Finite-size and Fluctuation Effects on Phase Transition and Critical Phenomena using Mean-Field Approach Based on Renormalized ϕ 4 Model: I. Theory
R. M. Keumo Tsiaze, S. E. Mkam Tchouobiap, A. J. Fotué, C. Kenfack Sadem, J. E. Danga, C. Lukong Faï, M. N. Hounkonnou
aa r X i v : . [ m a t h - ph ] F e b Finite-size and Fluctuation Effects on Phase Transition andCritical Phenomena using Mean-Field Approach Based onRenormalized φ Model: I. Theory
R. M. Keumo Tsiaze (1 , , S. E. Mkam Tchouobiap (2) , A. J. Fotu´e (3) ,C. Kenfack Sadem (3) , J. E. Danga (3) , C. Lukong Fa¨ı (3) and M. N. Hounkonnou* (4)(1) Laboratoire de Science des Mat´eriaux, Facult´e des Sciences, Universit´e de Yaound´e I,B.P. 812 Yaound´e, Cameroun. (2)
Laboratory of Research on Advanced Materials and Nonlinear Science (LaRAMaNS),Department of Physics, Faculty of Sciences, University of Buea, PO Box 63, Buea, Cameroon (3)
Mesoscopic and multilayer structure laboratory, University of Dschang,P.O.Box 479, Dschang, Cameroun. (4)
International Chair in Mathematical Physics and Applications,University of Abomey-Calavi, 72 BP50 Cotonou, Republic of Benin. ∗ Corresponding authors; email: [email protected], [email protected] 2, 2018
Abstract
An investigation of the spatial fluctuations and their manifestationsin the vicinity of the quantum critical point within the framework ofthe renormalized φ theory is proposed. Relevant features are reportedthrough the Ginzburg-Landau-Wilson (GLW)-based calculations, com-bined with an efficient non perturbative technique. Both the dimensionand size, but also microscopic details of the system, leading to criticalbehavior, and strongly deviating from the classical mean-field approachfar from the thermodynamic limit, are taken into account. Further, theimportant role that harmonic and anharmonic fluctuations and finite-size effects can play in the determination of the characteristic propertiesof corresponding various systems, involving phase transitions and criticalphenomena, is then discussed in detail with emphasis on the qualitativevalidity of the analysis. Keywords:
Phase transition, critical phenomena, finite-size effects, harmonicand anharmonic fluctuations, Gaussian and non-Gaussian approximation, Hartree-Fock decoupling, mean-field theory, φ model . PACS numbers(s): Introduction
Nowadays the theory of phase transitions and critical phenomena seems to bewell developed in general. It enables us to obtain both universal and non-universalproperties for many model systems. However, there remains a number of unsolvedimportant questions among which one can cite, for instance, the role of fluctuationson the stabilization of the physical and electronic systems, the fluctuations thatinduced or prevent phase transitions, etc. There also exist some physical systemshaving as a common characteristics the fact that the complex microscopic behaviorunderlies macroscopic effects.In simple cases the microscopic fluctuations average out when larger scalesare considered, and the averaged quantities satisfy classical continuum equation.Hydrodynamics is a standard example of this, where atomic fluctuations averageout and the classical hydrodynamic equations emerge. Unfortunately, there is amuch more difficult class of problems where fluctuations persist out to macroscopicwavelengths, and fluctuations on all intermediate length scales are important too.In the last category are the problems of critical phenomena. The critical phe-nomena in the thermodynamic limit is characterized by the divergence of thecorrelation length ξ near critical point. Nowadays, the experimental techniqueshave become so advanced that the correlation length ξ can be pushed up to severalthousand ˚A and the samples under study become comparable with ξ . As a conse-quence, the effects of finite size of the samples on the critical phenomena becomeincreasingly important. Generally such effects depend on the shape of the sample,the boundary condition, the dimension of the system and the number of compo-nents of the order parameter. In fully finite or quasi-one-dimensional systems thephase transition is smeared out, whereas in thin films of thickness L the criticaltemperature T c ( L ) is shifted with respect to the bulk T c .On the other hand, during the first half of the last century after the discoveryof superconductivity the problem of fluctuation smearing of the superconductingtransition was not even considered. In bulk samples of traditional superconductorsthe critical temperature T c sharply divides the superconducting and the normalphases. It is worth mentioning that such behavior of the physical characteris-tics of superconductors is in perfect agreement both with the Ginzburg-Landau(GL) phenomenological theory (1950) [1] and the BCS microscopic theory of su-perconductivity (1957) [2]. The characteristics of high temperature and organicsuperconductors, low dimensional and amorphous superconducting systems stud-ied today, strongly differ from those of the traditional superconductors discussedin textbooks. The transitions turn out to be much more smeared out. The ap-pearance of superconducting fluctuations above the critical temperature leads toprecursor effects of the superconducting phase occurring while the system is still inthe normal phase, sometimes far from T c . The conductivity, the heat capacity, thediamagnetic susceptibility, the sound attenuation, etc. may increase considerably n the vicinity of the transition temperature [3].Recently, we proposed a microscopic renormalized Gaussian approach to criticalfluctuations in the GLW model and finite-size scaling to describe the phase behav-ior and critical phenomena in very varied systems [4]. Within this more rigorousapproach the effects of fluctuations are examined beyond the standard Gaussianapproach (SGA) [5, 6] in more rigorous detail, and we are able to establish theinsufficiencies of the mean-field theory (MFT), and also estimate the width of thecritical region where corrections to MFT are important. The approach allowedus to obtain the effective functional of the Gaussian GLW Hamiltonian ( H GLW )expressed in terms of the collective variables without any procedure for furthersystematic improvement (e.g. by considering ”higher order” terms).In this paper, we construct the H GLW (in the space of the fluctuating fields)devoted to the study of harmonic and anharmonic fluctuations and their mani-festations in the near-critical region for the GLW model. In the former case [4]we took into account only the Gaussian approximation and showed that the ob-tained results were better and in good agreement with the experience as thosefound within the framework of the SGA method. Here we generalize the approachtaking into account the powers of field higher than the first one. We need to con-struct a renormalized GLW which includes the quartic term that also takes intoaccount the microscopic details of the systems. The importance and role of the φ interaction were emphasized by Fisher and Wilson [7, 8, 9, 10]. Its presence allowsa low-temperature behavior and its absence leads to the divergence of the Hamil-tonian. The main ingredient in the analysis is as follows: the Fourier transformof the φ field theory can be seen as an interaction between the Fourier compo-nents of the order parameter, which are in fact a combination of harmonic andanharmonic fluctuations modes of the order parameter. This makes it possible forthe coefficients of the φ and φ terms to be renormalized by the φ term coeffi-cient. The approach is a strategy for dealing with problems involving many lengthscales. The strategy is to tackle the problem in steps, one step for each lengthscale. In the case of critical phenomena, the problem is, technically, to carry outstatistical averages over thermal fluctuations on all size scales. The method is tointegrate out the fluctuations in sequence, starting with fluctuations on an atomicscale and then moving to successively larger scales until fluctuations on all scaleshave been averaged out. The integration of this deviation in the self-consistencyon the entire spectrum of the lattice vibration, conducted to a very improved selfconsistent energy. This improvement of the self consistent problem by the pro-cessed harmonic and anharmonic fluctuations goes, for instance, to reflect itselfon the thermodynamic quantities and electronic parameters, in the neighborhoodof the phase transition, in particular the singularities as function of the controlparameter, resulting notably in a substantial modification of the critical point andthermodynamic quantities. The resulting self consistent problem of this approachis solved analytically, what permits us to extract an effective theory, with notably mean-field critical temperature renormalized by anharmonic fluctuations. Thepresent work does not present any renormalization based method in the sense of arenormalization group theory approach but rather a self-consistent method (SCM)improving on Landau theory (LT). Apart from calculations of exponents and scal-ing functions, it is necessary to develop techniques for obtaining the correctionsto the asymptotic critical behavior, in terms of a small number of non universalparameters which can be fit by experimental results on different materials. Inthis way, it is hoped that a more rigorous confrontation between experiments andtheory can be achieved. The starting point of our investigation is the following H GLW functional, H GLW ≃ Z d d rξ h ( ∇ r φ ( r )) + F L i , (1)where F L is the Landau free energy given by F L = a ( T ) φ ( r ) + b ( T c ) φ ( r ) + u φ ( r ) + · · · . (2)Here, φ is the model order parameter characterizing the mode displacement. Thequadratic coefficient a ( T ) = a ( T − T c ) vanishes linearly as the temperature ap-proaches the mean-field critical temperature T c . For stability purposes, the co-efficient constant u is chosen such that u >
