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Hybrid cluster+RG approach to the theory of phase transitions in strongly coupled Landau-Ginzburg-Wilson model

It is argued that cluster methods provide a viable alternative to Wilson's momentum shell integration technique at the early stage of renormalization in the field-theoretic models with strongly coupled fields because these methods allow for systematic accounting of all interactions in the system irrespective of their strength. These methods, however, are restricted to relatively small spatial scales, so they ought to be supplemented with more conventional renormalization-group (RG) techniques to account for large scale correlations. To fulfil this goal a"layer-cake"renormalization scheme earlier developed for rotationally symmetric Hamiltonians has been generalized to the lattice case. The RG technique can be naturally integrated with an appropriately modified cluster method so that the RG equations were used only in the presence of large scale fluctuations, while in their absence the approach reduced to a conventional cluster method. As an illustrative example the simplest single-site cluster approximation together with the local-potential RG equation are applied to simple cubic Ising model to calculate several non-universal quantities such as the magnetization curve, the critical temperature, and some critical amplitudes. Good agreement with Monte Carlo simulations and series expansion results was found.


I. INTRODUCTION
Modern theory of critical phenomena in field-theoretic systems is based on the perturbative treatment of the Landau-Ginzburg-Wilson (LGW) model with weakly interacting fluctuating fields. [1][2][3][4][5] The theory is adequate in the critical region, at least in the space dimensions sufficiently close to four, where the interactions become weak at the late stages of the renormalization irrespective of whether the initial interactions were weak or strong. 1 In the latter case, however, the numerical values of the majority of non-universal quantities,-such as the critical temperatures and critical amplitudes,-remain largely unknown 4,5 because non-perturbative field-theoretic techniques that are necessary to treat the strong coupling case are much less reliable than the perturbation theory.
There exist, however, strongly-coupled LGW-type models of considerable practical importance. For example, the Ising model that is widely used in the description of alloy thermodynamics (see Ref. 6 and references therein) can be cast in the form of strongly coupled LGW model, as will be discussed in Section II below. Therefore, the quantitative theory of critical phenomena in strongly coupled LGW-type models could be used, in particular, in prediction of alloy's phase diagrams on the basis of ab initio band-structure calculations. [6][7][8][9][10][11] In terms of the RG theory a major obstacle hampering a fully quantitative description of critical phenomena in the strong coupling case lies in the absence of reliable methods of renormalization during the so-called transient regime that takes place at the early stage of the renormalization procedure. 1 The aim of the present paper is to suggest a technique that would combine two non-perturbative methods, namely, the cluster method of Refs. [12][13][14] and the "layer-cake" renormalization scheme in the local potential approximation (LPA) 15,16 appropriately modified in such a way that the RG treatment would be invoked only when the system develops long-range correlations while away from the criticality a simpler cluster technique would be sufficient.
The material below is distributed as follows. In the next section the models, basic equations and notation will be introduced; besides, with the help of the field-theoretic diffusion RG equations 17 qualitative picture of the renormalization in the strong-coupling regime will be discussed. In Section III the cluster plus RG (C+RG) technique will be introduced and the simplest single site cluster approximation (SSA) presented as an input to the "layercake" RG approach which is explained in Section IV. Furthermore, in Section IV the LPA RG equation will be derived as a natural complement of the SSA and in Section IV A the formulas for some thermodynamic quantities in the SSA+LPA approach will be derived. In Section V the LPA equation derived in the previous section is transformed to the form that facilitates its numerical solution. In Section V A the approach will be illustrated by the description in the framework of the SSA+LPA of the ferromagnetic ordering in the simple cubic (SC) Ising model in zero magnetic field. In Section VI brief discussion of the results obtained and some concluding remarks will be given.

