The renormalization-group (RG) functions for the three-dimensional nvector cubic model are calculated in the five-loop approximation. Highprecision numerical estimates for the asymptotic critical exponents of the three-dimensional impure Ising systems are extracted from the five-loop RG series by means of the Padé-Borel-Leroy resummation under n = 0. These exponents are found to be: γ = 1.325 ± 0.003, η = 0.025 ± 0.01, ν = 0.671 ± 0.005, α = −0.0125 ± 0.008, and β = 0.344 ± 0.006. For the correction-to-scaling exponent, the less accurate estimate ω = 0.32 ± 0.06 is obtained.

The critical thermodynamics of the two-dimensional N-vector cubic and MN models is studied within the field-theoretical renormalization group (RG) approach. The ␤ functions and critical exponents are calculated in the five-loop approximation and the RG series obtained are resummed using the Borel-Leroy transformation combined with the generalized Padé approximant and conformal mapping techniques. For the cubic model, the RG flows for various N are investigated. For N = 2 it is found that the continuous line of fixed points running from the XY fixed point to the Ising one is well reproduced by the resummed RG series and an account for the five-loop terms makes the lines of zeros of both ␤ functions closer to each another. For the cubic model with N ജ 3, the five-loop contributions are shown to shift the cubic fixed point, given by the four-loop approximation, towards the Ising fixed point. This confirms the idea that the existence of the cubic fixed point in two dimensions under N Ͼ 2 is an artifact of the perturbative analysis. For the quenched dilute O͑M͒ models (MN models with N =0) the results are compatible with a stable pure fixed point for M ജ 1. For the MN model with M , N ജ 2 all the nonperturbative results are reproduced. In addition a new stable fixed point is found for moderate values of M and N.

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For the three-dimensional cubic model, the nonlinear susceptibilities of the fourth, sixth, and eighth orders are analyzed and the parameters ␦ (i) characterizing their reduced anisotropy are evaluated at the cubic fixed point. In the course of this study, the renormalized sextic coupling constants entering the small-field equation of state are calculated in the four-loop approximation and the universal values of these couplings are estimated by means of the Padé-Borel-Leroy resummation of the series obtained. The anisotropy parameters are found to be ␦ (4) ϭ0.054Ϯ0.012, ␦ (6) ϭ0.102Ϯ0.02, and ␦ (8) ϭ0.144Ϯ0.04, indicating that the anisotropic ͑cubic͒ critical behavior predicted by the advanced higher-order renormalization-group analysis should be, in principle, visible in physical and computer experiments.

From the mid-1970s, the critical thermodynamics of three-dimensional impurity systems has been the object of intensive studies, both theoretical and experimental. Theoretical achievements, such as the determination of the mechanism governing the effect of impurities on the critical behavior, the formulation of the Harris criterion, the construction of the expansion, and the calculation of the critical indices and critical amplitude ratios in the framework of the perturbation theory [1][2][3][4][5][6][7][8][9][10][11][12][13], have stimulated subsequent studies, the development of which in the last few years acquired the character of an explosion. Advancement in this field of research was, to some extent, caused by the discovery of the fact that, for the systems under discussion, an increase in the order of the renormalized perturbation theory does not lead to stabilization of the numerical results for the critical indices and other universal physical quantities. This feature is in contradiction with the known properties of renormalized group expansions for pure systems, which allow one, by applying the appropriate resummation procedures, to determine the universal parameters with an accuracy progressively increasing from order to order [14][15][16][17][18][19][20][21][22]. Most likely, the aforementioned anomaly, which manifests itself only in the five-loop and six-loop approximations [23][24][25][26], reflects the much discussed Borel nonsummability of renormalization group expansions for impurity systems (see, e.g., [27][28][29] and recent reviews [30][31][32]).The absence of convergence of the iteration procedures based on the renormalization group theory of perturbations does not, however, preclude one from obtaining numerical estimates of the critical indices with an acceptable accuracy. The latter implies a relatively small scatter of the results obtained from different approximations, the insensitivity of the results to ⑀ changes in the resummation technique, and, evidently, a good agreement between the theoretical predictions and the results of physical and computer experiments. For example, for the critical index of susceptibility γ of the impurity three-dimensional Ising model, the four-, five-, and six-loop approximations yield the values 1.326-1.321 [10, 11], 1.325 [25], and 1.330 [26], respectively, and the variations of γ in passing from one resummation technique to another do not exceed 0.01. This suggests that the field-theoretical renormalization group method can be used for calculating other universal critical parameters of three-dimensional impurity systems.Below, we determine the nonlinear susceptibilities of the fourth ( χ 4 ) and sixth ( χ 6 ) orders and the effective coupling constant v 6 for a weakly disordered threedimensional Ising ferromagnet in the critical region. At T T c , these quantities, just like the linear susceptibility χ and other equilibrium parameters, take on universal asymptotic values, which can be measured with high accuracy in modern experiment.The free energy of a uni...