Physica A. Calculation of free energy of a three-dimensional Ising-like system in an external field with the use of the ρ 6 model

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1 Physia A 89 () 59 5 Contents lists available at SieneDiret Physia A journal homepage: Calulation of free energy of a three-dimensional Ising-lie system in an external field with the use of the ρ model I.V. Pylyu M.P. Kozlovsii Institute for Condensed Matter Physis of the National Aademy of Sienes of Uraine Svientsitsii Street 79 Lviv Uraine a r t i l e i n f o a b s t r a t Artile history: Reeived Marh Reeived in revised form 7 August Available online August Keywords: Ising-lie system External field Critial point Free energy The mirosopi approah to alulating the free energy of a three-dimensional Isinglie system in a homogeneous external field is developed in the higher non-gaussian approximation (the ρ model) at temperatures above the ritial value of T (T is the phase-transition temperature in the absene of an external field). The free energy of the system is found by separating the ontributions from the short- and long-wave spindensity osillation modes taing into aount both temperature and field flutuations of the order parameter. Our analytial alulations do not involve power series in the saling variable and are valid in the whole field temperature plane near the ritial point inluding the region in the viinity of the limiting field h whih divides external fields into the wea and strong ones (i.e. the rossover region). Elsevier B.V. All rights reserved.. Introdution The Ising model is one of the most studied models in the theory of the phase transitions not only beause it is onsidered as the prototype of statistial systems showing a non-trivial power-law ritial behaviour but also beause it desribes several physial systems []. Many systems haraterized by short-range interations and a salar order parameter undergo a transition belonging to the Ising universality lass. Despite the great suesses in the investigation of three-dimensional (D) Ising-lie systems made by means of various methods (see for example Ref. []) the statistial desription of the ritial behaviour of the mentioned systems in terms of both temperature and field variables and the alulation of saling funtions are still of interest []. This artile is a detailed presentation of the olletive variables (CV) method [ 5] developed for a D Ising-lie magnet in the higher non-gaussian approximation and in the presene of an external field. The CV method is non-perturbative and similar to the Wilson non-perturbative renormalization-group (RG) approah (integration on fast modes and onstrution of an effetive theory for slow modes) [ 8]. The term olletive variables is a ommon name for a speial lass of variables that are speifi for eah individual physial system []. The CV set ontains variables assoiated with order parameters. Beause of this the phase spae of CV is most natural for desribing a phase transition. For magneti systems the CV ρ are the variables assoiated with modes of spin-moment density osillations while the order parameter is related to the variable ρ in whih the subsript orresponds to the pea of the Fourier transform of the interation potential. The methods existing at present mae it possible to alulate universal quantities to a quite high degree of auray (see for example Ref. []). The advantage of the CV method lies in the possibility of obtaining and analysing thermodynami harateristis as funtions of the mirosopi parameters of the initial system [9 ]. Using the non-gaussian basis distributions of flutuations in alulating the free energy of the system does not reate the problem of the summation of various lasses of divergent (with respet to the Gaussian distribution) diagrams at the ritial point. A onsideration of the inreasing number of terms in the Corresponding address: Department for Statistial Theory of Condensed Systems Institute for Condensed Matter Physis of the National Aademy of Sienes of Uraine Svientsitsii Street 79 Lviv Uraine. Tel.: ; fax: address: piv@imp.lviv.ua (I.V. Pylyu). 78-7/$ see front matter Elsevier B.V. All rights reserved. doi:./j.physa..8.

