Free energy of 3 D Ising-like system near the phase transition point

An urgent problem in describing critical phenomena lies in the development of a generalized theory which allows one to obtain (except the calculation of critical exponents and some other universal characteristics) explicit expressions for physical quantities like in the case of classic theories [1,2]. The scaling theory turns out to be the most developed with respect to this problem. It is based on the similarity hypothesis proposed in [3–6] and on the scheme of constructing effective Kadanoff spin blocks [5]. We consider the system based on the Ising model on a simple cubic lattice having a lattice constant c. The initial lattice is split into blocks with linear sizes s · c, where s is an arbitrary number (s > 1). Then, instead of N initial sites with period c we get N1 sites (N1 = Ns ) with period c1(c1 = cs) each of which contains s d spins. When the system is near the phase transition point (PTP), the correlation length ξ is large and greatly exceeds C1. The free energy of such a system with effective spins Fs is related to the free energy of initial spins F by the known relation Fs(τ̃ , h̃) = s F (τ, h). (1)


Introduction
An urgent problem in describing critical phenomena lies in the development of a generalized theory which allows one to obtain (except the calculation of critical exponents and some other universal characteristics) explicit expressions for physical quantities like in the case of classic theories [1,2].The scaling theory turns out to be the most developed with respect to this problem.It is based on the similarity hypothesis proposed in [3][4][5][6] and on the scheme of constructing effective Kadanoff spin blocks [5].We consider the system based on the Ising model on a simple cubic lattice having a lattice constant c.The initial lattice is split into blocks with linear sizes s • c, where s is an arbitrary number (s > 1).Then, instead of N initial sites with period c we get N 1 sites (N 1 = N s −d ) with period c 1 (c 1 = cs) each of which contains s d spins.When the system is near the phase transition point (PTP), the correlation length ξ is large and greatly exceeds C 1 .The free energy of such a system with effective spins F s is related to the free energy of initial spins F by the known relation Here τ = (T − T c )/T c is a reduced temperature, h is external field.The variables τ and h related to τ and h by means of relations τ = s yτ τ, h = s y h h, where y τ and y h are some numbers defined by critical exponents Here ν is the critical exponent of the correlation length at the absence of the external field ξ τ = ξ ± |τ | −ν , and µ is the critical exponent of the same quantity at T = T c : ξ h = ξ (c) h −µ .It is known that µ = ν/βδ, where β and δ are critical exponents (temperature and field ones) of the order parameter.
The above expressions enable one to write down the scaling form of the singular part for free energy near PTP as F (τ, h) = s −d F s (s yt τ, s y h h) .
The theory parameter s is arbitrary but it cannot exceed the system correlation length since near the PTP there are two typical values which could be considered for the quantity s.The first one predicts the fulfillment of condition |s yt τ | = 1 that is equivalent to the parameter value s = s τ : In this case the equality (4) takes on the form The first multiplier describes the temperature behaviour of free energy of the system in the case of small values of the field or rather the quantity The second one f s is the so-called scaling function of the free energy.It depends on the ratio of the reduced external field h to the temperature field The function f s does not contain any explicit dependence on τ and h.
The second typical value of the parameter s related to the condition |s y h h| = 1, which corresponds to the relation At this value of the parameter we get an expression for the free energy (4) in the form It is evident that such a representation is valid for large values of the field or in other words for small values of the variable Note that quantities z and α 0 are mutually inverse z = α −1/δβ 0 .The free energy presentations (6) and (10) are basic in the scaling theory.Depending on the ratio between quantities τ and h one could use the first one or another.In particular, in the case of the absence of an external field, the dependence (6) should be used.At the presence of the external field (h = 0) with the temperature T tending to the critical point T c , the presentation (10) is more appropriate since an arbitrary small field becomes essential (the quantity z tends to zero).
In both cases of the RG transformation, this parameter (s τ or s h ) exceeds the value of the correlation length.One could find these results in papers [7,8] where an explicit dependence of the correlation length on the temperature and field is presented.Therefore, the above expressions for free energy (6) and (10) are formal.They are valid only in some limiting cases.The formula (6) could be used at h = 0 and the relation ( 10) is valid at T = T c only.As a consequence, the choice of the parameter s via conditions ( 5) and ( 9) does not enable one to describe the dependence of the free energy on both the field and the temperature parameters.