0. The LT of second order (fer-romagnetic) phase transitions, for example, amounts to postulating the existenceof a development of type b ( T c ) = k B T c T (cid:12)(cid:12)(cid:12) T = T c close to T c , what is not right for thesuperconductors where the microscopic relations between the GL parameters aresuch as a b ( T c ) = π ζ (3) ν, where ζ ( x ) is the Riemann zeta-function ( ζ (3) = 1.202...), k B = 1 . × − W s/K is Boltzmann’s constant and ν a parameter which willbe defined later. But just like the quadratic coefficient, the constant b ( T c ) mustbe to improve to take into account both the microscopic details, dimension andsize of the system. The real expressions of a ( T ) and b ( T ) within the frameworkof a renormalized theory will be defined later. ξ is the coherence length of thesample (interpolated down to T = 0 K). In superconductors, average extension ξ of a Cooper pair as a measure of the distance within which the correlation form-ing Cooper pairs is active. In magnetic systems, ξ represents the lowest length(microscopic) for spin correlations (spin waves), while in solids, it represents thatof the vibrational or phonon modes. It is in general of the order of the interac-tion range. However, fluctuations with wavelength ξ will be seen to be alwaysnegligible. d is the dimensionality of physical space.The polynomial in Eq. (2) originates from a power series expansion of somepotential V ( φ ) [11, 12]. The order to which its terms are kept depends on physical onsiderations given by renormalization theory [13] and on the type of symmetry-breaking effect [14] Eq. (2) is supposed to model. The odd powers in Eq. (2) aredropped as a result of e.g. time-reversal invariance. A ” φ ” model is of interestsince, on contrast to ” φ ” models it can describe both second ( b ( T ) >
0) and firstorder ( b ( T ) <
0) phase transitions. Moreover, it displays a ”butterfly” catastrophe[15] which is more complex than the cusp catastrophe of the ” φ ” model. The ” φ ”model allows the ground state to be up to triply degenerate; situations with nodegeneracy, double degeneracy or triple degeneracy can be studied by varyingparameters in the expansion. The case a ( T ) = b ( T ) = 0 is of particular interest inthe context of tricritical points on phase diagrams [16, 17].LT [11] implicitly assumes that analyticity is maintained as all space-dependentfluctuations are averaged out. The loss of analyticity arises only when averagingover the values of the overall average (order parameter) φ . It is this overall aver-aging, over exp( − βF L ), which leads to the rule that F L must be minimized over φ . At temperatures that are far below, or far above a critical point, the behaviorof the order parameter reassembles a tranquil ocean with no significant amountof thermal noise in its fluctuations. But fluctuations become increasingly impor-tant near the critical point as the correlation length diverges. At the second-orderphase transition, infinitely long-range ”critical fluctuations” develop in the orderparameter. The study of these fluctuations requires that we go beyond LT.The SGA to the problem posed by the H GLW is to decompose φ ( r ) into itsFourier components φ ( q ) according to φ ( r ) = L − d/ X | q | < Λ φ ( q ) e i q · r . (3)According to Eq. (3) the limit on wavelengths means that the integration over q isrestricted to values of q with | q | < Λ. Averaging over long-wavelength fluctuationsnow reduce to integrating over the variables φ ( q ), for all | q | < Λ [18]. There aremany such variables; normally this would lead to many coupled integrals to carryout, a hopeless task. Considerable simplifications will be made below in order tocarry out these integrations.We need an integrand for these integrations. The integrand is a constrainedsum of the Boltzmann factor k B over all atomic configurations. The constraintsare that all | q | < Λ are held fixed. this is a generalization of the constrained sumin the LT. We shall assume Landau’s analysis is still valid for the form of H GLW ,that is, H GLW is given by Eq. (1). However, the importance of long-wavelengthfluctuations means that the parameters a ( T ) and b ( T ) depend on Λ and thenon the dimension d of the system. The d dependence of a ( T ) and b ( T ) will bedetermined shortly. However the breakdown of analyticity at the critical point isa simple consequence of this d dependence. Details will also be discussed shortly.The change of variables to the Fourier modes, the H GLW expansion is given by he expression H GLW ≃ X q< Λ ¯ G − ( q ) φ ( q ) φ ( − q )+ b ( T c ) L d X { q } < Λ φ ( q ′ ) φ ( q ′′ ) φ ( q ′′′ ) φ ( − q ′ − q ′′ − q ′′′ )+ u L d X { q i } < Λ φ ( q ) φ ( q ) φ ( q ) φ ( q ) φ ( q ) φ ( − q − q − q − q − q ) (4)+ · · · where the q -mode function ¯ G − ( q ) = a ( T ) + cq , and L is the linear dimensionof the sample. Therefore, it is useful to express the partition function Z as afunctional integral of the wave vector fluctuations φ ( q ). Accordingly, Z generalizesand therefore factorizes into Z = Z Dφ · exp " − β ( X q ¯ G − ( q ) φ ( q ) φ ( − q )+ bL d X { q } < Λ φ ( q ′ ) φ ( q ′′ ) φ ( q ′′′ ) φ ( − q ′ − q ′′ − q ′′′ )+ u L d X { q i } < Λ φ ( q ) φ ( q ) φ ( q ) φ ( q ) φ ( q ) φ ( − q − q − q − q − q ) + · · · ) , (5)where Dφ is given by D φ = Y q (2 π ) − dφ ( q ) dφ ( − q ) , (6)and β = 1 /k B T . D φ is used to denote the measure of the functional integral.This quantity ensures that the total probability is normalized to unity through aconstant of proportionality 1 / (2 π )[13, 19]. (2 π ) − left out is formally divergentin the thermodynamic limit; it does not affect averages that are obtained fromderivatives of such integrals.In papers [20] and [21], a yet another approach to critical fluctuations hasbeen proposed. It is based on the fact that quartic term is dominant and nearcriticality since a ( T ) → T → T c . In the first paper [20] using mean-fieldapproximation (MFA), the influence of homogeneous fluctuations was examinedthrough the expansion H MFA ∼ = H GLW ( ¯ φ ) + λ ( φ − ¯ φ ) + λ ( φ − ¯ φ ) + · · · (7)where ¯ φ is the equilibrium value of the order parameter and H GLW is given by Eq.(1) without the Ginzburg term. In the second paper [21] expansion like that in q. (7) was studied for case other than spontaneous second-order transitions, i.e.,for field-induced transitions, first-order transitions, and liquid-vapor transitions.In both papers calculations were performed using non-Gaussian integral givenby[22, 23] Z ∞ φ mp − exp (cid:0) − λ φ m − λ φ m (cid:1) dφ = (2 m ) − (2 λ ) − p/ Γ( p ) D − p (cid:16) λ / p λ (cid:17) exp (cid:16) λ / λ (cid:17) , (8)and analyzed in all regimes including finite-size. D − p is the parabolic cylinderfunction [24] and Γ is usually so-called gamma function. The most interesting con-clusions were that Gaussian approximation fails for all values of parameters exceptfor in the thermodynamic limit ( V −→ ∞ ). As a result, power-law predictions ofthe Gaussian prediction are incorrect, except for at V = ∞ , and should thereforebe replaced by exponential asymptotic behavior as predicted by non-Gaussianmethods. This results, for example, in vastly different asymptotic predictionsfor finite-size scaling as was discussed at length in both papers [20, 21]. Thesetwo papers, however, dealt exclusively with mean-field properties of non-Gaussiancritical fluctuations, setting the Ginzburg term to zero. This, of course, neglecteda very important property of critical systems, i.e., their spatial inhomogeneity[1]. Nonetheless, interesting results were obtained [20, 21], leading asymptoticallyto the Gaussian approximation results and also providing finite-size scaling for T = T c . Therefore, we believe that the proposed non-Gaussian method offers largeregion of analyticity and possesses better convergence properties. The present pa-per is intended to provide another insight into the problem by using non-Gaussianway of averages calculation and keeping a significant part of the Ginzburg term.We believe the renormalized H GLW is capable of properly describing both the crit-ical and near-critical regimes. A non-Gaussian method of calculation, however,must be employed to adequately reveal the deviation from asymptotic behaviorand thus the crossover phenomena.
