Critical behaviour of a 3D Ising-like system in the ρ⁶ model approximation: Role of the correction for the potential averaging

The critical behaviour of systems belonging to the three-dimensional Ising universality class is studied theoretically using the collective variables (CV) method. The partition function of a one-component spin system is calculated by the integration over the layers of the CV phase space in the appro...

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Veröffentlicht in:	Condensed Matter Physics
Datum:	2012
Hauptverfasser:	Pylyuk, I.V., Ulyak, M.V.
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Sprache:	English
Veröffentlicht:	Інститут фізики конденсованих систем НАН України 2012
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spelling	Pylyuk, I.V. Ulyak, M.V. 2017-06-11T14:55:46Z 2017-06-11T14:55:46Z 2012 Critical behaviour of a 3D Ising-like system in the ρ⁶ model approximation: Role of the correction for the potential averaging / I.V. Pylyuk, M.V. Ulyak // Condensed Matter Physics. — 2012. — Т. 15, № 4. — С. 43006:1-10. — Бібліогр.: 20 назв. — англ. PACS: 05.50.+q, 64.60.F-, 75.10.Hk DOI:10.5488/CMP.15.43006 arXiv:1212.6139 https://nasplib.isofts.kiev.ua/handle/123456789/120304 The critical behaviour of systems belonging to the three-dimensional Ising universality class is studied theoretically using the collective variables (CV) method. The partition function of a one-component spin system is calculated by the integration over the layers of the CV phase space in the approximation of the non-Gaussian sextic distribution of order-parameter fluctuations (the ρ⁶ model). A specific feature of the proposed calculation consists in making allowance for the dependence of the Fourier transform of the interaction potential on the wave vector. The inclusion of the correction for the potential averaging leads to a nonzero critical exponent of the correlation function η and the renormalization of the values of other critical exponents. The contributions from this correction to the recurrence relations for the ρ⁶ model, fixed-point coordinates and elements of the renormalization-group linear transformation matrix are singled out. The expression for a small critical exponent η is obtained in a higher non-Gaussian approximation. З використанням методу колективних змiнних (КЗ) вивчається критична поведiнка систем, якi належать до класу унiверсальностi тривимiрної моделi Iзинга. Статистична сума однокомпонентної спiнової системи обчислюється шляхом iнтегрування за шарами фазового простору КЗ в наближеннi негаусового шестирного розподiлу флуктуацiй параметра порядку (модель ρ⁶). Особливiстю запропонованого розрахунку є прийняття до уваги залежностi фур’є-образу потенцiалу взаємодiї вiд хвильового вектора. Врахування поправки на усереднення потенцiалу приводить до вiдмiнного вiд нуля критичного показника кореляцiйної функцiї η i перенормування значень iнших критичних показникiв. Видiлено внески вiд цiєї поправки в рекурентнi спiввiдношення для моделi ρ⁶, координати фiксованої точки та елементи матрицi лiнiйного перетворення ренормалiзацiйної групи. Вираз для малого критичного показника η отримано у вищому негаусовому наближеннi. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Critical behaviour of a 3D Ising-like system in the ρ⁶ model approximation: Role of the correction for the potential averaging Критична поведiнка тривимiрної iзингоподiбної системи в наближеннi моделi ρ⁶: Роль поправки на усереднення потенцiалу Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	Critical behaviour of a 3D Ising-like system in the ρ⁶ model approximation: Role of the correction for the potential averaging
spellingShingle	Critical behaviour of a 3D Ising-like system in the ρ⁶ model approximation: Role of the correction for the potential averaging Pylyuk, I.V. Ulyak, M.V.
title_short	Critical behaviour of a 3D Ising-like system in the ρ⁶ model approximation: Role of the correction for the potential averaging
title_full	Critical behaviour of a 3D Ising-like system in the ρ⁶ model approximation: Role of the correction for the potential averaging
title_fullStr	Critical behaviour of a 3D Ising-like system in the ρ⁶ model approximation: Role of the correction for the potential averaging
title_full_unstemmed	Critical behaviour of a 3D Ising-like system in the ρ⁶ model approximation: Role of the correction for the potential averaging
title_sort	critical behaviour of a 3d ising-like system in the ρ⁶ model approximation: role of the correction for the potential averaging
author	Pylyuk, I.V. Ulyak, M.V.
author_facet	Pylyuk, I.V. Ulyak, M.V.
publishDate	2012
language	English
container_title	Condensed Matter Physics
publisher	Інститут фізики конденсованих систем НАН України
format	Article
title_alt	Критична поведiнка тривимiрної iзингоподiбної системи в наближеннi моделi ρ⁶: Роль поправки на усереднення потенцiалу
description	The critical behaviour of systems belonging to the three-dimensional Ising universality class is studied theoretically using the collective variables (CV) method. The partition function of a one-component spin system is calculated by the integration over the layers of the CV phase space in the approximation of the non-Gaussian sextic distribution of order-parameter fluctuations (the ρ⁶ model). A specific feature of the proposed calculation consists in making allowance for the dependence of the Fourier transform of the interaction potential on the wave vector. The inclusion of the correction for the potential averaging leads to a nonzero critical exponent of the correlation function η and the renormalization of the values of other critical exponents. The contributions from this correction to the recurrence relations for the ρ⁶ model, fixed-point coordinates and elements of the renormalization-group linear transformation matrix are singled out. The expression for a small critical exponent η is obtained in a higher non-Gaussian approximation. З використанням методу колективних змiнних (КЗ) вивчається критична поведiнка систем, якi належать до класу унiверсальностi тривимiрної моделi Iзинга. Статистична сума однокомпонентної спiнової системи обчислюється шляхом iнтегрування за шарами фазового простору КЗ в наближеннi негаусового шестирного розподiлу флуктуацiй параметра порядку (модель ρ⁶). Особливiстю запропонованого розрахунку є прийняття до уваги залежностi фур’є-образу потенцiалу взаємодiї вiд хвильового вектора. Врахування поправки на усереднення потенцiалу приводить до вiдмiнного вiд нуля критичного показника кореляцiйної функцiї η i перенормування значень iнших критичних показникiв. Видiлено внески вiд цiєї поправки в рекурентнi спiввiдношення для моделi ρ⁶, координати фiксованої точки та елементи матрицi лiнiйного перетворення ренормалiзацiйної групи. Вираз для малого критичного показника η отримано у вищому негаусовому наближеннi.
url	https://nasplib.isofts.kiev.ua/handle/123456789/120304
citation_txt	Critical behaviour of a 3D Ising-like system in the ρ⁶ model approximation: Role of the correction for the potential averaging / I.V. Pylyuk, M.V. Ulyak // Condensed Matter Physics. — 2012. — Т. 15, № 4. — С. 43006:1-10. — Бібліогр.: 20 назв. — англ.
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first_indexed	2025-11-24T16:32:36Z
last_indexed	2025-11-24T16:32:36Z
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fulltext	Condensed Matter Physics, 2012, Vol. 15, No 4, 43006: 1–10 DOI: 10.5488/CMP.15.43006 http://www.icmp.lviv.ua/journal Critical behaviour of a 3D Ising-like system in the ρ 6 model approximation: Role of the correction for the potential averaging I.V. Pylyuk1∗, M.V. Ulyak2 1 Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine, 1 Svientsitskii St., 79011 Lviv, Ukraine 2 Chervonograd State College of Mining Technologies and Economics, 17 Stus St., 80100 Chervonograd, Ukraine Received July 3, 2012, in final form September 12, 2012 The critical behaviour of systems belonging to the three-dimensional Ising universality class is studied theo- retically using the collective variables (CV) method. The partition function of a one-component spin system is calculated by the integration over the layers of the CV phase space in the approximation of the non-Gaussian sextic distribution of order-parameter fluctuations (the ρ6 model). A specific feature of the proposed calculation consists in making allowance for the dependence of the Fourier transform of the interaction potential on the wave vector. The inclusion of the correction for the potential averaging leads to a nonzero critical exponent of the correlation function η and the renormalization of the values of other critical exponents. The contributions from this correction to the recurrence relations for the ρ6 model, fixed-point coordinates and elements of the renormalization-group linear transformation matrix are singled out. The expression for a small critical exponent η is obtained in a higher non-Gaussian approximation. Key words: three-dimensional Ising-like system, critical behaviour, sextic distribution, potential averaging, small critical exponent PACS: 05.50.+q, 64.60.F-, 75.10.Hk 1. Method We shall use the approach of collective variables (CV) [1, 2], which allows us to calculate the expres- sion for the partition function of a system and to obtain complete expressions for thermodynamic func- tions near the phase-transition temperatureTc in addition to universal quantities (i.e., critical exponents). The CV approach is non-perturbative and similar to the Wilson non-perturbative renormalization-group (RG) approach (integration on fast modes and construction of an effective theory for slow modes) [3–5]. The term collective variables is a common name for a special class of variables that are specific for each individual physical system. The CV set contains variables associated with order parameters. Conse- quently, the phase space of CV is most natural in describing a phase transition. For magnetic systems, the CV ρk are the variables associated with the modes of spin-moment density oscillations, while the order parameter is related to the variable ρ0, in which the subscript “0” corresponds to the peak of the Fourier transform of the interaction potential. The methods available at present, make it possible to calculate universal quantities to a quite high degree of accuracy (see, for example, [6]). The advantage of the CV method lies in the possibility to obtain and analyse thermodynamic characteristics as functions of micro- scopic parameters of the original system. The use of the non-Gaussian basis distributions of fluctuations in calculating the partition function of a system does not bring about a problem of summing various classes of divergent (with respect to the Gaussian distribution) diagrams at the critical point. A considera- ∗E-mail: piv@icmp.lviv.ua © I.V. Pylyuk, M.V. Ulyak, 2012 43006-1 http://dx.doi.org/10.5488/CMP.15.43006 http://www.icmp.lviv.ua/journal I.V. Pylyuk, M.V. Ulyak tion of the increasing number of terms in the exponent of the non-Gaussian distribution is an alternative to the use of a higher-order perturbation theory based on the Gaussian distribution. The integration of partition function begins with the variables ρk having a large value of the wave vector k (of the order of the Brillouin half-zone boundary) and terminates