Topological defects in three dimensional altermagnets

The mean-field order parameter of an altermagnet corresponds to a minimum non-linear spin-orbit coupling that is quadratic in the electron momentum and is linear in the electron spin. It preserves inversion, but spontaneously breaks time-reversal and rotation symmetries. In three dimensions, the order parameter decomposes into the SO(3) irreducible p, d, and f wave components. The local Landau theory allows slow spatial modulations of the finite order parameter in the low temperature altermagnetic phase. We provide an exhaustive topological classification and characterization of such modulations. Examples include monopoles, vortices, three dimensional skyrmions, and Hopf textures.