Dzyaloshinskii-Moriya interaction beyond Lifshitz invariants

We consider the Dzyaloshinskii-Moriya interaction (DMI) in a generalized 2D Rashba ferromagnet. From electronic grand potential, we microscopically compute the micromagnetic energy density $w_D$ associated with DMI. We demonstrate that, even in the bulk of a system, $w_D$ cannot be expressed through Lifshitz invariants beyond the linear order in the spin-orbit coupling strength. Additional contributions to the DMI energy density $w_D$ arise due to the lack of spin-rotational invariance. These contributions are shown to affect the phase diagram of magnetic states and modify the spin wave dispersion. We also prove that the standard Bychkov-Rashba model leads to vanishing DMI at zero temperature when both spin sub-bands are partly occupied.