Eqs. (2.1), (3.5) and elsewhere depend, at least in principle, on Lo and can be denoted by T¢o(Lo) and T~(Lo). For Tco(Lo), the Lo-dependence is not obvious from this study, because the hamiltonian (2.1) is phenomenologically constructed. However, T¢o is the M F critical temperature and it is derived either by the integral transformation mentioned in Section 2 or by a direct application of the M F approximation to the corresponding microscopic model. In such derivations, one always obtains that T¢o depends on the size of the system, especially near the surface [28], but this dependence exponentially decreases with the distance from the surface and, hence, it can be ignored in our study of thick slabs with cyclic boundary conditions. In the bulk of the system, T¢o(Lo) is equal to the bulk value of the M F critical temperature T¢o(~v-=),that is, to the value corresponding to the infinite system. We shall therefore use the equalities Tco(Lo)= T ¢ ( ~ ) - T~o. These remarks are important for the correct derivation of the fractional shift. Moreover, they give the limitations of our consideration. In order to calculate the fractional shift

AtL = T c ( L o ) - T¢( ~ ) , T¢(~)

(3.22)

we write, with the help of (3.4), the equation for T~(Lo) as

T~(Lo) - T¢o = - 4 ( n + 2) Uo T~(Lo) AI(O), no

(3.23a)

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and for the equation for To(oo) we have T o ( ~ ) - T~o = - 4(n + 2) Uo T~(oo) A~°°)(0).

((3.23b)

(Zo

Subtracting (3.23b) from (3.23a) and using c = ~o 32, we obtain AttL4) _

n+ 2 Uo 3 a2(~oLo)2

(3.24a)

for D = 4 and, in the case of D = 5, AttLS) =

(n + 2)~(3) Uo n2 go2Lo(~oLo)2 •

(3.24b)

In general, we obtain AttL°) = --4(v + 2)Uo [A1 (0) - At°°)(0)] 0~o

~--"-

4(n + 2)((O - 2 ) r ( D - 2)Uo 2 0-4 2 ~oLo (~oLo)

(3.25)

In order to derive Eqs. (3.24) and (3.25) we have neglected terms of second order in Uo. The RHS of Eqs. (3.24) and (3.25) are dimensionless because Uo has always the dimension of [L°]; see Section 2. Eq. (3.24a) confirms previous calculations [7, 11, 12, 14], where the shift exponent 2, defined [2] by AtE ~ Lo x, is given by 2 = 2 for D = 4. These calculations have been made with the e( = 4 - d) expansion near D = 4; see Section 4. Here we have shown that the calculation of 2 -- 2 does not need the ~ expansions. The reason is that the value of the exponent 2 does not depend on any fixed point (FP) coordinate of RG equations. Moreover, we have shown that the value 2 = 2 is not universal for general D. In fact, the exponent 2 depends on D by the relation 2 = D - 2. Namely, Eq. (3.24b) yields fo(uo). At~L°) - L ~ - 2,

4(n + 2)((D - 2 ) r ( D - 2)Uo f o(uO)

=

2

2

~XO~0

,

(3.26)

where the constant f o ( u ) depends on D and the renormalized (u = uR) or the bare (u = Uo) value of the interaction constant (uR may appear from the RG and other nonperturbative studies). Having in mind the properties of the perturbation series, Eq. (3.26) should be valid for all D > 3, at least within the first-order perturbation expansion. Of course, Eq. (3.26) is valid for all D > 4 because no divergent perturbation terms exist in this case. The check of (3.26) for 3 < D < 4 in a more convincing way requires the RG or other nonperturbative investigations which work, in contrast to the epsilon expansions, directly at the dimensionality of interest, for example D --- 3.5 or D = 3.

