On the Stability of the O(N)-Invariant and the Cubic-Invariant Three-Dimensional N-Component Renormalization-Group Fixed Points in the Hierarchical Approximation

We compute renormalization-group fixed points and their spectrum in an ultralocal approximation. We study a case of two competing nontrivial fixed points for a three-dimensional real N-component field: the O(N)-invariant fixed point vs. the cubic-invariant fixed point. We compute the critical value N_c of the cubic φ⁴-perturbation at the O(N)-fixed point. The O(N)-fixed point is stable under a cubic φ⁴-perturbation below N_c; above N_c it is unstable. The Critical value comes out as 2.219435<N_c<2.219436 in the ultralocal approximation. We also compute the critical value of N at the cubic invariant fixed point. Within the accuracy of our computations, the two values coincide.