II. THE MODELS AND THE DIFFUSION RG EQUATIONS
In this paper we consider the standard LGW model and the Ising model as its concrete realization. The LGW Hamiltonian for the fluctuating field φ is where i, j are indexes of the lattice sites and U is the functional describing all interactions that are not written down explicitly in the equation. It also includes an arbitrary term of the form − 1 2 i,j φ i r ij φ j that also enters in the inverse of the pair correlation function (CF) in the first sum in Eq. (1) but with opposite sign, so that H does not depend on r. In the quasi-momentum representation the pair CF reads where for convenience the Fourier-transformed "bare" intersite interaction will be chosen in such a way that This can always be achieved by adding and subtracting a constant to/from the two terms in the denominator in Eq. (2); K in Eq. (3) is some proportionality constant. In the case of the Ising model with interaction J between nearest-neighbor (NN) spins on the SC lattice that we are going to use in our numerical calculations throughout the paper where Finally, the last term in Eq. (1) describes the interaction with both real and fictitious ("source") external fields. This slightly differs from the cluster approach of Refs. 12-14, and 18 where the physical field and the source field were treated separately, though formally they are indistinguishable because they enter into the formalism as the sum The reason for this modification of the formalism is that the present paper is devoted mainly to the RG approach though the cluster part will remain practically unchanged. But the magnetization in the RG part will be treated differently from the cluster part, namely, via the Legendre transform for the reasons explained below in section III A.
To simplify notation, the energy unit will be chosen in such a way that the Boltzmann factor 1/k B T were equal to unity. In this case the physical temperature in the system can be controlled by a value of some Hamiltonian parameter. In the calculations below we will use the inverse of the parameter K from Eqs. (3), (4), and (5) as our dimensionless temperature.
In this notation the generating functional of the CFs reads The partition function is obtained from Z[h] by setting h to be equal to the external field h ext . As is seen from Eq. (6), this is equivalent to the conventional route of setting the fictitious source field to zero. 13,15,18 .
Now with the use of the standard transformations [13][14][15]18,19 the generating functional can be cast in the form where the generating functional of the S-matrix is 17 For simplicity, in Eqs. (8) and (9) and below we omit summations over the site indexes by adopting vector-matrix notation in the N-dimensional space (N is the number of lattice sites) by treating fields as vectors and pair CFs as matrices.
As is easily seen from Eqs. (8) and (9), from the assumption that G is the exact correlation function of the system it follows that 12,13,18 Similar to the cluster approach of Refs. 12-14, and 18, this equation will constitute the selfconsistency condition for the calculation of r(k). In the present paper we restrict ourselves to the smallest one-site "clusters" and to the SSA in the cluster part of the C+RG approach; the RG part will be treated in the LPA-the approximation with quasimomentum-independent vertexes, so r in our explicit calculations will be just some non-negative number.

A. Diffusion RG equations
In the functional formalism 1,15,17,19 the renormalization (semi)group can be introduced by representing G in Eq. (9) as an integral over some evolution parameter t: (Here and below the subscripts corresponding to continuous variables will denote partial derivatives with respect to these variables.) With the use of the representation Eq. (11) it is easy to see that the functional Eq. (9) can be found as the solution of a Cauchy problem for the evolution equation for the group generator that in our case takes the form of the with the initial value In principle, any parametrization satisfying Eq. (11) can be used Eq. (12). For example, in Ref. 17 a straightforward parametrization was proposed consisting (in our notation) in so that G t = G. This time-independence of the "diffusion constant" allows one to represent the solution of the diffusion equation Eq. (12) as the convolution of the initial field distribution Eq. (13) with the fundamental solution For our purposes it is important to note that in the vicinity of the critical point G t = G(k) at small |k| is positive and large. As we know from the properties of the conventional diffusion, even the delta-function-like initial probability density distributions that in our case would describe infinitely strong interactions in Eq. (13) can be strongly smeared, given sufficient time and sufficiently large diffusion constant. In our case this would mean diminishing of the interaction in the renormalized effective interaction U in Eq. (13) to small values. It is to be stressed, however, that the smearing strongly depends on both the time of the diffusion and on the value of the diffusion constant, as can be qualitatively seen from the fundamental solution Eq. (15).
