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

Authors: M. P. Kozlovskii; I. V. Pylyuk
Journal: ISRN Condensed Matter Physics
 (2011)

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## Abstract

A three-dimensional Ising-like system in a homogeneous external field is
studied on the basis of the higher non-Gaussian measure density (theρ6 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.


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## Body

## 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 the3D 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 ρk 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 “0” corresponds to the peak of the Fourier transform of the interaction potential.The free energy of a3D 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 ρ2m models, there exists a preferred value of the RG parameter s=s* (s*=3.5862 for the ρ4 model, s*=2.7349 for the ρ6 model, s*=2.6511 for the ρ8 model, and s*=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 s* 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 m (which characterizes the order of the ρ2m model, m=2,3,4,5), changes insignificantly (see Figures 1 and 2). The point s≈s* in Figure 1 corresponds to the beginning of the ν(s) curve stabilization for each of the ρ2m models. The value of the exponent ν in Figure 2 is calculated for s=s*. The ρ2 model (Gaussian approximation) leads to the classical value ν=0.500. In the case when s=s*, 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 ρ2m model approximation, where the order of the model 2m≥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 2m≥6, and, in particular, at 2m=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.Figure 1
Evolution of the critical exponent of the correlation lengthν 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.Figure 2
Saturation of the critical exponentν with increasing order of the ρ2m 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 (theρ4 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 a3D Ising-like system on a simple cubic lattice with N sites and period c in a homogeneous external field h. Such a system is described by the Hamiltonian  (1)H=-12∑j,lΦ(rjl)σjσl-h∑jσj,
where rjl is the distance between particles at sites j and l, and σj is the operator of the z component of spin at the jth site, having two eigenvalues +1 and −1. The interaction potential has the form of an exponentially decreasing function  (2)Φ(rjl)=Aexp⁡(-rjlb).
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]:  (3)Φ̃(k)={Φ̃(0)(1-2b2k2),k≤B′,0,B′<k≤B,
where B is the boundary of the Brillouin half-zone (B=π/c),B′=(b2)-1,Φ̃(0)=8πA(b/c)3.In the CV representation for the partition function of the system, we have [2, 36]  (4)Z=∫exp⁡[12∑kβΦ̃(k)ρkρ-k+βhNρ0]J(ρ)(dρ)N.
Here, the summation over the wave vectors k is carried out within the first Brillouin zone, β=1/(kT) is the inverse temperature, the CV ρk are introduced by means of the functional representation for operators of spin-density oscillation modes ρ̂k=(N)-1∑lσlexp(-ikl),  (5)J(ρ)=2N∫exp⁡[2πi∑kωkρk+∑n≥1(2πi)2nN1-n×M2n(2n)!∑k1,…,k2nωk1⋯ωk2nδk1+⋯+k2n](dω)N
