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Skyrmions are topological solitons that emerge in many physical contexts. In magnetism, they appear as textures of the spin-density field stabilized by different competing interactions and characterized by a topological charge that counts the number of times the order parameter wraps the sphere. They behave as classical objects when the spin texture varies slowly on the scale of the microscopic lattice of the magnet. However, the fast development of experimental tools to create and stabilize skyrmions in thin magnetic films has led to a rich variety of textures, sometimes of atomistic sizes. In this paper, we discuss, in a pedagogical manner, how to introduce quantum interference in the translational dynamics of skyrmion textures, starting from the micromagnetic equations of motion for a classical soliton. We study how the nontrivial topology of the spin texture manifests in the semiclassical regime, when the microscopic lattice potential is treated quantum-mechanically, but the external driving forces are taken as smooth classical perturbations. We highlight close relations to the fields of noncommutative quantum mechanics, Chern–Simons theories, and the quantum Hall effect.

Electronic and dielectric properties are essential for understanding many functional materials, predicting their behavior and optimizing their performance across different shapes, geometries and scales. Several approaches were developed and explored to investigate more or less deeply the appropriate properties. One of the most appealing, accurate and efficient approach is first principle simulations based on modern theory of polarization. Especially with the increased availability of powerful computational resources and techniques. Building upon these advancements, our contribution aims to elucidate an efficient methodology for studying electronic and dielectric properties by applying the Berry phase and Maximally Localized Wannier functions methods. Our exploration will initially focus on a systematic study of the electronic, chemical bonding, ferroelectric and piezoelectric properties of the well-known prototypical bulk system PbTiO₃. Subsequently, we will extend our study to examine slab properties as surface termination and slab thickness effect on electronic properties, utilizing the robust Wannier-justified Tight Binding model.