Hidden magnetic octupoles: tensor decomposition and second-order magnetoelectric effect.*

We discuss how magnetic octupoles provide a basis for interpreting the second-order magnetoelectric effect, in which a bilinear electric field induces a linear magnetization, or the product of magnetic and electric fields induces a polarization. Magnetic octupoles are the next order terms, beyond the well-established magnetoelectric multipoles responsible for the linear magnetoelectric effect, in the multipole expansion of a non-uniform magnetization density, μ(r). They are represented by the rank-3 tensor M_ijk = ∫ μ_i(r) r_j r_k d³r, which we begin by decomposing into its irreducible spherical components. We then use Cr₂O₃ as a case study and show that, in addition to the linear magnetoelectric effect caused by its magnetoelectric multipoles, it has a hidden second-order local magnetic

response caused by the anti-ferroic ordering of its magnetic octupoles.