Thus, the problem of the free energy calculation with the possibility of unifying the dependences (6) and (10) appears to be actual.Considering the relation (4) it is easy to see that there is no such a value of s which could make it possible to unify the expressions ( 6) and (10).They are obtained via considering diametrically opposite limiting cases.It turned out that it is possible to find an explicit form for the free energy using the results presented in [7][8][9][10][11].Such a calculation employs the collective variables (CV) set [12] and it is valid for arbitrary values of the field and temperature.This is some combination of the expressions ( 6) and (10) and it is derived based on the mathematically rigorous microscopic approach.Using this method one could obtain an explicit form for the scaling function f s .

Method for calculation of the partition function near the second order phase transition point
Let us use the results of papers [7,9], where the method for calculating the partition function of the Ising-like model near the T c is proposed for the cases of small and large values of the external field.As was shown in these works, the distinction between small and large fields is related to the introduction of some temperature field h c = |τ | p0 , where τ = τ c k1 , f 0 are defined in [7].We also present some normalized field h where some constants are presented in [8].In the case of h h c , the free energy is defined by the field dependence (10) and at h h c the formula for free energy contains the temperature dependence only and is similar to (6).For free energy description the most problematic is the case of intermediate values of the field h ≈ h c .Therefore, the corresponding expressions obtained in [7] should be generalized.
As a result of step-by-step calculation for partition function for the Ising model in the external field we have the expression where in correspondence with [12,13] we have The variable x of the Weber's parabolic cylinder function U (0, x) has the form and for these functions the integral presentation could be written down as follows The quantities Q n are partial partition functions of the n-th level where N n = N 0 s −3n , s is the parameter of splitting the set of CV into subsets (s 1), For variables x n and y n we have the expressions where The quantity Φ(B n+1 , B n ) is the averaged value of the Fourier transform for interaction potential [8] in the region of wave vectors k ∈ B n \B n+1 .Here where c n = c 0 s n , N n = N nx N ny N nz is the sites number of n-th effective block structure, and N n = N 0 s −3n .For special functions U (t) and ϕ(t) we have the reccurance relations (RR) take place.Their explicit form was obtained in [14].By introducing notations where For initial values of quantities w n , r n and u n (at n = 0) we have (2.12) Here, the values of quantities a 2 , a 4 and Φ are presented in [8,9].The RR (2.11) differ from the ones obtained in [12] by the appearance of the additional equation for the quantity w n , which represents the existing external field.
In the expression (2.1) the quantity Z LGR is still undefined.It has the form where The number n p characterizes the effective block structure of spins with corresponding period c np , where The representation of the partition function in the form (2.1) is related to the presence of the new distance scale near PTP.When the system is far from this point, the lattice constant c plays the role of such a distance.However, in the vicinity of the critical point the characteristic distance is equal to the correlation length ξ = c np of the system.As far as the value of the effective lattice period c n is less than c np , the renormalization group (RG) symmetry takes place in the system [12] and for all n < n np , the general RR (2.11) could be substituted by approximated RR which corresponds to the linear deviations from the fixed point.In the case of c n > c np , the RG symmetry breaks down and for the following calculations of the contribution to the free energy one should use general RR.
The quantity n p is an important characteristic for a description of phase transitions.It defines the number of iterations at which the system is still in the scaling region.Note that the result of calculation (2.1) does not depend on the choice of n p .Indeed, the quantity n p splits the set of CV into two parts.The first one takes into account the contributions from CV η k with k ∈ B\B np+1 .The second part corresponds to the contributions of CV η k with k ∈ B np+1 , including the contribution from the "macroscopic" variable η 0 .A decrease or an increase of the quantity n p should not effect the total result of partition function calculation.The purpose of introducing this quantity is to optimize the mathematical evaluations near PTP.In the case of exact calculations, the choice of quantity n p would be arbitrary.However, linearization of the RR (2.11) near the fixed point predicts some restrictions to this quantity.
The general expression for the system exit point from the order parameter critical fluctuation regime at T > T c was found in [8,15].It is as follows: ) Quantities h 0 , f 0 and c k1 are presented in [7,16,17].We will use this expression in (2.1) for the following calculations of the partition function near PTP at T > T c .
Note that the calculations performed below concern the region of the temperatures |τ | < τ * , where τ * ∼ 10 −2 .We shall not apply any restrictions to the values of the field h as well.