Thus, in order the interactions at a given k were small, the effective time-diffusion con- can be obtained within our approach with the use of parametrization Integrating this from t = 0 to some sufficiently large value of t one gets In the critical region and at small k the denominator in the first term on the right hand side of Eq. (2) is small at small k, so in this case G(k, t) ≈ G(k). However, as ǫ(k) + r(k) grows towards and past the value of t, G(k, t) becomes small thus effectively cutting off short-wavelength fluctuations in G(k) qualitatively similarly to the Wilson's method. From This qualitative reasoning was meant to qualitatively substantiate the observation that the structure of strongly coupled LGW models is such that during the renormalization flow the system either enters into critical regime with weak coupling or retains the strong interactions but with only short-range correlations. This dichotomy should justify the need for the hybrid approach when strong coupling short-range behavior is treated within cluster approximations that are known to be well suited to this kind of problems while the critical behavior is dealt with in an RG technique that is the only viable approach to the problem in field-theoretic models.
Of course, not all LGW-type statistical models are strongly coupled but some practically important ones may have effective interactions even of infinite strength. For example, the Ising model can be transformed into a field-theoretic system as follows where the interaction can be represented as infinitely strong λφ 4 coupling if the deltafunctions in Eq. (18) are represented as (C λ is a normalization constant).
The Ising model is being widely used, inter alia, in the study of alloy thermodynamics. [6][7][8][9][10][11] Therefore, the development of efficient techniques of its solution with realistic interatomic/interspin interactions is a problem of considerable practical interest. This problem has been extensively studied in the past decades and sophisticated techniques has been

III. C+RG METHOD
In the functional cluster technique of Refs. 12-14, and 18 one can calculate with good accuracy only the critical temperature while the critical behavior in general is described very poorly (see, e. g., Fig. 1). This approach, however, is based on the field-theoretical formalism so in principle should be compatible with Wilson's RG techniques. The rest of the present paper will be devoted to the realization of this observation in application to the strongly coupled LGW model with the use of the SC Ising model in concrete calculations.
The cluster approach we will use is a systematic technique to treat lattice field-theoretic models and is well documented in previous publications, so here it will only be mentioned why it is insufficient to the description of the second order phase transitions. The essence of the approach is easier to explain using the coarse-grained version 14 of the cluster method though non-coarse-grained version may be more accurate 12,13 .
In the cluster approach one tries to approximate an infinite system of strongly coupled fields with a finite cluster of size N c . The reason is that the calculation of the partition function Eq. (7) can be approximately reduced to N c -dimensional integral that can be calculated exactly irrespective of the strength of the interactions. In Fig. 2 is drawn a cross-section of the CF G(k) in the first Brillouin Zone (BZ) of a d-dimensional cubic lattice along the k x axis. Let us imagine that the system is divided into clusters of size N c = 4 d sites. The BZ of such a system will shrink correspondingly (see vertical lines). But because the system remains the same we will keep the initial BZ divided into 4 d parts to simplify the discussion.
In the coarse-grained approximation G(k) is approximated by its average value within each sub-cell (the horizontal lines in Fig. 2) and thus N k-points in the BZ are replaced by a finite number N c of the points (for the details see Refs. 14,26 ). The accuracy of the cluster approximation can be estimated by the size of the deviation of the exact G(k) from its average values within the subcells. It is expected that with the growth of N c the cluster approximation will be converging to the exact solution as long as the Riemann sum over subcells will be converging towards the exact integral.
The Riemann sum, however, does not converge in the case of improper integrals, 27 in particular, when the integrand diverges, as is the case with G(k) at the critical point when G(k → 0) → ∞. In this case for an arbitrarily small integration cell surrounding the point k = 0, the deviation of the exact G(k) from the average CF will be arbitrarily large at a point k sufficiently close to zero. This necessitates introduction in the vicinity of the critical point of some other integration technique. To this end, let us first separate the singular part of G(k) near k ≈ 0 from the nonsingular behavior at other values of k as where Vertical dashed lines are the boundaries that divide the initial Brillouin zone into smaller parts when the system is divided into clusters having four cells in the x-direction.
(see Fig. 2). It is easy to see that by construction the variation ofḠ(k) is always bounded so the Riemann sum converge and the problem can be efficiently treated within the cluster approach. It is to be noted that far from criticality r in Eq. (2) can be is large and G(k) < t 0 .
