Phase transition in three-dimensional noncollinear magnetic systems with additional two-fold degeneracy

Using Monte Carlo simulations, we investigate the critical behavior in a three-dimensional frustrated helimagnet with an additional two-fold degeneracy, realized in a stacked-$J_1$-$J_2$-$J_3$ model on a cubic lattice. For the case of Heisenberg spins ($N=3$), a first-order transition is found. Using the renormalization group approach, the same result is also found for an arbitrary value $N$ of component number of the classical spin. The corresponding Ginzburg-Landau functional is obtained from the lattice model and analyzed in lower orders of the $4 - \varepsilon$ expansion. We argue that the qualitative result don't change if the higher orders of the expansion are taken into account.