Table of Contents
ISRN Condensed Matter Physics
VolumeΒ 2011, Article IDΒ 260750, 11 pages
http://dx.doi.org/10.5402/2011/260750
Research Article

Free Energy of Three-Dimensional Uniaxial Magnet in the Higher Non-Gaussian Approximation and in the Presence of an External Field

Institute for Condensed Matter Physics, National Academy of Sciences of Ukraine, 1 Svientsitskii Street, Lviv 79011, Ukraine

Received 18 August 2011; Accepted 25 September 2011

Academic Editor: A.Β Zaoui

Copyright Β© 2011 M. P. Kozlovskii and I. V. Pylyuk. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A three-dimensional Ising-like system in a homogeneous external field is studied on the basis of the higher non-Gaussian measure density (the 𝜌6 model). The presented solutions of recurrence relations for the coefficients of the effective measure densities and the generalized point of exit of the system from the critical regime are used for calculating the free energy of the system at temperatures 𝑇>𝑇𝑐 (𝑇𝑐 is the phase transition temperature in the absence of an external field). A calculation technique is based on the first principles of statistical physics and is naturally realized without any general assumptions and without any adjustable parameters. The obtained expression for the free energy does not involve series expansions in the scaling variable and is valid near the critical point not only in the regions of the so-called weak and strong external fields, but also in the crossover region between these fields, where power series in the scaling variable are not effective.

1. Introduction

The Ising model is one of the most studied models in the theory of the phase transitions, not only because it is considered as the prototype of statistical systems showing a nontrivial power-law critical behaviour, but also because it describes several physical systems [1]. Many systems characterized by short-range interactions and a scalar order parameter undergo a transition belonging to the Ising universality class. Despite the simplicity of the Ising model and great success in the research by means of various methods (see, e.g., [1]), the problem of analytic description of the Ising-like magnets in the three-dimensional (3𝐷) space near the critical point is still unsolved exactly even in the case of the absence of an external magnetic field. The presence of a magnetic field complicates this problem.

The description of the phase transitions in the 3𝐷 magnets, usually, is associated with the absence of exact solutions and with many approximate approaches for obtaining different system characteristics. In this paper, the behaviour of a 3𝐷 Ising-like system near the critical point in a homogeneous external field is studied using the collective variables (CVs) method [2–4]. The main peculiarity of this method is the integration of short-wave spin-density oscillation modes, which is generally done without using perturbation theory. The CV method is similar to the Wilson nonperturbative renormalization group (RG) approach (integration on fast modes and construction of an effective theory for slow modes) [5–7]. The term collective variables is a common name for a special class of variables that are specific for each individual physical system [2, 3]. The CV set contains variables associated with order parameters. Because of this, the phase space of CV is most natural for describing a phase transition. For magnetic systems, the CV 𝜌𝐀 are the variables associated with modes of spin-moment density oscillations, while the order parameter is related to the variable 𝜌0, in which the subscript β€œ0” corresponds to the peak of the Fourier transform of the interaction potential.

The free energy of a 3𝐷 Ising-like system in an external field at temperatures above 𝑇𝑐 is calculated using the non-Gaussian spin-density fluctuations, namely, the sextic measure density. The latter is represented as an exponential function of the CV whose argument includes the powers with the corresponding coupling constants up to the sixth power of the variable (the 𝜌6 model).

The present paper supplements the earlier works [8–11], in which the 𝜌6 model was used for calculating the free energy and other thermodynamic functions of the system in the absence of an external field. The 𝜌6 model provides a better quantitative description of the critical behaviour of a 3𝐷 Ising-like magnet than the 𝜌4 model [10]. For each of the 𝜌2π‘š models, there exists a preferred value of the RG parameter 𝑠=π‘ βˆ— (π‘ βˆ—=3.5862 for the 𝜌4 model, π‘ βˆ—=2.7349 for the 𝜌6 model, π‘ βˆ—=2.6511 for the 𝜌8 model, and π‘ βˆ—=2.6108 for the 𝜌10 model) nullifying the average value of the coefficient in the term with the second power in the effective density of measure at the fixed point. The values of 𝑠 close to π‘ βˆ— are optimal for the given method of calculations. The difference form of the recurrence relations (RRs) between the coefficients of effective non-Gaussian densities of measures operates successfully just in this region of 𝑠. It was established (see, e.g., [10, 11]) that as the form of the density of measure becomes more complicated, the dependence of the critical exponent of the correlation length 𝜈 on the RG parameter 𝑠 becomes weaker gradually, and, starting from the sextic density of measure, the value of the exponent 𝜈, having a tendency to saturate with increasing π‘š (which characterizes the order of the 𝜌2π‘š model, π‘š=2,3,4,5), changes insignificantly (see Figures 1 and 2). The point π‘ β‰ˆπ‘ βˆ— in Figure 1 corresponds to the beginning of the 𝜈(𝑠) curve stabilization for each of the 𝜌2π‘š models. The value of the exponent 𝜈 in Figure 2 is calculated for 𝑠=π‘ βˆ—. The 𝜌2 model (Gaussian approximation) leads to the classical value 𝜈=0.500. In the case when 𝑠=π‘ βˆ—, we have 𝜈=0.605 for the 𝜌4 model and 𝜈=0.637 for the 𝜌6 model. The value of the critical exponent 𝜈 for the 𝜌6 model agrees more closely with the other authors' data for the 3𝐷 Ising model than the estimate in the 𝜌4 model approximation, for example, with the values determined using the fixed-dimension perturbative RG (𝜈=0.6304(13) [12]), high-temperature series (𝜈=0.63002(23) [13]), and Monte Carlo simulations (𝜈=0.6296(7) [14]). The Ising model corresponds to the 𝜌2π‘š model approximation, where the order of the model 2π‘šβ‰₯4. The 𝜌4 model allows us to go beyond the classical analysis and to describe all qualitative aspects of the second-order phase transition. As is seen from Figures 1 and 2, the critical behaviour of a 3𝐷 Ising-like system within the CV method can be described quantitatively at 2π‘šβ‰₯6, and, in particular, at 2π‘š=6. It was shown in [10, 15] that the graphs of the temperature dependences of the order parameter (the spontaneous magnetization) and specific heat for the 𝜌6 model agree more closely with the Liu and Fisher's results [16] than the corresponding plots for the 𝜌4 model. The correctness of the choice of the 𝜌6 model for investigations is also confirmed in [17, 18], where Tsypin proved that the term with the sixth power of the variable in the effective potential plays an important role.

260750.fig.001
Figure 1: Evolution of the critical exponent of the correlation length 𝜈 with 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.
260750.fig.002
Figure 2: Saturation of the critical exponent 𝜈 with increasing order of the 𝜌2π‘š model.

The methods existing at present make it possible to calculate universal quantities to a quite high degree of accuracy (see, e.g., [1]). The advantage of the CV method lies in the possibility of obtaining and analysing thermodynamic characteristics as functions of the microscopic parameters of the initial system [10, 11, 15, 19, 20]. The results of calculations for a 3𝐷 Ising system on the basis of the 𝜌4 and 𝜌6 models are in accord with the results obtained by other authors. Comparison of the critical exponents and universal ratios of critical amplitudes with the data calculated within the field-theory approach [21–23] and high-temperature expansions [24–28] can be found in our articles [10, 15]. In [29], the scaling functions of the order parameter and susceptibility, calculated on the basis of the free energy for the 𝜌4 model, were graphically compared with other authors' data. Our results accord with the results obtained within the framework of the parametric representation of the equation of state [30] and Monte Carlo simulations [31].