A major limitation of the GLW theory is its incapacity to account for thecharacteristics of the intermediate mode between the adiabatic mode on the onehand and the non-adiabatic mode on the other hand. It is also and especiallyits incapacity to take account of the microscopic details and the dimensionalityof the systems. For instance, the GLW theory also predicts incorrect results,like an unphysical phase transition in 1D, or incorrect critical indices, in higherdimensions. However, Scalapino, Sears and Ferrell [25] showed that, at least in 1D,this failure is due to an improper treatment of fluctuations, so the complete successof the GLW theory will depend essentially on how fluctuations are taking intoaccount. Both mean-field theories and Gaussian-based power expansions as well s renormalisation-group calculations are concerned with the asymptotic propertiesof critical systems in the sense of infinite size and immediate proximity to T c . Thusthey predict singular power-law behavior of the systems with universal exponentsand scaling functions. As has been recently made abundantly clear, the asymptoticregime is indeed very small and agreement with experimental data deterioratesvery rapidly outside the near-critical region; in fact extrapolation from the critical-state equations fails outside criticality and vice-versa, and the analytic noncriticalequations of state do not reproduce the correct singular behavior at criticality.Thus there is a real challenge and a need to develop a method of calculation thatboth incorporates the asymptotic critical behavior and the crossover to regularregime.In the approximation that we propose, we wish to include those mode-modecoupling terms which involve balanced pairs of q and - q wave vectors, assumingthat the remaining combinations are less important as they may lead to numerouscancelations. We can analytically understand the effective free energy of the expo-nential argument in Eq. (5) as that containing the quadratic term for the modesof free fluctuations of the order parameter φ ( q ) (quadratic form), the quartic termfor the modes of harmonic fluctuations of the order parameter φ ( q ), and the sixth-order term as the anharmonic interactions between these modes. This conditionseems to be effectively justified if the sixth-order coefficient of the expansion issmall.Accordingly, it is important in this case to precise the following scenario:(i) The φ and φ terms are linked to the essential fluctuations. (ii) The φ term is linked to the redundant fluctuations. (iii) All other terms correspond tothe unessential fluctuations, i.e. contain no essential new physics and in fact are”irrelevant” (to the zero-temperature critical behavior in the sense described byWilson [6, 7, 8]).Therefore, the sixth-order term appears as and seems to play the role of inter-action terms between the Fourier components of the order parameter, which are infact the combination of harmonic and anharmonic fluctuation modes of the orderparameter. The sixth-order term can thus contribute to improve the quadratic andquartic coefficients. This makes it possible for the coefficients of the φ term tobe renormalized ”an-harmonically” and the φ term to be renormalized ”harmon-ically” by the φ term coefficient. It should keep in mind that the GLW theory isa phenomenological theory, founded on the intuition, with its own laws and rules.It accounts for the phenomena and it is that its justification.In this study, we will go beyond the GLW theory in order to evaluate theinfluence of the fluctuations on the critical phenomena and particularly criticaltemperature for systems with short-range interactions. We use a Hartree-Fockdecoupling for φ interactions in the consideration of the absence of long-rangeinteractions. This approximation will consist in considering that the Fourier com-ponents interact (in harmonical and anharmonical manner) only through the mean eld produced by other modes. In the case of continuous phase transition this ideaallows us to decouple the φ term into a sum of product of two quantities witheven powers:1 L d X { q i } < Λ φ ( q ) φ ( q ) φ ( q ) φ ( q ) φ ( q ) φ ( − q − q − q − q − q ) ≈ X q L d X { q i } < Λ D(cid:12)(cid:12) φ ( q ′ ) φ ( q ′′ ) φ ( q ′′′ ) φ ( − q ′ − q ′′ − q ′′′ )) (cid:12)(cid:12)E! φ ( q ) φ ( − q )+ 15 L d X { q i } L d X q< Λ D(cid:12)(cid:12) φ ( q ) φ ( − q ) (cid:12)(cid:12)E! φ ( q ′ ) φ ( q ′′ ) φ ( q ′′′ ) φ ( − q ′ − q ′′ − q ′′′ )) + · · · . (9)The factor 15 takes into account all possible contractions(statistical average of twoor four modes among six). The terms between brackets act like mean fields. Thesixth order term thus uncoupled consequently becomes a combination of quarticand quadratic terms, enabling us to obtain an effective non-Gaussian theory. Withall these important preliminaries and considerations, one may then convenientlywrite the effective H GLW in the absence of an external field as H eff [ φ ] = Z d d rξ (cid:20)(cid:0) ∇ r φ ( r ) (cid:1) + a ∗ ( T ) φ ( r ) + b ∗ ( T )) φ ( r ) + · · · (cid:21) , (10)and the result of applying Eq. (3) to Eq. (10) is H eff [ φ ] ≃ X q< Λ ¯ G ∗− ( q ) φ ( q ) φ ( − q )+ b ∗ ( T ) L d X { q } < Λ φ ( q ′ ) φ ( q ′′ ) φ ( q ′′′ ) φ ( − q ′ − q ′′ − q ′′′ ) + · · · (11)where the renormalized q -mode function ¯ G ∗− ( q ) = a ∗ ( T ) + cq .The quadratic coefficient a ∗ ( T ) and the quartic coefficient b ∗ ( T ) are now re-spectively given by a ∗ ( T ) = a ( T ) + Ω d ( T ) (12) b ∗ ( T ) = b ( T c ) + Θ d ( T ) . (13)The term Ω d ( T ) = 15 u L d X { q i } < Λ D(cid:12)(cid:12) φ ( q ′ ) φ ( q ′′ ) φ ( q ′′′ ) φ ( − q ′ − q ′′ − q ′′′ ) (cid:12)(cid:12)E , (14)includes a new contribution to the critical point and corresponds to the anharmonicvariance of the order parameter at a single point in space evaluated at temper-ature T. This term competes with Landau quadratic coefficient a ( T ). Therefore t determines the critical line and hence incorporates both the asymptotic criticalbehavior and the crossover to the regular regime.The renormalized quartic coefficient obtained within the renormalized GLWapproach is now given by Eq. (13) with the correction term Θ d ( T ). The termΘ d ( T ) = 15 u L d X q D(cid:12)(cid:12) φ ( q ) φ ( − q ) (cid:12)(cid:12)E , (15)then also includes another new contribution to the quartic coefficient. This termcompetes with Landau quartic coefficient b ( T c ) and defines the tricritical crossoverexponent. Both scaling densities Ω d ( T ) and Θ d ( T ) are relevant, i.e., exhibit criticalfluctuations. Now the quartic coefficient can be cancelled at some point highlight-ing the existence of a tricritical point. This fact represents the principal differencebetween the transitions corresponding to the tricritical Gaussian fixed point andthe ordinary second-order Gaussian fixed point. Hence the tricritical fixed pointcan be characterized as the simultaneous instability of the system to two types ofcritical fluctuations. Correction to the molecular-field tricritical behavior due tocritical fluctuations will be discussed in a forthcoming paper. It is found that theasymptotic tricritical form of some thermodynamic quantities or functions is nota power law but a power law multiplied by a fractional power of a logarithm.Taking a look at the influence of spatial fluctuations on the stability of therenormalized GLW analysis, the importance of fluctuations seems to be evidentand the divergence of the critical coefficient of the second-order term a ∗ ( T ) is nowrelated to the system size L and dimensionality d of the system. Therefore itis useful to determine the temperature at the critical point where the divergenceof quadratic coefficient should be observed. Accordingly, the modified criticaltemperature is given by T ∗ c = T c − Ω dc a (16)where Ω dc = Ω d ( T ∗ c ) is the solution of the self-consistent Eq. (25) that will bediscussed in the next subsection.Eq. (16) shows that T ∗ c can be neglected if the anharmonic fluctuations aretoo high. However, taking into consideration the φ term contained in the H GLW functional leads to the emergence of the limitations of the GLW approach andMFA related to the critical temperature T c , which is presented here as a charac-teristic scale of temperature related to both thermic fluctuations and finite-sizeeffects rather than the transition temperature [4]. Although that limitation is adisadvantage to the exactness of the GLW approach or MFA, its predictions arenot at all lacking interest and only the importance of the anharmonic term Ω dc should determine its degree of validity.Structural phase transition (for example) is accompanied by a change in struc-ture. Some of these changes in structure occur without macroscopic diffusion ofmatter in solids. They are initiated by local motions of atoms or molecular groups hich can distort the lattice in the high temperature phase to form structuresof lower symmetry at lower temperatures. These movements around equilibriumpositions do not occur instantaneously at temperature T c , and they are actuallyinitiated at a temperature lower or higher than the transition temperature [26]. Ina way, this pretransition or pretransformation phenomenon is the equivalent of thenucleation process. Eq. (16) can thus contribute to clarify this viewpoint, sinceΩ dc which is the solution of a n degree polynomial can be a positive or negativequantity. Ω d ( T ) , Θ d ( T ) and thermodynamic quan-tities A correct treatment of Ω d ( T ) shows that Θ d ( T ) is much more complex. Oncecritical fluctuations are not treated as the constants, one could imagine expandingorder parameter in a Taylor’s series about its value at some central location r .This means that H GLW could be a complicated functional of φ , an expression thatis hard to write down, with several parameters, instead of the simple GLW formwith only two parameters a ( T ) and b ( T c ). Consequently, a natural recourse is touse Gaussian approximation.Gaussian measures play a central role in many fields: in probability theory as aconsequence of the central limit theorem, in quantum mechanics, in quantum fieldtheory, in the