In this case the C+RG approach reduces to purely cluster approach of Refs. 12-14. But when the second term in Eq. (20) is non-zero, the cluster solution should be augmented with the RG part. The latter will be considered in the next section while in the rest of this section we will illustrate the cluster part of the technique with the simple example of the Ising model in zero external field in the SSA.

A. SSA in application to the Ising model
In the SSA the average over k is performed over the whole Brillouin zone and therefore the result is equal to the site-diagonal element of the truncated correlation function: In the present paper we will be interested mainly in the second order phase transition from the paramagnetic to ferromagnetic state in the absence of external magnetic fields.
The conventional SSA [12][13][14]18 consists in approximating matrix G in Eq. (9) by its diagonal elements in which case the calculation of R reduces to the calculation of 1D integrals. But before that one shifts the integration variables in Eq. (7) by the (unknown) value of the spontaneous magnetization m, so the effective potential in the SSA reads 13,14,18 where C does not depend on field φ and so its explicit value will not be needed in our present calculations. In case of necessity it can be easily derived or obtained from the formulas of In the ordered phase the SSA Eq. (23)  To be more specific, let us assume for the moment that the cut-off t 0 is infinitely large so the RG part is not necessary. In this case the problem is solved in the SSA as follows. 13,18 First, equating to zero the first derivative of Eq. (23) with respect to φ i the SSA equation for the equilibrium magnetization m 0 is obtained: then Eq. (10) for φ = 0 gives G ii + m 2 0 = 1 (25) and the SSA magnetization is shown in Fig. 1. As can be seen, when close to the saturation value m = 1, the SSA agrees well with the low-temperature expansion. It can be shown analytically that Eqs. (24)-(25) reproduce exactly its two leading terms.
From Fig. 1 one can see that close to the critical point, however, the SSA solution exhibits behavior that is very far from the correct one as seen, e. g., in the MC simulations. 24 Therefore, in this region the cluster approximation breaks down and should be replaced with more adequate treatment.
To apply the C+RG approach one has first to decide on the value of the cut-off t 0 to be used in the calculations. Assuming that SSA gives reasonably accurate solution in the non-critical region, let us use it in assessing the Ginzburg criterion by comparing the square of the long-wavelength fluctuations of the order parameter with the mean order parameter squared as 29 Assuming further that the critical region starts at the point where Gi ≈ 1, one finds with the use of the SSA solution that at this point (marked by the subscript 0) 1/K 0 ≈ 3.9 and t 0 = 1/r 0 = 0.65-the value we will use in calculations throughout the paper. In section V A we will show that the results obtained weakly depend on the choice of the parameter.

IV. LAYER CAKE RENORMALIZATION SCHEME AND THE LPA
Thus, having assumed that in the critical region the SSA can satisfactorily describe only those fluctuations whose correlation do not exceed some value t 0 , we also have to develop an RG approach that would take into account those fluctuations. Formally we need to calculate an expression similar to Eq. (9) but this time with the effective potential Eq. (23) renormalized by the SSA. To this end it is convenient to rewrite the diffusion RG Eq. (12) through the logarithm of the S-matrix 30 or the effective potential U[φ, t] = − ln R[φ, t] which we assume can be expanded in the functional Taylor series in the field φ as In terms of U Eq. (12) reads In the layer cake renormalization scheme 15 the CF G is represented as the sum of infinitesimal layers of thickness dt which when stacked together will form the function under consideration (see, e. g., Fig. 1 in Ref. 15). It is easy to see that this leads to where θ-function plays the role of the indicator function 31 and thus can be performed formally exactly in each term by simply substituting k by, say, n−1 j=1 k j in all factors entering the product. (The subscripts i, i ′ in Eq. (30) means that corresponding quasimomentum in the set is absent because it is set to be equal to ±k; these same terms reduce by unity the summations over j, j ′ in the arguments of the deltafunctions.) The first summation in Eq. (28), however, contains only one delta-function so that to carry out the summation one needs to know explicitly the dependences of the vertex functions on quasimomenta.