The expressions for the thermodynamic characteristics of the system in the presence of an external field have already been obtained on the basis of the simplest non-Gaussian measure density (the 𝜌4 model) in [32–35] using the point of exit of the system from the critical regime as a function of the temperature (the weak-field region) or of the field (the strong-field region). In [32, 33], the thermodynamic characteristics are presented in the form of series expansions in the variables, which are combinations of the temperature and field. Our calculations in the 𝜌4 model approximation were also performed for temperatures 𝑇>𝑇𝑐 [34] and 𝑇<𝑇𝑐 [35] without using similar expansions for the roots of cubic equations appearing in the theoretical analysis. In this paper, the free energy of a 3𝐷 uniaxial magnet within the framework of the more complicated 𝜌6 model is found introducing the generalized point of exit of the system from the critical regime. The expression for this point takes into account the temperature and field variables simultaneously. In our earlier article [29], the point of exit of the system from the critical regime was found in the simpler non-Gaussian approximation (the 𝜌4 model) using the numerical calculations. In contrast to [29], the point of exit of the system in the present paper is explicitly defined as a function of the temperature and field. This allows one to solve our problem, which consists in obtaining the free energy of a 3𝐷 Ising-like system in the higher non-Gaussian approximation and in the presence of an external field without involving numerical calculations and without using series expansions in the scaling variable.

2. General Relations

We consider a 3𝐷 Ising-like system on a simple cubic lattice with 𝑁 sites and period 𝑐 in a homogeneous external field β„Ž. Such a system is described by the Hamiltonian 1𝐻=βˆ’2𝐣,π₯Ξ¦ξ€·π‘Ÿπ£π₯ξ€ΈπœŽπ£πœŽπ₯ξ“βˆ’β„Žπ£πœŽπ£,(1) where π‘Ÿπ£π₯ is the distance between particles at sites 𝐣 and π₯, and 𝜎𝐣 is the operator of the 𝑧 component of spin at the 𝐣th site, having two eigenvalues +1 and βˆ’1. The interaction potential has the form of an exponentially decreasing function Ξ¦ξ€·π‘Ÿπ£π₯ξ€Έξ‚΅βˆ’π‘Ÿ=𝐴exp𝐣π₯𝑏.(2) Here 𝐴 is a constant and 𝑏 is the radius of effective interaction. For the Fourier transform of the interaction potential, we use the following approximation [2, 10, 11]: Φ(π‘˜)=Ξ¦(0)1βˆ’2𝑏2π‘˜2ξ€Έ,π‘˜β‰€π΅ξ…ž,0,π΅ξ…ž<π‘˜β‰€π΅,(3) where 𝐡 is the boundary of the Brillouin half-zone (𝐡=πœ‹/𝑐),π΅ξ…žβˆš=(𝑏2)βˆ’1,Φ(0)=8πœ‹π΄(𝑏/𝑐)3.

In the CV representation for the partition function of the system, we have [2, 36] ξ€œξƒ¬1𝑍=exp2𝐀𝛽Φ(π‘˜)πœŒπ€πœŒβˆ’π€βˆš+π›½β„Žπ‘πœŒ0𝐽(𝜌)(π‘‘πœŒ)𝑁.(4) Here, the summation over the wave vectors 𝐀 is carried out within the first Brillouin zone, 𝛽=1/(π‘˜π‘‡) is the inverse temperature, the CV 𝜌𝐀 are introduced by means of the functional representation for operators of spin-density oscillation modes Μ‚πœŒπ€βˆš=(𝑁)βˆ’1βˆ‘π₯𝜎π₯exp(βˆ’π‘–π€π₯), 𝐽(𝜌)=2π‘ξ€œξƒ¬ξ“exp2πœ‹π‘–π€πœ”π€πœŒπ€+𝑛β‰₯1(2πœ‹π‘–)2𝑛𝑁1βˆ’π‘›Γ—β„³2𝑛(2𝑛)!𝐀1,…,𝐀2π‘›πœ”π€1β‹―πœ”π€2𝑛𝛿𝐀1+β‹―+𝐀2π‘›βŽ€βŽ₯βŽ₯⎦(π‘‘πœ”)𝑁(5) is the Jacobian of transition from the set of 𝑁 spin variables 𝜎π₯ to the set of CV 𝜌𝐀, and 𝛿𝐀1+β‹―+𝐀2𝑛 is the Kronecker symbol. The variables πœ”π€ are conjugate to 𝜌𝐀, and the cumulants β„³2𝑛 assume constant values (see [2–4]).

Proceeding from (4) and (5), we obtain the following initial expression for the partition function of the system in the 𝜌6 model approximation: 𝑍=2𝑁2(π‘β€²βˆ’1)/2π‘’π‘Žβ€²0π‘β€²Γ—ξ€œβŽ‘βŽ’βŽ’βŽ’βŽ£expβˆ’π‘Žξ…ž1ξ€·π‘ξ…žξ€Έ1/2𝜌0βˆ’12ξ“π€π‘˜β‰€π΅β€²π‘‘ξ…ž(π‘˜)πœŒπ€πœŒβˆ’π€βˆ’3𝑙=2π‘Žξ…ž2𝑙𝑁(2𝑙)!ξ…žξ€Έπ‘™βˆ’1𝐀1,…,𝐀2π‘™π‘˜π‘–β‰€π΅ξ…žπœŒπ€1β‹―πœŒπ€2𝑙𝛿𝐀1+β‹―+𝐀2π‘™βŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦Γ—(π‘‘πœŒ)𝑁′.(6) Here 𝑁′=𝑁𝑠0βˆ’π‘‘ (𝑑=3 is the space dimension), 𝑠0√=𝐡/𝐡′=πœ‹2𝑏/𝑐, and π‘Žξ…ž1=βˆ’π‘ 0𝑑/2β„Žξ…ž,β„Žξ…ž=π›½β„Ž. The expressions for the remaining coefficients are given in [8–11]. These coefficients are functions of 𝑠0, that is, of the ratio of microscopic parameters 𝑏 and 𝑐. We shall use the method of β€œlayer-by-layer” integration of (6) with respect to variables 𝜌𝐀 described in [2–4]. The integration begins from the variables 𝜌𝐀 with a large value of the wave vector π‘˜ (of the order of the Brillouin half-zone boundary) and terminates at 𝜌𝐀 with π‘˜β†’0. For this purpose, we divide the phase space of the CV 𝜌𝐀 into layers with the division parameter 𝑠. In each 𝑛th layer, the Fourier transform of the interaction potential is replaced by its average value (the arithmetic mean in the given case).