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The integrals (3.4) and (3.19) are convergent for all D > 3, finite Lo and r ~ 0 and give a finite critical temperature. Clearly the shift exponent 2 decreases with the decrease of D for all D. In the limiting case Lo >> (, the dimensionalities D = 4 and D = 4 - e with e ,~ l are important both for the experiment and the theory because they describe the M F and the almost-MF behaviour of the system. For this reason the value 2 = 2 of the shift exponent seems to be of main interest. The dimensionality D = 3 is important because it is the physical dimensionality of most of the real systems. However, any finite size Lo, even Lo ~> ~, causes D = 2-dimensional effects, i.e. divergences in the three-dimensional system. Lowering the thickness Lo of the slab from a large initial value (Lo ~> () up to Lo ~ ~, these divergences should be observed as a rounding of the transition. The rounding will be described by the shift exponent 2 = 2 corresponding to D = 4, provided the experiment is performed outside the critical region, where the effective ("theoretical") dimensionality of the system is D = 4. The fluctuation and fractional shifts of Tc cannot be straightforwardly calculated for a D = 3-dimensional system, because the integrals (3.4) and (3.19) have a logarithmic divergence for finite Lo and r = 0. The direct calculation of the integral (3.19) for r > 0 shows terms of the type (1/Lo)exp( - (Lo) and (1/Lo)exp( - ALo); in this case the cutoff (A) corrections are larger or of the order of the cutoff-free value of the integral. It can be easily seen that the rounding At~L3) of Tc at D = 3 is of the type At~L3)= x ( l / L o ) e x p ( - ALo), but the numerical constant x is infinite. Even if the theory is regularized so that the constant K is finite, the exponential correction factor violates the power dependence in At~L3). Such exponential factors are typical for the dimensionality DL; here, DL = 3. They will not exist for any D > 3, where there is a power dependence of AttL3) on Lo.

4. RG in one-loop approximation Having in mind the discussion in Section 1 and 2, it is easy to justify the RG investigation of FS systems. It is convenient to choose the length units so that A = 1. Besides, a simple transformation of the field, v/~g0 ~ ~0,leads to a new definition of the parameters, ro/c --* ro and Uo/C 2 ~ u0; further we shall omit the index zero of ro and Uo, denoting them in this and the next sections by r and u. The first step in the R G investigation is to define the rescaling of the momentum q and the fields q~,(q) by the rescaling factor b > 1. Two ways of rescaling are possible: (a) rescaling of both ko and k, where 0 < k < A, k = Ik I, and (b) rescaling of k only. Within the scheme (b) we avoid the initial rescaling of ko but, instead, we obtain an additional RG equation, the equation for the k2-vertex and, hence, a transformation of Lo: Lo = bL~.

(4.1)

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This relation plays the role of an initial condition within the scheme (a). It is trivially derived within the scheme (b) to zeroth and first order of the epsilon expansion, but some efforts are needed to derive it to second order in e = 4 - d or eo = 3 - d; see Section 5. The next steps in the derivation of the R G equations are standard [15]. We shall show in this section that the schemes (a) and (b) are equivalent and either of them can be used in the t w o - l o o p a p p r o x i m a t i o n (Section 5). Within the scheme (a) we obtain the recursion relations

r' = bZ[r + 4(n + 2)uAl(r, b)]

(4.2a)

U' =

(4.2b)

and b4-D[u

--

4(n + 8)uZA2(r, b)],

where the difference between the integral A a(r, b) in (4.2a) and A l(r) in (3.4) is that we use here the bare p a r a m e t e r (ro -- r), c = 1 and the integration over k in Ax(r, b) and Az(r,b) is m a d e in the shell b - a < k < 1. The integral Az(r, b) is given by A 2 = -- OAa/Or. In this a p p r o x i m a t i o n the Fisher e x p o n e n t ~/is equal to zero. N o w we have to analyze Eqs. (4.2) and therefore to find the integrals A1 and A2. F o r the R G scheme in the notations used here this means to consider the limiting cases: (i) Lo ~ 1, and (ii) Lo ~> 1, which c o r r e s p o n d to the formal limits Lo ~ 0 and Lo ~ ~ discussed in the preceding sections. F o r the scheme (a) these cases are referred to as (a.i) and (a.ii). N o t e that the limiting case (i) could not be physically justified without our considerations in Section 2, where we have shown that A must be A = ~n/a with 7 ~ 1. Then in the original units Lo '~ 1 means a < Lo ~ a/n7 which, in view of the fact that ~ ~ 1, has the chance to be physically realized. F o r (a.i) we obtain Eqs. (4.2) in the form

and

v'= b5 °[v-4(n

+ 8)v2K41nb],

(4.3b)

where v = u/Lo and K4 = 1~8ha; Kn is given in Section 3, Eq. (3.9). T h e upper borderline dimensionality in the case (a.i) is dv = 4 because the integral Aa is logarithmically divergent for b ~ ~ at d = 4; see (4.2b) and (4.3b). Therefore Eqs. (4.3) are written for an e = 4 - d or, equivalently, an e = 5 - D expansion. In the case (a.ii), the integral Az is logarithmically divergent for d = 3 and, therefore, dv = 3. Here we must use the eo = 3 - d expansion; eo = 4 - D. The R G relations are

r' = bZ{r d- (n d- 2) uK3[(1 -- b -z) - r l n b ] }

(4.4a)

and U' = b 4 - O [ u

-(n

d-

8)uZK31nb].