Thus, Eq. (28) is a complicated nonlinear integro-differential equation with infinite number of variables and its solution presents a formidable task. It can be considerably simplified, however, in the LPA that consists in first setting to zero all φ k with k = 0 outside the contour defined by the level set G(k) = t. This is legitimate because in our problem only the homogeneous fields with k = 0 survives after the substitution φ = Gh ext in Eq. (10). After that we assume that the vertex functions u n are independent of the quasimomenta, so that the functional Eq. (27) can be mapped onto the function 15 The approximations should be good in the critical region where small quasimomenta dominate so that Eq. (32) approximates Eq. (27) as and where and n SC (E) is the integrated density of states with the dispersion law given by Eq. (4) (see Appendix A).
In Fig. 3  interactions that are usually considered to be the most important ones in alloys. [6][7][8][9][10][11]33 But the aim of the C+RG is to account for all cluster interactions in the system. The cluster part of the method is well suited for this purpose but the RG part needs some adjustment.
It is to be noted that only quasimomentum-dependent part of cluster interactions poses problems because the constant part is accounted for in the u n ({k j → 0}) local potential.
A straightforward approach would be to augment the LPA equation Eq.
so the smaller is the cutoff t 0 , the smaller isḠ ii hence, larger the interaction. For example, the coefficient strength of the λφ 4 interaction at the critical point 1/K c = 4.44 and cutoff whereḠ ii | t 0 =0.65 ≈ 0.6 and the 2K c comes from the renormalization of the field φ necessary to bring our quadratic term Kk 2 |φ k | 2 in our Hamiltonian (see Eqs. (1) and (3)) to the canonical form k 2 |φ k | 2 /2 1,2 . As we see, the coupling in the case t 0 = 0.65 is rather strong. This does not matter in the LPA that we use in the present paper because the RG equation was derived without any resort to the smallness of the couplings and thus can be applied to the strong coupling case of Eq. (38). However, the LPA is not easy to improve in the nonperturbative manner to exhibit, for example, more accurate critical behavior with improved indexes. 16 Therefore, it would be more practical to push the cutoff t 0 higher in which caseḠ ii will approach unity (see Eq. (25) with m 0 = 0) and the coupling in Eq. (38) will tend to much smaller value λ ≈ 0.4. This should facilitate application of perturbative techniques. [1][2][3][4][5] Of course, the omission of nonlocal contributions even at large t 0 /small k 0 values will still introduce some errors into the results. But it should be remembered that the cluster part of the calculations will always be of restricted accuracy because the complexity of the calculations grows exponentially with the cluster size, so the size will be bounded by some From Eqs. (23), (27), and (32) it is easy to derive the initial condition (we remind that in the RG part we set m = 0 both above and below the critical temperature).
The only peculiarity of our Cauchy problem is that it should be solved in a self-consistent manner to satisfy the condition Eq. (10): To calculate physical quantities from in the SSA+LPA we first need to find the per-site Gibbs potential With the use of Eq. (8) in the SSA+LPA approximation one gets To find solutions with spontaneously broken symmetry, however, we need the Helmholtz free energy that can be obtained from the Gibbs potential in a standard way via the Legendre transform: 19,28 where the dependence of h on magnetization m is found from the definition The magnetic susceptibility is where use has been made of the self-consistency condition Eq. (40). We note that in our approximation Eq. (45) agrees with another definition so that at this level of approximation the approach is consistent.
The critical amplitudes C ± describe the asymptotic behavior of the susceptibility in the vicinity of the critical point K c as 22,34,35 where γ is the critical index, and ± denotes whether the amplitude is being measured or calculated above (+) or below (-) the critical temperature. Now because r in Eqs.
where ν is another critical index. From this relation the behavior of r in the critical region can be derived as where a is a constant. Comparing this with Eq. (47) one can find a = 1/C ± and the relation between the exponents is This relation imply η = 0 22 and is a consequence of the LPA. 15 Finally, the amplitude B and the critical index β are defined from the behavior of the spontaneous magnetization near the critical point as These five amplitudes (C ± , f ± and B) together with the critical temperature will be the non-universal quantities that we will calculate within the C+RG approach in the present paper.
