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Novel results about magnetic fluctuation effects near the normal-to-superconducting phase transition in a zero magnetic field

A systematic treatment of the magnetic fluctuations effect on the properties of the normal-to-superconducting phase transition in a zero external magnetic field is given within the self-consistent approximation and the quasi-macroscopic Ginzburg-Landau model. New results for thin superconducting films are presented. Thermodynamic quantities having a direct experimental interest as the order parameter jump, latent heat, and specific heat are considered and numerically evaluated for bulk Al and thin Al films. The possibility for an experimental verification of the theoretical predictions is discussed.


Introduction
In 1974 Halperin, Lubensky and Ma (HLM) [1] showed that the magnetic fluctuations change the order of the superconducting phase transition in a zero external magnetic field (H 0 = | H 0 | = 0)), i.e., the order of the phase transition from normal-to-uniform (Meissner) superconducting state at T c0 = T c (H 0 = 0). In the mean-field approximation, when both magnetic and superconducting fluctuations are neglected, this phase transition is of second order; see, e.g., Refs. [2,3]. Moreover, the fluctuations ψ( x) = [ψ( x) − ψ( x) ] of the superconducting order parameter ψ( x) towards the statistical average ψ( x) are extremely small and can be safely ignored in usual low-temperature (T c0 < 20 K) superconductors. For a long time these superconductors have been considered as an excellent example of a standard phase transition of second order described by the mean-field approximation. When the magnetic fluctuations are taken into account in the Ginzburg-Landau (GL) free energy F (ψ, A) of superconductor [2], the same normal-to-superconducting phase transition in a zero (mean) external magnetic field ( H 0 = 0) is found to be a weaklyfirst order phase transition with a very small latent heat which cannot be observed by available experimental techniques [1]. The effect of a magnetic fluctuation change of the superconducting phase transition order, called HLM effect, is very weak in bulk (three dimensional, or 3D) superconductors even in Al where the GL number κ is very small (κ ≪ 1) -a circumstance which is in favor of the effect [1,4].
In this paper we shall investigate this fluctuation-induced first order phase transition in thin (quasi-2D) superconducting films. Bulk superconductors will be also discussed in order to compare them with the behavior of thin films. We shall use a self-consistent approximation [1], in which the fluctuations δψ of ψ are neglected but the magnetic fluctuations are completely taken into account. Note, that the so-called tree approximation [3] does not yield the HLM effect and the self-consistent, or mean-field-like, approximation, mentioned above, is the simplest analytical method for an investigation of this phenomenon.
The present study is intended to provide enough theoretical results about the behavior of measurable physical quantities directly related to the phase transition properties and in this way to ensure a theoretical basis for future experiments on the existence of the HLM effect. The need of an experimental observation of the HLM effect is very important because the effect remains a theoretical paradigm without a reliable experimental verification although its mechanism -the interaction of gauge fields in a quite universal Abelian-Higgs model -is of fundamental interest for different fields of physics as pure [5,9,10,11,12,13], and disordered [14,15,16,17,18,19,20,21] superconductors, quantum phase transitions [22,23], scalar electrodynamics [24], liquid crystals [25,26,27,28,29], and cosmology [30,31]. On the other hand, there are some theoretical studies, based on Monte Carlo simulations [32], the so-called dual model [33,34], and certain variants of the renormalization-group (RG) [35,36], in which no evidence of HLM effect was reported; for a discussion of this point, see, the review article [37]. Therefore, in the modern theory of phase transitions the problem for the existence of HLM effect is controversial and cannot be easily solved without a hint from the experiment. The experimental research of the effect in liquid crystals cannot be considered reliable although the reported results are in favor of its existence.
Recently, it has been shown [9] that the HLM effect is stronger in quasi-2D superconducting films than in bulk superconductors and the preliminary evaluation of the relevant physical quantities like the order parameter jump and the latent heat at the equilibrium point of the fluctuation-induced first order transition in superconducting films gives for them several orders bigger values than for those in bulk materials [10,11,12,13]. This result reopens the problem for an experimental search of HLM effect in type I supercon-ductors, in particular, in thin films of type I superconductors with relatively small GL parameter κ. Here we shall investigate this problem in a comprehensive way.
We shall neglect the fluctuations of the superconducting order parameter because their effect on the thermodynamics of the superconductor is very weak; see, e.g., Refs. [2,3]. Within this approximation, the problems in the scope of the work can be considered without the use of RG, as well as of numerous and quite interesting RG results available in the literature; for a review, see, e.g., Ref. [37].
In Sec. 2 we present a derivation of the effective free energy of a D-dimensional superconductor. In Sec. 3 we give the first thorough investigation of the effective free energy for bulk superconductors. In Sec. 4 the quasi-2D superconducting films and the validity of the Landau expansion are discussed. In Sec. 5 we summarize our main conclusions.

Model considerations
The GL free energy [2] of a D-dimensional superconductor of volume V D = (L 1 ...L D ) is given in the form In Eq. (1) the first Landau parameter a = α 0 (T − T c0 ) is expressed by the critical temperature T c0 = T c (H = 0) in a zero external magnetic field (H = | H|), b > 0 is the second Landau parameter and e ≡ |e| is the electron charge. The square B 2 of the magnetic induction B = ( H + 4π M ), is given by the vector potential A( x) = {A j ( x), j = 1, ..., D} in the form here the vector potential A( x) obeys the Coulomb gauge ∇· A( x) = 0. For a 3D superconductor the relation B = [∇ · A( x)] can be used and when B = B 0 is uniform along the zaxis, the Landau gauge A 0 ( x) = B 0 (−y/2, −x/2, 0) can be applied. This representation can be generalized for D > 2 -dimensional systems, where the magnetic induction B 0 is a second rank tensor: If we use the notation x = (x 1 , x 2 , r), where r is a (D − 2) -dimensional vector perpendicular to the plane (x 1 , x 2 ), in the 3D case we shall have r = (0, 0, z), and where ǫ jkl is the antisymmetric Levi-Civita symbol. The Landau gauge and Eqs. (3) -(4) can be used for uniform B = B 0 when δ B -fluctuations are neglected; see, e.g. Ref. [6].