2 I.V. Pylyu M.P. Kozlovsii / Physia A 89 () exponent of the non-gaussian distribution is an alternative to the use of a higher-order perturbation theory based on the Gaussian distribution. The free energy of a D Ising-lie system in an external field at temperatures above T is alulated using the non-gaussian spin-density flutuations namely the sexti measure density. The latter is represented as an exponential funtion of the CV whose argument inludes the powers with the orresponding oupling onstants up to the sixth power of the variable (the ρ model). In our alulations we use approximation for potential similar to the loal potential approximation (LPA) (see for example Refs. [8]) assuming that the orretion for the potential averaging is zero although it an be taen into aount if neessary (see Refs. [ 5]). The inlusion of this orretion leads to a nonzero value of the ritial exponent η haraterizing the behaviour of the pair-orrelation funtion for T = T. In the ρ model approximation we arrive at the result η. [55]. For omparison the exponents η =.5(5) η =.(8) and η =. were obtained within the framewor of the field-theory approah (7-loop alulations) [] Monte Carlo simulations [7] and the nonperturbative RG approah (the order of the derivative expansion) [8] respetively. Inluding the above-mentioned orretion and the related shift of the fixed point does not qualitatively hange the main thermodynami harateristis of the system. It should be noted that the onvergene of an expansion in field monomials φ n at η = has been studied in detail using Wilson RG tehniques [9 ]. Although the sum of the expansion does not onverge it appears [] that the asymptoti value of the ritial exponent of the orrelation length ν asympt =.95 for the Ising universality lass is reasonably losely approahed at n >. This value agrees with our result ν =.9 [5] obtained for the ρ model at the optimal RG parameter s = s =.8 and η =. The value of s = s (for eah of the ρ m models) orresponds to nullifying the average value of the oeffiient of the quadrati term in the effetive density of measure at the fixed point. It is expeted that the inlusion of a nonzero exponent η within the CV method will redue the ritial exponent of the orrelation length ν (as in the non-perturbative RG approah [8]). A tendeny to saturation of the ritial exponent ν with inreasing order of the ρ m model has been graphially illustrated in Refs. [95 7]. The present publiation supplements the earlier wors [989] in whih the ρ model was used for alulating the free energy and other thermodynami funtions of the system in the absene of an external field. The ρ model provides a better quantitative desription of the ritial behaviour of a D Ising-lie magnet than the ρ model [9]. The expressions for the thermodynami harateristis of the system in the presene of an external field have already been obtained on the basis of the simplest non-gaussian measure density (the ρ model) in Refs. [ ] using the point of exit of the system from the ritial regime as a funtion of the temperature (the wea-field region) or of the field (the strong-field region). In Refs. [] the thermodynami harateristis are presented in the form of series expansions in the variables whih are ombinations of the temperature and field. Our alulations in the ρ model approximation were also performed for temperatures T > T [] and T < T [] without using similar expansions for the roots of ubi equations appearing in the theoretial analysis. In this artile the free energy of a D uniaxial magnet within the framewor of the more ompliated ρ model is found without using series expansions introduing the generalized point of exit of the system from the ritial regime. This point taes into aount temperature and field variables simultaneously. In our earlier artile [] the point of exit of the system from the ritial regime was found in the simpler non-gaussian approximation (the ρ model) using the numerial alulations. In ontrast to [] the point of exit of the system in the present artile is expliitly defined as a funtion of the temperature and field. This allows one to obtain the free energy without involving numerial alulations.. Basis relations We onsider a D Ising-lie system on a simple ubi lattie with N sites and period in a homogeneous external field h. The Hamiltonian of suh a system has the form H = Φ(r jl )σ j σ l h σ j () jl j where r jl is the distane between partiles at sites j and l and σ j is the operator of the z omponent of spin at the jth site having two eigenvalues + and. The interation potential is an exponentially dereasing funtion Φ(r jl ) = A exp r jl. () b Here A is a onstant and b is the radius of effetive interation. For the Fourier transform of the interation potential we use the following approximation [9]: Φ() = Φ()( b ) B B < B where B is the boundary of the Brillouin half-zone (B = π/) B = (b ) Φ() = 8πA(b/). In the CV representation for the partition funtion of the system we have [5] Z = exp β Φ()ρ ρ + βh Nρ J(ρ) (dρ) N. () ()

3 59 I.V. Pylyu M.P. Kozlovsii / Physia A 89 () 59 5 Here the summation over the wave vetors is arried out within the first Brillouin zone β = /(T) is the inverse temperature the CV ρ are introdued by means of the funtional representation for operators of spin-density osillation modes ρˆ = ( N) σ l l exp(il) J(ρ) = N exp πi ω ρ + n (πi) n N n M n (n)!... n ω ω n δ + + n (dω) N (5) is the Jaobian of transition from the set of N spin variables σ l to the set of CV ρ and δ + + n is the Kroneer symbol. The variables ω are onjugate to ρ and the umulants M n assume onstant values (see Refs. [ 5]). Proeeding from Eqs. () and (5) we obtain the following initial expression for the partition funtion of the system in the ρ model approximation: Z = N (N )/ e a N exp a (N ) / ρ ()ρ ρ l= a l (l)!(n ) l... l i B d B ρ ρ l δ + +l (dρ) N. () Here N = Ns d (d = is the spae dimension) s = B/B = π b/ and a = sd/ h h = βh. The expressions for the remaining oeffiients are given in Refs. [989]. These oeffiients are funtions of s i.e. of the ratio of mirosopi parameters b and. The integration over the zeroth first seond... nth layers of the CV phase spae [ 59] leads to the representation of the partition funtion in the form of a produt of the partial partition funtions Q n of individual layers and the integral of the smoothed effetive measure density Z = N (N n+)/ Q Q Q n [Q (P n )] N n+ W (n+) (ρ) (dρ) N n+. (7) The expressions for Q n Q (P n ) are presented in Refs. [989] and N n+ = N s d(n+). The sexti measure density of the (n + )th blo struture W (n+) (ρ) has the form (ρ) = exp W (n+) a (n+) N / n+ ρ d n+()ρ ρ B n+ l= a (n+) l (l)!n l n+ ρ ρ l δ + +l... l i B n+ (8) where B n+ = B s (n+) d n+ () = a (n+) β Φ() a (n+) and a (n+) l are the renormalized values of the oeffiients a and a l after integration over n + layers of the phase spae of CV. The oeffiients a(n) = s n t n d n () = s n r n [appearing in the quantity d n () = d n () + β Φ()b ] a (n) = s n u n and a (n) = s n w n are onneted with the oeffiients of the (n + )th layer through the reurrene relations (RR) t n+ = s (d+)/ t n r n+ = s q + u / n Y(h n α n ) u n+ = s d u n B(h n α n ) w n+ = s d u / n D(h n α n ) whose solutions t n = t () s d/ h E n r n = r () + E n + w () (u() ) / E n + w () (u() ) E n u n = u () + w () (u() ) / E n + E n + w () (u() ) / E n w n = w () + w () u() E n + w () (u() ) / E n + E n () in the region of the ritial regime are used for alulating the free energy of the system. Here Y(h n α n ) = s d/ F (η n ξ n ) [C(h n α n )] / B(h n α n ) = s d C(η n ξ n ) [C(h n α n )] D(h n α n ) = s 7d/ N(η n ξ n ) [C(h n α n )] /. () (9)

4 I.V. Pylyu M.P. Kozlovsii / Physia A 89 () The quantity q = qβ Φ() determines the average value of the Fourier transform of the potential β Φ(B n+ B n ) = β Φ() q/s n in the nth layer (in this artile q = ( + s )/ orresponds to the arithmeti mean value of on the interval (/s ]). The basi arguments h n and α n are determined by the oeffiients of the sexti measure density of the nth blo struture. The intermediate variables η n and ξ n are funtions of h n and α n. The expressions for both basi and intermediate arguments as well as the speial funtions appearing in Eq. () are the same as in the absene of an external field (see Refs. [989]). The quantities E l in Eq. () are the eigenvalues of the matrix of the RG linear transformation t n+ t () r n+ r () u n+ u () w n+ w () = R R R R R R R R R R t n t () r n r () u n u () w n w (). () We have E = R = s (d+)/. Other nonzero matrix elements R ij (i = ; j = ) and the eigenvalues E E E oinide respetively with the quantities R i j (i = i ; j = j ) and E E E obtained in the ase of h =. The quantities f ϕ and ψ haraterizing the fixed-point oordinates t () = r () = f β Φ() u () = ϕ (β Φ()) w () = ψ (β Φ()) () as well as the remaining oeffiients in Eq. () are also defined on the basis of expressions orresponding to a zero external field.. Method of alulation Let us alulate the free energy F = T ln Z of a D Ising-lie system above the ritial temperature T. The basi idea of suh a alulation on the mirosopi level onsists in the separate inlusion of the ontributions from short-wave (F CR the region of the ritial regime) and long-wave (F the region of the limiting Gaussian regime) modes of spin-moment density osillations [ 5]: F = F + F CR + F. () Here F = TN ln is the free energy of N noninterating spins. Eah of three omponents in Eq. () orresponds to individual fators in the onvenient representation Z = N Z CR Z (5) for the partition funtion given by Eq. (7). The ontributions from short- and long-wave modes to the free energy of the system in the presene of an external field are alulated in the ρ model approximation aording to the sheme proposed in Refs. [989]. Short-wave modes are haraterized by an RG symmetry and are desribed by a non-gaussian measure density. The alulation of the ontribution from long-wave modes is based on using the Gaussian measure density as the basis one. Here we have developed a diret method of alulations with the results obtained by taing into aount the short-wave modes as initial parameters. The main results obtained in the ourse of deriving the omplete expression for the free energy of the system are presented below... Region of the ritial regime A alulation tehnique based on the ρ model for the ontribution F CR is similar to that elaborated in the absene of an external field (see for example Refs. [599]). Carrying out the summation of partial tree energies F n over the layers of the phase spae of CV we an alulate F CR : F CR = F + F CR F = TN [ln Q (M) + ln Q (d)] n F = p CR F n. n= An expliit dependene of F n on the layer number n is obtained using solutions () of RR and series expansions of speial funtions in small deviations of the basi arguments from their values at the fixed point. The main peuliar feature of the present alulations lies in using the generalized point of exit of the system from the ritial regime of order-parameter flutuations. The inlusion of the more ompliated expression for the exit point (as a funtion of both temperature and field variables) [] n p = ln( h + h ) ln E (7) leads to the distintion between formula () for F CR and the analogous relation at h = [99]. The quantity h = h /f is determined by the dimensionless field h while the quantity h = τ p is a funtion of the redued temperature τ = (T T )/T. Here τ = () τ/f p = ln E / ln E = (d + )ν/ () haraterizes the oeffiient in solutions ()

5 59 I.V. Pylyu M.P. Kozlovsii / Physia A 89 () 59 5 Table The eigenvalues E l and the exponents ν for the ρ model. E E E E ν () of RR ν = ln s/ ln E is the ritial exponent of the orrelation length. At h = n p beomes m τ = ln τ/ ln E (see Refs. [599]). At T = T (τ = ) the quantity n p oinides with the exit point n h = ln h/ ln E [7]. The limiting value of the field h is obtained by the equality of the exit points defined by the temperature and by the field (m τ = n h ). Having expression (7) for n p we arrive at the relations [8] E n p+ = ( h + h )/ τ E n p+ = H H = h/p ( h + h )/(p ) E n p+ = H H = ( h + h ) /(p ) E n p+ = H H = ( h + h ) /(p ) s (n p+) = ( h + h )/(d+) where = ln E / ln E and = ln E / ln E are the exponents whih determine the first and seond onfluent orretions respetively. Numerial values of the quantities E l (l = ) ν and for the optimal RG parameter s = s =.79 are ontained in Table. For omparison the other authors data alulated within the field-theory approah (ν =. =.98) and high-temperature expansions (ν =.8 =.5) are given in Refs. [9]. The values of the ritial exponent of the orrelation length ν =.() ν =.97(5) and ν =. are presented in the above-mentioned artiles [ 8] respetively. It should be noted that in omparison with =.55 and =.7 alulated for the ρ model at η = (see Table ) the exponents of the first and seond onfluent orretions =.8 and =.57 [7] for the ρ model agree more losely with the estimates.5... (for ) and.5... (for ) obtained using the Wilson RG equation in the LPA (see for example Ref. [9]). In the wea-field region ( h h ) quantities (8) an be alulated with the help of the following expansions: E n p+ = h h h = τ p h H = h p h H = h /p + h p h H = h /p + h p s (n p+) = h/(d+) h + d + h /p = τ h /p = τ h In the strong-field region ( h h ) these quantities satisfy the expressions E n p+ = h h /p H = ( h / h) h p h H = h /p + h p h H = h /p + h p h h h (8) h/(d+) = τ ν. (9)