Eq. (35) is easily solvable numerically in the symmetric phase. So if we were interested only in critical amplitudes, we could solve Eqs. (35) and (40) in this phase to find C + and f + and then utilized known values of the universal ratios to calculate C − , f − and even where Q c is a known universal quantity. 22 However, because our aim is to develop a method that would describe thermodynamics also beyond the critical region, we would like to solve the C+RG equations below the critical temperature as well. Unfortunately, Eq. (35) is not easily solvable in the ordered phase. Therefore, numerical calculations were performed with transformed equation that could be solved in both phases as explained below.
As is known, the Gaussian model is pathological in that at negative u 2 it is not bounded from below and so does not possess an ordered state. So if one attempts integration of Eq.
(35) with the initial u (G) (x, t 0 ) having negative value of u 2 (t 0 ), than the solution that can be obtained analytically in this case will have the form where c 0 is a negative constant such that 1/(2t 0 + c 0 ) = u 2 (t 0 ) < 0. As we see, at some  Equations similar to Eq. (57) were studied previously in LPA-RG approaches (see Refs. 16 and 32 and references therein) with some physical meaning being ascribed to the quantities similar to our v and y. In our approach, however, such a meaning is not easily discernible so we will consider these entities as purely auxiliary variables while physical quantities will be calculated with the use of thermodynamic potentials introduced in previous section.
As is easily seen, the arbitrary constant c in Eqs. (56) is convenient to chose to be equal to t 0 . In this case at t = t 0 y = x and v = u, so the initial function v is the same as in Eq.
where in the last equality use has been made of the self-consistency condition Eq. (58). The solution converged to reproducible results with the use of the following parameters.  The most difficult region of the solution was the jump in the inverse susceptibility that was observed in the data (see Fig. 4). In contrast to Ref. 38, the naive calculation of C + /C − ratio on the basis of our data from Table I gives value 5 in excellent agreement with the best estimates of this universal ratio. 22 This result, however, has to be taken with caution because of the large error in the calculated C − (see Table I). The reason for this was the self-consistency requirement Eqs. (58) and (60e) according to which r has to have the value equal to the jump in the inverse susceptibility. Geometrically this means that the value of the curve in Fig. 4 at the upper side of the step should be equal to r. But in fitting r to fulfill this requirement one meets with the problem that the position m 0 of the jump could change only in finite steps ∼ 2 · 10 −3 (see above). Because of this, it was impossible to make r(τ ) curve in Fig. 5 to be sufficiently smooth to ensure an accurate definition of C − . In this connection a more general question arises about the very existence of the jump.
It was found that its presence or absence depends strongly on the renormalization scheme used and on the system's dimensionality. 32,38 The calculations presented in Fig. 4 were performed at the largest value of r = 0.68 in our calculations in order to have the smallest integration time span ∆t = 1/r − t 0 ≈ 0.82. But even at such a short integration interval the jump was already sufficiently abrupt to fall within only one ∆m step. Besides, further properties of the numerical solution obtained were found to be in qualitative agreement with the known properties of the exact solution. 19,32,38 Namely, inside the mixed-phase region where magnetization varied from −m 0 to m 0 the free energy |f | varied on less than 3.5·10 −7 , |h| < 7.5 · 10 −7 , and |χ −1 | < 5.3 · 10 −5 . The inverse susceptibility variation was quite random and largely due to the computational errors. In the exact theory all these values on the horizontal line in Fig. 4  fits except one, both the amplitudes and the critical indexes were fitted to the data. The exception was made for r in the ordered phase where the scatter in the data (see Fig. 5) for the reasons explained above made the two-parameter fit unstable. Therefore, only the amplitude was fitted in Eq. (50) for this case while the critical index was given the known LPA value. In all other cases in sufficiently small region near the critical point (see Fig. 6) the indexes coincided to a good accuracy with the LPA values. The latter were derived from the known η LP A = 0 and ν LP A ≃ 0.65 15,16 and the scaling relations as γ LP A = 2ν LP A and To assess the sensitivity of the results to the choice of large t 0 , the SSA+LPA equations were solved in the disordered phase for t 0 = 2. As can be seen from Fig. 5, the dependence r(τ ) (hence, χ(τ )-see Eq. (45)) remained practically unchanged. The agreement of the calculated critical temperature K −1 c | t 0 =2 = 4.26 with the exact MC value worsened but only on ∼ 4% with respect to the t 0 = 0.65 case which is not too drastic a change taking into account that the cutoff t 0 changed more than three times. (50) for the data in the region τ < 10 −3 .