In the prevailing part of our study we shall apply the general Coulomb gauge of the field A( x) which does not exclude spatial dependent magnetic fluctuations δ B( x).
In nonmagnetic superconductors where the mean value M = ( M − δ M ) of the magnetization M is equal to zero in the normal state in zero external magnetic field, the magnetic induction in presence of external magnetic field takes the form: where H 0 is the (uniform) regular part of the external magnetic field and δ H is an irregular part of H created by uncontrollable effects. We neglect the irregular part δ H and set H 0 = 0, then B contains only a fluctuation part B ≡ δ B( x) = 4πδ M ( x) that describes the diamagnetic variations of M( x) around the zero value M = 0 due to fluctuations δψ( x) of the ordering field ψ( x) above (T > T c0 ) and below (T < T c0 ) the normal-tosuperconducting transition at T c0 . Note, that the non-fluctuation part corresponds to the regular part B 0 = ( H 0 + M ) = 0 of B in nonmagnetic superconductors ( M = 0) in a zero external magnetic field ( H 0 = 0). Then we can set A 0 ( x) = 0 and, hence, δ A( x) = A( x), so we have an entirely fluctuation vector potential A( x) which interacts with the order parameter ψ( x). This interaction can be of type ψ 2 A and ψ 2 A 2 and generates all effects discussed in the paper.
We accept periodic boundary conditions for the superconductor surface. This means to ignore the surface energy including the additional energy due to the penetration of the magnetic field in a surface layer of thickness equal to the London penetration depth λ(T ) = λ 0 |t 0 | −1/2 , t 0 = |T −T c0 |/T c0 ; λ 0 = (mc 2 b/8πe 2 α 0 T c0 ) 1/2 is the zero-temperature value of λ. This approximation is adequate for superconductors of thickness L 0 ≫ λ(T ) ≫ a 0 , where a 0 is the lattice constant and L 0 = min{L i , i = 1, ..., D}. As we suppose the external magnetic field to be zero (H 0 = 0) or very small in real experiments, the requirement L 0 ≫ λ(T ) cannot be satisfied and we take into account only the condition L 0 ≫ a 0 .
In microscopic models of periodic structures the periodic boundary conditions confine the wave vectors k i = {k i = (2πn i /L i ); i = 1, ..., D} in the first Brillouin zone [−(π/a 0 ) ≤ k i < (π/a 0 )] and the expansion of their values beyond this zone can be made either by neglecting the periodicity of the crystal structure or on the basis of the assumption that big wave numbers k = | k| have a negligible contribution to the calculated quantities. The last argument is widely accepted in the phase transitions theory where the long-wavelength (ka 0 ≪ 1) limit can be used. In particular, this argument is valid in the continuum limit (V D /a D 0 → ∞). Therefore, for both crystal and nonperiodic structures we can use a cutoff Λ ∼ (π/a 0 ) and afterwards to extend this cutoff to infinity provided the main contributions in the summations over k come from the relatively small wavenumbers (k ≪ Λ). Note, that here we make a quasimacroscopic description based on the GL functional (1) which means that the microscopic phenomena are excluded from our consideration.
The GL free energy functional takes into account phenomena with characteristic lengths ξ 0 and λ 0 or larger (ξ and λ) where λ(T ) is the London penetration length mentioned above and ξ(T ) = ξ 0 |t| −1/2 is the coherence length [2]; here ξ 0 = ( 2 /4mα 0 T c0 ) 1/2 is the zero-temperature coherence length. In low-temperature superconductors ξ 0 and λ 0 are much bigger than the lattice constant a 0 . Having in mind this argument we shall assume that in our investigation Λ ≪ (π/a 0 ). Whether the upper cutoff Λ is chosen to be either Λ ∼ 1/ξ 0 or Λ ∼ 1/λ 0 is a problem that has to be solved by additional arguments (see Sec. 3.3).
We shall use the Fourier expansion and where the Fourier amplitudes A j ( k) obey the relation A * j ( k) = A j (− k) and k. A( k) = 0. The Fourier amplitude ψ( k) is not equal to ψ * (− k) because ψ( x) is a complex function. For the same reason ψ(0) ≡ ψ( k = 0) is a complex number.

Approximations
The total ignoring of both superconducting and magnetic fluctuations in Eq. (1) leads to the familiar free approximation where the GL equations [2] should be solved. Note, that the free, or mean-field, approximation is the lowest order theory within the framework of the loop expansion, e.g., see [3,38]. The systematic treatment of the fluctuation effects in the asymptotic vicinity of the phase transition point can be given by RG.
The effect of the superconducting fluctuations δψ( x) on the phase transition properties is restricted in a negligibly small vicinity (|T − T c0 | ∼ 10 −12 ÷ 10 −16 ) of the temperature T c0 and we shall assume that δψ( x) = 0, i.e., ψ ≈ ψ( x) ; from now on we shall denote ψ( x) by ψ. So we apply a mean-field approximation with respect to the order parameter ψ( x). Within this approximation we shall take into account the δ A( x)-fluctuations for B 0 = 0, i.e., A( x) = δ A( x). Furthermore, the A( x)-fluctuations can be integrated out from the partition function, defined by: where the functional integral DA is defined by The integration is over all possible configurations of the field A( x); the δ-function takes into account the Coulomb gauge.