The integration over the zeroth, first, second,…, 𝑛th layers of the CV phase space leads to the representation of the partition function in the form of a product of the partial partition functions 𝑄𝑛 of individual layers and the integral of the β€œsmoothed” effective measure density 𝑍=2𝑁2(𝑁𝑛+1βˆ’1)/2𝑄0𝑄1⋯𝑄𝑛𝑄𝑃𝑛𝑁𝑛+1Γ—ξ€œπ’²6(𝑛+1)(𝜌)(π‘‘πœŒ)𝑁𝑛+1.(7) The expressions for 𝑄𝑛, 𝑄(𝑃𝑛) are presented in [8–11], and 𝑁𝑛+1=π‘ξ…žπ‘ βˆ’π‘‘(𝑛+1). The sextic measure density of the (𝑛+1)th block structure 𝒲6(𝑛+1)(𝜌) has the form 𝒲6(𝑛+1)⎑⎒⎒⎒⎣(𝜌)=expβˆ’π‘Ž1(𝑛+1)𝑁1/2𝑛+1𝜌0βˆ’12ξ“π€π‘˜β‰€π΅π‘›+1𝑑𝑛+1(π‘˜)πœŒπ€πœŒβˆ’π€βˆ’3𝑙=2π‘Ž(𝑛+1)2𝑙(2𝑙)!π‘π‘™βˆ’1𝑛+1𝐀1,…,𝐀2π‘™π‘˜π‘–β‰€π΅π‘›+1𝜌𝐀1β‹―πœŒπ€2𝑙𝛿𝐀1+β‹―+𝐀2π‘™βŽ€βŽ₯βŽ₯βŽ₯βŽ₯⎦,(8) where 𝐡𝑛+1=π΅ξ…žπ‘ βˆ’(𝑛+1),𝑑𝑛+1(π‘˜)=π‘Ž2(𝑛+1)ξ‚βˆ’π›½Ξ¦(π‘˜),π‘Ž1(𝑛+1) and π‘Ž(𝑛+1)2𝑙 are the renormalized values of the coefficients π‘Žξ…ž1 and π‘Žξ…ž2𝑙 after integration over 𝑛+1 layers of the phase space of CV. The coefficients π‘Ž1(𝑛)=π‘ βˆ’π‘›π‘‘π‘›,𝑑𝑛(0)=π‘ βˆ’2π‘›π‘Ÿπ‘› (appearing in the quantity 𝑑𝑛(π‘˜)=𝑑𝑛(0)+2𝛽Φ(0)𝑏2π‘˜2),π‘Ž4(𝑛)=π‘ βˆ’4𝑛𝑒𝑛, and π‘Ž6(𝑛)=π‘ βˆ’6𝑛𝑀𝑛 are connected with the coefficients of the (𝑛+1)th layer through the RR 𝑑𝑛+1=𝑠(𝑑+2)/2𝑑𝑛,π‘Ÿπ‘›+1=𝑠2ξ€Ίβˆ’π‘ž+𝑒𝑛1/2π‘Œξ€·β„Žπ‘›,𝛼𝑛,𝑒𝑛+1=𝑠4βˆ’π‘‘π‘’π‘›π΅ξ€·β„Žπ‘›,𝛼𝑛,𝑀𝑛+1=𝑠6βˆ’2𝑑𝑒𝑛3/2π·ξ€·β„Žπ‘›,𝛼𝑛(9) whose solutions 𝑑𝑛=𝑑(0)βˆ’π‘ 0𝑑/2β„Žβ€²πΈπ‘›1,π‘Ÿπ‘›=π‘Ÿ(0)+𝑐1𝐸𝑛2+𝑐2𝑀(0)12𝑒(0)ξ€Έβˆ’1/2𝐸𝑛3+𝑐3𝑀(0)13𝑒(0)ξ€Έβˆ’1𝐸𝑛4,𝑒𝑛=𝑒(0)+𝑐1𝑀(0)21𝑒(0)ξ€Έ1/2𝐸𝑛2+𝑐2𝐸𝑛3+𝑐3𝑀(0)23𝑒(0)ξ€Έβˆ’1/2𝐸𝑛4,𝑀𝑛=𝑀(0)+𝑐1𝑀(0)31𝑒(0)𝐸𝑛2+𝑐2𝑀(0)32𝑒(0)ξ€Έ1/2𝐸𝑛3+𝑐3𝐸𝑛4(10) in the region of the critical regime are used for calculating the free energy of the system. Here π‘Œξ€·β„Žπ‘›,𝛼𝑛=𝑠𝑑/2𝐹2ξ€·πœ‚π‘›,πœ‰π‘›πΆξ€·β„Žξ€Έξ€Ίπ‘›,π›Όπ‘›ξ€Έξ€»βˆ’1/2,π΅ξ€·β„Žπ‘›,𝛼𝑛=𝑠2π‘‘πΆξ€·πœ‚π‘›,πœ‰π‘›πΆξ€·β„Žξ€Έξ€Ίπ‘›,π›Όπ‘›ξ€Έξ€»βˆ’1,π·ξ€·β„Žπ‘›,𝛼𝑛=𝑠7𝑑/2π‘ξ€·πœ‚π‘›,πœ‰π‘›πΆξ€·β„Žξ€Έξ€Ίπ‘›,π›Όπ‘›ξ€Έξ€»βˆ’3/2.(11) The quantity π‘ž=ξ‚π‘žπ›½Ξ¦(0) determines the average value of the Fourier transform of the potential 𝛽Φ(𝐡𝑛+1,𝐡𝑛)=𝛽Φ(0)βˆ’π‘ž/𝑠2𝑛 in the 𝑛th layer (in this paper, π‘ž=(1+π‘ βˆ’2)/2 corresponds to the arithmetic mean value of π‘˜2 on the interval (1/𝑠,1]). The basic arguments β„Žπ‘› and 𝛼𝑛 are determined by the coefficients of the sextic measure density of the 𝑛th block structure. The intermediate variables πœ‚π‘› and πœ‰π‘› are functions of β„Žπ‘› and 𝛼𝑛. The expressions for both basic and intermediate arguments as well as the special functions appearing in (11) are the same as in the absence of an external field (see [8–11]). The quantities 𝐸𝑙 in (10) are the eigenvalues of the matrix of the RG linear transformation βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽπ‘‘π‘›+1βˆ’π‘‘(0)π‘Ÿπ‘›+1βˆ’π‘Ÿ(0)𝑒𝑛+1βˆ’π‘’(0)𝑀𝑛+1βˆ’π‘€(0)⎞⎟⎟⎟⎟⎟⎟⎠=βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽπ‘…110000𝑅22𝑅23𝑅240𝑅32𝑅33𝑅340𝑅42𝑅43𝑅44βŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽπ‘‘π‘›βˆ’π‘‘(0)π‘Ÿπ‘›βˆ’π‘Ÿ(0)π‘’π‘›βˆ’π‘’(0)π‘€π‘›βˆ’π‘€(0)⎞⎟⎟⎟⎟⎟⎟⎠.(12) We have 𝐸1=𝑅11=𝑠(𝑑+2)/2. Other nonzero matrix elements 𝑅𝑖𝑗 (𝑖=2,3,4; 𝑗=2,3,4) and the eigenvalues 𝐸2, 𝐸3, 𝐸4 coincide, respectively, with the quantities 𝑅𝑖1𝑗1 (𝑖1=π‘–βˆ’1; 𝑗1=π‘—βˆ’1) and 𝐸1, 𝐸2, 𝐸3 obtained in the case of β„Ž=0. The quantities 𝑓0, πœ‘0, and πœ“0 characterizing the fixed-point coordinates 𝑑(0)=0,π‘Ÿ(0)=βˆ’π‘“0𝛽𝑒Φ(0),(0)=πœ‘0𝛽Φ(0)2,𝑀(0)=πœ“0𝛽Φ(0)3,(13) as well as the remaining coefficients in (10), are also defined on the basis of expressions corresponding to a zero external field. The temperature-independent quantities 𝑓0, πœ‘0, and πœ“0 as well as the renormalized quantities 𝑓0=𝑓0/π‘ž,πœ‘0=πœ‘0/π‘ž2, and πœ“0=πœ“0/π‘ž3 independent of the potential averaging are presented in Table 1 for the optimal RG parameter 𝑠=π‘ βˆ—=2.7349.

tab1
Table 1: Quantities, which characterize the coordinates of the fixed point.

3. Contributions to the Free Energy of the System in the Presence of an External Field

Let us calculate the free energy 𝐹=βˆ’π‘˜π‘‡ln𝑍 of a 3𝐷 Ising-like system above the critical temperature 𝑇𝑐. The basic idea of such a calculation on the microscopic level consists in the separate inclusion of the contributions from short-wave (𝐹CR, the region of the critical regime) and long-wave (𝐹LGR, the region of the limiting Gaussian regime) modes of spin-moment density oscillations [2–4]: 𝐹=𝐹0+𝐹CR+𝐹LGR.(14) Here 𝐹0=βˆ’π‘˜π‘‡π‘ln2 is the free energy of 𝑁 noninteracting spins. Each of three components in (14) corresponds to individual factors in the convenient representation 𝑍=2𝑁𝑍CR𝑍LGR(15) for the partition function given by (7). The contributions from short- and long-wave modes to the free energy of the system in the presence of an external field are calculated in the 𝜌6 model approximation according to the scheme proposed in [8–11]. Short-wave modes are characterized by an RG symmetry and are described by the non-Gaussian measure density. The calculation of the contribution from long-wave modes is based on using the Gaussian measure density as the basis one. Here, we have developed a direct method of calculations with the results obtained by taking into account the short-wave modes as initial parameters. The main results obtained in the course of deriving the complete expression for the free energy of the system are presented below.