is the Jacobian of transition from the set of N spin variables σl to the set of CV ρk, and δk1+⋯+k2n is the Kronecker symbol. The variables ωk are conjugate to ρk, and the cumulants ℳ2n 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:  (6)Z=2N2(N′-1)/2ea0′N′×∫exp⁡[-a1′(N′)1/2ρ0-12∑kk≤B′d′(k)ρkρ-k-∑l=23a2l′(2l)!(N′)l-1∑k1,…,k2lki≤B′ρk1⋯ρk2lδk1+⋯+k2l]×(dρ)N′.
Here N′=Ns0-d (d=3 is the space dimension), s0=B/B′=π2b/c, and a1′=-s0d/2h′,h′=βh. 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 “layer-by-layer” integration of (6) with respect to variables ρk described in [2–4]. The integration begins from the variables ρk with 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, 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 “smoothed” effective measure density  (7)Z=2N2(Nn+1-1)/2Q0Q1⋯Qn[Q(Pn)]Nn+1×∫W6(n+1)(ρ)(dρ)Nn+1.
The expressions for Qn, Q(Pn) are presented in [8–11], and Nn+1=N′s-d(n+1). The sextic measure density of the (n+1)th block structure 𝒲6(n+1)(ρ) has the form  (8)W6(n+1)(ρ)=exp⁡[-a1(n+1)Nn+11/2ρ0-12∑kk≤Bn+1dn+1(k)ρkρ-k-∑l=23a2l(n+1)(2l)!Nn+1l-1∑k1,…,k2lki≤Bn+1ρk1⋯ρk2lδk1+⋯+k2l],
where Bn+1=B′s-(n+1),dn+1(k)=a2(n+1)-βΦ̃(k),a1(n+1) and a2l(n+1) are the renormalized values of the coefficients a1′ and a2l′ after integration over n+1 layers of the phase space of CV. The coefficients a1(n)=s-ntn,dn(0)=s-2nrn (appearing in the quantity dn(k)=dn(0)+2βΦ̃(0)b2k2),a4(n)=s-4nun, and a6(n)=s-6nwn are connected with the coefficients of the (n+1)th layer through the RR  (9)tn+1=s(d+2)/2tn,rn+1=s2[-q+un1/2Y(hn,αn)],un+1=s4-dunB(hn,αn),wn+1=s6-2dun3/2D(hn,αn)
whose solutions  (10)tn=t(0)-s0d/2h′E1n,rn=r(0)+c1E2n+c2w12(0)(u(0))-1/2E3n+c3w13(0)(u(0))-1E4n,un=u(0)+c1w21(0)(u(0))1/2E2n+c2E3n+c3w23(0)(u(0))-1/2E4n,wn=w(0)+c1w31(0)u(0)E2n+c2w32(0)(u(0))1/2E3n+c3E4n
in the region of the critical regime are used for calculating the free energy of the system. Here  (11)Y(hn,αn)=sd/2F2(ηn,ξn)[C(hn,αn)]-1/2,B(hn,αn)=s2dC(ηn,ξn)[C(hn,αn)]-1,D(hn,αn)=s7d/2N(ηn,ξn)[C(hn,αn)]-3/2.
The quantity q=q¯βΦ̃(0) determines the average value of the Fourier transform of the potential βΦ̃(Bn+1,Bn)=βΦ̃(0)-q/s2n in the nth layer (in this paper, q¯=(1+s-2)/2 corresponds to the arithmetic mean value of k2 on the interval (1/s,1]). The basic arguments hn and αn are determined by the coefficients of the sextic measure density of the nth block structure. The intermediate variables ηn and ξn are functions of hn 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  (12)(tn+1-t(0)rn+1-r(0)un+1-u(0)wn+1-w(0))=(R110000R22R23R240R32R33R340R42R43R44)(tn-t(0)rn-r(0)un-u(0)wn-w(0)).
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=i-1; j1=j-1) and E1, E2, E3 obtained in the case of h=0. The quantities f0, φ0, and ψ0 characterizing the fixed-point coordinates  (13)t(0)=0,r(0)=-f0βΦ̃(0),u(0)=φ0(βΦ̃(0))2,w(0)=ψ0(βΦ̃(0))3,
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, φ0, and ψ0 as well as the renormalized quantities f¯0=f0/q¯,φ¯0=φ0/q¯2, and ψ¯0=ψ0/q¯3 independent of the potential averaging are presented in Table 1 for the optimal RG parameter s=s*=2.7349.Table 1
Quantities, which characterize the coordinates of the fixed point.
f0φ0ψ0f¯0φ¯0ψ¯00.56680.56470.48701.00001.75762.6740