In Fig. 1  the exact low-temperature asymptotic behavior. 25 As is seen, the critical behavior smoothly passes into the low-temperature regime and no mean-field square-root like behavior can be discerned. Similar absence of the mean-field regime took place in our calculations above the critical temperature (see Fig. 6). It would be interesting to check this predictions with MC simulations or series expansions because it would allow to check the ability of our approach to describe the transient region between the critical point and the far from criticality region also in the symmetric phase.
The largest discrepancy in the SSA+LPA calculations was observed in the calculation of r at temperature K −1 1 (see Fig. 1 be calculated from the data with the use of the universal amplitude ratios known from the universality studies. 22 This may be useful in practice when only critical amplitudes are of interest but direct derivation of some of them from the C+RG solutions is difficult.

VI. CONCLUSION
The main result of the present paper is the development of the technique that allows for quantitative description of the second order phase transitions from known microscopic interactions of arbitrary strength. The layer-cake renormalization scheme allowed us to calculate the critical temperature with better than 2% accuracy and several critical amplitudes in good agreement with MC simulations and series expansions. The description of the critical regime in the LPA, however, was not perfect. 15 The improvement of the method should apparently lie in going beyond the LPA and accounting for non-local/quasimomentum-dependent contributions. This should also improve the critical indices that are known to be not very good because, for example, η LP A = 0 due to the locality 15,16,32 . But more important would be the possibility to account for the non-pair cluster interactions that are inherently nonlocal.
The main domain of application of the technique developed in the present paper is envisaged in the theories of phase transitions based on realistic interactions either calculated ab initio or extracted from experimental data. [6][7][8][9][10][11]33 Such interactions usually do not possess high symmetry and simplicity of theoretical models. Therefore, C+RG was designed to be able to account in a brute force manner for as many various interactions as possible. The cluster methods in principle are well suited to this goal but, unfortunately, they are restricted to rather large quasimomentum cutoffs. For example, the 4 d hypercubic cluster we used in the discussion is already highly unrealistic in 3D (∼ 2 64 terms would need be accounted in the cluster equations) though the quasimomentum cut-off provided by it will be only one fourth of the initial (maximum) cut-off. Therefore, some non-negligible nonlocalities may remain and it would be desirable to take them into account at the RG stage of the method.
It should be born in mind, however, that due to smaller quasimomenta and smaller renormalized coupling constants, the nonlocality may be strongly suppressed at the RG stage if sufficiently large clusters are used at the cluster stage. So apart from the incorrect critical indexes the technique can be quite accurate for large clusters even in its current form.
As to the critical behavior per se, it seems to be unrealistic to demand from a theory that aims to take into account as many realistic microscopic interactions (cluster, long-range, etc.) as possible, to be equally successful at the other extreme of the essentially macroscopic critical behavior. Presently to describe the critical region oversimplified models are being treated within the most sophisticated and rigorous series expansion techniques, special field-theoretic models are being designed to suppress undesirable contributions that obscure the behavior of interest, and yet disagreements and discrepancies between different studies continue to exist. 22 Therefore, instead of trying to solve some improved LPA equations in the critical region to obtain a priori inferior results, it seems more reasonable to integrate the equations up to some preasymptotic boundary to find the few parameters needed in the quantitative description of the critical behavior in the framework of some sophisticated field-theoretic approach. 4,5 Further work is necessary to develop practically efficient combination of cluster and RG methods to accurately describe thermodynamics of strongly coupled field-theoretic statistical models. The major conclusion that can be drawn from the results of the present study is that this task is quite feasible.