Is $\gamma_{KLS}$-generalized statistical field theory complete?

In this Letter we introduce some field-theoretic approach for computing the critical properties of $\gamma_{KLS}$-generalized systems undergoing continuous phase transitions, namely $\gamma_{KLS}$-statistical field theory. From this new approach emerges the new generalized O($N$)$_{\gamma_{KLS}}$ universality class, which is capable of encompassing nonconventional critical exponents for real imperfect systems known as manganites not described by standard statistical field theory. We compare the generalized results with those obtained from measurements in manganites. The agreement was satisfactory, where the relative errors are $<5\%$ for the most of manganites used. Although the present approach describes the aforementioned nonconventional critical indices, we show that it is not complete. For example, it does not explain the results for some other manganites, being explained only for nonextensive statistical field theory recently introduced in literature. So, $\gamma_{KLS}$-statistical field theory has to be discarded for statistical mechanics generalization purposes.


I. INTRODUCTION
Recently, a field-theoretic renormalization group method for describing the unconventional critical behavior of some real imperfect systems known as manganites [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] was proposed [21].These systems are complex and present a large number of strong interacting particles, defects, impurities, inhomogeneities, nonlinearities, competition etc.Such systems can not be understood by employing standard Gibbs-Boltzmann statistical field theory (SFT) formulated by Kenneth Wilson [22].Rather, we have to employ some generalized version of Wilson field-theoretic renormalization group, namely nonextensive statistical field theory (NSFT) [21].As it is known, there are many attempts of generalizing Gibbs-Boltzmann statistical mechanics [23][24][25][26][27].In this direction, we have to check what of these proposed generalizations satisfy to the all requirements needed for a consistent generalized theory.Some of these conditions are: they have to emerge from a maximum principle and a trace-form entropy, present decisivity, maximality, concavity, Lesche stability, positivity, continuity, symmetry and expansibility.Once all these conditions are satisfied, the corresponding proposed generalized theory has to be applied to all experimental situations not described by the standard theory.If there is, at least, one experimental situation in which some proposed generalized statistics does not describe the referred experimental results, this statistics is not general enough to represent a generalized theory.Then it has to be discarded as one candidate to generalize statistical mechanics.In fact, recently it was shown that it was the case for Kaniadakis attempt for generalizing statistical mechanics [28], i. e., this statistics, namely Kaniadakis statistics [24] does not describe the set of unconventional critical exponents values for some manganites as NSFT does [21].
The aim of this Letter is to search if there are some field-theoretic renormalization group generalizations sat-isfying all the requirements aforementioned other than NSFT (parameterized by the parameter q encoding some effective interaction [21]).For that, we employ the γ KLSgeneralized statistics proposed in Ref. [26].It is parameterized by the parameter γ [26,27].In the present context of critical exponents, we have to avoid to represent different concepts with the same letter, for example the γ parameter and the susceptibility critical exponent γ.Then we have to use γ KLS instead γ for representing that parameter.We will call such a theory as γ KLSgeneralized statistical field theory or γ KLS -SFT for short.The range of γ KLS is −1/2 < γ KLS < 1/2 [26,27].We expect to recover the nongeneralized critical exponents values obtained by Kenneth Wilson [22] in the limit γ KLS → 0. Also we expect the emergence of some γ KLSgeneralized universality class for γ KLS -generalized Isinglike systems, namely the O(N ) γKLS one.Now the critical indices will depend on the dimension d, N and symmetry of some N -component order parameter, if the interactions of the constituents of the systems are of shortor long-range type and γ KLS .From the physical interpretation of the results, it will emerge the corresponding physical interpretation of the γ KLS parameter.For a similar field-theoretic approach, see Ref. [29].

II. γKLS-SFT
We introduce the γ KLS -SFT by defining its generating functional by where is the γ KLS -exponential function [26,27].We can determine the constant N from the condition Z[J = 0] = 1.Now we can compute the static γ KLS -generalized critical exponents for O(N ) γKLS universality class for γ KLSφ 4 theory through six distinct and independent methods in d = 4 − ǫ dimensions.The two independent γ KLSgeneralized indices valid for all loop levels are given by The η and ν indices are the corresponding nongeneralized critical indices valid for all loop orders.The corresponding dynamic γ KLS -generalized critical index is given by where z is the nongeneralized index value valid for all loop levels.

γKLS-Heisenberg
βγ KLS γγ KLS La0.67Sr0.33MnO3[33] We interpret these results as follows: a given physical quantity, near the transition point, diverges.How much it diverges is measured by its associated critical exponent.In the case of susceptibility, for example, its inverse furnishes a measure of how much the material is susceptible to the changes in magnetic field.Higher (lower) values of the γ critical index indicates more (less) susceptible or weaker interacting (stronger) systems.These facts are in agreement with the form of the effective energy of the system, which can be obtained by the some expansion around γ KLS ≈ 0. In fact, taking the leading contribution to the energy of the system as E, in units of . So the effective energy E + 1 2 γ KLS E 2 increases with the increasing of γ KLS .Then higher (lower) values of γ KLS represent systems interacting weaker (stronger) or more (less) susceptible and thus possessing higher (lower) values of their critical exponents.Also higher (lower) values of γ KLS give higher (lower) values of E and then we have to furnish less (more) energy to attain the respective critical transition temperature so the critical transition temperatures assume lower (higher) values and decrease.Although γ KLS -SFT can explain the results for the Tables I-II, it does not explain the results of Table IV for some materiais, being explained only for NSFT (see Table IV) of Ref. [21].Then, γ KLS -SFT must be discarded as one trying to generalize statistical mechanics.For our knowledge, only NSFT of Ref. [21] remains a fully consistent or complete generalized formulation of statistical field theory.

IV. OTHER γKLS-MODELS
Now that γ KLS -SFT has been validated experimentally, we display the γ KLS -critical indices for other models.
E. γKLS-uniaxial systems with strong dipolar forces γ KLS -uniaxial systems with strong dipolar forces in the z-direction are described by the following Hamiltonian [50] where V µν ( x) is the short-range potential and γ is a parameter for controlling the dipolar forces intensity.In in d = 3 − ǫ dimensions, the γ KLS -critical exponents are given by where, η and ν are displayed in Ref. [50] up to two-loop level.

F. γKLS-spherical model
The γ KLS -spherical model [51] can be obtained by taking the limit N → ∞ [52] of the O(N ) γKLS model of the present Letter.After taking this limit, we obtain in where the corresponding nongeneralized values are exact, namely η = 0 and ν = 1/(2 − ǫ) [22].