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

Let us calculate the free energyF=-kTlnZ 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]:  (14)F=F0+FCR+FLGR.
Here F0=-kTNln2 is the free energy of N noninteracting spins. Each of three components in (14) corresponds to individual factors in the convenient representation  (15)Z=2NZCRZLGR
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 theρ6 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:  (16)FCR=F0′+FCR′,F0′=-kTN′[ln⁡Q(M)+ln⁡Q(d)],FCR′=∑n=1npFn.
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]  (17)np=-ln⁡(h̃2+h̃c2)2ln⁡E1-1
leads to the distinction between formula (16) for FCR′ and the analogous relation at h=0 [9, 10]. The quantity h̃=h′/f0 is determined by the dimensionless field h′, while the quantity h̃c=τ̃p0 is a function of the reduced temperature τ=(T-Tc)/Tc. Here τ̃=c̃1(0)τ/f0,p0=lnE1/lnE2=(d+2)ν/2, c̃1(0) characterizes the coefficient c1 in solutions (10) of RR, ν=lns/lnE2 is the critical exponent of the correlation length. At h=0,np becomes mτ=-lnτ̃/lnE2-1 (see [4, 9, 10]). At T=Tc (τ=0), the quantity np coincides with the exit point nh=-lnh̃/lnE1-1 [38]. The limiting value of the field h̃c is obtained by the equality of the exit points defined by the temperature and by the field (mτ=nh).Having expression (17) for np, we arrive at the relations [39]  (18)E1np+1=(h̃2+h̃c2)-1/2,τ̃E2np+1=Hc,Hc=h̃c1/p0(h̃2+h̃c2)-1/(2p0),E3np+1=H3,H3=(h̃2+h̃c2)Δ1/(2p0),E4np+1=H4,H4=(h̃2+h̃c2)Δ2/(2p0),s-(np+1)=(h̃2+h̃c2)1/(d+2),
where Δ1=-lnE3/lnE2 and Δ2=-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, Δ1, and Δ2 for the RG parameter s=s*=2.7349 are given in Table 2. In the weak-field  region (h̃≪h̃c), quantities (18) can be calculated with the help of the following expansions:  (19)E1np+1=h̃c-1(1-12h̃2h̃c2),h̃c-1=τ̃-p0,Hc=1-12p0h̃2h̃c2,H3=h̃cΔ1/p0(1+Δ12p0h̃2h̃c2),h̃cΔ1/p0=τ̃Δ1,H4=h̃cΔ2/p0(1+Δ22p0h̃2h̃c2),h̃cΔ2/p0=τ̃Δ2,s-(np+1)=h̃c2/(d+2)(1+1d+2h̃2h̃c2),h̃c2/(d+2)=τ̃ν.
In the strong-field region (h̃≫h̃c), these quantities satisfy the expressions  (20)E1np+1=h̃-1(1-12h̃c2h̃2),Hc=(h̃ch̃)1/p0(1-12p0h̃c2h̃2),H3=h̃Δ1/p0(1+Δ12p0h̃c2h̃2),H4=h̃Δ2/p0(1+Δ22p0h̃c2h̃2),s-(np+1)=h̃2/(d+2)(1+1d+2h̃c2h̃2).
It should be noted that the variables h̃/h̃c (the weak fields) and (h̃c/h̃)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 h=0 and τ≠0, (19) are defined as E1np+1=τ̃-p0,Hc=1,H3=τ̃Δ1,H4=τ̃Δ2,s-(np+1)=τ̃ν. At h≠0 and τ=0, we have h̃E1np+1=1,Hc=0, H3=h̃Δ1/p0,H4=h̃Δ2/p0,s-(np+1)=h̃2/(d+2) [see (20)].Table 2
The eigenvaluesEl and the exponents ν,Δ1, and Δ2 for the ρ6 model.
E1E2E3E4νΔ1Δ212.36954.84680.43670.00320.6370.5253.647We 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 E3np+1 (or H3) leads to the formation of the first confluent corrections in the expressions for thermodynamic characteristics of the system. The quantity E4np+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,h=0) is more significant than the small contribution from the second correction (h̃2+h̃c2≪1, Δ1≈0.5, and Δ2 is of the order of 3, see Table 2).Proceeding from an explicit dependence ofFn on the layer number n [4, 8, 9] and taking into account (18), we can now write the final expression for FCR (16):  (21)FCR=-kTN′(γ0(CR)+γ1τ+γ2τ2)+Fs,Fs=kTN′s-3(np+1)(γ¯3(CR)(0)++γ¯3(CR)(1)+c20(0)H3).
Here c20(0) characterizes c2 in solutions (10) of RR,  (22)γ¯3(CR)(0)+=fCR(0)1-s-3+fCR(1)φ0-1/2f0Hc1-E2s-3+fCR(7)φ0-1(f0Hc)21-E22s-3,γ¯3(CR)(1)+=fCR(2)φ0-11-E3s-3+fCR(4)φ0-3/2f0Hc1-E2E3s-3+fCR(8)φ0-2(f0Hc)21-E22E3s-3,
and the coefficients  (23)γ0(CR)=γ0(0)+δ0(0),γk=γ0(k)+δ0(k),k=1,2
are determined by the components of the quantities  (24)γ0=γ0(0)+γ0(1)τ+γ0(2)τ2,δ0=δ0(0)+δ0(1)τ+δ0(2)τ2.
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 h=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 (ZLGR(1)) corresponding to values of k close to Bnp, while the second is the Gaussian region (ZLGR(2)) corresponding to small values of wave vector (k→0). Thus, we have  (25)ZLGR=ZLGR(1)ZLGR(2).