3.1. Region of the Critical Regime

A calculation technique based on the 𝜌6 model for the contribution 𝐹CR is similar to that elaborated in the absence of an external field (see, e.g., [4, 9, 10]). Carrying out the summation of partial free energies 𝐹𝑛 over the layers of the phase space of CV, we can calculate 𝐹CR: 𝐹CR=πΉξ…ž0+πΉξ…žCR,πΉξ…ž0=βˆ’π‘˜π‘‡π‘ξ…ž[],𝐹ln𝑄(β„³)+ln𝑄(𝑑)ξ…žCR=𝑛𝑝𝑛=1𝐹𝑛.(16) An explicit dependence of 𝐹𝑛 on the layer number 𝑛 is obtained using solutions (10) of RR and series expansions of special functions in small deviations of the basic arguments from their values at the fixed point. The main peculiar feature of the present calculations lies in using the generalized point of exit of the system from the critical regime of order-parameter fluctuations. The inclusion of the more complicated expression for the exit point (as a function of both the temperature and field variables) [37] π‘›π‘ξ‚€ξ‚β„Ž=βˆ’ln2+ξ‚β„Ž2𝑐2ln𝐸1βˆ’1(17) leads to the distinction between formula (16) for πΉξ…žCR and the analogous relation at β„Ž=0 [9, 10]. The quantity ξ‚β„Ž=β„Žξ…ž/𝑓0 is determined by the dimensionless field β„Žξ…ž, while the quantity ξ‚β„Žπ‘=Μƒπœπ‘0 is a function of the reduced temperature 𝜏=(π‘‡βˆ’π‘‡π‘)/𝑇𝑐. Here Μƒπœ=̃𝑐1(0)𝜏/𝑓0,𝑝0=ln𝐸1/ln𝐸2=(𝑑+2)𝜈/2, ̃𝑐1(0) characterizes the coefficient 𝑐1 in solutions (10) of RR, 𝜈=ln𝑠/ln𝐸2 is the critical exponent of the correlation length. At β„Ž=0,𝑛𝑝 becomes π‘šπœ=βˆ’lnΜƒπœ/ln𝐸2βˆ’1 (see [4, 9, 10]). At 𝑇=𝑇𝑐 (𝜏=0), the quantity 𝑛𝑝 coincides with the exit point π‘›β„Žξ‚=βˆ’lnβ„Ž/ln𝐸1βˆ’1 [38]. The limiting value of the field ξ‚β„Žπ‘ is obtained by the equality of the exit points defined by the temperature and by the field (π‘šπœ=π‘›β„Ž).

Having expression (17) for 𝑛𝑝, we arrive at the relations [39] 𝐸𝑛𝑝1+1=ξ‚€ξ‚β„Ž2+ξ‚β„Ž2π‘ξ‚βˆ’1/2,ΜƒπœπΈπ‘›π‘2+1=𝐻𝑐,𝐻𝑐=ξ‚β„Ž1/𝑝0π‘ξ‚€ξ‚β„Ž2+ξ‚β„Ž2π‘ξ‚βˆ’1/(2𝑝0),𝐸𝑛𝑝3+1=𝐻3,𝐻3=ξ‚€ξ‚β„Ž2+ξ‚β„Ž2𝑐Δ1/(2𝑝0),𝐸𝑛𝑝4+1=𝐻4,𝐻4=ξ‚€ξ‚β„Ž2+ξ‚β„Ž2𝑐Δ2/(2𝑝0),π‘ βˆ’(𝑛𝑝+1)=ξ‚€ξ‚β„Ž2+ξ‚β„Ž2𝑐1/(𝑑+2),(18) where Ξ”1=βˆ’ln𝐸3/ln𝐸2 and Ξ”2=βˆ’ln𝐸4/ln𝐸2 are the exponents, which determine the first and second confluent corrections, respectively. Numerical values of the quantities 𝐸𝑙 (𝑙=1,2,3,4), 𝜈, Ξ”1, and Ξ”2 for the RG parameter 𝑠=π‘ βˆ—=2.7349 are given in Table 2. In the weak-field region (ξ‚ξ‚β„Žβ„Žβ‰ͺ𝑐), quantities (18) can be calculated with the help of the following expansions: 𝐸𝑛𝑝1+1=ξ‚β„Žπ‘βˆ’111βˆ’2ξ‚β„Ž2ξ‚β„Ž2𝑐ξƒͺ,ξ‚β„Žπ‘βˆ’1=Μƒπœβˆ’π‘0,𝐻𝑐1=1βˆ’2𝑝0ξ‚β„Ž2ξ‚β„Ž2𝑐,𝐻3=ξ‚β„ŽΞ”1/𝑝0𝑐Δ1+12𝑝0ξ‚β„Ž2ξ‚β„Ž2𝑐ξƒͺ,ξ‚β„ŽΞ”1/𝑝0𝑐=ΜƒπœΞ”1,𝐻4=ξ‚β„ŽΞ”2/𝑝0𝑐Δ1+22𝑝0ξ‚β„Ž2ξ‚β„Ž2𝑐ξƒͺ,ξ‚β„ŽΞ”2/𝑝0𝑐=ΜƒπœΞ”2,π‘ βˆ’(𝑛𝑝+1)=ξ‚β„Žπ‘2/(𝑑+2)11+ξ‚β„Žπ‘‘+22ξ‚β„Ž2𝑐ξƒͺ,ξ‚β„Žπ‘2/(𝑑+2)=Μƒπœπœˆ.(19) In the strong-field region (ξ‚ξ‚β„Žβ„Žβ‰«π‘), these quantities satisfy the expressions 𝐸𝑛𝑝1+1=ξ‚β„Žβˆ’111βˆ’2ξ‚β„Ž2π‘ξ‚β„Ž2ξƒͺ,𝐻𝑐=ξƒ©ξ‚β„Žπ‘ξ‚β„Žξƒͺ1/𝑝011βˆ’2𝑝0ξ‚β„Ž2π‘ξ‚β„Ž2ξƒͺ,𝐻3=ξ‚β„ŽΞ”1/𝑝0Δ1+12𝑝0ξ‚β„Ž2π‘ξ‚β„Ž2ξƒͺ,𝐻4=ξ‚β„ŽΞ”2/𝑝0Δ1+22𝑝0ξ‚β„Ž2π‘ξ‚β„Ž2ξƒͺ,π‘ βˆ’(𝑛𝑝+1)=ξ‚β„Ž2/(𝑑+2)11+ξ‚β„Žπ‘‘+22π‘ξ‚β„Ž2ξƒͺ.(20) It should be noted that the variables ξ‚ξ‚β„Žβ„Ž/𝑐 (the weak fields) and (ξ‚β„Žπ‘/ξ‚β„Ž)1/𝑝0 (the strong fields) coincide with the accepted choice of the arguments for scaling functions in accordance with the scaling theory. In the particular case of β„Ž=0 and πœβ‰ 0, (19) are defined as 𝐸𝑛𝑝1+1=Μƒπœβˆ’π‘0,𝐻𝑐=1,𝐻3=ΜƒπœΞ”1,𝐻4=ΜƒπœΞ”2,π‘ βˆ’(𝑛𝑝+1)=Μƒπœπœˆ. At β„Žβ‰ 0 and 𝜏=0, we have ξ‚β„ŽπΈπ‘›π‘1+1=1,𝐻𝑐=0, 𝐻3=ξ‚β„ŽΞ”1/𝑝0,𝐻4=ξ‚β„ŽΞ”2/𝑝0,π‘ βˆ’(𝑛𝑝+1)=ξ‚β„Ž2/(𝑑+2) [see (20)].

tab2
Table 2: The eigenvalues 𝐸𝑙 and the exponents 𝜈,Ξ”1, and Ξ”2 for the 𝜌6 model.

We shall perform the further calculations on the basis of (18), which are valid in the general case for the regions of small, intermediate (the crossover region), and large field values. The inclusion of 𝐸𝑛𝑝3+1 (or 𝐻3) leads to the formation of the first confluent corrections in the expressions for thermodynamic characteristics of the system. The quantity 𝐸𝑛𝑝4+1 (or 𝐻4) is responsible for the emergence of the second confluent corrections. The cases of the weak or strong fields can be obtained from general expressions by using (19) or (20). We disregard the second confluent correction in our calculations. This is due to the fact that the contribution from the first confluent correction to thermodynamic functions near the critical point (𝜏=0,β„Ž=0) is more significant than the small contribution from the second correction (ξ‚β„Ž2+ξ‚β„Ž2𝑐β‰ͺ1, Ξ”1β‰ˆ0.5, and Ξ”2 is of the order of 3, see Table 2).

