On the Stability of the O(N)-Invariant and the Cubic-Invariant 3-Dimensional $N$-Component Renormalization Group Fixed Points in the Hierarchical Approximation

Abstract

We compute renormalization group fixed points and their spectrum in an ultralocal approximation. We study a case of two competing non-trivial 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 ${\varphi }^4$-perturbation at the O(N)-fixed point. The O(N) fixed point is stable under a cubic ${\varphi }^4$-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.

27 pages Latex2e, 4 figures

References in corpus (2)

• Five-loop \sqrtε-expansions for random Ising model and marginal spin dimensionality for cubic systems
• The stability of the O(N) invariant fixed point in three dimensions

Cited by in corpus (1)

• Randomly dilute Ising model: A nonperturbative approach