at ρk with k → 0. For this purpose, we divide the phase space of the CV ρk into layers with the division parameter s. In each nth layer (corresponding to the region of wave vectors Bn+1 < k É Bn , Bn+1 = Bn /s, s > 1), the Fourier trans- form of the interaction potential is replaced by its average value. This simplified procedure leads to a zero value of the critical exponent η characterizing the behaviour of the pair-correlation function at the critical temperature Tc. 2. The setup The object of investigation is a three-dimensional (3D) Ising-like system. The Ising model, despite its simplicity, has, on the one hand, a wide scope of realistic applications, and, on the other hand, it can be considered as a model, which serves as a standard in studying other models possessing a much more complicated construction. In our previous calculations (for example, [7–10]), we assumed that the correction for the potential averaging is zero. As a result, we lost some information in the process of calculating the partition function of the system. In particular, the critical exponent ηwas equal to zero. In [11], the correction for the averaging of the Fourier transform of the potential is taken into account in the simplest non-Gaussian approximation (the ρ4 model based on the quartic fluctuation distribution). The inclusion of this correction gives rise to a nonzero value of the critical exponent η. Recurrence rela- tions (RR) between the coefficients of the effective distributions take another form as compared with the case when η = 0. A fixed point is shifted. Critical exponents of the correlation length ν, susceptibility γ and specific heat α are renormalized. Critical amplitudes are also modified. As is seen from table 1, the inclusion of a nonzero exponent ηwithin the CV method reduces the critical exponent ν (like in the non- Table 1. Estimates of the critical exponents for the ρ4 model and the RG parameter s = 4 in the case of the correction for the potential averaging not taken into account (∆Φ̃(k) = 0) and in the case of the correction for the potential averaging taken into account (∆Φ̃(k), 0). Condition η ν γ α ∆Φ̃(k) = 0 0 0.612 1.225 0.163 ∆Φ̃(k), 0 0.024 0.577 1.141 0.268 perturbative RG approach [12]). In order to obtain better quantitative estimates of ν and other critical exponents, it is necessary to use the distributions of fluctuations more complicated than the quartic dis- tribution. In the case of η= 0, the critical exponent of the correlation length for these distributions takes on larger values than ν for the ρ4 model (figure 1) [2, 8]. The results of calculations and their comparison with the other authors’ data show that the sextic distribution (the ρ6 model) provides a more adequate �� Figure 1. Evolution of the critical exponent of the correlation length ν with an increasing parameter of division of the CV phase space into layers s. Curves 1, 2, 3 and 4 correspond to the ρ4, ρ6, ρ8 and ρ10 models, respectively. 43006-2 Critical behaviour of a 3D Ising-like system quantitative description of the critical behaviour of a 3D Ising ferromagnet than the quartic distribu- tion [2, 13]. The sextic distribution for the modes of spin-moment density oscillations is presented as an exponential function of the CV whose argument includes the sixth power of the variable in addition to the second and the fourth powers. In the present publication, our aim is to investigate the effect of the above-mentioned correction for the potential averaging on the critical properties of a 3D Ising-like system and to elaborate a technique for calculating the small critical exponent η in the ρ6 model approximation. The analytic results, obtained in this higher non-Gaussian approximation, provide the basis for accurate analysis of the behaviour of the system near Tc with allowance for the exponent η. The developed approach permits to perform the calculations for a one-component spin system in real 3D space on the microscopic level without any adjustable parameters. The calculation technique for η is similar to that proposed in the case of the quartic distribution [11]. New special functions appearing in the construction of the phase-transition theory using the sextic distribution were considered in [2, 14]. In the case of the ρ6 model, we exploit the special functions with two arguments more complicated than the parabolic cylinder functions with one argument for the ρ4 model. 