Free energy calculation scheme near T c (T > T c )
The calculation of the one-component spin system free energy is performed near the PTP based on the expression for the partition function (2.1).In the case of the external field presence, the main difference of such an approach is that it is necessary to determine a generalized exit point from the critical regime of the order parameter fluctuations.In the case of T > T c coordinates of this point (on the plane filed-reduced temperature) are defined using the expression (2.16).Let us present the system free energy in the form of several terms Each of these terms is the contribution of a certain multiplier from expression (2.1).Particularly, corresponds to the expression for Z 0 and describes the free energy of noninteractive spins.The term CR is responsible for the contribution to the free energy from the critical regime of fluctuations.According to (2.1) it has the form where N n = N 0 • s −3n , and for function f n (x n , y n−1 ) the expression is valid.In comparison with the similar expression used in the case of the absence of the field [16] the difference of (3.3) is that we employ a more generalized expression for the exit point n p from the critical fluctuations regime.Quantities x n and y n , which belong to the (3.4), are presented in (2.7).
The term F LGR has the form where the expression for Z LGR is defined in (2.13).Note that Z np+1 from (2.14) can be presented in the form which is similar to the contribution from the critical regime of fluctuations, in other words, via the product of partial partition functions.Then for (3.5) we get Here where for f np+m we have (3.4) at n = n p + m, and for F we obtain The quantity n = n p + m 0 is the number of the spin blocks structure, and n > n p .The quantity Q(P n ) coincides with (2.6) at n = n .For Z n +1 we have expression (2.14), where n p should be substituted by n .Note the distinction in the type of the expressions for F Let us find the quantity CR from (3.3).In order to do that one should extract the explicit dependence of the quantity f n on n.Such calculations have been performed in [12].It was established that the variable y n takes on large values for arbitrary values of the temperature.Therefore, the asymptotic series expansions are used for Weber's parabolic cylinder functions U (0, y n ) and their combinations as well as for ϕ(y n ) from (2.10) We have a simplified expression for f n from (3.4) We restrict ourselves to the consideration of the case when the value of quantity x n in the fixed point reduces to zero (x * = 0).It is fulfilled at some value of the RG parameter s = s * , where s * = 3.5977.In this case for all n n p we have x n 1.The quantity (3.11) is represented in the form where f For coefficients A l we have where the quantity r 1 is expressed via parabolic cylinder functions U (0), ϕ(0) and their derivatives.In (3.12) we take into account the linear terms with respect to the x n only.The quadratic approximation is described in [12].In accordance with the results of work [17], we find Using an explicit solution of RR in the region n n p we obtain the expression for x n : Here, the terms which are proportional to E n 3 are not taken into account.(E 3 < 1).For coefficients B l we have Taking into account (3.16) from (3.11) we obtain where ).Let us use (3.17) to calculate (3.3).Note that due to (2.16) equalities are valid.Here notations are used.The quantity ∆ is the critical exponent for scaling correction , p 0 is the crossover critical exponent.For the model ρ 4 at s = s * they take on values1 ν = 0.605, ∆ = 0.465, p 0 = 1.512.
The result of calculating (3.3) is as follows: where for coefficients γ 0l one found expressions [17] which coincide with the result of the same calculations in the case of the absence of the field [12].The singular part F Here γl are constants which at s = s * take on the values: γ1 = 1.529, γ2 = −0.635,γ3 = −0.058.The singular part of the contribution F CR essentially differs from the same quantity at h = 0. Note that at the absence of the field we have At h = 0, the quantity H c < 1.For h h c we have H c → 0. The main contribution to γ+ is formed by quantity γ1 .One should note that similar calculations have been performed in [10].The exit point from CR was determined in [10] as a solution of some equation.Therefore, the results of calculations were presented in the form of figures.In this work the explicit expressions for the system free energy are obtained as functions of the temperature and field.
The contribution to the free energy F LGR in accordance with (3.6) contains two terms.In order to calculate the first one F (+) TR from (3.7) it is necessary to find an explicit dependence of the quantity f np+m on m.Let us use the solutions of RR.Taking into account (3.18) they take on the form 1 .The terms which are proportional to H 3 E m−1 3 are neglected since near PTP the quantity H 3 from (3.18) is small, and E m−1 3 at m 1 tends to zero (E 3 < 1).Such an approximation corresponds to scaling corrections being neglected.If necessary, such corrections could be taken into account by using the method suggested in [12].
Taking into account (3.26) from (2.7) we find where the notation is presented (3.28) The variable x np+m increases with m increasing and depends on the value of H c .The quantity H c is unambiguously defined by parameter α = h/h c (3.29) for every value of h.For small values of the fields the quantity H c tends to unity but with α increasing it reduces to zero.Such a behaviour of H c causes a different character of the dependence of the quantity x np+m from (3.27) on m at small and large values of the filed.
