Three-loop renormalization group analysis of a complex model with stable fixed point: Critical exponents up to $ε^3$ and $ε^4$

Abstract

The complete analysis of a model with three quartic coupling constants associated with an O(2N)--symmetric, a cubic, and a tetragonal interactions is carried out within the three-loop approximation of the renormalization-group (RG) approach in $D=4-2ε$ dimensions. Perturbation expansions for RG functions are calculated using dimensional regularization and the minimal subtraction (MS) scheme. It is shown that for $N\ge 2$ the model does possess a stable fixed point in three dimensional space of coupling constants, in accordance with predictions made earlier on the base of the lower-order approximations. Numerical estimate for critical (marginal) value of the order parameter dimensionality $N_c$ is given using Padé-Borel summation of the corresponding $ε$--expansion series obtained. It is observed that two-fold degeneracy of the eigenvalue exponents in the one-loop approximation for the unique stable fixed point leads to the substantial decrease of the accuracy expected within three loops and may cause powers of $\sqrtε$ to appear in the expansions. The critical exponents $γ$ and $η$ are calculated for all fixed points up to $ε^3$ and $ε^4$, respectively, and processed by the Borel summation method modified with a conformal mapping. For the unique stable fixed point the magnetic susceptibility exponent $γ$ for N=2 is found to differ in third order in $ε$ from that of an O(4)--symmetric point. Qualitative comparison of the results given by $ε$--expansion, three-dimensional RG analysis, non-perturbative RG arguments, and experimental data is performed.