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.
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.
The field-theoretical renormalization group (RG) approach in three dimensions is used to estimate the universal critical values of renormalized coupling constants g6 and g8 for the O(n)-symmetric model. The RG series for g6 and g8 are calculated in the four- and three-loop approximations, respectively, and then resummed by means of the Padé-Borel-Leroy technique. Under the optimal value of the shift parameter b providing the fastest convergence of the iteration procedure, numerical estimates for g*6 are obtained with an accuracy no worse than 0.3%. The RG expansion for g8 demonstrates a stronger divergence, and results in considerably cruder numerical estimates.
We analyze the critical behavior of two-dimensional N-vector spin systems with noncollinear order within the five-loop renormalization-group (RG) approximation. The structure of the RG flow is studied for different N leading to the conclusion that the chiral fixed point governing the critical behavior of physical systems with N=2 and N=3 does not coincide with that given by the 1/N expansion. We show that the stable chiral fixed point for Nless than or equal toN(*), including N=2 and N=3, turns out to be a focus. We give a complete characterization of the critical behavior controlled by this fixed point, also evaluating the subleading crossover exponents. The spiral-like approach of the chiral fixed point is argued to give rise to unusual crossover and near-critical regimes that may imitate varying critical exponents seen in numerous physical and computer experiments
Higher-order vertices at zero external momenta for the scalar field theory, describing the critical behaviour of the Ising model, are studied within the field-theoretical renormalization group (RG) approach in three dimensions. Dimensionless six-point g 6 and eight-point g 8 effective coupling constants are calculated in the three-loop approximation. Their numerical values, universal at criticality, are estimated by means of the Pade and Pade-Borel summation of the RG expansions found and by putting the renormalized quartic coupling constant equal to its universal fixed-point value known from six-loop RG calculations. The values of g * 6 obtained are compared with their analogs resulting from the ǫ-expansion, Monte Carlo simulations, the Wegner-Houghton equations and the linked cluster expansion series. The field-theoretical estimates for g * 6 are shown to be in a good agreement with each other, differing considerably from the values given by other methods.
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