Magnetoelectric multipoles and spin textures in real-space and reciprocal space*

In condensed matter systems, the multipoles provide a quantitative measure of the complex angular distribution of the charge and magnetization densities in a broken-symmetry phase of a solid. Magnetoelectric multipoles, that break both space-inversion I and time-reversal T symmetries, quantify the antisymmetric magnetization density of a solid. In my talk, I will discuss a general theory to classify magnetic skyrmions and related spin textures in terms of their magnetoelectric multipoles [1]. Since magnetic skyrmions are now established in insulating materials, where the magnetoelectric multipoles govern the linear magnetoelectric response, our classification provides a recipe for manipulating the magnetic properties of skyrmions using applied electric fields. Taking the examples of skyrmions, antiskyrmions, and bimerons of different helicities, we show that the nonzero components of the magnetoelectric multipole and magnetoelectric response tensors are uniquely determined by the topology, helicity, and geometry of the spin texture. We have proposed straightforward linear magnetoelectric response measurements as an alternative to Lorentz microscopy for characterizing insulating skyrmionic textures. Similarly, to the description of real-space spin textures by real-space magnetoelectric multipoles, momentum-space spin textures can also be described by k-space magneto-electric multipoles [2]. These k-space magnetoelectric multipoles are reciprocal to the real-space charge dipoles associated with the broken inversion symmetry. Using the prototypical ferroelectric PbTiO₃ as an example material, I will discuss the tuning of the k-space magnetoelectric multipoles and their possible detection using designed magnetic Compton scattering techniques.