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 $\phi^{ 4}$-perturbation at the O(N)-fixed point. The O(N) fixed point is stable under a cubic $\phi^{ 4}$-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.
Categories: cond-mat.stat-mech