6 s (n p+) = h/(d+) + d + I.V. Pylyu M.P. Kozlovsii / Physia A 89 () h h. /p It should be noted that the variables h/ h (the wea fields) and ( h / h) (the strong fields) oinide with the aepted hoie of the arguments for saling funtions in aordane with the saling theory. In the partiular ase of h = and τ expressions (9) are defined as E n p+ = τ p H = H = τ H = τ s (n p +) = τ ν. At h and τ = we n p + have he = H = H = h /p H = h /p s (n p +) = h/(d+) [see Eq. ()]. We shall perform the further alulations on the basis of Eq. (8) whih are valid in the general ase for the regions of small intermediate (the rossover region) and large field values. The inlusion of E n p+ (or H ) leads to the formation of the first onfluent orretions in the expressions for thermodynami harateristis of the system. The quantity E n p+ (or H ) is responsible for the emergene of the seond onfluent orretions. The ases of the wea or strong fields an be obtained from general expressions by using Eqs. (9) or (). We disregard the seond onfluent orretion in our alulations. This is due to the fat that the ontribution from the first onfluent orretion to thermodynami funtions near the ritial point (τ = h = ) for various values of s is more signifiant than the small ontribution from the seond orretion ( h + h is of the order of.5 and > see Table [8]). Proeeding from an expliit dependene of F n on the layer number n [589] and taing into aount Eq. (8) we an now write the final expression for F CR (): F CR = TN γ (CR) + γ τ + γ τ + F s F s = TN s (n p+) γ (CR)()+ + γ (CR)()+ () H Here () haraterizes in solutions () of RR γ (CR)()+ = f () CR s + f γ (CR)()+ = f () CR ϕ E s + f and the oeffiients γ (CR) = γ () + δ () () CR ϕ/ f H + f (7) CR ϕ (f H ) E s E s (). () () CR ϕ/ f H E E s + f (8) CR ϕ (f H ) E E () s γ = γ () + δ () = () are determined by the omponents of the quantities γ = γ () + γ () τ + γ () τ δ = δ () + δ () τ + δ() τ. The omponents δ (i) (i = ) satisfy the earlier relations [589] obtained in the ase of a zero external field. The omponents γ (i) are given by the orresponding expressions at h = under ondition that the eigenvalues E E and E should be replaed by E E and E respetively. Let us now alulate the ontribution to the free energy of the system from the layers of the CV phase spae beyond the point of exit from the ritial-regime region. The alulations are performed aording to the sheme proposed in Refs. [59]. As in the previous study while alulating the partition funtion omponent Z from Eq. (5) it is onvenient to single out two regions of values of wave vetors. The first is the transition region (Z () ) orresponding to values of lose to B np while the seond is the Gaussian region (Z () ) orresponding to small values of wave vetor ( ). Thus we have Z = Z () Z ()... Transition region This region orresponds to m layers of the phase spae of CV. The lower boundary of the transition region is determined by the point of exit of the system from the ritial-regime region (n = n p + ). The upper boundary orresponds to the layer n p + m +. We use for m the integer losest to m. The ondition for obtaining m is the equality [99] () (5) h np + m = A s where A is a large number (A ). ()

7 59 I.V. Pylyu M.P. Kozlovsii / Physia A 89 () 59 5 The free energy ontribution F () = TN n p + f (m) = ln π m m= s m f (m) + ln ln C(η n p +m ξ np +m) + ln I (h np +m+ α np +m+) + ln I (η np +m ξ np +m) (7) orresponding to Z () from Eq. (5) is alulated by using the solutions of RR. The basi arguments in the (n p + m)th layer h np +m = (r np +m + q)(/u np +m) / α np +m = 5 w n p +m/u / n p +m an be presented using the relations t np +m = s d/ f E m h( h + h )/ r np +m = β Φ() f + f H E m + () H ϕ / w () Em u np +m = (β Φ()) ϕ + f H ϕ / w () Em + () H E m w np +m = (β Φ()) ψ + f H ϕ w () Em + () H ϕ / w () Em obtained on the basis of Eqs. () and (8). We arrive at the following expressions: h np +m = h () n p +m + h() n p +m () H h () n p +m = h () n p +m = E m α np +m = α () n p +m q f + f