### 3.2. Transition Region

This region corresponds tom̃0 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+m̃0+1. We use for m̃0 the integer closest to m̃0′. The condition for obtaining m̃0′ is the equality [9, 10]  (26)|hnp+m̃0′|=A01-s-3,
where A0 is a large number (A0≥10).The free energy contribution(27)FLGR(1)=-kTNnp+1∑m=0m̃0s-3mfLGR1(m),fLGR1(m)=ln⁡(2π)+14ln⁡24-14ln⁡C(ηnp+m,ξnp+m)+ln⁡I0(hnp+m+1,αnp+m+1)+ln⁡I0(ηnp+m,ξnp+m)
corresponding to ZLGR(1) from (25) is calculated by using the solutions of RR.The basic arguments in the (np+m)th layer  (28)hnp+m=(rnp+m+q)(6unp+m)1/2,αnp+m=615wnp+munp+m3/2
can be presented using the relations  (29)tnp+m=-s0d/2f0E1m-1h̃(h̃2+h̃c2)-1/2,rnp+m=βΦ̃(0)(-f0+f0HcE2m-1+c20(0)H3φ0-1/2w12(0)E3m-1),unp+m=(βΦ̃(0))2(φ0+f0Hcφ01/2w21(0)E2m-1+c20(0)H3E3m-1),wnp+m=(βΦ̃(0))3(ψ0+f0Hcφ0w31(0)E2m-1+c20(0)H3φ01/2w32(0)E3m-1)
obtained on the basis of (10) and (18). We arrive at the following expressions:  (30)hnp+m=hnp+m(0)(1+h¯np+m(1)c20(0)H3),hnp+m(0)=6q¯-f0+f0HcE2m-1(φ0+f0Hcφ01/2w21(0)E2m-1)1/2,h¯np+m(1)=E3m-1(φ0-1/2w12(0)q¯-f0+f0HcE2m-1-121φ0+f0Hcφ01/2w21(0)E2m-1),αnp+m=αnp+m(0)(1+α¯np+m(1)c20(0)H3),αnp+m(0)=615ψ0+f0Hcφ0w31(0)E2m-1(φ0+f0Hcφ01/2w21(0)E2m-1)3/2,α¯np+m(1)=E3m-1(φ01/2w32(0)ψ0+f0Hcφ0w31(0)E2m-1-321φ0+f0Hcφ01/2w21(0)E2m-1).
In contrast to Hc, the quantity H3 in expressions (30) for hnp+m and αnp+m as well as in expression (21) for Fs takes on small values with the variation of the field h̃ (see Figure 3). The quantity Hc at h̃→0 and near h̃c is close to unity, and series expansions in Hc are not effective here.Figure 3
Dependence of quantitiesHc and H3 on the ratio h̃/h̃c for the RG parameter s=s*=2.7349 and the reduced temperature τ=10-4.Power series in small deviations(hnp+m-hnp+m(0)) and (αnp+m-αnp+m(0)) for the special functions appearing in the expressions for the intermediate arguments  (31)ηnp+m=(6sd)1/2F2(hnp+m,αnp+m)[C(hnp+m,αnp+m)]-1/2,ξnp+m=615s-d/2N(hnp+m,αnp+m)[C(hnp+m,αnp+m)]-3/2
allow us to find the relations  (32)ηnp+m=ηnp+m(0)[1-(η¯1(np+m)hnp+m(0)h¯np+m(1)+η¯2(np+m)αnp+m(0)α¯np+m(1)η¯1(np+m)hnp+m(0)h¯np+m(1))c20(0)H3η¯1(np+m)hnp+m(0)h¯np+m(1)],ξnp+m=ξnp+m(0)[1-(ξ¯1(np+m)hnp+m(0)h¯np+m(1)+ξ¯2(np+m)αnp+m(0)α¯np+m(1))c20(0)H3].
The quantities ηnp+m(0), η¯1(np+m),η¯2(np+m), and ξnp+m(0),ξ¯1(np+m),ξ¯2(np+m) are functions of F2l*(np+m)=I2l*(np+m)/I0*(np+m), where(33)I2l*(np+m)=∫0∞x2lexp⁡(-hnp+m(0)x2-x4-αnp+m(0)x6)dx.Proceeding from expression (27) for fLGR1(m), we can now write the following relation accurate to within H3:  (34)fLGR1(m)=fLGR1(0)(m)+f¯LGR1(1)(m)c20(0)H3,fLGR1(0)(m)=ln⁡(2π)+14ln⁡24-14ln⁡C(ηnp+m(0),ξnp+m(0))+ln⁡I0(hnp+m+1(0),αnp+m+1(0))+ln⁡I0(ηnp+m(0),ξnp+m(0)),f¯LGR1(1)(m)=φ1(np+m)hnp+m(0)h¯np+m(1)+φ2(np+m)αnp+m(0)α¯np+m(1)+φ3(np+m+1)hnp+m+1(0)h¯np+m+1(1)+φ4(np+m+1)αnp+m+1(0)α¯np+m+1(1),φk(np+m)=bk(np+m)+P4k(np+m)4,k=1,2,φ3(np+m+1)=-F2*(np+m+1),φ4(np+m+1)=-F6*(np+m+1).
The quantities bk(np+m),P4k(np+m) depend on F2l*(np+m) as well as on F2l**(np+m)=I2l**(np+m)/I0**(np+m), where(35)I2l**(np+m)=∫0∞x2lexp⁡(-ηnp+m(0)x2-x4-ξnp+m(0)x6)dx.The final result forFLGR(1) (see (27) and (34)) assumes the form  (36)FLGR(1)=-kTN′s-3(np+1)(f¯TR(0)+f¯TR(1)c20(0)H3),f¯TR(0)=∑m=0m̃0s-3mfLGR1(0)(m),f¯TR(1)=∑m=0m̃0s-3mf¯LGR1(1)(m).
On the basis of (26) and (30), it is possible to obtain the quantity m̃0′ determining the summation limit m̃0 in formulas (36):  (37)m̃0′=ln⁡L0-ln⁡Hcln⁡E2+1,L0=A1+(A12-A2)1/2,A1=1-q¯f0+A02φ01/2w21(0)12f0(1-s-3)2,A2=1-2q¯f0+(q¯f0)2-A02φ06f02(1-s-3)2.Let us now calculate the contribution to the free energy of the system from long-wave modes in the range of wave vectors(38)k≤B′s-np′,np′=np+m̃0+2
using the Gaussian measure density.