theory of phase transitions in statistical physics. In this section,calculations will be performed and analyzed in all regimes including finite sizes,using the Gaussian integrals in the form [24], Z ∞ y n exp( − py ) dy = (2 n − p ) n r πpp > , n = 0 , , , · · · ; (2 n − · · · · · (2 n − . (17)We can formulate the Hamiltonian by averaging the new contributions. The pro-cedure for identifying the variational parameter of the quadratic and quartic coeffi-cients, Ω d ( T ) and Θ d ( T ), and for determining the value of T ∗ c and thermodynamicsquantities is as follows. First the Hamiltonian is derived in the MFA and the ob-tained Ω d ( T ) is substituted for the self-consistent equation. The derived equationis used to determine both the value of T ∗ c and thermodynamic quantities.As a result, it is shown that the value of T ∗ c and thermodynamic quantitiesstrongly deviate from the classical MFA but thermodynamic quantities adoptmean-field critical exponents provided by the classical Landau approach. We findthat the variational parameter of the Hamiltonian obtained by renormalizing thatof the quadratic term, corresponding to the square of angular frequency, is pro-portional to T ∗ c − T . he thermal average of a physical quantity y ( φ ) is given by h y i = Z ∞ y ( φ ) exp[ − βH eff ] dφ , Z ∞ exp[ − βH eff ] dφ. (18)The obvious difficulty that the transformation given by Eq. (3) brings relatesto the mode-mode coupling present in the last term in Eq. (11). Consequently, anatural recourse is to use the Gaussian approximation where H eff [ φ ] is truncatedto H [ φ ] = X q< Λ ¯ G ∗− ( q ) φ ( q ) φ ( − q ) (19)This then conveniently factorizes the partition function as Z = Y q< Λ s πβ (cid:16) a ( T − T c ) + Ω d + q (cid:17) . (20)By making a transition to continuum through X q ( ... ) = L d (2 π ) − d Z ∞−∞ ( ... ) d d q, (21)the free energy is F = − k B T L d π ) d Z ∞ ln " πβ (cid:16) a ( T − T c ) + Ω d + q (cid:17) d d q (22)The heat capacity of the system is given by C ∼ = − T ∂ F ∂T = κ d h TT c i (cid:16) a + ∂ Ω d ∂T (cid:17) (cid:16) ǫ + Ω d a T c (cid:17) d/ − + less singular terms , (23)where κ d = k B L d ( ξ ) − d d − π d/ a Γ( d/ Z + ∞ x d − dx (1 + x ) , ǫ = T − T c T c . (24)The element volume is ℜ d q d − dq , with ℜ d = π d/ Γ( d/ the sphere unit surface in R d space. κ d is an integral correction constant which is lattice dimension dependent.The behavior of the integral correction constant changes dramatically at d = 4.For d > x and is dominated by the upper cut-off Λ, while for d <
4, the integral is convergent in both limits. Although theexpression of the specific heat is presented in not so complicated form, it is notquite transparent to know what type of behavior it will exhibit depending ondimension. Only the nature of Ω d will make it possible to describe its behavior. he dominant behavior of C close to T ∗ c is through C ∗ ∼ | ǫ ∗ | dν − , where thereduced and renormalized temperature ǫ ∗ = ǫ + Ω dc a T c and ν = 1 /
2, so that weobtain α = 2 − dν as in the SGA . When thermal fluctuations can be neglected,as it appears to be the case in conventional superconductors, the specific heatexhibits at the transition temperature T c ( L → ∞ ) a step discontinuity. Thisdiffers drastically from the behavior when thermal fluctuations dominate. Dueto the finite-size effect, the specific heat peak occurs at a temperature T ∗ c ( L )shifted from the homogeneous system by an amount proportional to L − /ν , andthe magnitude of the peak located at temperature T ∗ c ( L ) scales as L α/ν [27]. Thisresult seems to be qualitatively in good agreement with the well-known result since C ∼ | T − T ⋆c | dν − [6]. Ω d ( T ) , Θ d ( T ) The structural phase transition has been, hiterto, treated as follow[28]: In the”quasiharmonic” approximation [29, 30], it was assumed that harmonic frequencyfor the unstable mode is purely imaginary, namely, ω < ω = ω + c h u i ∝ T − T c where u denote the displacement from the interatomic distance. Other soft-modetheories are LT and MFT [31, 32]. In the expansion of free energy, the normalcoordinate was adopted as the order parameter. It was shown that the criticalcoefficient of the second-order term, a ( T ), equals the square of the soft-modefrequency, a ( T ) = ω ∝ T c − T . The LT gives a qualitatively correct view of thesoft mode, but cannot explain how it occurs; furthermore, this theory is not validclose to T c where critical fluctuations can no longer be neglected. So its completesuccess will depend essentially on how fluctuations are taken into account.In the present work, we show the approach to this problem from the micro-scopic point of view, both in terms of the order parameter and self-correlationfunctions h φ i and h φ i . From Eqs. (17), (18) and (21), we obtain the result thatthe temperature dependence of the variational parameters, Ω d ( T ) and Θ d ( T ), cor-responding to the anharmonic and harmonic fluctuations of the order parameter,are presented by Ω d = K d V (cid:20) TT c (cid:21) (cid:18) ǫ + Ω d a T c (cid:19) d − where V = L d , (25)and Θ d = K d (cid:20) TT c (cid:21)(cid:18) ǫ + Ω d a T c (cid:19) d − . (26) K d and K d are the dimension-dependent constants, respectively, and explicitly stablished as follows: K d = 45 u k B ξ − d d − π d/ a Γ( d/ Z + ∞ x d − dx (1 + x ) , (27)and K d = 15 u k B ξ − d d − π d/ a Γ( d/ Z + ∞ x d − dx (1 + x ) . (28)The equations ((25) and (26)) are derived taking into account the expected factthat Ω d ( T ) and Θ d ( T ) do not explicitly depend on wave number q . The behaviorof K d is the same one as that of κ d . For certain values of d, the integral K d divergesat large x . This is not a first time when such problem arises and we know how todeal with it: this ultra-violet (UV) divergence is related to the restrictions on theapplicability of the GLW functional for | q | & ξ − , so the integral has to be cut offat ξ · | q | = x c ∼ d <
4) have alarge cut-off limit after a simple renormalization, that is, after one has taken thedeviation from the critical temperature as a parameter. The field amplitude renor-malizations are finite. In the critical phenomena situation instead, the fluctuations,which is related to microscopic parameters of the theory, is fixed. This means that,after the introduction by rescaling of the cut-off Λ, quadratic coefficient remainsfinite when Λ → ∞ for T = T ∗ c .As we are concerned with phase fluctuations, Eqs. (25) and (26) show thattheir effect is dimension d dependent. It has been demonstrated that the inclusionof phase fluctuations leads to a reduction in the degree of order in d > d ≤
2; for d >
2, the phase fluctuations are finitewhile they become asymptotically large for d ≤ Mermin-Wagner theorem [34]: itstates that there is no spontaneous breaking of a continuous symmetry in systemswith short-range interaction in dimension d ≤
2, and as a corollary the borderlinedimensionality of 2, known as the lower critical dimension d l has to be treatedcarefully. With regards to Eq. (26) dimension d = 2 seems to play a crucial role inagreement with the Mermin-Wagner theorem . Indeed, when d = 2, the quantity d/ − d (Eq. (26)) is reduced to a linear function of T with a positive coefficient asΘ d = K d h TT c i . On the other hand, above d = 4, the MFT predicts correctly theuniversal quantities, whereas it is definitely not valid for dimension 4 and below.(Eq. (25) seems to arise this aspect with the quantity d/ − d = 4, and the self-consistent equation for the anharmonic correction termΩ d (Eq. (25)) is reduced to a function of T with a positive coefficient as Ω d = K d h TT c i . Although the expressions of the correction terms are presented in not socomplicated form, it is not quite transparent to know what type of behavior they ill exhibit depending on dimension, temperature and the size of the system. Thepossibility that Θ d or Ω d increases with decreasing temperature could highlightthe quantum character of the system according to both the dimensionality andthe size. However, the numerical aspect which will be approached in anotherwork, will enable us to better clarify the behavior of Θ d and Ω d and to give thema suitable physical direction.The renormalized critical coefficient of the second-order term a ∗ ( T ), equals thesquare of the renormalized soft-mode frequency, a ∗ ( T ) = ω ∗ ∝ T ∗ c − T , but thecritical temperature now takes into account both the dimension of the system, itssize and critical fluctuation. To determine the degree of validity of this approach, let us formulate theGinzburg criterion [35, 36], which usually tells us quantitatively when MFT isvalid. It is clear that the fluctuations become more and more pronounced as thetemperature approaches the true critical point T ∗ c . The Ginzburg criterion in-dicates in a semi-quantitative manner the temperature range where the distancefrom the SCM is important.Therefore, the critical Ginzburg width [35, 37] in the vicinity of the criticaltemperature is then given by∆ t G = (cid:12)(cid:12)(cid:12) T ∗ c − T c T c (cid:12)(cid:12)(cid:12) = Ω dc a T c . (29)In the last formula, Ω dc = Ω d ( T ∗ c ) is the solution of the following self-consistentequation (obtained in the same spirit as Eq. (19) in [4]): (cid:16) d − (cid:17) ln h | ǫ | + Ω dc a T c i = ln h a T c Ω dc K d ( a T c − Ω dc ) i (30)Here, T ∗ c also determines the temperature under which the description of fluctua-tions goes beyond the independence of the MFA modes due to precise agreement.∆ t G corresponds to the width of the critical region about the real transition tem-perature for which both the Landau and the GL theories are valid. Further bytaking into account the effective GLW Hamiltonian, this quantity also defines thedomain of validity of the classical critical behavior [38]. The classical descriptionfails for | ∆ t | ≡ | T /T ∗ c − | ≪ | ∆ t G | . This finding is corroborated by the Ginzburgcriterion and can be interpreted as [13, 35]. Thus, the Ginzburg criterion allowsus to restore some credibility to the MFT in those cases. Eq. (25) shows that theanharmonic fluctuation Ω d → V → ∞ involving a ∗ ( T ) → a ( T ). In thethermodynamic limit, the renormalized quadratic coefficient a ∗ ( T ) is the same asin the LT. It is well established that power-law predictions of the LT are correctin the thermodynamic limit, T ∗ c → T c when V → ∞ . s it is well established (for V = ∞ ), no Gaussian approximation even renor-malized in some way can describe precisely critical phenomena near the transitionpoint. Also, within this approach, the effects of spatial fluctuations are stronglydependent on lattice dimensionality and we can now appreciate how fluctuationsand correlations modify the macroscopic thermodynamic properties.From the thermodynamic definition, the inverse renormalized susceptibility isgiven by the following analytic expression: χ ∗− = a ( T − T c ) + Ω d . (31)Taking a look at the influence of spatial fluctuations on the stability of the renor-malized mean-field analysis, the importance of fluctuations seems to be evidentand the divergence of the susceptibility is related to the system size L and dimen-sionality d .The correlation length gives information about the distance for which the orderparameter φ varies in the space. In order to calculate the correlation range, wefirst evaluate ∂ G ∗ ( q ) /∂q where G ∗ ( q ) is the renormalized q -mode autocorrelationfunction. Taking into account Eq. (20), the function G ∗ ( q ) is obtained as G ∗ ( q ) = h| φ ( q ) | i = (cid:20) β (cid:18) a ( T ) + Ω d + q (cid:19)(cid:21) − . (32)Accordingly, as T −→ T ∗ c within the limit q −→