H E m (ϕ + f H ϕ / w () Em ϕ / w () q f + f H E m + ᾱ () n p +m () H ) / ϕ + f H ϕ / w () Em α () ψ + f H ϕ w () n p +m = Em () 5 (ϕ + f H ϕ / / ᾱ () n p +m = E m w () Em ) ϕ / w () ψ + f H ϕ w () Em ϕ + f H ϕ / w () Em In ontrast to H the quantity H in expressions () for h np +m and α np +m as well as in expression () for F s taes on small values with the variation of the field h (see Fig. ). The quantity H at h and near h is lose to unity and series expansions in H are not effetive here. Power series in small deviations (h np +m h () n p +m) and (α np +m α () n p +m) for the speial funtions appearing in the expressions for the intermediate arguments η np +m = (s d ) / F (h np +m α np +m) C(h np +m α np +m) / ξ np +m = 5 sd/ N(h np +m α np +m) C(h np +m α np +m) / () allow us to find the relations η np +m = η () n p +m η (n p+m) h () () n p +m h n p +m + η (n p+m) α () n p +mᾱ () n p +m () H ξ np +m = ξ () n p +m ξ (n p+m) h () () n p +m h n p +m + ξ (n p+m) α () n p +mᾱ () n p +m () H. () The quantities η () n p +m η (n p+m) η (n p+m) and ξ () n p +m ξ (n p+m) ξ (n p+m) are funtions of F (n p+m) l I (n p+m) l = ;. = I (n p+m) l /I (n p+m) where x l exp(h () n p +m x x α () n p +m x ) dx. () (8) (9)

8 I.V. Pylyu M.P. Kozlovsii / Physia A 89 () Fig.. Dependene of quantities H and H on the ratio h/ h for the RG parameter s = s =.79 and the redued temperature τ =. Proeeding from expression (7) for f (m) we an now write the following relation aurate to within H : f (m) = f () () (m) + f (m) () H f () (m) = ln + π ln ln C(η() n p +m ξ () n p +m) + ln I (h () n p +m+ α() ) + n p +m+ ln I (η () n p +m ξ () n p +m) f () (m) = ϕ (n p+m) h () () n p +m h n p +m + ϕ (n p+m) α () n p +mᾱ () n p +m + ϕ (n p+m+) h () () n p +m+ h + p+m+) n p +m+ ϕ(n α () n p +m+ᾱ() n p +m+ ϕ (n p+m) = b (n p+m) + P (n p+m) / = ϕ (n p+m+) = F (n p+m+) ϕ (n p+m+) = F (n p+m+). () The quantities b (n p+m) I (n p+m) l = P (n p+m) The final result for F () depend on F (n p+m) l as well as on F (n p+m) l = I (n p+m) l /I (n p+m) where x l exp(η () n p +m x x ξ () n p +m x ) dx. (5) [see Eqs. (7) and ()] assumes the form F () = TN s (n p+) f () TR + f () TR () H f () TR f () TR = m m= = m m= s m f () (m) s m f () (m). On the basis of Eqs. () and () it is possible to obtain the quantity m determining the summation limit m in formulas (): m = ln L ln H ln E + L = A + (A A ) / A = q + A ϕ/ w () f f ( s ) A = q q A + ϕ f f ( s ). (7) f Let us now alulate the ontribution to the free energy of the system from long-wave modes in the range of wave vetors B s n p n p = n p + m + (8) using the Gaussian measure density. ()

9 598 I.V. Pylyu M.P. Kozlovsii / Physia A 89 () Region of small values of wave vetor ( ) The free energy omponent F () = T Nn p ln P (n p ) + B n p ln dn p () N(h ) (9) d n p () = orresponding to Z () from Eq. (5) is similar to that presented in Refs. [599]. The alulations of the first and seond terms in Eq. (9) are assoiated with the alulations of the quantities where P (n p ) = h n p F (h n p α n p ) d n p () = P (n p ) d n p (B n p B n p ) + β Φ(B n B p n p) β Φ() () d n p (B n p B n p ) = s (n p ) (r n p + q) () and r n p h n p = h () n + h() p n p () H α n p = α () n + ᾱ () p n p () H satisfy the orresponding expressions from Eqs. (9) and () at m = m +. Introduing the designation p = h n p F (h n p α n p ) and presenting it in the form p = p ( + p () H ) we obtain the following relations for the oeffiients: p = h () n p p(n p ) p = h() n p (n p ) p h () n + p (n p ) p α () n pᾱ() n. () p The quantities p (n p ) = F (n p ) p (n p ) p (n p ) = F (n 8 p ) F (n p) determine the funtion F (h n p α n p ) = p (n p ) = F (n Here F (n p ) l = I (n p ) l /I (n p ) where I (n p ) l = p ) F (n p) F (n p ) F (n p ) (5) p (n p ) h () () h n p n + p p(n p ) α () n pᾱ() n p () () () H. () x l exp(h () n p x x α () n p x ) dx. (7) Taing into aount Eqs. () and () we rewrite formulas () as P (n p ) = p ) β Φ()p ( q f + f H E m s(n d n p () = s (n p ) β Φ() G + β Φ()b ) + ϕ / w () E m q f + f H E m + p () H G = g ( + ḡ () H ) g = (f + f H E m )p + (p ) q ḡ = p p ( q f + f H E m g ) + ϕ/ w () E m. (8)