### 3.3. Region of Small Values of Wave Vector (k→0)

The free energy component(39)FLGR(2)=12kT[Nnp′ln⁡P2(np′-1)+∑k=0Bnp′ln⁡d̃np′(k)-N(h′)2d̃np′(0)]
corresponding to ZLGR(2) 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  (40)P2(np′-1)=2hnp′-1F2(hnp′-1,αnp′-1)×[dnp′-1(Bnp′,Bnp′-1)]-1,d̃np′(k)=[P2(np′-1)]-1+βΦ̃(Bnp′,Bnp′-1)-βΦ̃(k),
where  (41)dnp′-1(Bnp′,Bnp′-1)=s-2(np′-1)(rnp′-1+q),
and rnp′-1,hnp′-1=hnp′-1(0)(1+h¯np′-1(1)c20(0)H3),αnp′-1=αnp′-1(0)(1+α¯np′-1(1)c20(0)H3) satisfy the corresponding expressions from (29) and (30) at m=m̃0+1.Introducing the designation(42)p=hnp′-1F2(hnp′-1,αnp′-1)
and presenting it in the form  (43)p-1=p0(1+p¯1c20(0)H3),
we obtain the following relations for the coefficients:  (44)p0=[hnp′-1(0)p20(np′-1)]-1,p¯1=-h¯np′-1(1)(1-p21(np′-1)hnp′-1(0))+p22(np′-1)αnp′-1(0)α¯np′-1(1).
The quantities  (45)p20(np′-1)=F2*(np′-1),p21(np′-1)=F4*(np′-1)F2*(np′-1)-F2*(np′-1),p22(np′-1)=F8*(np′-1)F2*(np′-1)-F6*(np′-1)
determine the function  (46)F2(hnp′-1,αnp′-1)=p20(np′-1)[1-(p21(np′-1)hnp′-1(0)h¯np′-1(1)+p22(np′-1)αnp′-1(0)α¯np′-1(1))×c20(0)H3].
Here F2l*(np′-1)=I2l*(np′-1)/I0*(np′-1), where(47)I2l*(np′-1)=∫0∞x2lexp⁡(-hnp′-1(0)x2-x4-αnp′-1(0)x6)dx.Taking into account (41) and (43), we rewrite formulas (40) as  (48)P2(np′-1)={(φ0-1/2w12(0)E3m̃0q¯-f0+f0HcE2m̃0+p¯1)12s-2(np′-1)βΦ̃(0)p0(q¯-f0+f0HcE2m̃0)×[1+(φ0-1/2w12(0)E3m̃0q¯-f0+f0HcE2m̃0+p¯1)c20(0)H3]}-1,d̃np′(k)=s-2(np′-1)βΦ̃(0)G̃+2βΦ̃(0)b2k2,G̃=g0(1+g¯1c20(0)H3),g0=12[(-f0+f0HcE2m̃0)p0+(p0-2)q¯],g¯1=12p0g0[p¯1(q¯-f0+f0HcE2m̃0)+φ0-1/2w12(0)E3m̃0].
The second term in (39) is defined by the expression  (49)12∑k=0Bnp′ln⁡d̃np′(k)=Nnp′{12ln⁡(G̃+s-2)+ln⁡s-np′ln⁡s+12ln⁡(βΦ̃(0))-13+G̃s2-(G̃s2)3/2arctan[(G̃s2)-1/2]}.Relations (48) and (49) make it possible to find the component FLGR(2) in the form  (50)FLGR(2)=-kT[N(h′)2γ¯4+βΦ̃(0)N′s-3(np+1)(f¯(0)′+f¯(1)′c20(0)H3)+N(h′)2γ¯4+βΦ̃(0)s2(np+1)(1-g¯1c20(0)H3)],f¯(0)′=s-3(m̃0+1)f(0),f¯(1)′=s-3(m̃0+1)f¯(1),f(0)=-12ln⁡(s-2+g0g0+q¯)+13-g0′[1-g0′arctan(1g0′)],f¯(1)=12(g0g¯1g0+q¯-g¯1(g0′)-1+1-g0′g¯1(g0′)-1+1)-g0′g¯1[1-32g0′arctan(1g0′)],g0′=s2g0,γ¯4+=s2m̃0(2g0).On the basis of (36) and (50), we can write the following expression for the general contribution FLGR=FLGR(1)+FLGR(2) to the free energy of the system from long-wave modes of spin-moment density oscillations:  (51)FLGR=-kT[N(h′)2γ¯4+βΦ̃(0)N′s-3(np+1)(f¯LGR(0)+f¯LGR(1)c20(0)H3)+N(h′)2γ¯4+βΦ̃(0)s2(np+1)(1-g¯1c20(0)H3)],f¯LGR(l)=f¯TR(l)+f¯(l)′,l=0,1.