3. Basic relations We consider a 3D Ising-like system on a simple cubic lattice with N sites and period c. The Hamilto- nian of such a system has the form H =− 1 2 ∑ i,j Φ ( rij ) sisj −h ∑ i si , (3.1) where si is the operator of the z-component of spin at the ith site, having two eigenvalues +1 and −1. The interaction potential is an exponentially decreasing function Φ ( rij ) = A exp ( −rij/b ) . (3.2) Here, A is a constant, rij is the interparticle distance, and b is the radius of effective interaction. In the representation of the CV ρk, the partition function of the system in the absence of an external field H (h =µBH = 0, µB is the Bohr magneton) can be written in the form Z = 2N 2(N1−1)/2Q0[Q(P )]N1 ∫ exp { − 1 2 ∑ kÉB1 [ d ′(k)−d ′ (B1,B ′)]ρkρ−k } × ( 1+ ∆̂g +·· · ) exp [ − 1 2 R2 ∑ kÉB1 ρkρ−k − 3 ∑ l=2 R2l (2l)!N l−1 1 ∑ k1,...,k2lÉB1 ρk1 · · ·ρk2l δk1+···+k2l ] (dρ)N1 , (3.3) where B1 = B ′/s, N1 = N ′s−3, B ′ = ( b p 2 )−1 , N ′ = N s−3 0 , s0 = B/B ′ , B = π/c is the boundary of Brillouin half-zone, and δk1+···+k4 is the Kronecker symbol. For the coefficient d ′(k), we have d ′(k) = a′ 2 −βΦ̃(k). (3.4) Here, β = 1/(kT ) is the inverse temperature. For the Fourier transform of the interaction potential, we use the following approximation [8, 15]: Φ̃(k) = { Φ̃(0) ( 1−2b2k2 ) , k É B ′, 0, B ′ < k É B. (3.5) The quantities Q0, Q(P ) and R2l are ultimate functions of the initial coefficients a′ 2l (l = 0,1,2,3) [2, 16]. The coefficients a′ 2l are determined by special functions of two arguments and are dependent on the 43006-3 I.V. Pylyuk, M.V. Ulyak ratio of the effective interaction radius b to the lattice constant c, i.e., on the microscopic parameters of the system (see, for example, [17]). The correction, which is introduced by the operator ∆̂g , is considered in the linear approximation in ∆Φ̃(k) = q −2b2βΦ̃(0)k2. (3.6) The quantity ∆Φ̃(k) corresponds to the deviation βΦ̃(k) from the average value βΦ̃ ( B1,B ′ ) . Here, q = q̄βΦ̃(0), q̄ defines the geometric mean value of k2 on the interval (1/s,1]. In the above-mentioned ap- proximation, we arrive at the expression ∆̂ (1) g = 1 2 ∑ k1,...,k6ÉB1 [ 4C (h,α) a′ 4 ]2 ∂6 ∂ρk1 · · ·∂ρk6 (N ′)−4 ∑ B1<kÉB ′ ∆g (k) × ∑ l1,l2 exp[−i (k1 +k2 +k3 +k) l1 − i (k4 +k5 +k6 −k) l2] , (3.7) which defines the operator ∆̂g accurate to within the term proportional to ∂6 ∂ρk1 ···∂ρk6 (the terms propor- tional to the higher orders of operators ∂/∂ρk are not taken into account). The summation over the sites l1, l2 in (3.7) is carried out for the lattice with period c ′ =πb p 2. The role of∆g (k) is played by the quantity ∆g (k) = ∆Φ̃(k) 1−S2(2π)−2∆Φ̃(k) , (3.8) where S2 = (2π)2 ( 24/a′ 4 )1/2 F2(h,α). The forms of the functions C (h,α) and F2(h,α) as well as of their ar- guments h and α are presented in the next section. We assume that ∆̂ (1) g operates only on exp ( − 1 2 R2 ∑ kÉB1 ρkρ−k ) in (3.3). This assumption is associated with small contributions from R4 and R6 in comparison with the contribution from R2 (in particular, R4/ ( 6R2 2 ) ∼ 10−4 [11]). 4. Partition function of the system with allowance for the correction for the potential averaging The integration over the zeroth, first, second, . . . , nth layers of the CV phase space leads to the rep- resentation of the partition function in the form of a product of the partial partition functions Q̃n of individual layers and the integral of the “smoothed” effective distribution of fluctuations. We have Z = 2N 2(Nn+1−1)/2Q̃0Q̃1 · · ·Q̃n[Q(Pn)]Nn+1 ∫ W̃ (n+1) 6 (ρ)(dρ)Nn+1 . (4.1) The effective sextic distribution of fluctuations in the (n+1)th block structure is written as follows: W̃ (n+1) 6 (ρ)=exp [ − 1 2 ∑ kÉBn+1 d̃n+1(k)ρkρ−k − 3 ∑ l=2 ã(n+1) 2l (2l)!N l−1 n+1 ∑ k1,...,k2l ÉBn+1 ρk1 · · ·ρk2l δk1+···+k2l ] . (4.2) Here, Bn+1 = B ′s−(n+1), Nn+1 = N ′s−3(n+1), and d̃n+1(k), ã(n+1) 4 , ã(n+1) 6 are the renormalized values of coefficients d ′(k), a′ 4, a′ 6 after integration over n+1 layers of the phase space of CV. In comparison with the results obtained earlier without taking into account the correction for the potential averaging (see, for example, [2, 8, 16, 17]), the expression (4.1) includes new quantities. In par- ticular, the function f (hn ,αn ) =−48s9/2C 1/2(hn ,αn )F 3 2 (ηn ,ξn ) qn tn √ ã(n) 4 F0 (4.3) appearing in Q̃n characterizes the correction to the partial partition functions. The basic arguments hn = d̃n (Bn+1,Bn ) ( 6/ã(n) 4 )1/2 and αn = p 6ã(n) 6 /[ 15 ( ã(n) 4 )3/2] are expressed in terms of the coefficients d̃n(Bn+1,Bn ) (the