The partition function Z(ψ) corresponds to an effective free energy F The magnetic fluctuations will be completely taken into account, if only we are able to solve exactly the integral (8). The exact solution can be done for a uniform order parameter ψ. The uniform value of ψ is different from the mean-field value of ψ because the uniform fluctuations of ψ( x) always exist, so we should choose one of these two possibilities. The problem for this choice arises after the calculation the integral (8) at a next stage of consideration when the effective free energy F D is analyzed and the properties of the superconducting phase (ψ > 0) are investigated. The effective free energy is a particular case of the effective thermodynamic potential in the phase transition theory [3,38] and we must treat the uniform ψ in the way prescribed in the field theory of phase transitions. It will become obvious from the next discussion that we shall use a loop-like expansion which can be exactly summed up to give a logarithmic dependence on |ψ| 2 .
Because of the spontaneous symmetry breaking of the continuous symmetry in the ground state, the ordered phase ψ > 0, i.e., the effective free energies discussed in this paper depend on the modulus |ψ| of the complex number ψ = |ψ|e iθ but not on the phase angle θ which remains arbitrary. That is why we shall consider the modulus |ψ| as an "effective order parameter" because the angle θ does not play any role in the phenomena investigated in the paper. The quantity |ψ| remains undetermined up to the stage when we define the equilibrium order parameter |ψ 0 | by the equation of state [∂F D (ψ)/∂ψ] = 0. This equation gives the equilibrium value ψ 0 of ψ and the difference δψ 0 = (ψ 0 -ψ) can be treated as the uniform (zero dimensional) fluctuation of the field ψ( x). The xdependent fluctuations δψ( x) have been neglected because of the uniformity of ψ. The solution ψ 0 will be stable towards the uniform fluctuation δψ provided the same solution ψ 0 = |ψ 0 |e iθ 0 corresponds to a stable (normal or superconducting) phase; the phase angle θ 0 remains unspecified. Therefore, we begin our investigation setting ψ uniform but at some stage of consideration we shall also ignore the uniform fluctuation δψ and deal only with the equilibrium value ψ 0 of ψ. The equilibrium value will be calculated after taking into account magnetic fluctuations, so it will be different from the usual result when both magnetic and superconducting fluctuations are ignored. This simplest approximation for the equilibrium value of ψ is obtained from the GL free energy (1) provided e = 0 and the gradient term is neglected. Hereafter we shall keep the symbol |ψ 0 | for the equilibrium order parameter in the more general case when the magnetic fluctuations are not neglected and shall denote the same quantity for e = 0 by η ≡ |ψ 0 (e = 0)| = (|a|/b) 1/2 .
The above described approximation neglects the saddle point solutions of GL equations, where ψ( x) is x-dependent. Therefore, the vortex state that is stable in type II superconductors cannot be achieved. This is consistent with the choice of a zero external magnetic field, where the vortex state cannot occur in any type superconductor. These arguments can be easily verified with the help of GL equations [2] for a zero external magnetic field; the only nonzero solution for ψ in this case is given by η = (|a|/b) 1/2 although the magnetic fluctuations A( x) = δ A( x) are properly considered.
In conclusion we can argue that the described method will be convenient for both type I and type II superconductors in a zero external magnetic field, provided the ψ-fluctuations have a negligibly small effect on phase transition properties T c0 = T c (H 0 = 0), where T c denotes the phase transition line for any H 0 ≥ 0. For type II superconductors in H 0 > 0, two lines T c1 (H 0 ) and T c2 (H 0 ) should be defined, usually given by H c1 (T ) and H c2 (T ) [2].

Derivation of effective free energy
When the order parameter ψ is uniform the functional (1) is reduced to and Here ρ = ρ 0 |ψ| 2 and ρ 0 = (8πe 2 /mc 2 ). It is convenient to calculate the partition function Z(ψ) and the effective free energy F D (ψ) in the k-space, where Eqs. (9) and (13) take the form and Here and note, that we have used the Coulomb gauge k. A( k) = 0.
Then the partition function (8) will be where with F A (ψ) given by (17) and the functional integration is defined by the rule (14). With the help of Eqs. (10) - (19) the effective free energy F D (ψ) becomes where F 0 (ψ) is given by Eq. (12) and is the ψ-dependent fluctuation part of F (ψ). In Eq. (20) the ψ-independent fluctuation energy {−k B T ln [Z A (0)]} has been omitted. This energy should be ascribed to the normal state of the superconductor which, by convention, is set equal to zero.
Defining the statistical averages we can write Eq. (21) in the form Eq. (23) is a good starting point for the perturbation calculation of F f (ψ). We expand the exponent in Eq. (23) and also take into account the effect of the logarithm on the infinite series [3] and obtain in result where ... c denotes connected averages [3]. Now we have to calculate averages of the type Here we shall use the Wick theorem and the correlation function of form where The calculation of lowest order terms (l = 1, 2, 3) in Eq. (24) with the help of (25) - (27) is straightforward. The infinite series (24) can be exactly summed up and the result is the following logarithmic function The same result for F f (ψ) can be obtained by a direct calculation of the Gaussian functional integral (9). This is done using the integral representation of δ-function in (9) or (14) but it introduces an additional functional integration that should be carried out after the integration over A j ( x).