10 I.V. Pylyu M.P. Kozlovsii / Physia A 89 () The seond term in Eq. (9) is defined by the expression B n p ln dn p () = N n p ln( G + s ) + ln s n p ln s + ln(β Φ()) = + Gs ( Gs ) / artan ( Gs ) /. (9) Relations (8) and (9) mae it possible to find the omponent F () in the form [ F () = T N s (np+) () ( f () + f () H ) + N(h ) γ + ] β Φ() s(n p+) ( ḡ () H ) f () = s ( m +) f () () f = s ( m +) f () f () = s ln + g + g + q g g artan g f () = g ḡ g + q ḡ (g ) + gḡ (g ) + g ḡ g artan g g = s g γ + = s m /(g ). On the basis of Eqs. () and (5) we an write the following expression for the general ontribution F = F () + F () the free energy of the system from long-wave modes of spin-moment density osillations: [ F = T N s (np+) () ( f + () f () H ) + N(h ) γ + ] s (np+) ( ḡ () β Φ() H ) f (l) = (l) f + (l) TR f l =. (5) to (5). Total free energy of the system at T > T The total free energy of the system is alulated taing into aount Eqs. () () and (5). Colleting the ontributions to the free energy from all regimes of flutuations at T > T in the presene of an external field and using the relation for s (np+) from Eq. (8) we obtain [ F = TN γ + γ τ + γ τ + ( γ ()+ + γ ()+ () H)( h + h )/5 + γ + ] (h ) β Φ() ( ḡ () H)( h + h )/5 γ = ln + s γ (CR) γ = sγ γ = sγ γ (l)+ = s (CR)(l)+ (l) ( γ + f ) l =. (5) The oeffiients γ (CR) γ γ are defined by Eq. () ḡ is presented in Eq. (8) and γ + is given in Eq. (5). The oeffiients of the non-analyti omponent of the free energy F [see Eq. (5)] depend on H. The terms proportional to H determine the onfluent orretions by the temperature and field. As is seen from the expression for F the free energy of the system at h = and τ = in addition to terms proportional to τ ν (or h/5 ) and h/5 ontains the terms proportional to τ ν+ and h /5+ /p respetively. At h and τ the terms of both types are present. It should be noted that > /p. At h = h we have τ ν+ = h/5+ /p and the ontributions to the thermodynami harateristis of the system from both types of orretions beome of the same order. When h (or h ) the order parameter of the system is nonzero both for τ > (or τ > ) and for τ < (or τ < ) and τ = is not physially distintive. The free energy for h and τ (the strong-field region) has no singularity with /p respet to τ and must therefore have an expansion in integral powers of this variable [or the saling variable ( h / h) ]. The onept of the wea field presupposes that τ. For a given nonzero value of τ a zero field is not a singularity of the thermodynami funtions. Hene the free energy for τ and h (the wea-field region) an be expanded in integral powers of the variable h [or h/ h ]. The mentioned series expansions in the saling variables for the regions of the wea and strong fields an be obtained using Eqs. (9) and () respetively.

11 5 I.V. Pylyu M.P. Kozlovsii / Physia A 89 () 59 5 The advantage of the method presented in this artile is the possibility of deriving analyti expressions for the freeenergy oeffiients as funtions of the mirosopi parameters of the system (the lattie onstant and parameters of the interation potential i.e. the effetive radius b of the potential the Fourier transform Φ() of the potential for = ). 5. Conlusions An analyti method for alulating the total free energy of a D Ising-lie system (a D uniaxial magnet) near the ritial point is developed on the mirosopi level in the higher non-gaussian approximation based on the sexti distribution for modes of spin-moment density osillations (the ρ model). The simultaneous effet of the temperature and field on the behaviour of the system is taen into aount. An external field is introdued in the Hamiltonian of the system from the outset. In ontrast to previous studies on the basis of the asymmetri ρ model [] the field in the initial proess of alulating the partition funtion of the system is not inluded in the Jaobian of transition from the set of spin variables to the set of CV. Suh an approah leads to the appearane of the first seond fourth and sixth powers of CV in the expression for the partition funtion and allows us to simplify the mathematial desription beause the odd part is represented only by the linear term. In the ase of another approah when the field is inluded in the transition Jaobian the measure density involves the odd powers of CV in addition to the even powers. The oeffiients (whih proportional to different powers of the field) and RR for the asymmetri ρ model have already been onsidered in Ref. []. The theory is being built ab initio beginning from the Hamiltonian of the system up to the expression for the