average value of d̃n (k) in the nth layer of the CV phase space), ã(n) 4 and ã(n) 6 . The in- termediate variables ηn = ( 6s3 )1/2 F2(hn ,αn ) / C 1/2(hn ,αn ) and ξn = p 6s−3/2N (hn ,αn ) /[ 15C 3/2(hn ,αn ) ] 43006-4 Critical behaviour of a 3D Ising-like system are functions of hn and αn . The special functions C (hn ,αn ) =−F4(hn ,αn )+3F 2 2 (hn ,αn) and N (hn ,αn ) = F6(hn ,αn ) − 15F4(hn ,αn )F2(hn ,αn ) + 30F 3 2 (hn ,αn ) are combinations of the functions F2l (hn ,αn) = I2l (hn ,αn ) / I0(hn ,αn ), where I2l (hn ,αn ) = ∫∞ 0 t 2l exp ( −hn t 2 − t 4 −αn t 6 ) dt . The relation (4.3) for f (hn ,αn), except qn = q 1+α′ 0 s2 1+α′ 1 s2 · · · 1+α′ n−1 s2 , (4.4) contains the factor tn = √ ã(n) 4 24 1 F2(hn ,αn) s2 1+α′ 0 s2 1+α′ 1 · · · s2 1+α′ n−1 t (n) 0 1 βΦ̃(0) . (4.5) For the quantity α′ n , which defines the correction for the averaging of the Fourier transform of the poten- tial in the nth layer of the phase space of CV, we obtain α′ n = 144π2s6F 4 2 (ηn ,ξn )q̄ tnB0 . (4.6) The expressions for F0 (appearing in f (hn ,αn )), t (n) 0 (appearing in tn ) and B0 (appearing in α′ n ) are presented in [2, 11]. At large values of the RG parameter s, there emerge large intervals of wave vectors, in which Φ̃(k) is averaged. In this case (s > 5), the correction βΦ̃(k)−βΦ̃(Bn+1,Bn ) is substantial, so that its accounting in the linear approximation is incorrect. 5. Analysis of recurrence relations for the ρ 6 model. Critical exponent η The coefficients d̃n+1(k), ã(n+1) 4 and ã(n+1) 6 in (4.2) satisfy the following RR: d̃n+1(Bn+2,Bn+1) = (ã(n) 4 )1/2Ỹ (hn ,αn )−q 1+α′ 0 s2 1+α′ 1 s2 · · · 1+α′ n−1 s2 ( 1− 1+α′ n s2 ) , ã(n+1) 4 = ã(n) 4 s−3B̃(hn ,αn ), ã(n+1) 6 = ( ã(n) 4 )3/2 s−6D̃(hn ,αn ). (5.1) The contributions to the functions Ỹ (hn ,αn ) = Y (hn ,αn ) [1−G(hn ,αn )A0] , B̃(hn ,αn ) = B(hn ,αn ) [1+K (hn ,αn )C0] , D̃(hn ,αn ) = D(hn ,αn ) [1−L (hn ,αn )D0] (5.2) from the potential averaging are given by the terms G(hn ,αn )A0,K (hn ,αn )C0 andL (hn ,αn )D0. Here, Y (hn ,αn ) = s3/2F2(ηn ,ξn)C−1/2(hn ,αn ), B(hn ,αn ) = s6C (ηn ,ξn )C−1(hn ,αn ), D(hn ,αn ) = s21/2N (ηn ,ξn )C−3/2(hn ,αn ), (5.3) and G(hn ,αn ) = 288s9/2F 3 2 (ηn ,ξn )C 1/2(hn ,αn ) qtn p ũn , K (hn ,αn ) = 1728s3/2 F 5 2 (ηn ,ξn) C (ηn ,ξn ) C 1/2(hn ,αn ) qtn p ũn , L (hn ,αn ) = 17280s−3/2 F 6 2 (ηn ,ξn ) N (ηn ,ξn ) C 1/2(hn ,αn ) qtn p ũn . (5.4) 43006-5 I.V. Pylyuk, M.V. Ulyak The quantities A0, C0 and D0 as well as the quantities F0 and B0 mentioned above appear due to the inclusion of the averaging correction. They are calculated with the help of the summation over the dis- tances to the particles located at the lattice sites (see [2, 11]). In terms of the variables r̃n = s2 1+α′ 0 s2 1+α′ 1 · · · s2 1+α′ n−1 d̃n(0), ũn = s4 ( 1+α′ 0 )2 s4 ( 1+α′ 1 )2 · · · s4 ( 1+α′ n−1 )2 ã(n) 4 , w̃n = s6 ( 1+α′ 0 )3 s6 ( 1+α′ 1 )3 · · · s6 ( 1+α′ n−1 )3 ã(n) 6 , (5.5) the RR (5.1) assume the forms r̃n+1 = s2 1+α′ n [ −q + (ũn )1/2 Ỹ (hn ,αn ) ] , ũn+1 = s ( 1+α′ n )2 ũn B̃(hn ,αn ), w̃n+1 = 1 ( 1+α′ n )3 (ũn )3/2D̃(hn ,αn ). (5.6) There are two essential distinctions between the RR (5.6) and those obtained without involving the correction for ∆Φ̃(k) [2, 14, 18]. The first of them consists in the specific substitution of variables (5.5), which differs from the corresponding substitution without the correction by including the factors ( 1+α′ 0 )( 1+α′ 1 ) · · · ( 1+α′ n−1 ) . The second distinction concerns the transformation of special functions Y (hn ,αn ), B(hn ,αn ) andD(hn ,αn ) (5.3) into the functions Ỹ (hn ,αn ), B̃(hn ,αn ) and D̃(hn ,αn ) (5.2). This distinction is associated with a shift of the fixed-point coordinates and with corrections to the critical exponents of thermodynamic functions. A particular solution of RR (5.6) is a new fixed point (r̃ , ũ, w̃ ), which differs at∆Φ̃(k), 0 from the fixed point (r (0),u(0), w (0)) for the case of ∆Φ̃(k) = 0 [2, 14, 18]. The coordinates of the fixed point of RR (5.6) can be expressed as follows: r̃ =− f̃ βΦ̃(0), ũ = ϕ̃[βΦ(0)]2, w̃ = ψ̃[βΦ(0)]3. (5.7) Here, f̃ = q̄ [ Ỹ ( h̃,α̃ ) − h̃/ p 6 ][ Ỹ ( h̃,α̃ ) − ( 1+α′(0) ) h̃ /( s2 p 6 )]−1 , ϕ̃ = q̄2 [ 1− ( 1+α′(0) ) s−2 ]2 [ Ỹ ( h̃,α̃ ) − ( 1+α′(0) ) h̃ /( s2 p 6 )]−2 , ψ̃ = ( 1+α′(0) )−3 ( ϕ̃ )3/2 D̃ ( h̃,α̃ ) , (5.8) and h̃ = p 6 r̃ +q (ũ)1/2 , α̃ = p 6 15 w̃ (ũ)3/2 , α′(0) = 144π2s6F 4 2 ( η(0),ξ(0) ) q̄ t (0) B0 , t (0) = tn(u(0),h(0),α(0)) . (5.9) The quantities h(0), α(0) and η(0), ξ(0) describe, respectively, the basic (hn , αn ) and intermediate (ηn , ξn) arguments at the fixed point obtained without involving the correction for the potential averaging. In the 43006-6 Critical behaviour of a 3D Ising-like system linear approximation in ∆Φ̃(k), we have f̃ = f0 ( 1+ {[ Y ′ h ( h(0),α(0) ) h(0) / p 6−Y ( h(0),α(0) )/ p 6 ] ∆h+Y ′ α ( h(0),α(0) ) h(0) / p 6∆α −Y ( h(0),α(0) ) G ( h(0),α(0) ) A0h(0) /p 6 } ( 1− s−2 ) {[ Y ( h(0),α(0) ) −h(0) /p 6 ] × [ Y ( h(0),α(0) ) −h(0) /( s2 p 6 ) ]}−1 +α′(0)h(0) /( s2 p 6 ) [ Y ( h(0),α(0) ) −h(0) /( s2 p 6 ) ]−1 ) , ϕ̃ = ϕ0 ( 1+2 { − [ Y ′ h ( h(0),α(0) ) −1 /( s2 p 6 ) ] ∆h−Y ′ α ( h(0),α(0) ) ∆α+Y ( h(0),α(0) ) G(h(0),α(0))A0 +α′(0)h(0) /( s2 p 6 ) }[ Y ( h(0),α(0) ) −h(0) /( s2 p 6 ) ]−1 −2α′(0) s−2 /( 1− s−2 ) ) , ψ̃ = ψ0 [ 1+3 ( {[ Y ( h(0),α(0) ) −h(0) /( s2 p 6 ) ] D ′ h ( h(0),α(0) )/ 3− [ Y ′ h ( h(0),α(0) ) −1 /( s2 p 6 ) ] ×D ( h(0),α(0) ) } ∆h+ {[ Y ( h(0),α(0) ) −h(0) /( s2 p 6 ) ] D ′ α ( h(0),α(0) )/ 3 −Y ′ α ( h(0),α(0) ) D ( h(0),α(0) ) } ∆α+Y ( h(0),α(0) ) D ( h(0),α(0) ) G ( h(0),α(0) ) A0 +α′(0)h(0)D(h(0),α(0)) /( s2 p 6 ) ) {[ Y ( h(0),α(0) ) −h(0) /( s2 p 6 ) ] D ( h(0),α(0) ) }−1 −3α′(0) /( 1− s−2 ) −L ( h(0),α(0) ) D0 ] . (5.10) The quantities f0,ϕ0 andψ0 characterize the fixed-point coordinates in the case when η= 0. Formulas for the derivatives Y ′ h ( h(0),α(0) ) , Y ′ α ( h(0),α(0) ) , D ′ h ( h(0),α(0) ) and D ′ α(h(0),α(0)) can be written using series expansions of the corresponding functions in the vicinity of the fixed point [14]. The differences ∆h = h̃ −h(0) and ∆α = α̃−α(0) determine the displacements of the basic arguments hn and αn at the fixed points (r̃ , ũ, w̃) and ( r (0),u(0), w (0) ) . The RR (5.6)make it possible to find the elements of the RG linear transformationmatrix. Thesematrix elements R̃i j (i = 1,2,3; j = 1,2,3) can be presented in the following forms (the linear approximation in ∆Φ̃(k)): R̃11 = R11 ( 1−α′(0) ) +R(1h) 11 ∆h+R(1α) 11 ∆α+R(2) 11 A0 , R̃22 = R22 ( 1−2α′(0) ) +R(1h) 22 ∆h+R(1α) 22 ∆α+R(2) 22 C0 , R̃33 = R33 ( 1−3α′(0) ) +R(1h) 33 ∆h+R(1α) 33 ∆α+R(2) 33 D0 , R̃i j = R̃(0) i j (ũ)(i− j )/2 , i , j , R̃(0) 1k1 = R(0) 1k1 ( 1−α′(0) ) +R(1h) 1k1 ∆h+R(1α) 1k1 ∆α+R(2) 1k1 A0 , k1 = 2,3, R̃(0) 2k2 = R(0) 2k2 ( 1−2α′(0) ) +R(1h) 2k2 ∆h+R(1α) 2k2 ∆α+R(2) 2k2 C0 , k2 = 1,3, R̃(0) 3k3 = R(0) 3k3 ( 1−3α′(0) ) +R(1h) 3k3 ∆h+R(1α) 3k3 ∆α+R(2) 3k3 D0 , k3 = 1,2. (5.11) It should be noted that formulas for the quantities Ri i and R(0) i j (i , j ) from (5.11) coincide with the corresponding expressions for the matrix elements obtained without taking into account the correction for the potential averaging [2]. The contributions to the matrix elements R̃i j from terms R(1h) i j ∆h and R(1α) i j ∆α correspond to a fixed-point shift due to the inclusion of the dependence of the Fourier transform of the interaction potential on the wave vector. The terms like R(2) i j A0, R (2) i j C0 and R(2) i j D0 describe a direct contribution to R̃i j from the correction for averaging. The explicit solutions of RR r̃n = r̃ + c̃1Ẽ n 1 + c̃2w̃ (0) 12 (ũ)−1/2Ẽ n 2 + c̃3w̃ (0) 13 (ũ)−1Ẽ n 3 , ũn = ũ + c̃1w̃ (0) 21 (ũ)1/2Ẽ n 1 + c̃2Ẽ n 2 + c̃3w̃ (0) 23 (ũ)−1/2Ẽ n 3 , w̃n = w̃ + c̃1w̃ (0) 31 ũẼ n 1 + c̃2w̃ (0) 32 (ũ)1/2Ẽ n 2 + c̃3Ẽ n 3 (5.12) 43006-7 I.V. Pylyuk, M.V. Ulyak in terms of (5.5) read d̃n (Bn+1,Bn ) = s−2n [ n−1 ∏ m=0 ( 1+α′ m ) ] [ r̃ +q + c̃1Ẽ n 1 + c̃2w̃ (0) 12 (ũ)−1/2Ẽ n 2 + c̃3w̃ (0) 13 (ũ)−1Ẽ n 3 ] , ã(n) 4 = s−4n [ n−1 ∏ m=0 ( 1+α′ m )2 ] [ ũ + c̃1w̃ (0) 21 (ũ)1/2Ẽ n 1 + c̃2Ẽ n 2 + c̃3w̃ (0) 23 (ũ)−1/2Ẽ n 3 ] , ã(n) 6 = s−6n [ n−1 ∏ m=0 ( 1+α′ m )3 ] [ w̃ + c̃1w̃ (0) 31 ũẼ n 1 + c̃2w̃ (0) 32 (ũ)1/2Ẽ n 2 + c̃3Ẽ n 3 ] . (5.13) Here, r̃ , ũ and w̃ are given in (5.7) and (5.10). The coefficients c̃l are obtained from the initial condi- tions at n = 0. The temperature-independent quantities w̃ (0) i l determine the eigenvectors of the RG linear transformation matrix, and Ẽl are the eigenvalues of this matrix. The main distinction between the solutions (5.13) and the solutions of RR in the absence of the cor- rection for the potential averaging [2, 8] lies in the availability of factors of the type ( 1+α′ m ) . Taking into account the fact that limm→∞α′ m (Tc) = α′(0) at T = Tc, we obtain the following asymptotics in n for the quantities d̃n , ã(n) 4 and ã(n) 6 from (5.13): d̃n(Bn+1,Bn ) = (r̃ +q)s−n(2−η), ã(n) 4 = ũs−2n(2−η), ã(n) 6 = w̃ s−3n(2−η). (5.14) The quantity η is given by the formula η= α′(0) ln s (5.15) and corresponds to the critical exponent