Eqs. (10), (20) and (28) give the effective free energy density in the form where and Eqs. (20) and (29) -(32) are the basis of our further considerations. We should mention that the fluctuation contribution ∆f D (ψ) to f (ψ) transforms to a convergent integral in the continuum limit where K D = 2 1−D π −D/2 /Γ(D/2) for all spatial dimensionalities D ≥ 2. But the terms in the expansion of the logarithm in (32) are power-type divergent with the exception of several low-order terms in certain dimensionalities D. Therefore, we shall work with a finite sum of an infinite series of infinite terms. In our further calculations we shall keep the cutoff Λ finite for all relevant terms in ∆f D (ψ). This is the condition to obtain correct results.

Particular dimensions
For purely 2D superconductor consisting of a single atomic layer, we can use Eqs. (29)-(32) setting D = 2 and calculate ∆f 2 (ψ) with the help of the rule (33): The first term of this free energy can be expanded in powers of |ψ| 2 : Thus we obtain the result from Ref. [39]. This case is of special interest because of the logarithmic term in the Landau expansion for f (ψ) but it has no practical application for the lack of ordering in purely 2D superconductors.
For quasi-2D superconductors we assume that (2π/Λ) > L 0 ≫ a 0 , where L 0 is the thickness of the superconducting film and a more precise choice of the upper cutoff Λ ≪ (1/a 0 ) for the wave numbers k i is a matter of an additional investigation [9] (see Sec. 2.1 and 2.5).
In order to justify this definition of a quasi-2D system we consider the more general case of a 3D system of volume V = (L 1 L 2 L 0 ), where we can take the continuum limit along the large dimensions (L 1 and L 2 ) of the film because of the assumption L α ≫ (2π/Λ), (α = 1, 2). The summation over the wave number k 0 = (2πn 0 /L 0 ) cannot be substituted with an integration because L 0 ≪ L α and the dimension L 0 does not obey the conditions, valid for L α [40,41,42]. Therefore, for such 3D system we must sum over k 0 and integrate over two other components (k 1 and k 2 ) of the wave vector k. This gives an opportunity for a systematic description of the 2D-3D crossover in superconductors [13,41,42,43,44] which fully justifies the application of more simple treatment for a 0 ≪ L 0 < (2π/Λ).
We consider the conditions (2π/Λ) > L 0 ≫ a 0 as a definition of a quasi-2D films of thickness L 0 . The condition (2π/Λ) > L 0 means that the sum in Eq. (32) contains only terms with (k 0 = 0). The summation over k = (k 1 , k 2 , 0) gives a correct description of quasi-2D films of thickness L 0 and this can be shown as a limiting case of the more general 2D-3D crossover described in Refs. [13,41,42,43,44]. Therefore, for a quasi-2D film we have the expression; where ∆f 2 (ψ) is given by Eq. (34).
For the bulk (3D) superconductor we obtain: For the Landau expansion in powers of |ψ| this form of f 3 (ψ) confirms the respective results in Refs. [1,4] and moreover correctly gives a term of type ρ 2 0 |ψ| 4 which was supposed small and neglected in these preceding papers. This problem will be discussed in Sec. 3.
The above expression for ∆f 4 (ψ) can be also expanded in powers of |ψ| to show that it contains a term of the type |ψ| 4 ln ( √ ρ 0 |ψ|/Λ) which produces a first order phase transition; this case is considered in the scalar electrodynamics [24]. In our further investigation we shall focus our attention on 3D and quasi-2D superconductors.
The free energy density ∆f D (ψ) can be expanded in powers of |ψ| but the Landau expansion can be done only in an incomplete way for even spatial dimensions. Thus f 2 (ψ), f 4 (ψ), and f (ψ) -the free energy density corresponding to the quasi-2D films, contain logarithmic terms which should be kept in their original form in the further treatment of the function ∆f D (ψ) in the Landau expansion. We shall do our analysis in two ways: with and without Landau expansion of ∆f D (ψ). These variants of the theory will be called "exact" theory (ET) and "Landau" theory (LT), respectively. We shall show that these two ways of investigation give the same results in all cases except for quasi-2D films with relatively small thicknesses (L 0 ≪ ξ 0 ). It seems important to establish the differences between two variants of the theory because the HLM effect is very small and any incorrectness in the theoretical analysis may be a cause for an incorrect result. By same arguments we shall investigate the effect of the factor T in ∆f D (ψ) on the thermodynamics of quasi-2D films. This factor can be represented as T = T c0 (1 + t 0 ) and one may expect that the usual approximation T ≈ T c0 , which is well justified in the Landau theory of phase transitions [2,3], may be applied. We shall show for both 3D and quasi-2D superconductors, that this way of approximation can be made by neglecting terms in the thermodynamic quantities smaller than the leading ones. On the other hand practical calculations lead to the conclusion that this approximation cannot be made without a preliminary examination because for some quasi-2D films it produces a substantial error of about 10%. LT, in which the factor T is substituted by T c0 , will be called a "simplified Landau expansion" -SLT.

Validity
The general result (29) -(32) for the effective free energy f (ψ) has the same domain of validity [2] as the GL free energy functional in a zero external magnetic field. When we neglect a sub-nano interval of temperatures near the phase transition point we can use Eq. (1) provided |t 0 | = |T − T c0 |/T c0 < 1, or in the particular case of type I superconductors, |t 0 | < κ 2 [2]. Note, that the latter inequality does not appear in the general GL approach. It comes as a condition for the consistency of this approach with the microscopic BCS theory for type I superconductors [2].
