The critical behaviour of helimagnets and stacked triangular antiferromagnets is analyzed in (4 − ǫ) dimensions within three-loop approximation. Numerical estimates for marginal values of the order parameter dimensionality N obtained by resummation of corresponding ǫ-expansions rule out the possibility of continuous chiral transitions in magnets with Heisenberg or planar spins.

The detailed analysis of the global structure of the renormalization-group ͑RG͒ flow diagram for a model with isotropic and cubic interactions is carried out in the framework of the massive field theory directly in three dimensions ͑3D͒ within an assumption of isotropic exchange. Perturbative expansions for RG functions are calculated for arbitrary N up to four-loop order and resummed by means of the generalized Padé-Borel-Leroy technique. Coordinates and stability matrix eigenvalues for the cubic fixed point are found under the optimal value of the transformation parameter. Critical dimensionality of the model is proved to be equal to N c ϭ2.89Ϯ0.02 that agrees well with the estimate obtained on the basis of the five-loop expansion ͓H. Kleinert and V. Schulte-Frohlinde, Phys. Lett. B 342, 284 ͑1995͔͒ resummed by the above method. As a consequence, the cubic fixed point should be stable in 3D for Nу3, and the critical exponents controlling phase transitions in three-dimensional magnets should belong to the cubic universality class. The critical behavior of the random Ising model being the nontrivial particular case of the cubic model when Nϭ0 is also investigated. For all physical quantities of interest the most accurate numerical estimates with their error bounds are obtained. The results achieved in the work are discussed along with the predictions given by other theoretical approaches and experimental data.

A new approach to summation of divergent field-theoretical series is suggested. It is based on the Borel transformation combined with a conformal mapping and does not imply the exact asymptotic parameters to be known. The method is tested on functions expanded in their asymptotic power series. It is applied to estimating the critical exponent values for an N -vector field model, describing magnetic and structural phase transitions in cubic and tetragonal crystals, from five-loop ǫ expansions.PACS: 64.60.Ak, 11.15.Bt, 11.15.Me, 64.60.Fr.Recently Kleinert and Schulte-Frohlinde calculated renormalization group (RG) functions for the cubic model in (4 − ǫ)-dimensions within the five-loop approximation of the renormalized perturbation theory [1]. The critical (marginal) dimensionality N c of the order parameter field found on the base of those expansions turned out to be smaller then three in three dimensions. This means that the critical behavior of the model should be controlled by the cubic fixed point with a specific set of critical exponents. Thus, calculation of the critical exponent values for the cubic model is an actual problem. Since the model of concern describes the critical behavior of cubic and tetragonal crystals undergoing magnetic and structural phase transitions, critical exponent estimates can be compared with experimental data.The field-theoretical RG series are known to be badly divergent. To extract the physical information relevant to predicting the critical behavior of real systems they should be processed by a proper resummation procedure. The series for N c obtained in Ref.[1] proved to be alternating in signs that allowed to resumm it by the simple Padé 1

The critical behavior of a complex N -component order parameter Ginzburg-Landau model with isotropic and cubic interactions describing antiferromagnetic and structural phase transitions in certain crystals with complicated ordering is studied in the framework of the four-loop renormalization group (RG) approach in (4 − ε) dimensions. By using dimensional regularization and the minimal subtraction scheme, the perturbative expansions for RG functions are deduced and resummed by the Borel-Leroy transformation combined with a conformal mapping. Investigation of the global structure of RG flows for the physically significant cases N = 2 and N = 3 shows that the model has an anisotropic stable fixed point governing the continuous phase transitions with new critical exponents. This is supported by the estimate of the critical dimensionality N c = 1.445(20) obtained from six loops via the exact relation N c = 1 2 n c established for the complex and real hypercubic models.

The global structure of the renormalization-group flows of a model with isotropic and cubic interactions is studied using the massive field theory directly in three dimensions. The four-loop expansions of the β-functions are calculated for arbitrary N. The critical dimensionality N c = 2.89 ± 0.02 and the stability matrix eigenvalues estimates obtained on the basis of the generalized Padé-Borel-Leroy resummation technique are shown to be in a good agreement with those found recently by exploiting the five-loop ε-expansions.