## 3.1. Region of the Critical Regime

A calculation technique based on theρ6 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:  (16)FCR=F0′+FCR′,F0′=-kTN′[ln⁡Q(M)+ln⁡Q(d)],FCR′=∑n=1npFn.
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]  (17)np=-ln⁡(h̃2+h̃c2)2ln⁡E1-1
leads to the distinction between formula (16) for FCR′ and the analogous relation at h=0 [9, 10]. The quantity h̃=h′/f0 is determined by the dimensionless field h′, while the quantity h̃c=τ̃p0 is a function of the reduced temperature τ=(T-Tc)/Tc. Here τ̃=c̃1(0)τ/f0,p0=lnE1/lnE2=(d+2)ν/2, c̃1(0) characterizes the coefficient c1 in solutions (10) of RR, ν=lns/lnE2 is the critical exponent of the correlation length. At h=0,np becomes mτ=-lnτ̃/lnE2-1 (see [4, 9, 10]). At T=Tc (τ=0), the quantity np coincides with the exit point nh=-lnh̃/lnE1-1 [38]. The limiting value of the field h̃c is obtained by the equality of the exit points defined by the temperature and by the field (mτ=nh).Having expression (17) for np, we arrive at the relations [39]  (18)E1np+1=(h̃2+h̃c2)-1/2,τ̃E2np+1=Hc,Hc=h̃c1/p0(h̃2+h̃c2)-1/(2p0),E3np+1=H3,H3=(h̃2+h̃c2)Δ1/(2p0),E4np+1=H4,H4=(h̃2+h̃c2)Δ2/(2p0),s-(np+1)=(h̃2+h̃c2)1/(d+2),
where Δ1=-lnE3/lnE2 and Δ2=-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, Δ1, and Δ2 for the RG parameter s=s*=2.7349 are given in Table 2. In the weak-field  region (h̃≪h̃c), quantities (18) can be calculated with the help of the following expansions:  (19)E1np+1=h̃c-1(1-12h̃2h̃c2),h̃c-1=τ̃-p0,Hc=1-12p0h̃2h̃c2,H3=h̃cΔ1/p0(1+Δ12p0h̃2h̃c2),h̃cΔ1/p0=τ̃Δ1,H4=h̃cΔ2/p0(1+Δ22p0h̃2h̃c2),h̃cΔ2/p0=τ̃Δ2,s-(np+1)=h̃c2/(d+2)(1+1d+2h̃2h̃c2),h̃c2/(d+2)=τ̃ν.
In the strong-field region (h̃≫h̃c), these quantities satisfy the expressions  (20)E1np+1=h̃-1(1-12h̃c2h̃2),Hc=(h̃ch̃)1/p0(1-12p0h̃c2h̃2),H3=h̃Δ1/p0(1+Δ12p0h̃c2h̃2),H4=h̃Δ2/p0(1+Δ22p0h̃c2h̃2),s-(np+1)=h̃2/(d+2)(1+1d+2h̃c2h̃2).
It should be noted that the variables h̃/h̃c (the weak fields) and (h̃c/h̃)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 h=0 and τ≠0, (19) are defined as E1np+1=τ̃-p0,Hc=1,H3=τ̃Δ1,H4=τ̃Δ2,s-(np+1)=τ̃ν. At h≠0 and τ=0, we have h̃E1np+1=1,Hc=0, H3=h̃Δ1/p0,H4=h̃Δ2/p0,s-(np+1)=h̃2/(d+2) [see (20)].Table 2
The eigenvaluesEl and the exponents ν,Δ1, and Δ2 for the ρ6 model.
E1E2E3E4νΔ1Δ212.36954.84680.43670.00320.6370.5253.647We 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 E3np+1 (or H3) leads to the formation of the first confluent corrections in the expressions for thermodynamic characteristics of the system. The quantity E4np+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,h=0) is more significant than the small contribution from the second correction (h̃2+h̃c2≪1, Δ1≈0.5, and Δ2 is of the order of 3, see Table 2).Proceeding from an explicit dependence ofFn on the layer number n [4, 8, 9] and taking into account (18), we can now write the final expression for FCR (16):  (21)FCR=-kTN′(γ0(CR)+γ1τ+γ2τ2)+Fs,Fs=kTN′s-3(np+1)(γ¯3(CR)(0)++γ¯3(CR)(1)+c20(0)H3).
Here c20(0) characterizes c2 in solutions (10) of RR,  (22)γ¯3(CR)(0)+=fCR(0)1-s-3+fCR(1)φ0-1/2f0Hc1-E2s-3+fCR(7)φ0-1(f0Hc)21-E22s-3,γ¯3(CR)(1)+=fCR(2)φ0-11-E3s-3+fCR(4)φ0-3/2f0Hc1-E2E3s-3+fCR(8)φ0-2(f0Hc)21-E22E3s-3,
and the coefficients  (23)γ0(CR)=γ0(0)+δ0(0),γk=γ0(k)+δ0(k),k=1,2
are determined by the components of the quantities  (24)γ0=γ0(0)+γ0(1)τ+γ0(2)τ2,δ0=δ0(0)+δ0(1)τ+δ0(2)τ2.
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 h=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 (ZLGR(1)) corresponding to values of k close to Bnp, while the second is the Gaussian region (ZLGR(2)) corresponding to small values of wave vector (k→0). Thus, we have  (25)ZLGR=ZLGR(1)ZLGR(2).