of the correlation function. Thus, the correction for the potential averaging being involved in calculating the partition function of the system, leads to a change of the asymptotics for the coefficients d̃n , ã(n) 4 and ã(n) 6 at T = Tc (in contrast to the case of ∆Φ̃(k) = 0, the exponents of these coefficients contain the quantity η). The renormalization of the critical exponent of the correlation length ν= ln s/ln Ẽ1, compared to the case of η= 0, is associated with a change of the larger eigenvalue (Ẽ1 > 1) of the RG linear transformation matrix. In contrast to ν, the critical exponent of the susceptibility γ= (2−η)ν explicitly depends on η. The specific heat of the system is characterized by the exponent α= 2−3ν, the expression for which contains a renormalized critical exponent of the correlation length ν. 6. Discussion and conclusions Within the CV approach, an analytic method for calculating the free energy of a 3D Ising-like system near the critical point was elaborated in [11] with the allowance for the simplest non-Gaussian fluctu- ation distribution (the ρ4 model) and the critical exponent η. The critical exponent of the correlation function η= 0.024 was found in [11] by including the correction for the potential averaging in the course of calculating the partition function of the system. This value of η is in accord with the other authors’ data. For comparison, the exponents η = 0.0335(25), η = 0.0362(8) and η = 0.033 were obtained within the framework of the field-theory approach (7-loop calculations) [19], Monte Carlo simulations [20] and non-perturbative RG approach (the order ∂4 of the derivative expansion) [12], respectively. Some differ- ence between the value of η = 0.024 and other authors’ data can be connected with the approximations in calculations, in particular, with the approximations which are made within the CV method (the sim- plest non-Gaussian distribution is used for obtaining η = 0.024; the correction inserted by the operator ∆̂g is considered in the linear approximation in ∆Φ̃(k); the terms proportional to the higher orders of operators ∂/∂ρk are not taken into account in the expression for ∆̂g ; the operator ∆̂g in (3.3) acts only on the first term in the exponent). As is established in [11], the inclusion of a nonzero exponent η within the CV method leads to a reduction of the value of the critical exponent for the correlation length, ν (in 43006-8 Critical behaviour of a 3D Ising-like system comparison with the case of η= 0). For better quantitative estimates of ν and other renormalized critical exponents, it is necessary to use the non-Gaussian approximations higher than the quartic distribution (e.g., the sextic one). This paper supplements the previous study [11] based on the ρ4 model. In the present publication, the correction for the averaging of the Fourier transform of the interaction potential is taken into account using the sextic fluctuation distribution (the ρ6 model). The effect of the mentioned correction on the critical behaviour of a 3D uniaxial magnet is investigated in the linear approximation. Extending the method for the layer-by-layer integration of the partition function of the system to the case of the ρ6 model, the RR for the coefficients of the effective distributions are written and analysed. It is shown that the inclusion of the correction for the potential averaging gives rise to a change of the asymptotics for the RR solutions at T = Tc. The critical exponents of the correlation length, susceptibility and specific heat are renormalized due to the above-mentioned correction. Our analytic representations acquire a more general and complete character as compared with the case when η = 0. An explicit expression (4.1) for the partition function allows us to calculate the free energy and other thermodynamic characteristics of the system near Tc taking into account the non- Gaussian sextic distribution and the small critical exponent η. An analytic procedure for the calculation of the critical exponent of the correlation function devel- oped in this paper on the basis of the ρ6 model for a one-component spin system may be generalized to the case of a system with an n-component order parameter. References 1. Yukhnovskii I.R., Phase Transitions of the Second Order. Collective Variables Method, World Scientific, Singa- pore, 1987. 2. Yukhnovskii I.R., Kozlovskii M.P., Pylyuk I.V., Microscopic Theory of Phase Transitions in the Three-Dimensional Systems, Eurosvit, Lviv, 2001 (in Ukrainian). 