Proceeding from an explicit dependence of 𝐹𝑛 on the layer number 𝑛 [4, 8, 9] and taking into account (18), we can now write the final expression for 𝐹CR (16): 𝐹CR=βˆ’π‘˜π‘‡π‘ξ…žξ‚€π›Ύ(CR)0+𝛾1𝜏+𝛾2𝜏2+𝐹𝑠,𝐹𝑠=π‘˜π‘‡π‘ξ…žπ‘ βˆ’3(𝑛𝑝+1)𝛾(CR3)(0)++𝛾(CR3)(1)+𝑐(0)20𝐻3.(21) Here 𝑐(0)20 characterizes 𝑐2 in solutions (10) of RR, 𝛾(CR3)(0)+=𝑓(0)CR1βˆ’π‘ βˆ’3+𝑓(1)CRπœ‘0βˆ’1/2𝑓0𝐻𝑐1βˆ’πΈ2π‘ βˆ’3+𝑓(7)CRπœ‘0βˆ’1𝑓0𝐻𝑐21βˆ’πΈ22π‘ βˆ’3,𝛾(CR3)(1)+=𝑓(2)CRπœ‘0βˆ’11βˆ’πΈ3π‘ βˆ’3+𝑓(4)CRπœ‘0βˆ’3/2𝑓0𝐻𝑐1βˆ’πΈ2𝐸3π‘ βˆ’3+𝑓(8)CRπœ‘0βˆ’2𝑓0𝐻𝑐21βˆ’πΈ22𝐸3π‘ βˆ’3,(22) and the coefficients 𝛾(CR)0=𝛾0(0)+𝛿0(0),π›Ύπ‘˜=𝛾0(π‘˜)+𝛿0(π‘˜),π‘˜=1,2(23) are determined by the components of the quantities 𝛾0=𝛾0(0)+𝛾0(1)𝜏+𝛾0(2)𝜏2,𝛿0=𝛿0(0)+𝛿0(1)𝜏+𝛿0(2)𝜏2.(24) The components 𝛿0(𝑖)(𝑖=0,1,2) satisfy the earlier relations [4, 8, 9] obtained in the case of a zero external field. The components 𝛾0(𝑖) are given by the corresponding expressions at β„Ž=0 under condition that the eigenvalues 𝐸1, 𝐸2, and 𝐸3 should be replaced by 𝐸2, 𝐸3, and 𝐸4, respectively.

Let us now calculate the contribution to the free energy of the system from the layers of the CV phase space beyond the point of exit from the critical regime region. The calculations are performed according to the scheme proposed in [2, 4, 10, 11]. As in the previous study, while calculating the partition function component 𝑍LGR from (15), it is convenient to single out two regions of values of wave vectors. The first is the transition region (𝑍(1)LGR) corresponding to values of 𝐀 close to 𝐡𝑛𝑝, while the second is the Gaussian region (𝑍(2)LGR) corresponding to small values of wave vector (π‘˜β†’0). Thus, we have 𝑍LGR=𝑍(1)LGR𝑍(2)LGR.(25)

3.2. Transition Region

This region corresponds to ξ‚π‘š0 layers of the phase space of CV. The lower boundary of the transition region is determined by the point of exit of the system from the critical regime region (𝑛=𝑛𝑝+1). The upper boundary corresponds to the layer 𝑛𝑝+ξ‚π‘š0+1. We use for ξ‚π‘š0 the integer closest to ξ‚π‘šξ…ž0. The condition for obtaining ξ‚π‘šξ…ž0 is the equality [9, 10] |||β„Žπ‘›π‘+ξ‚π‘šβ€²0|||=𝐴01βˆ’π‘ βˆ’3,(26) where 𝐴0 is a large number (𝐴0β‰₯10).

The free energy contribution 𝐹(1)LGR=βˆ’π‘˜π‘‡π‘π‘›π‘+1ξ‚π‘š0ξ“π‘š=0π‘ βˆ’3π‘šπ‘“LGR1𝑓(π‘š),LGR1ξ‚€2(π‘š)=lnπœ‹ξ‚+14βˆ’1ln244ξ‚€πœ‚ln𝐢𝑛𝑝+π‘š,πœ‰π‘›π‘+π‘šξ‚+ln𝐼0ξ‚€β„Žπ‘›π‘+π‘š+1,𝛼𝑛𝑝+π‘š+1+ln𝐼0ξ‚€πœ‚π‘›π‘+π‘š,πœ‰π‘›π‘+π‘šξ‚(27) corresponding to 𝑍(1)LGR from (25) is calculated by using the solutions of RR.

The basic arguments in the (𝑛𝑝+π‘š)th layer β„Žπ‘›π‘+π‘š=ξ‚€π‘Ÿπ‘›π‘+π‘šξ‚ξƒ©6+π‘žπ‘’π‘›π‘+π‘šξƒͺ1/2,𝛼𝑛𝑝+π‘š=√6𝑀15𝑛𝑝+π‘šπ‘’π‘›3/2𝑝+π‘š(28) can be presented using the relations 𝑑𝑛𝑝+π‘š=βˆ’π‘ 0𝑑/2𝑓0𝐸1π‘šβˆ’1ξ‚β„Žξ‚€ξ‚β„Ž2+ξ‚β„Ž2π‘ξ‚βˆ’1/2,π‘Ÿπ‘›π‘+π‘šξ‚ξ‚€=𝛽Φ(0)βˆ’π‘“0+𝑓0𝐻𝑐𝐸2π‘šβˆ’1+𝑐(0)20𝐻3πœ‘0βˆ’1/2𝑀(0)12𝐸3π‘šβˆ’1,𝑒𝑛𝑝+π‘š=𝛽Φ(0)2ξ‚€πœ‘0+𝑓0π»π‘πœ‘01/2𝑀(0)21𝐸2π‘šβˆ’1+𝑐(0)20𝐻3𝐸3π‘šβˆ’1,𝑀𝑛𝑝+π‘š=𝛽Φ(0)3ξ‚€πœ“0+𝑓0π»π‘πœ‘0𝑀(0)31𝐸2π‘šβˆ’1+𝑐(0)20𝐻3πœ‘01/2𝑀(0)32𝐸3π‘šβˆ’1(29) obtained on the basis of (10) and (18). We arrive at the following expressions: β„Žπ‘›π‘+π‘š=β„Žπ‘›(0)𝑝+π‘šξ‚€1+β„Žπ‘›(1)𝑝+π‘šπ‘(0)20𝐻3,β„Žπ‘›(0)𝑝+π‘š=√6π‘žβˆ’π‘“0+𝑓0𝐻𝑐𝐸2π‘šβˆ’1ξ‚€πœ‘0+𝑓0π»π‘πœ‘01/2𝑀(0)21𝐸2π‘šβˆ’11/2,β„Žπ‘›(1)𝑝+π‘š=𝐸3π‘šβˆ’1ξƒ©πœ‘0βˆ’1/2𝑀(0)12π‘žβˆ’π‘“0+𝑓0𝐻𝑐𝐸2π‘šβˆ’1βˆ’121πœ‘0+𝑓0π»π‘πœ‘01/2𝑀(0)21𝐸2π‘šβˆ’1ξƒͺ,𝛼𝑛𝑝+π‘š=𝛼𝑛(0)𝑝+π‘šξ‚€1+𝛼𝑛(1)𝑝+π‘šπ‘(0)20𝐻3,𝛼𝑛(0)𝑝+π‘š=√6πœ“150+𝑓0π»π‘πœ‘0𝑀(0)31𝐸2π‘šβˆ’1ξ‚€πœ‘0+𝑓0π»π‘πœ‘01/2𝑀(0)21𝐸2π‘šβˆ’13/2,𝛼𝑛(1)𝑝+π‘š=𝐸3π‘šβˆ’1ξƒ©πœ‘01/2𝑀(0)32πœ“0+𝑓0π»π‘πœ‘0𝑀(0)31𝐸2π‘šβˆ’1βˆ’321πœ‘0+𝑓0π»π‘πœ‘01/2𝑀(0)21𝐸2π‘šβˆ’1ξƒͺ.(30) In contrast to 𝐻𝑐, the quantity 𝐻3 in expressions (30) for β„Žπ‘›π‘+π‘š and 𝛼𝑛𝑝+π‘š as well as in expression (21) for 𝐹𝑠 takes on small values with the variation of the field ξ‚β„Ž (see Figure 3). The quantity 𝐻𝑐 at ξ‚β„Žβ†’0 and near ξ‚β„Žπ‘ is close to unity, and series expansions in 𝐻𝑐 are not effective here.

