Critical exponents for the 3D O(n)-symmetric model with n > 3 are estimated on the base of six-loop renormalisation-group (RG) expansions. Simple Padé-Borel technique is used for resummation of RG series and Padé approximants [L/1] areshown to give rather good numerical results for all calculated quantities. For large n, the fixed point location g c and critical exponents are also determined directly from six-loop expansions, without addressing to resummation procedure. Analysis of numbers obtained shows that resummation becomes unnecessary when n exceeds 28 provided an accuracy about 0.01 is adopted as satisfactory for g c and critical exponents. Further, results of the calculations performed are used to estimate numerical accuracy of the 1 n -expansion. The same value, n = 28, is shown to play a role of lower boundary of the domain where this approximation provides high-precision estimates for critical exponents.

The critical behavior of a model with N-vector complex order parameters and three quartic coupling constants that describes phase transitions in unconventional superconductors, helical magnets, stacked triangular antiferromagnets, superBuid He, and zero-temperature transitions in fully frustrated Josephson-junction arrays is studied within the field-theoretical renormalization-group (RG) approach in three dimensions. To obtain qualitatively and quantitatively correct results perturbative expansions for P functions and critical exponents are calculated up to three-loop order and resummed by means of the generalized Pade-Borel procedure. Fixed-point coordinates, critical-exponent values, RG Qows, etc. , are found for the physically interesting cases of N = 2 and 3. Critical (marginal) values of N at which the topology of the Bow diagram changes are determined as well. It is argued, on the basis of several independent criteria, that the accuracy of the numerical results obtained is about 0.01, an order of magnitude better than that given by resummed two-loop RG expansions. In most cases the systems mentioned are shown to undergo Buctuation-driven first-order phase transitions. Continuous transitions are allowed in hexagonal d-wave superconductors, in planar helical magnets (into sinusoidal linearly polarized phase), and in triangular antiferromagnets (into simple unfrustrated ordered states) with critical exponents p = 1.336, v = 0.677, o = -0.030, P = 0.347, and g = 0.026, which are hardly believed to be experimentally distinguishable from those of the three-dimensional XY model. The chiral fixed point of RG equations is found to exist and possess some domain of attraction provided N ) 4. Thus, magnets with Heisenberg (N = 3) and XYlike (N = 2) spins should not demonstrate chiral critical behavior with unusual values of critical exponents; they can approach the chiral state only via first-order phase transitions.

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.

By analyzing the dielectric non-linearity with the Landau thermodynamic expansion, we find a simple and direct way to assess the importance of the eighth order term. Following this approach, it is demonstrated that the eighth order term is essential for the adequate description of the para/ferroelectric phase transition of BaTiO 3 . The temperature dependence of the quartic coefficient β is accordingly reconsidered and is strongly evidenced by the change of its sign above 165 o C. All these findings attest to the anomalously strong polarization anharmonicity of this material, which is unexpected for classical displacive ferroelectrics.

The paper is devoted to the calculation of renormalization-group (RG) functions in the O(n)symmetry two-dimensional model of the λϕ 4 type in the five-loop approximation and to an analysis of the critical behavior of systems described by this model. Five-loop expansions for the β function and the critical indices are determined in bulk theory. They are summed up using the Padé-Borel and Padé-Borel-Le Roy methods, making it possible to optimize the summation procedure and to estimate the accuracy of the obtained numerical values. It is shown that in the Ising (n = 1) case, as well as in other cases, the inclusion of the fiveloop contribution to the β function displaces the coordinate of the Wilson fixed point only insignificantly, leaving it outside the interval formed by the results of computations on lattices; even "spreads" of the error in the renormalization group and lattice estimates do not overlap. This discrepancy is attributed to the effect of the nonanalytic component of the β function, which cannot be determined in perturbation theory. A computation of critical indices proves that, although the inclusion of the fiveloop terms in the corresponding RG expansion slightly improves the concordance with the exact results, the nonanalytic contributions are apparently also significant in this case.