3. Berges J., Tetradis N., Wetterich C., Phys. Rep., 2002, 363, No. 4–6, 223; doi:10.1016/S0370-1573(01)00098-9. 4. Tetradis N., Wetterich C., Nucl. Phys. B, 1994, 422, No. 3, 541; doi:10.1016/0550-3213(94)90446-4. 5. Bagnuls C., Bervillier C., Phys. Rep., 2001, 348, No. 1–2, 91; doi:10.1016/S0370-1573(00)00137-X. 6. Pelissetto A., Vicari E., Phys. Rep., 2002, 368, No. 6, 549; doi:10.1016/S0370-1573(02)00219-3. 7. Pylyuk I.V., Kozlovskii M.P., Izv. Akad. Nauk SSSR, Ser. Fiz., 1991, 55, 597 (in Russian). 8. Yukhnovskii I.R., Kozlovskii M.P., Pylyuk I.V., Phys. Rev. B, 2002, 66, No. 13, 134410; doi:10.1103/PhysRevB.66.134410. 9. Kozlovskii M.P., Pylyuk I.V., Prytula O.O., Nucl. Phys. B, 2006, 753, No. 3, 242; doi:10.1016/j.nuclphysb.2006.07.006. 10. Pylyuk I.V., Kozlovskii M.P., Physica A, 2010, 389, No. 23, 5390; doi:10.1016/j.physa.2010.08.022. 11. Yukhnovskii I.R., Kozlovskii M.P., Pylyuk I.V., Ukr. J. Phys., 2012, 57, No. 1, 80 [Ukr. Fiz. Zh., 2012, 57, No. 1, 83 (in Ukrainian)]. 12. Canet L., Delamotte B., Mouhanna D., Vidal J., Phys. Rev. B, 2003, 68, No. 6, 064421; doi:10.1103/PhysRevB.68.064421. 13. Pylyuk I.V., Low Temp. Phys., 1999, 25, No. 12, 953; doi:10.1063/1.593847 [Fiz. Nizk. Temp., 1999, 25, No. 12, 1271 (in Russian)]. 14. Pylyuk I.V., Ukr. Fiz. Zh., 1996, 41, No. 9, 885 (in Ukrainian). 15. Yukhnovs’kii I.R., Riv. Nuovo Cimento, 1989, 12, No. 1, 1; doi:10.1007/BF02740597. 16. Pylyuk I.V., Theor. Math. Phys., 1998, 117, No. 3, 1459; doi:10.1007/BF02557185 [Teor. Mat. Fiz., 1998, 117, No. 3, 442 (in Russian)]. 17. Kozlovskii M.P., Pylyuk I.V., Dukhovii V.V., Condens. Matter Phys., 1997, 11, 17. 18. Kozlovskii M.P., Theor. Math. Phys., 1989, 78, No. 3, 300; doi:10.1007/BF01017668 [Teor. Mat. Fiz., 1989, 78, No. 3, 422 (in Russian)]. 19. Guida R., Zinn-Justin J., J. Phys. A, 1998, 31, No. 40, 8103; doi:10.1088/0305-4470/31/40/006. 20. Hasenbusch M., Int. J. Mod. Phys. C, 2001, 12, No. 7, 911; doi:10.1142/S0129183101002383. 43006-9 http://dx.doi.org/10.1016/S0370-1573(01)00098-9 http://dx.doi.org/10.1016/0550-3213(94)90446-4 http://dx.doi.org/10.1016/S0370-1573(00)00137-X http://dx.doi.org/10.1016/S0370-1573(02)00219-3 http://dx.doi.org/10.1103/PhysRevB.66.134410 http://dx.doi.org/10.1016/j.nuclphysb.2006.07.006 http://dx.doi.org/10.1016/j.physa.2010.08.022 http://dx.doi.org/10.1103/PhysRevB.68.064421 http://dx.doi.org/10.1063/1.593847 http://dx.doi.org/10.1007/BF02740597 http://dx.doi.org/10.1007/BF02557185 http://dx.doi.org/10.1007/BF01017668 http://dx.doi.org/10.1088/0305-4470/31/40/006 http://dx.doi.org/10.1142/S0129183101002383 I.V. Pylyuk, M.V. Ulyak Критична поведiнка тривимiрної iзингоподiбної системи в наближеннi моделi ρ6: Роль поправки на усереднення потенцiалу I.В. Пилюк1, М.В. Уляк2 1 Iнститут фiзики конденсованих систем НАН України, вул. Свєнцiцького, 1, 79011 Львiв, Україна 2 Державний вищий навчальний заклад “Гiрничо-економiчний коледж”, вул. Стуса, 17, 80100 Червоноград, Україна З використанням методу колективних змiнних (КЗ) вивчається критична поведiнка систем, якi належать до класу унiверсальностi тривимiрної моделi Iзинга. Статистична сума однокомпонентної спiнової си- стеми обчислюється шляхом iнтегрування за шарами фазового простору КЗ в наближеннi негаусового шестирного розподiлу флуктуацiй параметра порядку (модель ρ6). Особливiстю запропонованого розра- хунку є прийняття до уваги залежностi фур’є-образу потенцiалу взаємодiї вiд хвильового вектора. Вра- хування поправки на усереднення потенцiалу приводить до вiдмiнного вiд нуля критичного показника кореляцiйної функцiї η i перенормування значень iнших критичних показникiв. Видiлено внески вiд цiєї поправки в рекурентнi спiввiдношення для моделi ρ6, координати фiксованої точки та елементи матрицi лiнiйного перетворення ренормалiзацiйної групи. Вираз для малого критичного показника η отримано у вищому негаусовому наближеннi. Ключовi слова: тривимiрна iзингоподiбна система, критична поведiнка, шестирний розподiл, усереднення потенцiалу, малий критичний показник 43006-10 Method The setup Basic relations Partition function of the system with allowance for the correction for the potential averaging Analysis of recurrence relations for the rho6 model. Critical exponent eta Discussion and conclusions

Critical behaviour of a 3D Ising-like system in the ρ⁶ model approximation: Role of the correction for the potential averaging

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