## 3.2. Transition Region

This region corresponds tom̃0 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+m̃0+1. We use for m̃0 the integer closest to m̃0′. The condition for obtaining m̃0′ is the equality [9, 10]  (26)|hnp+m̃0′|=A01-s-3,
where A0 is a large number (A0≥10).The free energy contribution(27)FLGR(1)=-kTNnp+1∑m=0m̃0s-3mfLGR1(m),fLGR1(m)=ln⁡(2π)+14ln⁡24-14ln⁡C(ηnp+m,ξnp+m)+ln⁡I0(hnp+m+1,αnp+m+1)+ln⁡I0(ηnp+m,ξnp+m)
corresponding to ZLGR(1) from (25) is calculated by using the solutions of RR.The basic arguments in the (np+m)th layer  (28)hnp+m=(rnp+m+q)(6unp+m)1/2,αnp+m=615wnp+munp+m3/2
can be presented using the relations  (29)tnp+m=-s0d/2f0E1m-1h̃(h̃2+h̃c2)-1/2,rnp+m=βΦ̃(0)(-f0+f0HcE2m-1+c20(0)H3φ0-1/2w12(0)E3m-1),unp+m=(βΦ̃(0))2(φ0+f0Hcφ01/2w21(0)E2m-1+c20(0)H3E3m-1),wnp+m=(βΦ̃(0))3(ψ0+f0Hcφ0w31(0)E2m-1+c20(0)H3φ01/2w32(0)E3m-1)
obtained on the basis of (10) and (18). We arrive at the following expressions:  (30)hnp+m=hnp+m(0)(1+h¯np+m(1)c20(0)H3),hnp+m(0)=6q¯-f0+f0HcE2m-1(φ0+f0Hcφ01/2w21(0)E2m-1)1/2,h¯np+m(1)=E3m-1(φ0-1/2w12(0)q¯-f0+f0HcE2m-1-121φ0+f0Hcφ01/2w21(0)E2m-1),αnp+m=αnp+m(0)(1+α¯np+m(1)c20(0)H3),αnp+m(0)=615ψ0+f0Hcφ0w31(0)E2m-1(φ0+f0Hcφ01/2w21(0)E2m-1)3/2,α¯np+m(1)=E3m-1(φ01/2w32(0)ψ0+f0Hcφ0w31(0)E2m-1-321φ0+f0Hcφ01/2w21(0)E2m-1).
In contrast to Hc, the quantity H3 in expressions (30) for hnp+m and αnp+m as well as in expression (21) for Fs takes on small values with the variation of the field h̃ (see Figure 3). The quantity Hc at h̃→0 and near h̃c is close to unity, and series expansions in Hc are not effective here.Figure 3
Dependence of quantitiesHc and H3 on the ratio h̃/h̃c for the RG parameter s=s*=2.7349 and the reduced temperature τ=10-4.Power series in small deviations(hnp+m-hnp+m(0)) and (αnp+m-αnp+m(0)) for the special functions appearing in the expressions for the intermediate arguments  (31)ηnp+m=(6sd)1/2F2(hnp+m,αnp+m)[C(hnp+m,αnp+m)]-1/2,ξnp+m=615s-d/2N(hnp+m,αnp+m)[C(hnp+m,αnp+m)]-3/2
allow us to find the relations  (32)ηnp+m=ηnp+m(0)[1-(η¯1(np+m)hnp+m(0)h¯np+m(1)+η¯2(np+m)αnp+m(0)α¯np+m(1)η¯1(np+m)hnp+m(0)h¯np+m(1))c20(0)H3η¯1(np+m)hnp+m(0)h¯np+m(1)],ξnp+m=ξnp+m(0)[1-(ξ¯1(np+m)hnp+m(0)h¯np+m(1)+ξ¯2(np+m)αnp+m(0)α¯np+m(1))c20(0)H3].
The quantities ηnp+m(0), η¯1(np+m),η¯2(np+m), and ξnp+m(0),ξ¯1(np+m),ξ¯2(np+m) are functions of F2l*(np+m)=I2l*(np+m)/I0*(np+m), where(33)I2l*(np+m)=∫0∞x2lexp⁡(-hnp+m(0)x2-x4-αnp+m(0)x6)dx.Proceeding from expression (27) for fLGR1(m), we can now write the following relation accurate to within H3:  (34)fLGR1(m)=fLGR1(0)(m)+f¯LGR1(1)(m)c20(0)H3,fLGR1(0)(m)=ln⁡(2π)+14ln⁡24-14ln⁡C(ηnp+m(0),ξnp+m(0))+ln⁡I0(hnp+m+1(0),αnp+m+1(0))+ln⁡I0(ηnp+m(0),ξnp+m(0)),f¯LGR1(1)(m)=φ1(np+m)hnp+m(0)h¯np+m(1)+φ2(np+m)αnp+m(0)α¯np+m(1)+φ3(np+m+1)hnp+m+1(0)h¯np+m+1(1)+φ4(np+m+1)αnp+m+1(0)α¯np+m+1(1),φk(np+m)=bk(np+m)+P4k(np+m)4,k=1,2,φ3(np+m+1)=-F2*(np+m+1),φ4(np+m+1)=-F6*(np+m+1).
The quantities bk(np+m),P4k(np+m) depend on F2l*(np+m) as well as on F2l**(np+m)=I2l**(np+m)/I0**(np+m), where(35)I2l**(np+m)=∫0∞x2lexp⁡(-ηnp+m(0)x2-x4-ξnp+m(0)x6)dx.The final result forFLGR(1) (see (27) and (34)) assumes the form  (36)FLGR(1)=-kTN′s-3(np+1)(f¯TR(0)+f¯TR(1)c20(0)H3),f¯TR(0)=∑m=0m̃0s-3mfLGR1(0)(m),f¯TR(1)=∑m=0m̃0s-3mf¯LGR1(1)(m).
On the basis of (26) and (30), it is possible to obtain the quantity m̃0′ determining the summation limit m̃0 in formulas (36):  (37)m̃0′=ln⁡L0-ln⁡Hcln⁡E2+1,L0=A1+(A12-A2)1/2,A1=1-q¯f0+A02φ01/2w21(0)12f0(1-s-3)2,A2=1-2q¯f0+(q¯f0)2-A02φ06f02(1-s-3)2.Let us now calculate the contribution to the free energy of the system from long-wave modes in the range of wave vectors(38)k≤B′s-np′,np′=np+m̃0+2
using the Gaussian measure density.


