Topological insulators and semimetals in classical magnetic systems

Abstract

Pursuing topological phases in natural and artificial materials is one of the central topics in modern physical science and engineering. In classical magnetic systems, spin waves (or magnons) and magnetic solitons (such as domain wall, vortex, skyrmion, etc.) represent two important excitations. Recently, the topological insulator and semimetal states in magnon- and soliton-based crystals (or metamaterials) have attracted growing attention owing to their interesting dynamics and promising applications for designing robust spintronic devices. Here, we give an overview of current progress of topological phases in structured classical magnetism. We first provide a brief introduction to spin waves and their band structure in periodic lattices. Then, we elaborate typical topological invariants and pedagogical models that are important to understand the topological nature of magnons, such as the magnon Hall effect, topological magnon insulators, Dirac (Weyl) magnon semimetals, topological magnon polarons (magnon–phonon​ hybrid excitations), and higher-order topological magnons. Appealing proposal of topological magnonic devices is also highlighted. We then review the collective-coordinate approach for describing the dynamics of magnetic soliton lattice. We focus on the topological properties of magnetic solitons, by theoretically analyzing the first-order topological insulating phases in low dimensional systems and higher-order topological states in breathing crystals. Finally, we discuss the experimental realization and detection of the edge states in both magnonic and solitonic crystals. We remark the challenges and future prospects before concluding this article.