(4.4b)

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505

The exponent of the rescaling factor in (4.4b) is given by e, = 4 - D, i.e. e,o = 3 - d, and K 3 =

1/2n2.

The analysis of the R G equations (4.3) and (4.4) is very easy to accomplish. F o r example, the F P values of u and t, are u*

27t2g° n+8

and

v*

2nag n+8

(4.5)

These two F P s of Heisenberg type describe the crossover from a d-dimensional to a D = (d + 1)-dimensional critical behaviour. The exponent v of the correlation length has the universal value 1 n+2 v = ~ -~ 4(n + 8~) g'

(4.6)

where g = (e, Co). This exponent together with q = 0 defines the universality classes (n, D) and (n, D - 1) corresponding to the n u m b e r s O and D - 1 of the infinite spatial dimensions. The asymptotic critical behaviour of a d-dimensional "infinite" slab (Lo ~> 1) is equivalent to the asymptotic critical behaviour of a D = (d + 1)-dimensional finite slab (Lo ,~ 1). The systematic way to obtain the fractional shift in powers o f g is to use the m e t h o d of direct calculation of the critical exponents [15]. Within this method, one has to calculate the integral A~(r)=Al(r)-A~l~t(r), see Eqs. (3.22), (3.23), at the corresponding upper borderline dimensionality dv = Du - 1. F o r Lo ~> 1, du = 3 and Du = 4, and we can use our previous result (3.24a) for D = 4. Substituting there Uo with u* from (4.5) and setting c = 1 in the correlation length ~o, we obtain [7, 11, 12, 14] A?c41(e'°)=-

2g 2 n + 2 ~.o 3 n + 8 L o 2"

(4.7a)

Here we have used %¢2 = c = 1. This result shows the behaviour of the fractional shift in a system with D = 4 - eo spatial dimensionalities and a large finite size (Lo > ~). F o r Lo '~ l, dv = 4 and Du = 5. The result (3.24a) for D = 5 cannot be used, because it is valid for Lo ~> 1. In order to obtain the fractional shift At~5~(e) in a system of dimensionality D = 5 - ~ (e <~ l) and a small FS (Lo <~ ~), we must first calculate Ate,m from Eq. (3.22) for D = 5 and Lo <~ ~. Here we shall use Eq. (3.9) for the fluctuation shift Atc(Lo). The second term in (3.9) yields exactly ( - At~D~). In this result we have to put v* from (4.5) and to set D = 5. Then At~LS)(E)is At"5~(e) -

1 n+2

2

48 n + 8 e[Lo + O(Lo4)];

A = c = 1.

(4.7b)

Eq. (4.7b) describes the fractional shift in a system with a physical dimensionality D = 5 - e, where the finite size Lo is relatively small (Lo "~ ~). O n e can use the second term in (3.9) with Vo = uo/Lo as a general expression of ( - At[°)) for all D > 2; L ~ ~.

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506

Despite some doubts [11] the negative sign of the fractional shift coincides with the conventional understanding of the way in which the fluctuations act. They always depress the critical temperature Tc in comparison with the M F one (Too). This effect becomes stronger if the dimensionality D is decreased. The fractional shift is a fluctuation effect. Once Lo is lowered from infinity to a finite value, the slab behaves as a system of dimensionality d' between d and D. So, effectively, we have d' < D and, hence, the fractional shift will be negative. This explains Eq. (4.7a). As Lo is enlarged from Lo ~ a (a true d-dimensional system) to a ,~ L ,~ 3, the system begins to exhibit the properties of its physical dimensionality D = d + 1, that is, to depress the relatively stronger d-dimensional fluctuations. This explains the negative sign in Eq. (4.7b). Within the scheme (b) we obtain again the relation (4.2a) for r', but the relation for u' is somewhat different: u '=b4_a

u _4(n+8)~oA2(r,b)

.

(4.8)

Moreover, an addition relation for ko2 appears: (k~)' = b 2 k~.