260750.fig.003
Figure 3: Dependence of quantities 𝐻𝑐 and 𝐻3 on the ratio ξ‚ξ‚β„Žβ„Ž/𝑐 for the RG parameter 𝑠=π‘ βˆ—=2.7349 and the reduced temperature 𝜏=10βˆ’4.

Power series in small deviations (β„Žπ‘›π‘+π‘šβˆ’β„Žπ‘›(0)𝑝+π‘š) and (𝛼𝑛𝑝+π‘šβˆ’π›Όπ‘›(0)𝑝+π‘š) for the special functions appearing in the expressions for the intermediate arguments πœ‚π‘›π‘+π‘š=ξ€·6𝑠𝑑1/2𝐹2ξ‚€β„Žπ‘›π‘+π‘š,𝛼𝑛𝑝+π‘šπΆξ‚€β„Žξ‚ξ‚ƒπ‘›π‘+π‘š,𝛼𝑛𝑝+π‘šξ‚ξ‚„βˆ’1/2,πœ‰π‘›π‘+π‘š=√6𝑠15βˆ’π‘‘/2π‘ξ‚€β„Žπ‘›π‘+π‘š,𝛼𝑛𝑝+π‘šπΆξ‚€β„Žξ‚ξ‚ƒπ‘›π‘+π‘š,𝛼𝑛𝑝+π‘šξ‚ξ‚„βˆ’3/2(31) allow us to find the relations πœ‚π‘›π‘+π‘š=πœ‚π‘›(0)𝑝+π‘šξ‚ƒξ‚€1βˆ’πœ‚(𝑛𝑝1+π‘š)β„Žπ‘›(0)𝑝+π‘šβ„Žπ‘›(1)𝑝+π‘š+πœ‚(𝑛𝑝2+π‘š)𝛼𝑛(0)𝑝+π‘šπ›Όπ‘›(1)𝑝+π‘šξ‚π‘(0)20𝐻3ξ‚„,πœ‰π‘›π‘+π‘š=πœ‰π‘›(0)𝑝+π‘šξ‚ƒξ‚€1βˆ’πœ‰(𝑛𝑝1+π‘š)β„Žπ‘›(0)𝑝+π‘šβ„Žπ‘›(1)𝑝+π‘š+πœ‰(𝑛𝑝2+π‘š)𝛼𝑛(0)𝑝+π‘šπ›Όπ‘›(1)𝑝+π‘šξ‚π‘(0)20𝐻3ξ‚„.(32) The quantities πœ‚π‘›(0)𝑝+π‘š, πœ‚(𝑛𝑝1+π‘š),πœ‚(𝑛𝑝2+π‘š), and πœ‰π‘›(0)𝑝+π‘š,πœ‰(𝑛𝑝1+π‘š),πœ‰(𝑛𝑝2+π‘š) are functions of πΉβˆ—(𝑛𝑝+π‘š)2𝑙=πΌβˆ—(𝑛𝑝+π‘š)2𝑙/πΌβˆ—(𝑛𝑝0+π‘š), whereπΌβˆ—(𝑛𝑝+π‘š)2𝑙=ξ€œβˆž0π‘₯2𝑙expβˆ’β„Žπ‘›(0)𝑝+π‘šπ‘₯2βˆ’π‘₯4βˆ’π›Όπ‘›(0)𝑝+π‘šπ‘₯6𝑑π‘₯.(33)

Proceeding from expression (27) for 𝑓LGR1(π‘š), we can now write the following relation accurate to within 𝐻3: 𝑓LGR1(π‘š)=𝑓(0)LGR1(π‘š)+𝑓(1)LGR1(π‘š)𝑐(0)20𝐻3,𝑓(0)LGR1ξ‚€2(π‘š)=lnπœ‹ξ‚+141ln24βˆ’4ξ‚€πœ‚ln𝐢𝑛(0)𝑝+π‘š,πœ‰π‘›(0)𝑝+π‘šξ‚+ln𝐼0ξ‚€β„Žπ‘›(0)𝑝+π‘š+1,𝛼𝑛(0)𝑝+π‘š+1+ln𝐼0ξ‚€πœ‚π‘›(0)𝑝+π‘š,πœ‰π‘›(0)𝑝+π‘šξ‚,𝑓(1)LGR1(π‘š)=πœ‘(𝑛𝑝1+π‘š)β„Žπ‘›(0)𝑝+π‘šβ„Žπ‘›(1)𝑝+π‘š+πœ‘(𝑛𝑝2+π‘š)𝛼𝑛(0)𝑝+π‘šπ›Όπ‘›(1)𝑝+π‘š+πœ‘(𝑛𝑝3+π‘š+1)β„Žπ‘›(0)𝑝+π‘š+1β„Žπ‘›(1)𝑝+π‘š+1+πœ‘(𝑛𝑝4+π‘š+1)𝛼𝑛(0)𝑝+π‘š+1𝛼𝑛(1)𝑝+π‘š+1,πœ‘(π‘›π‘π‘˜+π‘š)=𝑏(π‘›π‘π‘˜+π‘š)+𝑃(𝑛𝑝+π‘š)4π‘˜4πœ‘,π‘˜=1,2,(𝑛𝑝3+π‘š+1)=βˆ’πΉβˆ—(𝑛𝑝2+π‘š+1),πœ‘(𝑛𝑝4+π‘š+1)=βˆ’πΉβˆ—(𝑛𝑝6+π‘š+1).(34) The quantities 𝑏(π‘›π‘π‘˜+π‘š),𝑃(𝑛𝑝+π‘š)4π‘˜ depend on πΉβˆ—(𝑛𝑝+π‘š)2𝑙 as well as on πΉβˆ—βˆ—(𝑛𝑝+π‘š)2𝑙=πΌβˆ—βˆ—(𝑛𝑝+π‘š)2𝑙/πΌβˆ—βˆ—(𝑛𝑝0+π‘š), whereπΌβˆ—βˆ—(𝑛𝑝+π‘š)2𝑙=ξ€œβˆž0π‘₯2𝑙expβˆ’πœ‚π‘›(0)𝑝+π‘šπ‘₯2βˆ’π‘₯4βˆ’πœ‰π‘›(0)𝑝+π‘šπ‘₯6𝑑π‘₯.(35)

The final result for 𝐹(1)LGR (see (27) and (34)) assumes the form 𝐹(1)LGR=βˆ’π‘˜π‘‡π‘ξ…žπ‘ βˆ’3(𝑛𝑝+1)𝑓(0)TR+𝑓(1)TR𝑐(0)20𝐻3,𝑓(0)TR=ξ‚π‘š0ξ“π‘š=0π‘ βˆ’3π‘šπ‘“(0)LGR1(π‘š),𝑓(1)TR=ξ‚π‘š0ξ“π‘š=0π‘ βˆ’3π‘šπ‘“(1)LGR1(π‘š).(36) On the basis of (26) and (30), it is possible to obtain the quantity ξ‚π‘šξ…ž0 determining the summation limit ξ‚π‘š0 in formulas (36): ξ‚π‘šξ…ž0=ln𝐿0βˆ’ln𝐻𝑐ln𝐸2+1,𝐿0=𝐴1+𝐴21βˆ’π΄2ξ€Έ1/2,𝐴1=1βˆ’π‘žπ‘“0+𝐴20πœ‘01/2𝑀(0)2112𝑓0ξ€·1βˆ’π‘ βˆ’3ξ€Έ2,𝐴2=1βˆ’2π‘žπ‘“0+ξ‚΅π‘žπ‘“0ξ‚Ά2βˆ’π΄20πœ‘06𝑓20ξ€·1βˆ’π‘ βˆ’3ξ€Έ2.(37)

Let us now calculate the contribution to the free energy of the system from long-wave modes in the range of wave vectorsπ‘˜β‰€π΅ξ…žπ‘ βˆ’π‘›β€²π‘,π‘›ξ…žπ‘=𝑛𝑝+ξ‚π‘š0+2(38) using the Gaussian measure density.