In our calculations we often use another parameter µ τ = (1/πτ κ) 2 and, in particular, µ ≡ µ 1 = (1/πκ) 2 and in terms of µ the condition for the validity of expansion of f D (ψ) becomes µ|t 0 | ≪ 1, or, more generally, µ τ |t 0 | ≪ 1. Choosing τ = 1/π we obtain the BCS criterion for the validity of the GL free energy of type I superconductors [2]. The choice τ = (ξ 0 /πλ 0 ) corresponds to the cutoff Λ τ = 1/λ 0 . As we shall see in Sec. 3 and 4 the thermodynamics near the phase transition point has no substantial dependence on the value of the cutoff Λ τ but it should be chosen in a way that is consistent with the mean-field-like approximation.
Alternatively, the inequality (ρ/Λ 2 ) ≪ 1 may be investigated with the help of the reduced order parameter ϕ defined by ϕ = |ψ|/η 0 , where η 0 ≡ η(T = 0) = (α 0 T c0 /b) 1/2 is the socalled zero-temperature value of the order parameter within the GL free energy f 0 (ψ), given by Eq. (31); see also Sec. 2.2. The reduced order parameter ϕ will be equal to |t 0 | for t 0 < 0, if only the magnetic fluctuations are ignored, i.e., when |ψ| = η. Using the notation ϕ, we obtain the condition (ρ/Λ 2 ) ≪ 1 in the form µ τ ϕ 2 ≪ 1. This condition seems to be more precise because it takes into account the effect of magnetic fluctuations on the order parameter ψ.
We shall just outline the analysis of the above free energy. It can be shown by both analytical and numerical calculations [10] that |ψ| 6 -term has no substantial effect on the thermodynamics, described by the free energy (39). That is why we ignore this term and do the analysis in the standard way [3]. The possible phases |ψ 0 | are found as a solution of the equation of state: There always exists a normal phase |ψ 0 | = 0 which gives a minimum of f 3 (ψ) for a 3 > 0. The possible superconducting phases are given by Having in mind the existence and stability conditions of |ψ 0 | ± -phases [3], we obtain that the |ψ 0 | + -phase exists for (16a 3 b 3 ) ≤ 9q 2 3 and this region of existence always corresponds to a minimum of f 3 (ψ). The |ψ 0 | − -phase exists for 0 < a 3 < (9q 2 3 /16b 3 ) and this region of existence always corresponds to a maximum of f 3 (ψ), i.e., this phase is absolutely unstable. For a 3 = 0, |ψ 0 | − = 0 and hence, coincides with the normal phase. For 9q 2 3 The equilibrium temperature T eq of the first order phase transition is defined by the equation f (|ψ 0 | + ) = 0 which gives the following result: These results are confirmed by numerical calculations of the effective free energy (39) [10]; there also the influence of the |ψ| 6 -term is evaluated.

Entropy and specific heat capacity
The equilibrium entropy jump is ∆S = V ∆s and ∆s = −(df 3 (|ψ|)/dT ) can be calculated with the help of Eq. (39) and the equation of state (44): where Φ(|ψ 0 |) is the following function: The specific heat capacity per unit volume ∆C = T (∂∆s/∂T ) is obtained from (47) The quantities ∆s(T ) and ∆C(T ) can be evaluated at the equilibrium phase transition point T eq which is found from Eq. (46): provided |∆T c | = |T c0 − T eq | ≪ T c0 . Further we shall see that the condition |∆T c | ≪ T c0 is valid in real substances. The second term in r.h.s. of Eq. (50) is a typical negative fluctuation contribution whereas the positive third term in r.h.s. of the same equality is typical for first-order transitions [3].
To obtain the jumps ∆s and ∆C at T eq we have to put the solution |ψ 0 | + found from Eq. (45) in Eqs. (47) -(49). The result will be: and where b 3c and q 3c are the parameters b 3 and q 3 at T = T eq . As |∆T c | = |T c0 − T eq | ≪ T c0 we can set T eq ≈ T c0 in r.h.s. of Eqs. (51) and (52) and obtain q 3c ≡ q 3 (T = T eq ) ≈ q 3 (T c0 ) and b 3c ≈ b 3 (T c0 ).
The latent heat Q = −V T eq ∆s of the first order phase transition at T eq can be calculated from Eq. (51). If we neglect the charge (e = 0) which means to set ρ 0 = q 3 = 0 and b c3 = b in Eqs. (51) -(52) we shall get the result from Ref. [1] for the ratio Here we should mention that Eq. (52) gives the jump ∆C at the equilibrium phase transition point of the first order phase transition, described by |ψ| 3 term [3], while ∆C calculated in Ref. [1] is equal to the specific heat jump at the standard second order transition ∆C = (α 2 0 T c0 /b) and is four times smaller. Therefore, we obtain (∆T ) eq four times smaller than the respective value in Ref. [1].

Numerical values for Al
In order to do the numerical estimates we represent the Landau parameters α 0 and b with the help of the zero-temperature coherence length ξ 0 and the zero-temperature critical magnetic field H c0 . The connection between them is given by formulae of the standard GL theory of superconductivity [2]: The expression for the zero-temperature penetration depth λ 0 = ( c/2 √ 2eH c0 ξ 0 ) is obtained from the above relation and λ 0 = (b/α 0 T c0 ρ 0 ) 1/2 . We shall use the following experimental values of T c0 , H c0 and ξ 0 for Al: T c0 = 1.19K, H c0 = 99Oe, ξ 0 = 1.6µm, κ = 0.01 [1,45]. The experimental values for T c0 , H c0 and ξ 0 vary about 10-15% depending on the method of measurement and the geometry of the samples (bulk material or films) but such deviations do not affect the results of our numerical investigations.