(4.9)

Eq. (4.9) merely confirms the relation (4.1). Using Eqs. (4.2a), (4.8) and (4.9), it is easy to verify that the schemes (a) and (b) of the RG investigation are equivalent:

5. Two-loop approximation The verification of the FSC in Section 4 has been easy because no q-dependent contributions to the fluctuation self-energy appear in the one-loop approximation. The two-loop approximation in its variant of g expansions has the advantage of giving more precise values for the critical exponents and confirming the picture of the possible FPs of the RG equations predicted in the first-order g analysis. However, qualitatively new results from the second-order g analysis such as appearance of new FPs of the RG equations, and, hence, a new type of phase transition, should not be taken seriously but rather as an artifact of the theory. The reason is that the expansions are asymptotic. In this section we shall confirm the results from the first-order g expansions for FS systems. In the two-loop approximation the RG equations are quite complicated and the proof of the FSC is not so straightforward. Here we shall use the scheme (b) of rescaling explained in Section 4. We derive from the q-dependent part of the self-energy the following RG equations: b" = 1 - 32(n + 2)u2AB(k)

(5.1)

(ko2)' = b2-~k2[1 - 32(n + 2)u2ABo(ko)],

(5.2)

and

D.L Uzunov, M. S u z u k i / P h y s i c a A 216 (1995) 4 8 9 - 5 1 0

507

where AB(k) = k- 2 [B(k, 0, 0) - B(0, 0, 0)]

(5.3a)

ABo(ko) = k02 [B(0, ko, 0) - B(0, 0, 0)]

(5.3b)

and

are obtained from

B(k, ko, r) =

J~''d~k2s(2rt) ~k~f a

~ B(k 1, k2, k, ko, r), ~zrt)-

(5.4a)

with 1

1

Sn = L2

(5.4b)

(a 2 + k01) [a2 + (ko, + k02)2] [a 2 + (ko + k02)2]

In Eq. (5.4a) the prime of the integrals means that the RG restrictions on the integrations must be fulfilled. In Eq. (5.4b), a2=k 2+r,

a2 = ( k , +kz) 2 + r ,

and

a3=(k2+k) 2+r.

(5.5)

The analysis of Eq. (5.1) for q and Eq. (5.2) for ko2 can be done after the summation in (5.4b) and the integration in (5.4a) are performed. The summation in (5.4b) is performed with the help of a simple trick by which terms of type l/(y 2 + n 2) are represented as the sum of two terms of the type 1/(y + in). Then the summation formula (3.3) can be modified to sum up terms of the type 1/(y +_ in). The result for SB can be written in the form

SB -- 16a

la~a%a%(i al +a2 +a3 At+)Bt+) a2 + a 3 - a l al +a2 +a3) 2 +k~ -~ (a2 +a3 - a l ) 2 +k~ At+~BI-~ \

+ (al

a l + a3 -- a2 + a3 - a 2 ) 2 +

At_~Ct+ ) k2

+ (as

a 3 - - a 2 - - a~ - a2 - a a ) 2 +

At_~Ct_)) koz ,' (5.6)

where A t -+) = coth(½ Loa2) + coth(½ Loa3), B t +-~ = coth(½ LoaO +_ coth[½ Lo(az + as)],

C I +-~= coth(½Loax) + coth[½Lo(a3 - a2)].

15.7)

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For the limit Lo ~ 0 we obtain

1

[

SB -- (Loaia2a3) 2 1 -

k~(.

1

1

4 \ ( a l + a2 + a3) 2 + k~ + (az + a3 - al) 2 + k 2

1

1

'7

+ (al + a3 -- az) z + k 2 -+ (al + ae -- a3) 2 + k 2 ) j '

(5.8)

And, for the limit Lo ~ o% we have

1

)

S , = 4ala2aa(al + a2 + a3)

(al + a2 + a3) 2 "

(5.9)