0, it diverges according to G ∗ ( q , T = T ∗ c ) ∼ q η − , with a small positive value η and the critical exponent η = 0. Therefore, ¯ G − ( q ) appears as the reduced inverse q-mode autocorrelationfunction. According to Eq. (32), the renormalized correlation length is obtainedas ξ ⋆d ( T ) = (cid:20) − lim q → G ⋆ − ( q ) (cid:18) ∂ G ⋆ ( q ) ∂q (cid:19)(cid:21) / = ξ +0 T / c h T − T ⋆c + a (Ω d − Ω dc ) i − / , for T > T ⋆c ξ − T / c h T ⋆c − T + a (Ω dc − Ω d ) i − / , for T < T ⋆c (33)which generally sets the characteristic length scale of fluctuations. The universalexponents and amplitude ratios are again recovered from this equation with regardto the critical temperature T ∗ c . Hence, the corresponding critical exponents are ν = ν ′ = 1/2. The temperature dependence of the renormalized correlation length ξ ∗ d ( T ) based on the above formula depends on the dimensionality d and the size L of the physical space and behaves differently in function of them.For finite-size systems (in d = 2,3) it has been recognized [9, 39] that the systemsize L ”scales” with the correlation length of the bulk system. Thus it is convenient o define a reduced length of the system: l ∗ = L/ξ ∗ = l T − / c h T ∗ c − T + 1 a (Ω dc − Ω d ) i / , for T < T ⋆c . (34)The value of l ∼ ξ − depends only on the nature of the substance under study. Infact both l ∗ and l are dimensionless, and they are the ratio of the real thicknessof the system to certain characteristic length. Indeed, if l ∗ ≫
1, no significantfinite-size effects should be observed. On the other hand, for l ∗ ≤
1, the systemsize will cut-off long-distance correlations so that an appreciable finite-size round-ing of critical-point singularities is to be expected. This result is not surprisingconsidering the physical meaning of the correlation length which can be regardedas an indication of the influence range of the boundary condition. One shouldnotice that what plays a role is not the real length L of the system but the reduceone l ∗ which depends also on the deviation from the bulk critical point. Detailswill be discussed later in the section reserved for the applications. This showsthat if the reduced length of the system is too short, either due to the small sizeof the system or due to its closeness to the bulk critical point, the influence ofthe boundary will strongly dominate. As l ∗ becomes large, the renormalized H becomes independent of details of the system at the atomic level. This leads to anexplanation of the universality of critical behavior for different kinds of systems atthe atomic level. Liquid-gas transitions, magnetic transitions, alloys transitions,etc., all show the same critical exponents experimentally; theoretically this can beunderstood from the hypothesis that the same ”fixed point” interaction describesall these systems.In order to illustrate the meaning of the correlation length, it is perhaps worthcalculating the space-dependent correlation function G ∗ d ( X ) defined by the Fouriertransform of the q-mode autocorrelation function as G ∗ d ( X ) = L − d/ Z exp( iqX ) G ∗ ( q ) d d q, (35)which can also be considered here in the sense of the entanglement in the modelsystem. The propagator G ∗ d ( X ) should always go to zero at large distances, so thata measure of order in the system is the long-length behaviour of the propagator.If the propagator goes to zero then the system can only have short-range order. Ifthe propagator goes to a non-zero constant at large distances, then we must havea non-zero order parameter, and the system has long-range order.The integral (35) is well behaved at small q . At large q without the exponentialthe integral would only be convergent for d = 1. For d = 2 there would be a logdivergence, while for higher d there would be an algebraic divergence. Clearly thisdivergence is cut off by the exponential factor. As the exponential goes to 1 when X goes to zero, the correlation function should ”diverge” at small distances. Infact the divergence will be cut off by the discretization length Λ (discreteness of nderline lattice). The choice of scale depends on the situation we want to study.First of all, in a numerical attempt to describe experimental data, it is clear thatfor most practical cases the cutoff Λ must not be chosen too large, since this wouldrequire a knowledge of the lattice dispersion way beyond the parabolic approxima-tion corresponding to the simple gradient correction in real space. However, forthe study of phase transitions, this is not a severe restriction since the anomaliesappearing in such transitions result from long range correlations, i.e. the behaviorof the system at small q -vectors. For large distances X > ξ , the exponentialis oscillating much faster than any other variations, and the correlation functionshould fall rapidly to zero (in fact exponentially). This behaviour is demonstratedif we explicitly calculate the integral for dimensions one, two and three.Using Eq. (35) and due to the fact that the space-dependent correlation func-tion assumes various forms depending on the dimensionality of the physical space[5], G ⋆d ( X ) satisfies G ∗ d ( X ) = (cid:16) xξ ∗ (cid:17) exp h − (cid:16) xξ ∗ (cid:17)i , for d = 1 π K (cid:16) ρξ ∗ (cid:17) , for d = 2 (cid:16) π r (cid:17) exp h − (cid:16) rξ ∗ (cid:17)i for d = 3 , (36)where ρ = ( x + y ) / , and r = ( x + y + z ) / . K ( r ) is the modified Besselfunction (that is logarithmic for small arguments and exponentially decaying forlarge arguments).To keep the discussion self-contained, in the remainder of this subsection wereview the predictions for the static critical exponents. First, we define the reducedtemperature ǫ ∗ = ( T − T ∗ c ) /T c . The exponents α , β , γ , η and ν describe thesingular behavior of the theory with strictly zero renormalized quadratic coefficientas ǫ ∗ →
0. For the specific heat, taking into account Eq. (23) one finds C ( T ) ∼ | ǫ ∗ | − α + less singular terms (37) η and ν describe the behavior of the correlation length ξ , where G αβ ( r ) ≡ h φ ( r ) α φ (0) β i − h φ α ih φ β i (38)and the exponent η is defined through the behavior of the Fourier transform of thecorrelation function: G αβ ( q → ∼ q − η . (39)The correlation length exponent ν is defined by ξ ∼ | ǫ ∗ | − ν (40) he exponent β will be defined later. The last exponent, δ, is related to thebehavior of the system in a small magnetic field h which explicitly breaks the O (4) symmetry. The six critical exponents defined above are related by fourscaling relations [6]. α = 2 − dν, α + β (1 + δ ) = 2 = α + 2 β + γ, γ = β ( δ − . (41) φ theory All perturbative approaches are based on the division of the free energy into aGaussian term and higher order perturbative terms. In fact, the coupling constantwith φ model of the perturbation is not necessary small, so that the convergenceof the perturbation expansion cannot be ensured. Thus some more effective ap-proaches to the calculation are needed. For systems with boundaries, one shouldconsider the influences of the boundaries on the thermal properties near the bulkcritical point. In additions, the spatial distributions of the order parameter shouldbe taken into account for finite-size systems. As is well known for finite-size sys-tem, however, the spatial distribution cannot be considered as uniform any longerdue to the influence of the boundary though the condition of minimum free energywould prefer a smooth distribution. Different thermodynamic phases are characterized by certain macroscopic, usu-ally extensive state variables called order parameters; examples are the magneti-zation in ferromagnetic systems, polarization in ferroelectrics, and the macroscop-ically occupied ground-state wave function for superfluids and superconductors.We shall henceforth set our order parameter to vanish in the high temperaturedisordered phase, and to assume a finite value in the low-temperature orderedphase. Landau’s basic construction of a general mean-field description for phasetransitions relies on an expansion of the free energy (density) in terms of the or-der parameter, naturally constrained by the symmetries of the physical systemunder consideration [40]. For example, consider a scalar order parameter ψ withdiscrete inversion or Z symmetry that in the ordered phase may take either oftwo degenerate values ψ ± = ±| ψ | [41]. We shall see that the following genericexpansion (with real and renormalized coefficients) indeed describes a continuousor second-order phase transition: F ∗ L = a ∗ ( T ) ψ + b ∗ ( T ) ψ + ... − hψ, (42)if the temperature-dependent parameter a ∗ changes sign at T ∗ c . For simplicity, andagain in the spirit of a regular Taylor expansion, we let a ∗ ( T ) = a ( T − T ∗ c ), where T ∗ c denotes the critical temperature from mean-field approach with renormalized φ odel. The free energy is of almost the same as the Landau functional, except forthe presence of the fluctuations-dependent in the quadratic and quartic coefficients,which contain the essential information about the microscopic nature of the system,its size and dimension. Our functional therefore describes a set of interacting,weakly-GLW-damped excitations.Stability requires that b ∗ > b ∗ at T ∗ c . Details will be discussed shortly. Note that theexternal field h, thermodynamically conjugate to the order parameter, explicitlybreaks the assumed Z symmetry ψ → − ψ . Minimizing the free energy with re-spect to ψ then yields the thermodynamic ground state. Thus, from ∂F ∗ L /∂ψ = 0we immediately infer the equation of state h ( T, ψ ) = 2 a ∗ ψ + 4 b ∗ ψ (43)and the minimization or stability condition reads 0 < ∂ F ∗ L /∂ψ = 2 a ∗ + 12 b ∗ ψ .At T = T ∗ c , Eq. (43) reduces to the critical isotherm h ( T ∗ c , ψ ) = 4 b ∗ ψ . For a ∗ ( T ) >