Taking into consideration the solutions (3.26) one could claim that the value of the quantity m 0 = 1 causes the quitting of the large values of the variable x np+m0+1 .It is a sufficient condition for using Gauss distribution of fluctuations for arbitrary ρ k with | k| ∈ B np+2 .Thus, the contribution to the system free energy (3.7) from the transition region contains just one term, which has the form The coefficient f np+1 is presented as To evaluate the last contribution F to the system free energy from (3.6) we calculate the expression for Z from (3.8).At m 0 = 1 we have where (3.33) For coefficients d np+2 and a (np+2) 4 we get = s −4(np+2) u np+2 . (3.34) Note that the method for calculation of (3.33) in the case of h = 0 is developed in [12].It is based on the condition that the quantity r np+2 (h = 0) u np+2 (h = 0), since it permits to use the Gauss approximation in calculating the (3.33).However, at the presence of the field, the coefficient r np+2 decreases with the field increasing and takes on the negative values at strong fields.Therefore, improvements are required to the method developed earlier.

Extracting the macroscopic part of the order parameter
The calculation of the explicit expression for quantity Z np+2 from (3.33) depends both on the presence of the field and on the system state above or bellow critical temperature.Such a situation is conditioned by the behaviour of the coefficients r np+2 and u np+2 , for which we have the expressions The coefficient u np+2 is always positive for arbitrary values of the field h and temperature τ h.It provides a convergence of the integration in (3.33).The coefficient r np+2 is positive and greatly exceeds u np+2 at small values of the field.(h c h).To calculate (3.33) at these temperatures one can use Gauss approximation which is discussed in detail in [12].However, for large values of the field (h c h) the coefficient r np+2 is small and takes on negative values with the temperature decreasing.In this temperature region the use of the Gauss approximation is groundless.It can be improved by performing the substitution of variables in (3.33) where σ + is some constant.In the case of T < T c , the presence of the spontaneous order parameter in the system is the reason for substitution of variable ρ 0 (average value of which is related to the order parameter).A similar situation takes place for temperatures above T c at the presence of the external field.In this case the order parameter induced by the field exists.As a result of substitution of variables (4.2) the expression (3.33) takes on the form Here For the coefficients A 0 , d(k), b and ā4 we have expressions In order to carry out the following calculations one should define the quantity σ + .Since the expressions (4.3)-(4.5)are valid for arbitrary values of σ + , we find this quantity via using the condition ∂E 0 (σ Taking into account (4.4), we get the equation the solution of which will be found in the form For quantity σ 0 we obtain the cubic equation where for coefficients p and q we have expressions In a general case, the quantities p and q are functions of temperature and field.The form of the solutions (4.8) depends on the discriminant sign It is known that for positive Q we have one real solution and in the case of Q < 0 the equation (4.8) has three real solutions.It is evident that at T > T c the quantity Q is positive at arbitrary values of the field.We consider some partial cases for solutions of (4.8).When h = 0, then q = 0 for arbitrary τ = 0 and corresponding solutions (4.8) have the form The sign of the quantity p depends on the localization of the system above or below T c .Taking into account equalities (3.18) and (4.1), in the case of T > T c we obtain Using equalities (4.11), from (4.9) we find that the value of the quantity is positive.In this case (h = 0, T > T c ) the equation (4.8) has no real solutions.The condition T = T c is another partial case for solutions of equation (4.8).Corresponding quantities p and q from (4.9) become independent of the field.Indeed at T = T c we have H c = 0, and Then, from (4.9) we get The quantity Q (c) = Q(τ = 0) has the form where The sign Q (c) is defined by square brackets of expression (4.15).The quantity Q (c) takes on positive values only due the existence of the small multiplier s −5 .Note that Q (c) does not depend on the magnitude of the field and is defined by microscopic parameters of the system.For their following values [17] s 0 = 2, b/c = 0.3, h 0 = 0.760 (4.17) we find p (c) = −0.389,q (c) = −2.790,Q (c) = 1.944.(4.18)One can prove that for other values of the system parameters s 0 and b/c the quantity Q (c) is positive only.At T = T c the solutions of equation (4.8) have the form where In the case of (4.17At T = T c the magnitude of the displacement σ + from (4.2) is described by the expression: In general case the solution of the equation (4.8) depends both on the field and on the temperature.For arbitrary τ > 0 at h = 0 we find the solution for equation (4.8) using the Cardano's method σ 0 = A + B, where The dependence σ 0 (τ ) is presented in figure 1 for different values of the field.Note that the value σ 0 = 1.500 at τ = 0 does not depend on the field magnitude.
