Renormalization-group analysis of critical modes at the ferromagnetic Lifshitz point in crystalline systems

Using a renormalization-group $\epsilon$-expansion approach, the critical behavior expected at all possible Lifshitz points associated with an instability of a ferromagnetic order parameter is analyzed. It is found that Lifshitz points are not likely to occur in cubic systems, or in tetragonal systems in which the spontaneous magnetization lies in the basal plane, as the $O\left(\epsilon \right)$ recursion relations exhibit no stable fixed points. In all other cases (i.e., when the ordering is Ising-like and, for most hexagonal and rhombohedral systems, also when the ordering is in the basal plane), there exists a stable fixed point which characterizes the critical behavior at the Lifshitz point.