free energy. The main distintive feature of the proposed method is the separate inlusion of the ontributions to the free energy from the short- and long-wave spin-density osillation modes. The generalized point of exit of the system from the ritial regime ontains both temperature and field variables. The form of temperature and field dependenes for the free energy of the system is determined by solutions of RR near the fixed point. The expression for the free energy F obtained at temperatures T > T without using power series in the saling variable and without any adjustable parameters an be employed in the field region near h (the rossover region between the so-alled wea and strong external fields). The limiting field h satisfies the ondition of the equality of sizes of the ritial-regime region by the temperature and field (the effet of the temperature and field on the system in the viinity of the ritial point is equivalent) [7]. The interesting rossover region is diffiult for the analyti treatment sine the saling variable alulated here is of the order of unity and power series in this variable are not effetive. We hope that the proposed method as well as our analyti representations may provide useful benhmars in studying the effet of an external magneti field on the ritial behaviour of D Ising-lie systems within the framewor of the higher non-gaussian approximation. Proeeding from the expression for the free energy whih involves the leading terms and the terms determining temperature and field onfluent orretions we an find other thermodynami harateristis (the average spin moment suseptibility entropy and speifi heat) by diret differentiation of F with respet to field or temperature. Referenes [] A. Pelissetto E. Viari Phys. Rep. 8 () 59. [] J. Engels L. Fromme M. Seniuh Nulear Phys. B 55 (FS) () 77. [] I.R. Yuhnovsii Phase Transitions of the Seond Order. Colletive Variables Method World Sientifi Singapore 987. [] I.R. Yuhnovsii Riv. Nuovo Cimento (989). [5] I.R. Yuhnovsii M.P. Kozlovsii I.V. Pylyu Mirosopi Theory of Phase Transitions in the Three-Dimensional Systems Eurosvit Lviv (in Urainian). [] J. Berges N. Tetradis C. Wetterih Phys. Rep. (). [7] N. Tetradis C. Wetterih Nulear Phys. B (FS) (99) 5. [8] C. Bagnuls C. Bervillier Phys. Rep. 8 () 9. [9] I.R. Yuhnovsii M.P. Kozlovsii I.V. Pylyu Phys. Rev. B (). [] I.R. Yuhnovsii I.V. Pylyu M.P. Kozlovsii J. Phys.: Condens. Matter (). [] I.R. Yuhnovsii I.V. Pylyu M.P. Kozlovsii J. Phys.: Condens. Matter () 7. [] M.P. Kozlovsii I.V. Pylyu Phys. Status Solidi B 8 (99). [] M.P. Kozlovsii I.V. Pylyu Z.E. Usateno Phys. Status Solidi B 97 (99) 5. [] L. Canet B. Delamotte D. Mouhanna J. Vidal Phys. Rev. D 7 () 5. [5] I.R. Yuhnovsii M.P. Kozlovsii I.V. Pylyu Preprint ITF-88-5R Inst. Teor. Fiz. Kiev 988 (in Russian). [] R. Guida J. Zinn-Justin J. Phys. A (998) 8. [7] M. Hasenbush Internat. J. Modern Phys. C () 9. [8] L. Canet B. Delamotte D. Mouhanna J. Vidal Phys. Rev. B 8 (). [9] T.R. Morris Phys. Lett. B (99) 55. [] K.-I. Aoi K. Moriawa W. Souma J.-I. Sumi H. Terao Prog. Theor. Phys. 95 (99) 9. [] K.-I. Aoi K. Moriawa W. Souma J.-I. Sumi H. Terao Prog. Theor. Phys. 99 (998) 5. [] D.F. Litim Phys. Rev. D () 57. [] D.F. Litim Preprint arxiv:hep-th/59. [] D.F. Litim Nulear Phys. B () 8. [5] I.V. Pylyu Preprint ITF-88-7R Inst. Teor. Fiz. Kiev 988 (in Russian). [] M.P. Kozlovsii Theoret. and Math. Phys. 78 (989). [7] M.P. Kozlovsii I.V. Pylyu Ur. Fiz. Zh. 5 (99) (in Urainian). [8] M.P. Kozlovsii I.V. Pylyu V.V. Duhovii Condens. Matter Phys. (997) 7. [9] I.V. Pylyu Low Temp. Phys. 5 (999) 877. [] I.V. Pylyu M.P. Kozlovsii O.O. Prytula Ferroeletris 7 (5). [] M.P. Kozlovsii I.V. Pylyu O.O. Prytula Phys. Rev. B 7 () 7.

12 [] M.P. Kozlovsii I.V. Pylyu O.O. Prytula Physia A 9 () 5. [] M.P. Kozlovsii I.V. Pylyu O.O. Prytula Condens. Matter Phys. 8 (5) 79. [] M.P. Kozlovsii I.V. Pylyu O.O. Prytula Nulear Phys. B 75 (FS) (). [5] V.V. Duhovii M.P. Kozlovsii I.V. Pylyu Theoret. and Math. Phys. 7 (99) 5. [] M.P. Kozlovsii Phase Transit. 8 (7). [7] M.P. Kozlovsii I.V. Pylyu O.O. Prytula Condens. Matter Phys. 7 (). [8] I.V. Pylyu Phase Transit. 8 (7). [9] C. Bervillier A. Jüttner D.F. Litim Nulear Phys. B 78 (FS) (7). [] I.V. Pylyu J. Magn. Magn. Mater. 5 (). [] I.V. Pylyu M.P. Kozlovsii Condens. Matter Phys. () 5. I.V. Pylyu M.P. Kozlovsii / Physia A 89 ()