  We explore the ferromagnetic quantum critical point in a three-dimensional
semimetallic system with upward- and downward-dispersing bands touching at the
Fermi level. Evaluating the static spin susceptibility to leading order in the
coupling between the fermions and the fluctuating ferromagnetic order
parameter, we find that the ferromagnetic quantum critical point is masked by
an incommensurate, longitudinal spin density wave phase. We first analyze an
idealized model which, despite having strong spin-orbit coupling, still
possesses O(3) rotational symmetry generated by the total angular momentum
operator. In this case, the direction of the incommensurate spin density wave
propagation can point anywhere, while the magnetic moment is aligned along the
direction of propagation. Including symmetry-allowed anisotropies in the
fermion dispersion and the coupling to the order parameter field, however, the
ordering wavevector instead breaks a discrete symmetry and aligns along either
the [111] or [100] direction, depending on the signs and magnitudes of these
two types of anisotropy.