3.3. Region of Small Values of Wave Vector (π‘˜β†’0)

The free energy component 𝐹(2)LGR=12βŽ‘βŽ’βŽ’βŽ£π‘π‘˜π‘‡π‘›β€²π‘ln𝑃(𝑛′𝑝2βˆ’1)+π΅π‘›β€²π‘ξ“π‘˜=0𝑑lnπ‘›β€²π‘π‘ξ€·β„Ž(π‘˜)βˆ’ξ…žξ€Έ2𝑑𝑛′𝑝(⎀βŽ₯βŽ₯⎦0)(39) corresponding to 𝑍(2)LGR from (25) is similar to that presented in [4, 9, 10]. The calculations of the first and second terms in (39) are associated with the calculations of the quantities 𝑃(𝑛′𝑝2βˆ’1)=2β„Žπ‘›β€²π‘βˆ’1𝐹2ξ‚€β„Žπ‘›β€²π‘βˆ’1,π›Όπ‘›β€²π‘βˆ’1ξ‚Γ—ξ‚ƒπ‘‘π‘›β€²π‘βˆ’1𝐡𝑛′𝑝,π΅π‘›β€²π‘βˆ’1ξ‚ξ‚„βˆ’1,𝑑𝑛′𝑝𝑃(π‘˜)=(𝑛′𝑝2βˆ’1)ξ‚„βˆ’1Φ𝐡+𝛽𝑛′𝑝,π΅π‘›β€²π‘βˆ’1ξ‚ξ‚βˆ’π›½Ξ¦(π‘˜),(40) where π‘‘π‘›β€²π‘βˆ’1𝐡𝑛′𝑝,π΅π‘›β€²π‘βˆ’1=π‘ βˆ’2(π‘›β€²π‘βˆ’1)ξ‚€π‘Ÿπ‘›β€²π‘βˆ’1+π‘ž,(41) and π‘Ÿπ‘›β€²π‘βˆ’1,β„Žπ‘›β€²π‘βˆ’1=β„Žπ‘›(0)β€²π‘βˆ’1(1+β„Žπ‘›(1)β€²π‘βˆ’1𝑐(0)20𝐻3),π›Όπ‘›β€²π‘βˆ’1=𝛼𝑛(0)β€²π‘βˆ’1(1+𝛼𝑛(1)β€²π‘βˆ’1𝑐(0)20𝐻3) satisfy the corresponding expressions from (29) and (30) at π‘š=ξ‚π‘š0+1.

Introducing the designation 𝑝=β„Žπ‘›β€²π‘βˆ’1𝐹2ξ‚€β„Žπ‘›β€²π‘βˆ’1,π›Όπ‘›β€²π‘βˆ’1(42) and presenting it in the form π‘βˆ’1=𝑝0ξ‚€1+𝑝1𝑐(0)20𝐻3,(43) we obtain the following relations for the coefficients: 𝑝0=ξ‚ƒβ„Žπ‘›(0)β€²π‘βˆ’1𝑝(π‘›β€²π‘βˆ’1)20ξ‚„βˆ’1,𝑝1=βˆ’β„Žπ‘›(1)β€²π‘βˆ’1ξ‚€1βˆ’π‘(π‘›β€²π‘βˆ’1)21β„Žπ‘›(0)β€²π‘βˆ’1+𝑝(π‘›β€²π‘βˆ’1)22𝛼𝑛(0)β€²π‘βˆ’1𝛼𝑛(1)β€²π‘βˆ’1.(44) The quantities 𝑝(π‘›β€²π‘βˆ’1)20=πΉβˆ—(𝑛′𝑝2βˆ’1),𝑝(π‘›β€²π‘βˆ’1)21=πΉβˆ—(𝑛′𝑝4βˆ’1)πΉβˆ—(𝑛′𝑝2βˆ’1)βˆ’πΉβˆ—(𝑛′𝑝2βˆ’1),𝑝(π‘›β€²π‘βˆ’1)22=πΉβˆ—(𝑛′𝑝8βˆ’1)πΉβˆ—(𝑛′𝑝2βˆ’1)βˆ’πΉβˆ—(𝑛′𝑝6βˆ’1)(45) determine the function 𝐹2ξ‚€β„Žπ‘›β€²π‘βˆ’1,π›Όπ‘›β€²π‘βˆ’1=𝑝(π‘›β€²π‘βˆ’1)20𝑝1βˆ’(π‘›β€²π‘βˆ’1)21β„Žπ‘›(0)β€²π‘βˆ’1β„Žπ‘›(1)β€²π‘βˆ’1+𝑝(π‘›β€²π‘βˆ’1)22𝛼𝑛(0)β€²π‘βˆ’1𝛼𝑛(1)β€²π‘βˆ’1×𝑐(0)20𝐻3ξ‚„.(46) Here πΉβˆ—(π‘›β€²π‘βˆ’1)2𝑙=πΌβˆ—(π‘›β€²π‘βˆ’1)2𝑙/πΌβˆ—(𝑛′𝑝0βˆ’1), whereπΌβˆ—(π‘›β€²π‘βˆ’1)2𝑙=ξ€œβˆž0π‘₯2𝑙expβˆ’β„Žπ‘›(0)β€²π‘βˆ’1π‘₯2βˆ’π‘₯4βˆ’π›Όπ‘›(0)β€²π‘βˆ’1π‘₯6𝑑π‘₯.(47)

Taking into account (41) and (43), we rewrite formulas (40) as 𝑃(𝑛′𝑝2βˆ’1)=⎧βŽͺ⎨βŽͺ⎩12π‘ βˆ’2(π‘›β€²π‘βˆ’1)𝛽Φ(0)𝑝0ξ‚΅π‘žβˆ’π‘“0+𝑓0π»π‘πΈξ‚π‘š02ξ‚ΆΓ—βŽ‘βŽ’βŽ’βŽ£βŽ›βŽœβŽœβŽπœ‘1+0βˆ’1/2𝑀(0)12πΈξ‚π‘š03π‘žβˆ’π‘“0+𝑓0π»π‘πΈξ‚π‘š02+𝑝1βŽžβŽŸβŽŸβŽ π‘(0)20𝐻3⎀βŽ₯βŽ₯⎦⎫βŽͺ⎬βŽͺβŽ­βˆ’1,𝑑𝑛′𝑝(π‘˜)=π‘ βˆ’2(π‘›β€²π‘βˆ’1)𝛽ΦΦ(0)𝐺+2𝛽(0)𝑏2π‘˜2,𝐺=𝑔0ξ‚€1+𝑔1𝑐(0)20𝐻3,𝑔0=12ξ‚Έξ‚΅βˆ’π‘“0+𝑓0π»π‘πΈξ‚π‘š02𝑝0+𝑝0ξ€Έβˆ’2π‘žξ‚Ή,𝑔1=12𝑝0𝑔0𝑝1ξ‚΅π‘žβˆ’π‘“0+𝑓0π»π‘πΈξ‚π‘š02ξ‚Ά+πœ‘0βˆ’1/2𝑀(0)12πΈξ‚π‘š03ξ‚Ή.(48) The second term in (39) is defined by the expression 12π΅π‘›β€²π‘ξ“π‘˜=0𝑑ln𝑛′𝑝(π‘˜)=𝑁𝑛′𝑝12ln𝐺+π‘ βˆ’2+lnπ‘ βˆ’π‘›ξ…žπ‘1ln𝑠+2ξ‚€π›½ξ‚ξ‚βˆ’1lnΞ¦(0)3+𝐺𝑠2βˆ’ξ‚€ξ‚πΊπ‘ 23/2arctan𝐺𝑠2ξ‚βˆ’1/2.ξ‚Ήξ‚Ό(49)

