Superconducting diode effect due to higher order magnetic toroidal moments

The superconducting diode effect (SDE), by which the critical current along a certain direction is different from the critical current along the reciprocal direction, has been widely discussed in systems that experience an effective electric field and an external magnetic field. This combination of fields can give rise to a magnetic toroidal dipole moment, which in turn ensures the existence of a Lifshitz invariant in the superconducting Ginzburg-Landau free energy. Here, we consider instead the case of magnetically ordered systems that host magnetic toroidal octupole moments. By analyzing the Ginzburg-Landau free energies for single-component as well as multicomponent superconducting order parameters, we find a cubic-in-momentum term that can stabilize both the SDE and a pair density wave. We also discuss the magnetic symmetries responsible for this behavior.