Chiral ferromagnetism beyond Lifshitz invariants

We consider a contribution w(ch), to the micromagnetic energy density that is linear with respect to the first spatial derivatives of the local magnetization direction. For a generalized two-dimensional Rashba ferromagnet, we present a microscopic analysis of this contribution and, in particular, demonstrate that it cannot be expressed through Lifshitz invariants beyond the linear order in the spin-orbit coupling (SOC) strength. Terms in w(ch), beyond Lifshitz invariants emerge as a result of spin rotation symmetry breaking caused by SOC. The effects of these terms on the phase diagram of magnetic states and spin-wave dispersion are discussed. Finally, we present a classification of terms in w(ch), allowed by symmetry, for each crystallographic point group.