We write down expressions for renormalized coefficients (4.5) in the form For calculation of expression (4.3) we use the Gauss approximation.It is related to the small values of coefficients v r and u r in comparison with r R .Performing the substitution of variables in (4.3) where The quantities x n are small in comparison with the unity and decrease at T → T c .Therefore, the expression (4.25) is reduced to the product of the simple integrals The contribution to the system free energy from (4.24) has the form Here where the following notation is introduced: In (4.28) the sum with respect to the k ∈ B np+2 can be calculated using the transition to the integral which could be calculated via method [19].
Taking into account the expressions (3.32) and (4.24), we write down the corresponding part of the free energy (3.32) as the sum of two contributions corresponds to extracting the macroscopic part of the order parameter.For F G we have The coefficient f G has the form Here the quantities r R and y np+2 are defined in (4.23) and (2.7) respectively, and for f G we have the expression where .36)

System free energy in the external field near the phase transition point at T > T c
Let us collect the contributions to the system free energy from different fluctuation processes which are present near the second order PTP.In accordance with the free energy representation, we have a few types of contributions.Taking into account (3.2) one can see that the expression for F 0 contains the analytic dependence on the field h only.The term F CR exhibits the same type of dependence in expression (3.20).Unifying these contributions we present the notes This is the analytic part of the free energy.It has the form where 3) The free energy contributions F and can be represented in the form where Thus, instead of presenting the free energy in the form of (5.1) we have an equivalent expression where the analytic part F a is written down as (5.2), and for F . (5.8) The formulas for coefficients e 0 and e 2 are presented in (4.30).The last term of expression (5.8) is the same as the for quantity F s .Therefore, the free energy can be written down in a more simple form where F a given in (5.2), and s − e 2 . (5.10) The free energy contribution F h plays the main role in describing the effect of the external field on the system behaviour near the critical point.

Conclusions
In this work the method for free energy calculation near the second order phase transition point is suggested.The explicit analytical expression for free energy of the Ising-like system as a function of the temperature and field is found.It could be reduced to the well-known formulas [20,21] represented by ( 6) and (10) via proceeding to limits T → T c or h → 0 correspondingly.
It is demonstrated that for such an evaluation one should take into account the presence of two different fluctuation regions.One of them (region CR) is formed by the order parameter fluctuation modes, the wavelength of which does not exceed the system correlation length at a given temperature and external field.In this fluctuation region the RG symmetry takes place.It is responsible for forming the nonclassic values of critical exponents.The second fluctuation region (LGR) is formed by the order parameter fluctuations the wavelength of which exceeds the system correlation length.It is characterized by the Gauss fluctuation distribution and does not effect the values of the critical exponents.Each of these fluctuation regions gives the corresponding contribution to the system free energy.
The evaluation of the expression for free energy also includes the presence of the contribution F TR from the transition region (between regions CR and LGR) as well as the contribution F 0 from the macroscopic part of the average value related to the order parameter.The contribution F TR is formed by the order parameter fluctuations with wavelengths which are measurable with the system correlation length.The contribution F + 0 is one of the crucial in describing the critical behaviour of the system.It represents some microscopic analogue of Landau free energy and allows one to qualitatively describe all characteristics of the system near the phase transition point.This is some variant of the series expansion for the system free energy in the order parameter.Corresponding coefficients of this expansion are defined as functions of the field and temperature.
from (3.3) and F (+) TR from(3.7).Despite the same functional form of these expressions, the values of variables x n and y n are different in magnitude.Partially, for F (+) CR (region n < n p ) these quantities are close to their values near the fixed point (x n ≈ x * , y n ≈ y * ).Nevertheless performing the calculations for F (+) TR (region n p < n n p + m 0 ) one should take into account their deviation from the fixed point.

Figure 1 .
Figure 1.Dependence of the solution σ0 for equation (4.8) on the temperature.The curve 1 corresponds to the value h = 0.0001, the curve 2 presents the case of h = 0.00005, and curve 3 describes the case of h = 0.00001.
30) and F (4.31) and (4.32) contain the nonanalytic dependence on the temperature τ and field h.We write down their sum in the form of the two types of terms.The first one F (+) 0 is related to the displacement of the variable ρ 0 and has the form presented in (4.31).The second type F (+) s is presented by the sum of the other nonanalytic contributions (the region T > T c )