0, the spontaneous order parameter at zero external field h = 0 vanishes;for a ∗ ( T ) <
0, one obtains ψ ± = ±| ψ | , where ψ = p | a ∗ | / b ∗ = f ( T ) · φ (44) φ = p | a | / b is the order parameter in LT. The behavior of the renormalizedorder parameter defines β : h| ψ |i ∼ | ǫ ∗ | β for ǫ ∗ < ψ enters in theamplitude prefactor f ( T ) defined by f ( T ) = s | d /a ( T ) | d /b ( T c ) . (46)The SCM (taking into account both the dimension, the finite-size effects and thetemperature dependence of f ( T )) reveals a competition between three scales ofenergies that are in competition: the thermal energy a ( T ) ∝ k B T versus the en-ergy resulting to anharmonic instabilities Ω d on one hand, and the anharmonicinstabilities versus the harmonic instabilities on the other hand. These competi-tions determine both the existence of the transition, but also the adiabatic regimeof fluctuations of the ”pre-transition”.As in many case, it is also possible to write the generalized GLW functional ina form which preserves certain transformations of ψ ( r ) like either of the three.reversal ψ ( r ) → − ψ ( r ) (47) hange of phase ψ ( r ) → e iθ ψ ( r ) (48)rotation ψ ( r ) → U ψ ( r ) (49)(U is a rotation matrix). This holds for example for the Ising model, superfluidhelium and the Heisenberg model respectively. The amount of ψ is uniquely de-fined, but the sign, phase and direction of ψ ( r ), are not defined. It depends on thehistory (preparation in an external field which removes the symmetry Eq. (48) oraccidental fluctuations) of the system.Note the emergence of characteristic power laws in the thermodynamic functionsthat describe the properties near the renormalized critical point located at T = T ∗ c ,h = 0. Inserting Eq. (44) into the Landau free energy Eq. (42) one finds for T < T ∗ c and h = 0 F ∗ L ( ψ ± ) = a ∗ ψ = − a ∗ b ∗ . (50)Because of our assumptions about a ∗ and b ∗ , the renormalized free energy is pro-portional to ( T ∗ c − T ) . This is characteristic of all second-order transitions, andconsequently for the specific heat C ∗ h =0 = − L d T (cid:18) ∂ F ∗ L ∂T (cid:19) h =0 , (51)whereas by construction F ∗ L (0) = 0 and C ∗ h =0 = 0 in the disordered phase. Thus,Landau’s renormalized MFT also predicts a critical point discontinuity∆ C ∗ h =0 = L d T ∗ c a b ∗ ∂ Ω d ∂T (cid:12)(cid:12)(cid:12) T = T ∗ c ! , (52)for the specific heat. Experimentally, one indeed observes singularities in thermo-dynamic observables and power laws at continuous phase transitions, but oftenwith critical exponents that differ from the above mean-field predictions. Indeed,the divergence of the order parameter susceptibility indicates violent fluctuations,inconsistent with any mean-field description that entirely neglects such fluctua-tions and correlations. Let us mention that the jump of the heat capacity wasobtained because of the system volume was taken to infinity first, and after thisthe reduced temperature ǫ ∗ was set equal to zero.Within the framework of this approach, taking into account the fact that ∂ Ω d ∂T (cid:12)(cid:12)(cid:12) T = T ∗ c = 0 and b ∗ ( T ∗ c ) = b ( T c ), the anomalous part of the specific heat isreally given by ∆ C ∗ h =0 = L d T c a b ( T c ) − Ω dc a T c ! . (53)Then, it can be concluded that (taking into account new fluctuating quantities) ifthe transition takes place at low temperatures, the anomalous part of the specific eat could be negligible or important compared to Debey’s specific heat stemmingfrom the acoustic phonon distribution [4]. Our approach suggests the first casewhere this anomalous part could be too small to be detected. Eq. (53) is anindication of the decreased anomalous part of the specific heat as the anharmonicfluctuation of the system increases.A close look at the heat capacity transition in optimally doped YBCO samplesshows that it starts several degrees above T c , and presents a rather sharp peak,with an increasing slope ( dC ∗ /dT ) as T c is approached [42]. These are strongindications that thermodynamical fluctuations are playing an important role inthe transition. We can use Eq. (29) to estimate the width of the critical region ina typical cuprate.A different deviation from mean field behavior is also seen in the transition ofhigh T c cuprates, as shown in [42]. In addition to some broadening observed above T c the shape of the main transition is modified. Instead of a jump, it looks moreas a narrow peak. Qualitatively, it reminds of the specific heat peak seen at thetransition of superfluid Helium.An even more radically different form of heat capacity transition is observedin Bi Sr CaCu δ . There is no more heat capacity jump at the transition, butrather a cusp. The transition is better fitted by a Bose-Einstein condensation thanby a BCS one [43]. More details will be discussed shortly. In order to test the theory of critical phenomena it is important to have accurateexperiments on well characterized systems, very close to the critical point. Suchexperiments exist in magnetism, ferroelectric thin films and superconductors.
Magnetism is caused at the atomic level by unpaired electron with magneticmoments, and in a ferromagnet, a pair of nearby electron with moments alignedhas a lower energy than if the moments are antialigned [18]. The Curie point ofa ferromagnet will be used as a specific example of a critical point. Below theCurie temperature T c , the ideal ferromagnet exhibits spontaneous magnetization( φ = 0) in the absence of an external field; the direction of the magnetizationdepends on the history of the magnet. Above the Curie temperature, there is nospontaneous magnetization. This ferromagnetism is observed in certain metals likeiron, nickel and cobalt. Just below the Curie temperature the mean field magneticsusceptibility is observed to behave as χ m = C | T − T c | γ , γ = 1 (54) hat is the Curie-Weiss law. C is the Curie constant [44, 45]. Experimentally,we observe that γ is about 4/3. On the other hand, the reduced magnetization¯ φ ≡ φ/ ( N µ B ) (that is also the local spin density) is observed to behave as¯ φ LT ∝ (cid:26) T > T c ( T c − T ) β T < T c (55)i.e. the exponent β is 1/2, which disagrees with the evidence, experimental andtheoretical, that β is about 1/3.As we showed above, the approach that we propose preserves the mean-fieldcritical exponents; all the critical mean-field exponents are suitable. Here we do notrenormalize the critical exponent from the renormalization-group viewpoint, butwe take into account the explicit fact that the mean-field critical temperature T c is in reality a characteristic scale of temperature linked to the thermal fluctuationsrather than to the transition temperature. Ferromagnetism and the Curie temperature were explained by Weiss in termsof a huge internal ”molecular field” proportional to the magnetization. The theoryis applicable both to localized and delocalized electrons. No such magnetic fieldreally exists, but it is a useful way of approximating the effect of the interatomicCoulomb interaction in quantum mechanics. When the distance between magneticmoments is small, the Pauli exclusion principle, which states that two identicalfermions may not have the same quantum states, results in interaction betweenmagnetic moments. Heisenberg introduced a model to describe this exchangeinteraction on microscopic scale. The Heisenberg exchange Hamiltonian may bewritten in the form H exch = − X i 0, so that the limited system will stay in the isordered phase until the temperature becomes absolute zero (It should be keptin mind that T ∗ c = 0 for 2D- and 3D-system [4]. Fluctuations in two dimensionsare not expected to destroy the ordering as in the case of an isotropic Heisenbergferromagnet because of the anisotropy of the coupling.). Thus there exists a lowerbound for the size of the system above which there may exist phase transition.A deviation of the T ∗ c ( L )-versus-L was experimentally found in Refs [61, 62] formagnetic films below a critical thickness (note that such deviation carries impor-tant information, such as, e.g., the possible predominant role of fluctuations andsurface on the transition temperature of ultrathin films).The traditional wisdom for ferroelectric thin films has been that for film thick-nesses smaller than 100 nm the depolarization field will destroy any switchablepolarization making small particles or thin films non-ferroelectric [57, 58]. Thegeneral prediction was that the transition temperature T c will decrease with de-creasing size and ferroelectricity will vanish below a minimum critical thickness.Recent thin film experiments [63] have however shown that switchable ferro-electric films can be made down to 0.9 nm for a crystalline Langmuir-Blodgettdeposited random copolymer of vinylidene fluoride with trifluoroethylene, (PVDF-TrF70:30) on graphite. The minimum thickness is just two mono-layers. The crit-ical size for small spherical lead zirconate-titanate (PZT) particles was calculatedto be 25 ˚A[64]. Similarly it has been