## 3.3. Region of Small Values of Wave Vector (k→0)

The free energy component(39)FLGR(2)=12kT[Nnp′ln⁡P2(np′-1)+∑k=0Bnp′ln⁡d̃np′(k)-N(h′)2d̃np′(0)]
corresponding to ZLGR(2) 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  (40)P2(np′-1)=2hnp′-1F2(hnp′-1,αnp′-1)×[dnp′-1(Bnp′,Bnp′-1)]-1,d̃np′(k)=[P2(np′-1)]-1+βΦ̃(Bnp′,Bnp′-1)-βΦ̃(k),
where  (41)dnp′-1(Bnp′,Bnp′-1)=s-2(np′-1)(rnp′-1+q),
and rnp′-1,hnp′-1=hnp′-1(0)(1+h¯np′-1(1)c20(0)H3),αnp′-1=αnp′-1(0)(1+α¯np′-1(1)c20(0)H3) satisfy the corresponding expressions from (29) and (30) at m=m̃0+1.Introducing the designation(42)p=hnp′-1F2(hnp′-1,αnp′-1)
and presenting it in the form  (43)p-1=p0(1+p¯1c20(0)H3),
we obtain the following relations for the coefficients:  (44)p0=[hnp′-1(0)p20(np′-1)]-1,p¯1=-h¯np′-1(1)(1-p21(np′-1)hnp′-1(0))+p22(np′-1)αnp′-1(0)α¯np′-1(1).
The quantities  (45)p20(np′-1)=F2*(np′-1),p21(np′-1)=F4*(np′-1)F2*(np′-1)-F2*(np′-1),p22(np′-1)=F8*(np′-1)F2*(np′-1)-F6*(np′-1)
determine the function  (46)F2(hnp′-1,αnp′-1)=p20(np′-1)[1-(p21(np′-1)hnp′-1(0)h¯np′-1(1)+p22(np′-1)αnp′-1(0)α¯np′-1(1))×c20(0)H3].
Here F2l*(np′-1)=I2l*(np′-1)/I0*(np′-1), where(47)I2l*(np′-1)=∫0∞x2lexp⁡(-hnp′-1(0)x2-x4-αnp′-1(0)x6)dx.Taking into account (41) and (43), we rewrite formulas (40) as  (48)P2(np′-1)={(φ0-1/2w12(0)E3m̃0q¯-f0+f0HcE2m̃0+p¯1)12s-2(np′-1)βΦ̃(0)p0(q¯-f0+f0HcE2m̃0)×[1+(φ0-1/2w12(0)E3m̃0q¯-f0+f0HcE2m̃0+p¯1)c20(0)H3]}-1,d̃np′(k)=s-2(np′-1)βΦ̃(0)G̃+2βΦ̃(0)b2k2,G̃=g0(1+g¯1c20(0)H3),g0=12[(-f0+f0HcE2m̃0)p0+(p0-2)q¯],g¯1=12p0g0[p¯1(q¯-f0+f0HcE2m̃0)+φ0-1/2w12(0)E3m̃0].
The second term in (39) is defined by the expression  (49)12∑k=0Bnp′ln⁡d̃np′(k)=Nnp′{12ln⁡(G̃+s-2)+ln⁡s-np′ln⁡s+12ln⁡(βΦ̃(0))-13+G̃s2-(G̃s2)3/2arctan[(G̃s2)-1/2]}.Relations (48) and (49) make it possible to find the component FLGR(2) in the form  (50)FLGR(2)=-kT[N(h′)2γ¯4+βΦ̃(0)N′s-3(np+1)(f¯(0)′+f¯(1)′c20(0)H3)+N(h′)2γ¯4+βΦ̃(0)s2(np+1)(1-g¯1c20(0)H3)],f¯(0)′=s-3(m̃0+1)f(0),f¯(1)′=s-3(m̃0+1)f¯(1),f(0)=-12ln⁡(s-2+g0g0+q¯)+13-g0′[1-g0′arctan(1g0′)],f¯(1)=12(g0g¯1g0+q¯-g¯1(g0′)-1+1-g0′g¯1(g0′)-1+1)-g0′g¯1[1-32g0′arctan(1g0′)],g0′=s2g0,γ¯4+=s2m̃0(2g0).On the basis of (36) and (50), we can write the following expression for the general contribution FLGR=FLGR(1)+FLGR(2) to the free energy of the system from long-wave modes of spin-moment density oscillations:  (51)FLGR=-kT[N(h′)2γ¯4+βΦ̃(0)N′s-3(np+1)(f¯LGR(0)+f¯LGR(1)c20(0)H3)+N(h′)2γ¯4+βΦ̃(0)s2(np+1)(1-g¯1c20(0)H3)],f¯LGR(l)=f¯TR(l)+f¯(l)′,l=0,1.


## 4. Total Free Energy of the System atT>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 s-(np+1) from (18), we obtain  (52)F=-kTN[γ¯4+(h′)2βΦ̃(0)γ0′+γ1′τ+γ2′τ2+(γ¯3(0)++γ¯3(1)+c20(0)H3)(h̃2+h̃c2)3/5+γ¯4+(h′)2βΦ̃(0)(1-g¯1c20(0)H3)(h̃2+h̃c2)-2/5],γ0′=ln⁡2+s0-3γ0(CR),γ1′=s0-3γ1,γ2′=s0-3γ2,γ¯3(l)+=s0-3(-γ¯3(CR)(l)++f¯LGR(l)),l=0,1.
The coefficients γ0(CR), γ1, and  γ2 are defined by (23), g¯1 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 h̃=0 and τ̃=0, in addition to terms proportional to τ̃3ν (or h̃c6/5) and h̃6/5, contains the terms proportional to τ̃3ν+Δ1 and h̃6/5+Δ1/p0, respectively. At h̃≠0 and τ̃≠0, the terms of both types are present. It should be noted that Δ1>Δ1/p0. At h̃=h̃c, we have τ̃3ν+Δ1=h̃6/5+Δ1/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 constantc and parameters of the interaction potential, that is, the effective radius b of the potential, the Fourier transform Φ̃(0) of the potential for k=0).


## 5. Conclusions

A3D 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 tn 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 energyF (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 h̃c (the crossover region). The limiting field h̃c 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 h̃c, 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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*Source: 260750-2011-12-27.xml*