Eqs. (5.8) and (5.9) together with (5.3) and (5.4) can be used to analyze the RG equations (5.1) and (5.2). For this aim, one has to write down the RG equations for r and u/Lo as well and to investigate them in the large-b limit [ 15, 29]. Here we shall not present these RG equations, because they have the same structure as in the usual D-dimensional case [15]. The perturbation integrals in the equations for r and u/Lo are obtained from B(k, ko, r) setting k = ko = 0 and differentiating A~(r) and B(0, 0, r) with respect to r. All perturbation contributions to the recursion relations for u/L o have a factor 1/Lo; see, for example the second term on the RHS of Eq. (4.8). This fact and the special dependence of the perturbation integrals on Lo enable us again to use the parameter Vo as an effective expansion constant for the limiting case Lo ,~ 1. The RG treatment of the equations for r and u/Lo in the large-b limit is performed without any technical problems within both e and eo expansions. Let us focus our attention on Eqs. (5.1) and (5.2). The results for the exponent r/and the scaling transformation of ko ",~ 1/Lo, which should be obtained from these equations, are crucial for the verification of the d - D crossover. In order to obtain these results we need Eqs. (4.5) as well. The integrals A B ( K ) and ABo(ko) are calculated by a precise account of RG restrictions on the limits of integration. This point is important in order to avoid divergences. In the limit Lo ~ 0 the integral difference AB(k) is given by AB(k) -

In b 256n4Lo2,

(5.10)

where terms of order O (1) have been ignored. Substituting this result in (5.1) we obtain the universal value of the exponent q: r / = (n + 2) ~2/2(n + 8) 2. In the limit Lo ---,0 the integral ABo(ko) has precisely the same value as AB(k). As a result, the u 2 perturbation contribution in (5.2) is totally compensated by the factor b " and, hence, the relation (4.1) is proven up to second order in e2 = 4 - d. When Li--* Go, the calculation of the integrals is performed for d = 3. This calculation is lengthy but the result is simple: AB(k) = ABo(ko) -

lnb 256n 4

(5.11)

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The exponent r/has the universal value q = (n + 2) eo2/2(n + 8) 2 corresponding to the upper critical dimensionality du = 3, and, again, the scaling relation (4.1) does not acquire any epsilon corrections. We have therefore proven the asymptotic FSC by both e and eo expansions. The results for q and the relation (4.1) can be obtained by the method of direct calculation of the critical exponents to second order in the g expansions. In this case the technique of calculation of the integrals is different, but the results for r/and (4.1) remain the same. These calculations are similar to those performed for a quantum model of displacive structural phase transitions [19].

6. Concluding remarks We have presented the first thorough description of the most important nonuniversal features of second-order phase transitions in FS systems - the width of the critical region, the fluctuation and the fractional (fluctuation FS) shifts of the critical temperature. These quantities essentially depend on the FS (thickness) Lo of the D-dimensional slabs investigated here as well as on the interrelation between Lo and the other characteristic lengths of the system: A, ~.o and ~. The fluctuation shift of Tc is larger than the fractional one and the difference between these two shifts is Lo-independent. The correct interpretation of these results relies very much on the analysis of the applicability of the LG model and its relation to microscopic lattice models performed in Section 2. Our results for the width of the critical region and the fluctuation shifts mentioned above are in accordance with the intuitive notion that the fluctuation effects are enhanced as the effective spatial dimensionality of the system is lowered. The interesting case of D = 3-dimensional systems has not been investigated because of the divergences of the perturbation integrals. The three-dimensional FS systems need another investigation. We have established the relation 2 = D - 2 between the shift exponent 2 and the physical dimensionality D of the system which is valid for D ~> 4. Another method is needed to show whether this relation is correct for 3 < D < 4. We have investigated the universal (asymptotic) features of the second-order phase transitions in FS systems with a slab geometry. The essentially new results from our RG investigations in Sections 4 and 5 are the calculation of the fractional shift (4.7b) and the proof of the FSC up to the second order in the g expansions. Two limiting cases of small (Lo ~ ~) and large (Lo >> ~) thickness of the slab have been considered. The study can be extended to describe the crossover region (Lo ~ ~) by numerical calculations. We have investigated systems with short-range microscopic interactions (see Section 2). For long-range microscopic interactions the q-dependence in the spectrum of the free (noninteracting) fluctuations is of the type q~ and this leads to an additional dimensional crossover (D~ = D6 + 4 - 2 ~ ) for Lo-~ @. This effect has been extensively studied in quantum systems [15,20]. In systems with a < 2 the

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divergences at D = 3 discussed above will not occur. Except for the effective shift Du ~ D6 of the effective dimensionality and the more difficult calculations this case does not offer any novelty. In discussions of real systems with long-range interactions the dimensional shift (Du ~ D6) must be held in mind.

Acknowledgements One of us (D.I.U.) thanks RIKEN and the Department of Physics of Tokyo University where this work has been performed for their hospitality. The financial support by RIKEN and a research grand (F205) by the National Science Foundation (Sofia) are also acknowledged.

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