The temperature shift t eq = t 0 (T eq ) for bulk Al can be estimated with the help of Eq. (50). We obtain that this shift is negative and very small: t eq ∼ −10 −4 . Note, that the second term in the r.h.s. of Eq. (50) is of order 10 −4 provided Λ ∼ (1/ξ 0 ) whereas the third term in the r.h.s. of the same equality is of order 10 −5 . Once again the change of the cutoff Λ to values much higher than (π/ξ 0 ) will take the system outside the temperature interval where the BCS condition for Al is valid. Let us note, that in Ref. [10] the parameter t corresponds to our present notation t 0 . But the numerical calculation of the free energy function f 3 (ψ) in Ref. [10] was made for the SLT variant of the theory and the shifted parameter (t 0 + 0.972 × 10 −4 ) was incorrectly identified with t and this lead to the wrong conclusion for its positiveness at the equilibrium phase transition point T eq . As a matter of fact, the shifted parameter (t 0 + 0.972 × 10 −4 ) is positive at T eq but t eq ≡ t 0 (T eq ) is negative.
Having in mind these remarks, when we evaluate ∆s and ∆C for bulk Al we can use simplified versions of (51) and (52) which means to consider only the first terms in the r.h.s and to take q 3c ≈ q 3 and b 3c ≈ b at T c0 . In this way we obtain and ∆C = 2.62 × 10 3 erg cm 3 .
The results are consistent with an evaluation of ∆C for Al as a jump (∆C = α 2 0 T c0 /b) at the second order superconducting transition point [1] that, as we mentioned above, is four times smaller than the jump ∆C given by Eq. (57).
A complete numerical evaluation of the function f 3 (ψ) and the jump of the order parameter at T eq for bulk Al was presented for the first time in Ref. [10]. The results there confirm that the order parameter jump and Q for bulk type I superconductors are very small and can hardly be observed in experiments.
We shall finish the presentation of bulk Al with a discussion of the ratio (53). It can be also written in the form (∆T ) eq = 32π 9 and it differs by a factor 1/4 from the respective result in Ref. [1]. This difference is due to the fact that we take ∆C as the jump at the first order transition temperature T eq while in the above cited paper [1] the authors define ∆C as a hypothetic jump (∆C) at the standard second order phase transition point. From Eq. (56) we obtain and multiplying the number coefficient in the above expression by 4 we can obtain Eq. (10) from Ref. [1].

QUASI-2D FILMS
Thin quasi-2D films (a 0 ≪ L 0 < 2π/Λ) can be investigated with the help of the respective free energy density f (ψ) given by Eqs. (30) and (31), ∆f D (ψ) is taken from Eqs. (34) and (36). The free energy of quasi-2D superconducting films was derived and analyzed for the first time in Ref. [9] using the Landau expansion of ∆f 2 (ψ) in powers of |ψ| 2 ; see Eq. (35). As is shown for the first time by Lovesey [39] in the simple 2D case the fluctuation contribution ∆f D (ψ), of form given by Eq. (35), leads to a fluctuation-induced first order phase transition. In contrast to 3D superconductors where the first order of the phase transition is generated by |ψ| 3 -term in ∆f 3 (ψ), in 2D superconductors the first order of the phase transition is a result of the presence of |ψ| 2 ln|ψ| in Eq. (35). But the Meissner phase cannot occur in 2D (single atomic layer) superconductors because of the strong fluctuations and hence this case is of no interest. In quasi-2D films, where the Meissner phase does occur for properly chosen thickness of the film (L 0 ≪ 2π/Λ) [9], the change of the order of normal-to-superconducting phase transition is better pronounced than in bulk superconductors. This is well illustrated in the above cited paper [9] by numerical data for Al films with thickness L 0 = 0.1µm. Following Refs. [9,11] and the arguments presented in Sec. 3.3 we shall choose the cutoff Λ = π/ξ 0 .
The expansion of the respective free energy in powers of |ψ| leads to somewhat clumsy analysis and for this reason we shall use the approach in Ref. [11,12] where the quasi-2D films have been investigated with the help of the general form of ∆f (ψ) described by Eq. (34). In both variants of the theory (ET and LT; see Sec. 2.4) the thickness L 0 of the quasi-2D film has an effect on the thermodynamic behavior, that is similar to the influence of material parameters α 0 and b. This is very well seen in the Landau expansion (35) of the free energy f (ψ) given by (36), where the parameters a and b acquire a fluctuation contribution that depends on L 0 . The influence of L 0 on the thermodynamic properties can be considered as a characteristic feature of quasi-2D systems [42,43], a feature, absent in purely 2D films [39]. It is unambiguously demonstrated by several theoretical studies of the 2D-3D crossover in systems with slab geometry [13,42,43,44] that the L 0 -dependence as given in Eq. (36) correctly describes quasi-2D films.
Following Refs. [11,12] and having in mind the above discussion we can present the free energy density f (ψ) = (F (ψ)/L 1 L 2 ) in the form where Γ(y) = (1 + y) ln (1 + y) − y ln y, In Eqs. (60) -(62) we have set Λ = (π/ξ 0 ) and introduced the notation ϕ = |ψ|/η 0 ; the quantity η 0 is defined in Sec. 2.5. Some of the properties of free energy (60) were analyzed in Ref. [11] for Al films and in Ref. [12] for films of Tungsten (W), Indium (In), and Aluminium (Al). Here we shall summarize and justify the preceding results and, moreover, we shall present new results about the properties of the Landau expansion of effective free energy. Note, that the function Γ(y) cannot be fully expanded in powers of y because of the term of type (y lny) in Eq. (61).