Relations (48) and (49) make it possible to find the component 𝐹(2)LGR in the form 𝐹(2)LGR𝑁=βˆ’π‘˜π‘‡ξ…žπ‘ βˆ’3(𝑛𝑝+1)𝑓(0)β€²+𝑓(1)′𝑐(0)20𝐻3ξ‚Ά+π‘ξ€·β„Žξ…žξ€Έ2𝛾+4𝛽𝑠Φ(0)2(𝑛𝑝+1)ξ‚€1βˆ’π‘”1𝑐(0)20𝐻3,𝑓(0)ξ…ž=π‘ βˆ’3(ξ‚π‘š0+1)𝑓(0),𝑓(1)ξ…ž=π‘ βˆ’3(ξ‚π‘š0+1)𝑓(1),𝑓(0)1=βˆ’2𝑠lnβˆ’2+𝑔0𝑔0+π‘žξ‚Ά+13βˆ’π‘”ξ…ž0βŽ‘βŽ’βŽ’βŽ’βŽ£ξ”1βˆ’π‘”ξ…ž0βŽ›βŽœβŽœβŽœβŽ1arctanξ”π‘”ξ…ž0⎞⎟⎟⎟⎠⎀βŽ₯βŽ₯βŽ₯⎦,𝑓(1)=12𝑔0𝑔1𝑔0+π‘žβˆ’π‘”1ξ€·π‘”ξ…ž0ξ€Έβˆ’1βˆ’π‘”+1ξ…ž0𝑔1ξ€·π‘”ξ…ž0ξ€Έβˆ’1ξƒͺ+1βˆ’π‘”ξ…ž0𝑔1⎑⎒⎒⎒⎣31βˆ’2ξ”π‘”ξ…ž0βŽ›βŽœβŽœβŽœβŽ1arctanξ”π‘”ξ…ž0⎞⎟⎟⎟⎠⎀βŽ₯βŽ₯βŽ₯⎦,π‘”ξ…ž0=𝑠2𝑔0,𝛾+4=𝑠2ξ‚π‘š0ξ€·2𝑔0ξ€Έ.(50)

On the basis of (36) and (50), we can write the following expression for the general contribution 𝐹LGR=𝐹(1)LGR+𝐹(2)LGR to the free energy of the system from long-wave modes of spin-moment density oscillations: 𝐹LGR𝑁=βˆ’π‘˜π‘‡ξ…žπ‘ βˆ’3(𝑛𝑝+1)𝑓(0)LGR+𝑓(1)LGR𝑐(0)20𝐻3+π‘ξ€·β„Žξ…žξ€Έ2𝛾+4𝛽𝑠Φ(0)2(𝑛𝑝+1)ξ‚€1βˆ’π‘”1𝑐(0)20𝐻3,𝑓(𝑙)LGR=𝑓(𝑙)TR+𝑓(𝑙)ξ…ž,𝑙=0,1.(51)

4. Total Free Energy of the System at 𝑇>𝑇𝑐

The total free energy of the system is calculated taking into account (14), (21), and (51). Collecting the contributions to the free energy from all regimes of fluctuations at 𝑇>𝑇𝑐 in the presence of an external field and using the relation for π‘ βˆ’(𝑛𝑝+1) from (18), we obtain 𝛾𝐹=βˆ’π‘˜π‘‡π‘ξ…ž0+π›Ύξ…ž1𝜏+π›Ύξ…ž2𝜏2+𝛾3(0)++𝛾3(1)+𝑐(0)20𝐻3ξ‚β„Žξ‚ξ‚€2+ξ‚β„Ž2𝑐3/5+𝛾+4ξ€·β„Žξ…žξ€Έ2𝛽Φ(0)1βˆ’π‘”1𝑐(0)20𝐻3ξ‚β„Žξ‚ξ‚€2+ξ‚β„Ž2π‘ξ‚βˆ’2/5ξƒ­,π›Ύξ…ž0=ln2+𝑠0βˆ’3𝛾(CR)0,π›Ύξ…ž1=𝑠0βˆ’3𝛾1,π›Ύξ…ž2=𝑠0βˆ’3𝛾2,𝛾3(𝑙)+=𝑠0βˆ’3ξ‚€βˆ’π›Ύ(CR3)(𝑙)++𝑓(𝑙)LGR,𝑙=0,1.(52) The coefficients 𝛾(CR)0, 𝛾1, and 𝛾2 are defined by (23), 𝑔1 is presented in (48), and 𝛾+4 is given in (50). The coefficients of the nonanalytic component of the free energy 𝐹 [see (52)] depend on 𝐻𝑐. The terms proportional to 𝐻3 determine the confluent corrections by the temperature and field. As is seen from the expression for 𝐹, the free energy of the system at ξ‚β„Ž=0 and Μƒπœ=0, in addition to terms proportional to Μƒπœ3𝜈 (or ξ‚β„Žπ‘6/5) and ξ‚β„Ž6/5, contains the terms proportional to Μƒπœ3𝜈+Ξ”1 and ξ‚β„Ž6/5+Ξ”1/𝑝0, respectively. At ξ‚β„Žβ‰ 0 and Μƒπœβ‰ 0, the terms of both types are present. It should be noted that Ξ”1>Ξ”1/𝑝0. At ξ‚ξ‚β„Žβ„Ž=𝑐, we have Μƒπœ3𝜈+Ξ”1=ξ‚β„Ž6/5+Ξ”1/𝑝0 and the contributions to the thermodynamic characteristics of the system from both types of corrections become of the same order.

The advantage of the method presented in this paper is the possibility of deriving analytic expressions for the free-energy coefficients as functions of the microscopic parameters of the system (the lattice constant 𝑐 and parameters of the interaction potential, that is, the effective radius 𝑏 of the potential, the Fourier transform Φ(0) of the potential for π‘˜=0).

5. Conclusions

A 3𝐷 Ising-like system (a 3𝐷 uniaxial magnet) in a nonzero external field is investigated in the higher non-Gaussian approximation based on the sextic distribution for modes of spin-moment density oscillations (the 𝜌6 model). The simultaneous effect of the temperature and field on the critical behaviour of the system is taken into account. An external field is introduced in the Hamiltonian of the system from the outset. In contrast to previous studies on the basis of the asymmetric 𝜌4 model [32, 33, 40], the field in the initial process of calculating the partition function of the system is not included in the Jacobian of transition from the set of spin variables to the set of CV. Such an approach leads to the appearance of the first, second, fourth, and sixth powers of CV in the expression for the partition function and allows us to simplify the mathematical description because the odd part is represented only by the linear term. When the field is included in the transition Jacobian, the measure density involves the odd powers of CV in addition to the even powers.

An initial expression for the partition function of the system is constructed in the form of a functional with explicitly known coefficient functions [see (6)]. The partition function is integrated over the layers of the CV phase space. The RR and their solutions near the critical point are written for the 𝜌6 model containing the field term. The presence of the field caused appearance of the additional relation for 𝑑𝑛 in (10). The fixed-point coordinates and the eigenvalues of the RG linear transformation matrix as well as the exponents of the confluent corrections obtained in the higher non-Gaussian approximation are given.

The main distinctive feature of the presented method for calculating the total free energy of the system is the separate inclusion of the contributions to the free energy from the short- and long-wave spin-density oscillation modes. The expression for the generalized point of exit of the system from the critical regime contains both the temperature and field variables. The form of the temperature and field dependences for the free energy of the system is determined by solutions of RR near the fixed point.

The expression for the free energy 𝐹 (52) obtained at temperatures 𝑇>𝑇𝑐 without using power series in the scaling variable and without any adjustable parameters can be employed in the field region near ξ‚β„Žπ‘ (the crossover region). The limiting field ξ‚β„Žπ‘ satisfies the condition of the equality of sizes of the critical regime region by the temperature and field (the effect of the temperature and field on the system in the vicinity of the critical point is equivalent) [32, 33, 38, 40]. In the vicinity of ξ‚β„Žπ‘, the scaling variable is of the order of unity, and power series in this variable are not effective. Proceeding from the expression for the free energy, which involves the leading terms and terms determining the temperature and field confluent corrections, we can find other thermodynamic characteristics (the average spin moment, susceptibility, entropy, and specific heat) by direct differentiation of 𝐹 with respect to field or temperature.

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