reported that switchable ferroelectric filmscan be made of PZT down to 3 or 4 nm [65] in disagreement with earlier theoriesbut in agreement with recent theoretical calculations. This shows that finite-sizeeffects impose no practical limitation on thin film ferroelectric memory capacitorsthough for some designs tunneling currents may become too large. The lateralwidth of the memory element as well does not represent a limitation at present asno change in the coercive field has been observed when the lateral size of the PZTcell was decreased from 1 µ to 0.1 µ . The voltage necessary for polarization rever-sal in such a 100 nm × nm × nm cell is in the technologically accessiblerange of 5 V [66]. The corresponding hysteresis curve has been measured via theatomic force microscope (AFM) in the piezoelectric mode [67].Taking into account Eq. (16), Eq. (63) can be put in the following form T ∗ c ( l ) = T c (cid:16) − Ω dc a T c (cid:17)h − (cid:18) πl (cid:19) i . (64)With regard to this last equation, a competition is to be envisaged between vari-ous types of fluctuations (thermal fluctuations and fluctuations due to finite-sizeeffects) and its consideration in the self-consistent theory can show that finite-sizeeffects impose no practical limitation on thin film ferroelectric, in agreement withrecent theoretical calculations and certain effects observed by now [66].It should be mentioned that finite-size effects such as depressions of T c , andreductions in P S (the spontaneous alignment of dipoles in the ferroelectrics), haveindeed been observed in some nano-crystals as small as 250 ˚A in diameter [68]. It is owever known that at such sizes this is not an electrostatic phenomenon but seemsto be due to surface strains or to inhomogeneity effects [69]. One should also stressthat the local polarization P as a function of the depth z may actually increase asthe surface is approached as in the case of PZT [70] leading to an increase in T c , asthe thickness d decreases. Alternatively it may decrease resulting in a depressionof T c , with decreasing d . Both effects have been observed by now. This can bedescribed by the fact that the extrapolation length can have either sign leading toan increase of a decrease of P at the surface [66]. In renormalized GLW theory the macroscopic wave function of the super-conducting state ψ ( r ) = ψ ( r ) exp [ − iϕ ( r )] (65)serves as the order parameter with the amplitude squared | ψ | = n s being the den-sity of the superconducting particles. We use here the macroscopic wave functionthat is a characteristics of the superfluid state of helium and of superconductivity.Using the variation method, the two GLW renormalized equations12 m ( − i ~ ∇ + 2 e A ) ψ + a ∗ ψ + b ∗ | ψ | ψ = 0 . (66) j s = ie ~ m ( ψ ∗ ∇ ψ − ψ ∇ ψ ∗ ) − e m | ψ | A . (67)are found. In the first term of Eq. (66), magnetic-field effects are included bymaking the usual replacement ∇ → ∇ + 2 e A . For the charge of the Cooper pairshere we have written -2 e , although in the original formulation of the theory theelectronic charge - e was used.If we consider a specimen with dimensions much greater than the penetrationdepth, the magnetic field vanishes inside the sample, and ψ = const. Therefore,only the last two terms of Eq. (66) are relevant, leading to | ψ | = − a ∗ / b ∗ . Ofcourse, this result is identical to Eq. (44). Inserting this expression in Eq. (67),we obtain for the current density j s = 2 e m | a ∗ | b ∗ A . (68)Obviously, this expression for the supercurrent is identical to the second Londonequation. However the statistically appearing (thermal and voluminal) fluctuationsresult in an additional current density (or additional electrical conductivity).Taking into account Eq. (12), the last equation can roughly be written in theform of two contributions: j s = j GL + j fluct . (69) he first one, j GL , just reproduces the London expression without fluctuations.The second term, i.e., the fluctuation part of the supercurrent j fluct ≃ e mb ∗ Ω d A (70)has a more sophisticated nature. In order to carry out the j fluct term the an-harmonic contributions in the GLW functional, originating from the fourth orderterm, have to be taken into account. Details will be discussed shortly. Conse-quently, the GLW renormalized expressions for the penetration depth λ ∗ and thecoherence length ξ GLW are, respectively, given by λ ∗ = s mb ∗ µ e | a ∗ | (71) ξ ∗ GLW = ~ p m | a ∗ | (72)Both characteristic length scales, λ ∗ and ξ ∗ GLW , have the same dependence on a ∗ .Since their ratio κ GLW = λ ∗ /ξ ∗ GLW is a function of b ∗ only, it is temperature,dimension and size dependent, given by κ ∗ GLW = s m b ∗ µ ~ e = κ s d b ( T c ) . (73)Practically, the mean-field parameter κ is the so-called GL parameter that allowsa distinction between type I and type II superconductors. This could not betrue in high-temperature superconductors if we must take into account both thefluctuations and finite-size effects. The penetration depth, the coherence length,and the critical fields are intimately connected.As we saw, T c is a characteristic scale of temperature related to thermal fluctua-tions and finite-size effects rather than the transition temperature, that also meansthat thermal fluctuations of the equilibrium state also exist above the real criticaltemperature T ∗ c . In the normal conducting state the deviation from equilibriumcan lead to the transient appearance of the superconducting state within certainregions, i. e., to the formation of ”puddles” of Cooper pairs. These deviationsfrom equilibrium are not stable, and they will disappear more or less quickly. Thestatistical appearance of Cooper pairs will become more and more rare the higherthe temperature, since with increasing temperature the normal conducting statebecomes more and more stable compared to the superconducting state. There-fore, with increasing temperature, larger and larger deviations from equilibriumare needed to generate the superconducting state. If we note further that thepuddles of Cooper pairs represent perfectly conducting regions, we understand im-mediately that already above T ∗ c due to the fluctuations in the normal conducting tate the statistically appearing puddles of Cooper pairs result in an additionalelectrical conductivity, which must strongly increase on approaching T ∗ c .This influence of the thermal fluctuations can be clearly detected for a numberof superconductors. In [71], for example, it shows the transition curve of a bis-muth film near the mean-field critical temperature T c . One sees there clearly thatthe full normal resistance is reached only at temperatures considerably above T c and the electrical conductance is plotted instead of the resistance, the additionalconductance σ ∗ of the Cooper pair puddles, statistically appearing and vanishingagain, is particularly clearly visible.The additional conductance due to the Cooper pairs (that is well confirmed byexperiment and that is due to the fluctuations.) can be calculated from the existingtheories of superconductivity in combination with the theory of fluctuations [72,73]. T > T ∗ c The fluctuations of the order parameter for T > T ∗ c can also contribute to theelectrical conductivity, and the pretransitional rise in the conductivity is referred toas paraconductivity. The Fourier transform of the nonlocal electrical conductivitycan be calculated with the aid of the Kubo formula σ ∗ ( k ) = 12 k B T Z + ∞−∞ dt h ˆ j k ( t )ˆ j k (0) i (74)where the bracket implies a combined quantum mechanical and statistical average.Using the plane wave decomposition of ψ ( r ) given in Eq. (3) and the definitionof the GLW current operator given in Eq. (67) without magnetic-field effets, wehave j ( r ) = e ~ m X q,q ′ ( q + q ′ ) ψ q ψ ∗ q ′ exp[ i ( q − q ′ ) · r ] (75)Fourier transforming Eq. (75) in exp( iq · r ) yields j ( k ) = e ~ m X q (2 q + k ) ψ ∗ q ψ q + k (76)Substituting Eq. (76) into Eq. (74) and limiting ourselves to the k = 0, we obtain σ ∗ (0) = 4 e ~ k B T m Z + ∞−∞ dt X q,q ′ qq ′ h| ψ q ( t ) | | ψ q ′ (0) | i . (77)We assume terms with q = q ′ are statistically independent and average to zero;thus σ ∗ ij (0) = 4 e ~ k B T m Z + ∞−∞ dt X q q i q j |h ψ ⋆q ( t ) ψ q (0) i| (78) f we assume an exponential decay, the correlation function entering Eq. (78) maybe written h ψ ⋆q ( t ) ψ q (0) i = 2 m ~ k B Tq + ξ ∗− exp( − t/τ q ) , (79)which reduces to the Ornstein-Zernicke form Eq. (32) for t = 0 in the case ofsuperconductors. The relaxation time, τ q will be calculated from the Landau-Khalatnikov model. Note that while we had an equation of motion involving thephase of the order parameter, we did not introduce an equation of motion for themagnitude, | ψ | . The time scale for achieving phase variations is necessarily slow(hydrodynamic) whereas | ψ | may change rapidly (by the conversion of supercon-ductor into normal conductor and vice versa). An exception is near the secondorder phase transition where the free energy becomes ”soft” with respect to a vari-ation of the magnitude of the order parameter and the relaxational dynamics slowsdown. This intuitive idea was quantified by Landau and Khalatnikov (1954) intheir discussion of relaxation phenomena in superfluid He near the lambda point;they made the ansatz [74] ∂ψ q ∂t = − Υ ~ m (cid:0) q + ξ ∗− (cid:1) ψ q , (80)where Υ is a rate constant (Υ > ψ q ( t ) = ψ q (0) exp( − t/τ q ) , where τ q = 2 m Υ ~ q + ξ ∗− . (81)Thermodynamic systems have characteristic microscopic (rapid) relaxation timesand the relaxation of the vast majority of their internal degrees of freedom proceedson these time scales. There are two important exceptions: (i) modes involvingdegrees of freedom for which conservation laws exist; and (ii) additional modesinvolving a broken symmetry of the system. Additional equations of motion existwhen the system spontaneously breaks some symmetry. Taking into account Eq.