The extensive investigations [11,12] of films of W, Al and In with thicknesses from 0.05 µm to 2 µm confirm the intuitive notion that the HLM effect is stronger for smaller values of L 0 . The numerical analysis shows that type I superconductors with relatively small GL parameter κ and relatively high critical field H c0 may be the best candidates for the experimental observation of the effect. The best material from the above enumerated substances seems to be Al; tungsten has an extremely small GL parameter but also a small critical field that makes it inconvenient for experiments. The relatively high H c0 of In results in relatively large latent heat, Q ∼ 4.0 (erg/cm 3 ) but for films with L 0 ∼ 0.05 µm the order parameter jump |ψ| eq = ϕ eq η 0 at T eq is twice smaller than that for the respective Al films [12]: |ψ| eq = 0.05 × 10 −11 for In and |ψ| eq = 0.1 × 10 −11 for Al. We have to stress the role of critical magnetic field H c0 , a fact established for the first time in [12] and the present paper.
With the help of data from Refs. [9,10,11,12] we compared the thermodynamic quantities near the first order phase transition point in bulk Al and Al films of L 0 ∼ 0.1µm. They are given in Table 1, where t eq = t 0 (T eq ), ϕ eq = ϕ(T eq ) is the equilibrium jump of the reduced order parameter and |ψ| eq = ϕ eq η 0 is the order parameter jump at T eq . The shift t eq of the equilibrium transition temperature due to magnetic fluctuations is very small in both bulk Al and thin Al films so the difference (T eq − T co ) can be neglected in all calculations of thermodynamic quantities near T eq . The equilibrium jump ϕ eq , or, equivalently, |ψ| eq , is one order of magnitude higher in the film with L 0 = 0.1µm than in bulk Al but the latent heat Q is 10 2 times bigger for films. These values are almost one order of magnitude higher for L 0 ∼ 0.05 µm than for L 0 = 0.1 µm. The numerical data in Table 1 are obtained by SLT for the bulk Al samples and by ET for the Al film of thickness L 0 = 0.1 µm; for the abbreviations SLT, LT and ET, see Sec. 2.4. The difference in the numerical results obtained from ET, LT and SLT will be discussed in the remainder of the paper.
The investigation of bulk superconductors yields the same results irrespective of whether we analyze the free energy f 3 (ψ) by ET, LT or SLT. The situation in quasi-2D superconductors is however different; the three different variants of treatment of the free energy give different results, in particular, for relatively small thicknesses (L 0 ≪ ξ 0 ). This feature of free energy, Eq. (60), is illustrated in Fig. 1, where three curves for three different variants of f (ψ) are shown for Al films of thickness L 0 = 0.4 µm. The solid line corresponds to ET, the line of crosses represents the LT result, and the line of circles stands for SLT. All three curves are calculated for t eq ≡ t 0 (T eq ) = −0.00057, which is the ET equilibrium phase transition temperature T eq = 0.9994T c0 . Note, that ϕ eq is the nonzero global minimum of f (ϕ) and the function f (ϕ) depicted in Fig. 1 has only one minimum for ϕ > 0 because all curves are calculated in the thermodynamic regime corresponding to the stable Meissner phase.
The main conclusion that can be made from Fig. 1 is that the two variants of the Landau expansion give approximately the same quantitative results and therefore, the factor (1 + t 0 ) in (60) can always be substituted by unity, though the present investigation is intended to quite small physical effects. This conclusion is consistent with the argument [11] that allows to use the same approximation (1 + t 0 ) ≈ 1 for the calculation of Q and ∆C in both variants of the theory: ET and LT. Besides, the Fig. 1 shows that both variants of LT give slightly higher equilibrium phase transition temperatures T eq and substantially higher equilibrium jumps ϕ eq than ET. Thus, using LT for film thicknesses a 0 ≪ L 0 < 0.1µm one may obtain up to 10 times higher value of ϕ eq and up to 10 2 times bigger latent heat Q than the respective values in Table  1. The problem is whether these higher values predicted by LT are reliable. Fig. 2 shows the free energy drawn in the three variants (ET, LT and SLT) of f (ϕ) for Al films of thickness L 0 = 0.1µm. In Fig. 2, the curves f (ϕ) are drawn at their respective equilibrium phase transition points. The variation in t eq (−0. 00148 for the solid line, −0. 00115 for "+" -line and −0. 00046 for •-line) are of order of the typical values of t eq itself so the differences in t eq due to the way of calculation cannot be neglected. Although for both variants of the Landau expansion (LT and SLT), the quantity ϕ eq is again practically the same, the difference in t eq is more pronounced for smaller thickness of Al film and moreover, both variants of Landau expansion are not so good approximation to the result of the exact calculation (ET) as for L 0 = 0.4µm. The conclusions, we have already drawn from the results shown in Fig. 1, are completely confirmed by the form of the curves from Fig. 2 and, moreover, we see that the deviation of the results of LT from those of ET becomes bigger with the decrease of film thickness L 0 . On the basis of the above observations we may conclude that in all cases when ET and the respective Landau expansions give different results, the Landau expansion yields a better established first order transition, with a higher jump ϕ eq , and hence, bigger values of Q and ∆C. In order to establish where LT is a good approximation we have made systematic numerical calculations for Al films of different thicknesses L 0 = 0.05 ÷ 3µm. When L 0 is lowered beginning with 3µm the quantitative differences between the two variants of theory, with and without Landau expansion, respectively, become substantial about L 0 ≈ 0.4µm. Bearing in mind the condition for the validity of the Landau expansion (see, Sec. 2.5) and the requirement for the equilibrium jump of the order parameter, µϕ 2 eq < 1 we may suppose that the predictions done with the help of the Landau expansions do not satisfy this inequality for Al films with thicknesses L 0 ≤ 0.1µm. For this relatively small L 0 -size, ET is absolutely reliable. The numerical data show that films with L 0 > 0.1µm are described well quantitatively by the Landau expansion. The differences between the curves in Figs. 1 and 2 can be neglected in numerical calculations intended to give theoretical predictions for experiments. This general result is supported by the following simple argument. In the Landau expansion of free energy (60) for quasi-2D films the parameter a acquires a cutoff (Λ−) independent contribution of the form ∆a = k B T ρ 0 4πL 0 .
