Topological insulators and semimetals in classical magnetic systems

Pursuing topological phases in natural and artificial materials is one of the\ncentral topics in modern physical science and engineering. In classical\nmagnetic systems, spin waves (or magnons) and magnetic solitons (such as domain\nwall, vortex, skyrmion, etc) represent two important excitations. Recently, the\ntopological insulator and semimetal states in magnon- and soliton-based\ncrystals (or metamaterials) have attracted growing attention owing to their\ninteresting dynamics and promising applications for designing robust spintronic\ndevices. Here, we give an overview of current progress of topological phases in\nstructured classical magnetism. We first provide a brief introduction to spin\nwave, and discuss its topological properties including magnon Hall effects,\ntopological magnon insulators, and Dirac (Weyl) magnon semimetals. Appealing\nproposal of topological magnonic devices is also highlighted. We then review\nthe collective-coordinate approach for describing the dynamics of magnetic\nsoliton lattice. Pedagogical topological models such as the\nSu-Schrieffer-Heeger model and the Haldane model and their manifestation in\nmagnetic soliton crystals are elaborated. Then we focus on the topological\nproperties of magnetic solitons, by theoretically analyzing the first-order\ntopological insulating phases in low dimensional systems and higher-order\ntopological states in breathing crystals. Finally, we discuss the experimental\nrealization and detection of the edge states in both the magnonic and solitonic\ncrystals. We remark the challenges and future prospects before concluding this\narticle.\n