(81) and inserting Eq. (79) into the k = 0 limit of Eq. (78), carrying out theintegration over time, and writing q i = d q , we obtain σ ∗ (0) = 16 d k B T e m Υ ~ (2 π ) d Z d d q q ( q + ξ ∗− ) . (82)The integration over negative time is accomplished by replacing exp( − t/τ q ) withexp( −| t | /τ q ).Inserting the expression for Υ obtained within the framework of time-dependentGLW renormalized theory in the vicinity of the renormalized critical temperature[72, 74] a Υ = 8 k B ~ π , (83) nd taking into account the dependence of the correlation length ξ ∗ GLW on thetemperature using Eqs. (12), (25) and (72), the singular component expression ofthe additional conductance is obtained as σ ∗ d ( T ) ≃ ℵ d e ξ − dGLW (0) ~ (cid:16) − Ω dc a T c (cid:17)(cid:16) ǫ + Ω d ( T ) a T c (cid:17) d − (84)where ℵ d = − d π − d/ d Γ( d/ R + ∞ x d +1 dx (1+ x ) is a constant integral which is dimension depen-dent and ξ GLW (0) = ~ √ ma T c . In low dimensions d < 4, we set x = qξ to renderthe fluctuation integral, which is UV-finite, dimensionless.In Eq. (84) we have indicated the additional conductance for d-dimensionalsamples. • In the case of three-dimensional superconductors, the thickness, the width,and the length of the sample are large compared to ξ GLW (0). σ ∗ D ( T ) ≃ e ~ ξ GLW (0) (cid:16) − Ω c a T c (cid:17)(cid:16) ǫ + Ω D ( T ) a T c (cid:17) − . (85) • For a thin film with length L small compared to ξ GLW (0), we may neglect thefluctuations along the film normal and the system becomes effectively 2D. σ ∗ D ( T ) ≃ e L ~ (cid:16) − Ω c a T c (cid:17)(cid:16) ǫ + Ω D ( T ) a T c (cid:17) − . (86) • In the case of one-dimensional superconductors, the length of the sample is largecompared to ξ GLW (0), the thickness ℓ and the width ̺ are small compared to ξ GLW (0) σ ∗ D ( T ) ≃ πe ξ GLW (0)16 ℓ̺ ~ (cid:16) − Ω c a T c (cid:17)(cid:16) ǫ + Ω D ( T ) a T c (cid:17) − . (87)Ω dc is the the solution of the self-consistent Eq. (30) for one-, two- and three-dimensional sample. Qualitatively, we can easily understand that the sample di-mensions must influence the magnitude of the fluctuations, since the Cooper pairdensity can vary only on a length scale of about ξ ∗ GLW . More rapid spatial varia-tions require relatively high energies and, hence, practically do not appear. Withina sample that is large in all three spatial directions, the Cooper pair density canvary spatially in all directions. All these possible configurations must be taken intoaccount in the calculation of the additional conductance. For a two-dimensionalsample, along the shortest extension the Cooper pair density is always constantspatially. Hence, averaging over all possible spatial configurations of the Cooperpair density along this direction is not necessary. For a one-dimensional sam-ple, averaging is unnecessary along both directions in which the sample is smallcompared to ξ ∗ GLW . We see that the statistics is restricted because of the samplegeometry. This results in various expressions for the additional conductance. xperience shows that the transition curves of three-dimensional samples, say,of wires with a diameter large compared to ξ ∗ GLW , are very sharp, i. e., the effectswe have just discussed cannot be observed. The reason is not the absence offluctuations, but rather the comparatively high residual conductance of the three-dimensional sample. The quantity f (Ω dc ) = (cid:16) − Ω dc a T c (cid:17) of Eqs. (85-87) is anindication of the decreased additional conductance as the fluctuation effects of thesystem increase.So far we have only discussed how the fluctuations affect the electrical conduc-tance. However, if puddles of Cooper pairs appear statistically above T ∗ c , this mustbe noticed also in other properties. We know that below T ∗ c a superconductor ex-pels small magnetic fields out of its interior, i. e., it turns into an ideal diamagnet[73]. We expect that, similar to the effect of the fluctuations on the conductance,some part of this diamagnetic property also appears above T ∗ c . The puddles ofCooper pairs should result in a characteristic temperature dependence of the dia-magnetic behavior of the superconductor above T ∗ c . Only a few hundredths of adegree away from T ∗ c the additional diamagnetism is already very small and corre-sponds to the expulsion of just a few flux quanta. However, it has been possible todetect this effect clearly [75] by utilizing a superconducting quantum interferome-ter. The fluctuations should also lead to an increase of the specific heat C alreadyabove T ∗ c . This effect could also be experimentally demonstrated [73, 76]. The considerable success of the mean-field theory in conventional supercon-ductors originates from the low value (∆ t G = Ω dc /a T c ∼ − ) of the criticalGinzburg width [1, 18, 45]. Accordingly, Eq. (16) obviously leads to T ⋆c ≃ T c , where T c is the mean field (BCS) value of the transition temperature. This impliesthat it is impossible to detect with actual experimental precision the deviationsfrom the mean-field theory. Indeed, one observes that the specific heat does notpresent any divergence, but rather a compatible jump with the predictions ofEq. (3) in [4]. Practically, critical phenomena should not be observable in usualconventional superconductors, which could not be true in high-temperature super-conductors and particularly high- T c superconductors [3, 77].In compounds of the BiSrCaCuO, LaSrCaCuO and TlSrCaCuO type, estima-tions around ξ ∼ ξ ≈ ˚A [3, 4]. These values haveto approximately equal the size of the units that undergo ordering at the phasetransition. This can question the validity of the mean-field theory for supercon-ductors. But, we think that such a distance is related to the existence of largefluctuations and finite-size effects present in those compounds. The increase ofthe critical temperature in non-conventional superconductors also means that the ond energy of the Cooper pair increases at the same rate. When the energy bondis higher, the bond is mostly confined in space and ξ then reduces at the samelevel. It is for instance the case in La − x Sr x CuO where ∆ t G can reach the value10 − , and in Y Ba Cu O − y compounds where ∆ t G can reach the values typicallyin the range 10 − to 10 − [4]. These values seem to be accessible in experimentand suggest principal deviations compared to the mean-field theory observable.Also, the newer ceramic high-temperature superconductors have a much smallercoherence length of ξ , which can indeed help to show some effects of fluctuations[78, 79, 80]. We have described the fundamentals of renormalized φ theory on the basis ofa GLW calculations, combined with an efficient no perturbative technique, thattakes into account both dimension, size and microscopic details of the system,and which leads to critical behavior, strongly deviating from the classical MFA farfrom the thermodynamic limit. Within this more rigorous approach the effects offluctuations are examined in more rigorous detail, and we are able to establish theinsufficiencies of the MFT, and also estimate the width of the critical region wherecorrections to MFT are important. We find a consistent interpretation relating thedimension, the size of the system and the spatial fluctuations which can give aninterpretation to the degeneracy of energy levels. We have also calculated asymp-totic expressions of thermodynamic observables as a function of temperature.This theory does not present any renormalization based method in the sense ofa renormalization group approach but rather a SCM improving on LT. The SCMis a strategy for dealing with problems involving many length scales. The strategyis to tackle the problem in steps, one step for each length scale. In the case ofcritical phenomena, the problem, technically, is to carry out statistical averagesover thermal fluctuations on all size scales. The SCM is to integrate out the fluctu-ations in sequence, starting with fluctuations on an atomic scale and then movingto successively larger scales until fluctuations on all scales have been averaged out.The integration of this deviation in the self-consistency on the entire spectrum ofthe lattice vibration, conducted to a very improved self consistent energy. Thisimprovement of the self consistent problem by the processed harmonic and anhar-monic fluctuations goes for example to reflect itself on the thermodynamic andelectronic parameters, resulting notably in a substantial reduction of the criticaltemperature.The resulting self consistent problem of this approach has been solved ana-lytically, what permitted us to extract an effective theory, with notably a meanfield critical temperature renormalized by fluctuations. An interesting point withthis renormalized critical temperature is its dependence on the quantity Ω dc a whichaccording to the text, characterizes the importance of the fluctuations and is a unction of both dimension and size of the system.We have revealed the thickness dependence of the Curie temperature for varioustypes of thin and ultrathin films. In agreement with recent theoretical calculationsit is found that the relation of Eq. (64) reproduces rather well such dependence forthickness down to l , in contrast with the usual finite-size scaling law that breaksdown below L and that does not take into account both the fluctuations and themicroscopic details of the system. We are confident that the present work providesa deeper knowledge of nanoscience, phase transitions, and critical behaviors indipolar systems. Furthermore, the parameters characterizing this approach dependstrongly on the dimension and the size of the sample, but in different ways thanthe ones obtained in the MFA. Thus, this SCM seems to be appropriate for modelcalculations in a number of critical systems. For this reason, analysis has also beenperformed on other systems such as non-conventional superconductors, localizedmagnetism and can be extended to all system, whose transitions indeed apparentlybelong to the same universality class, in order to lend a significant level of supportto this theoretical approach.By comparing the thermal energy and the energy resulting to structural insta-bilities, we introduced the report ν d , which could allow to estimate the quantumfluctuation importance. 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