Ferroically Ordered Magnetic Octupoles in d-Wave Altermagnets

We show that time -reversal symmetry -broken, centrosymmetric antiferromagnets with nonrelativistic spin splitting of d -wave symmetry-the so-called d -wave altermagnets-are conveniently described in terms of the ferroic ordering of magnetic octupoles. The magnetic octupoles are the lowest -order ferroically ordered magnetic quantity in this case and so are the natural order parameter for the transition into the magnetically ordered state. They provide a unified description of the broken time -reversal symmetry and the nonrelativistic spin splitting, as well as a platform for manipulating the latter, and account for other phenomena, such as piezomagnetism, characteristic of this class of antiferromagnets. Unusually for antiferromagnets, we show that the magnetic octupoles cause a nonzero magnetic Compton scattering, providing a route for their direct experimental detection. We illustrate these concepts using densityfunctional and model calculations for the prototypical nonrelativistic spin -split antiferromagnet, rutilestructure manganese difluoride MnF2.