When we compare ∆a with the bare parameter value |a| = (α 0 T c0 |t 0 |) for Al, it is easily obtained that |t 0 | will not exceed 10 −4 for thicknesses L 0 which are of order 1 µm or larger. Therefore, for L 0 ∼ 0.1 µm the Landau expansion gives results which are quantitatively different from those obtained by ET.
We have studied the dependence of the free energy density f (ϕ) of Al films with L 0 = 0.1µm on the cutoff Λ. Fig. 3 shows the free energy density f (ϕ) for three values of the cutoff: Λ = π/λ 0 , Λ = 1/λ 0 , and Λ = π/ξ 0 . As the cutoff increases from (π/ξ 0 ) to π/λ 0 ∼ 10 2 (π/ξ 0 ), the equilibrium jump ϕ eq increases, too. We have already mentioned in Sec. 3.3 that the increase of the cutoff Λ for type I superconductors up to the value (π/λ 0 ) is inconsistent in the present theory. The numerical result for Al films shown in Fig. 3 is, therefore, a demonstration of the validity of our arguments about the choice of the cutoff Λ presented in Sec. 3.3. If we take the cutoff Λ ≫ (π/ξ 0 ) we shall go beyond the scope of validity of our theory.
There is a similarity between the breakdown of the present theory for cutoffs Λ ≫ (π/ξ 0 ) and the breakdown of the condition (ρ/Λ 2 ) ≪ 1 for the validity of LT at small thicknesses L 0 . In both cases, when there is an inconsistency of the theory, we obtain enhanced values of the characteristic jumps of thermodynamic quantities at the equilibrium point of the first-order phase transition.

CONCLUSION
We did a detailed analysis of the HLM effect in bulk (3D-) superconductors and quasi-2D superconducting films within the self-consistent approximation introduced in Refs. [1,4,24]. We have studied for a first time the validity of this approximation and calculated thermodynamic quantities of direct experimental interest like the equilibrium jumps of the order parameter, entropy and specific heat at the point of the fluctuation-induced first-order phase transition to superconducting state in a zero external magnetic field. Our investigation is supported by numerical calculations for bulk Al and Al films.
We have presented for a first time a comprehensive analysis of the effective free energy of the superconductor in a zero external magnetic field and on the basis of this analysis we compared the results from the investigation of the effective free energy without a partial Landau expansion with those from the Landau expansion of the effective free energy of quasi-2D films. For quasi-2D films of type I superconductors, the Landau expansion leads to reliable results, provided the film thickness is above some value depending on the characteristic lengths ξ 0 and λ 0 , i.e., on the material parameters. For Al films the results from the Landau expansion become unreliable below film thicknesses L 0 ∼ 0.1µm. For quasi-2D films of type II superconductors the Landau expansion of the effective free energy can be used for any thickness above the level of destruction of superconductivity (L 0 ∼ 10 −3 µm).
Our investigation provides a reliable theoretical basis for a future experimental search of the HLM effect in thin films of type I superconductors, where the effect is much stronger than in bulk materials. In accord with preceding works [9,10,11,12,13] we have justified earlier results which indicate that the HLM effect will be better pronounced in films of materials with relatively high values of the critical magnetic field H c0 and relatively small thicknesses L 0 . We cannot be certain which superconducting material provides the best experimental conditions for transport or caloric measurements of the jumps of the thermodynamic quantities at the point of the fluctuation-induced first order transition, but from the data for Al, W and In available from recent studies [11,12] and the present paper, the most suitable substance seems to be Al. But investigations of other superconductors may put forward materials which are even better candidates for the experimental test of the HLM effect.
Looking for the most convenient material for the experimental search of the HLM effect we should have in mind a number of other experimental requirements which are not related to the results from the present theoretical investigation. Here we shall briefly discuss the problem of the external magnetic field and the possible change of the superconductivity from type I to type II with the decrease of the film thickness [46]. This change for film thicknesses L 0 , which are convenient for the experimental study of the HLM effect, is not a great problem because we have shown in our investigation that the effect could be observed also in films of type II superconductors, provided the external magnetic field H 0 is very low so the effect of the vortex phase and the magnetic energy jump (H 2 0 /8π) at the phase transition point is negligible. The magnetic energy jump (H 2 0 /8π) may obscure the HLM effect on the latent heat also in type I superconductors and, therefore the experimental problem for the elimination of the residual laboratory external magnetic field H 0 is common for both type I and type II superconducting films. If we take as a basis the latent heat of order 1 [erg/cm 3 ] in Al films with L 0 ∼ 0.1µm, as reported in Sec. 4, the magnetic field which ensures the ratio (H 2 0 /8πQ) ≪ 1 will be obviously about 1 Oe. In thinner films of convenient materials this experimental condition may become H 0 ∼ 10 Oe but no more.