Topology manifests itself in some of the major discoveries in condensed-matter physics of the past 50 years, including the quantum Hall effect (see the article by Joseph Avron, Daniel Osadchy, and Ruedi Seiler, Physics Today, August 2003, page 38), topological insulators (see Physics Today, April 2009, page 12), and the research honored by the 2016 Nobel Prize in Physics (see Physics Today, December 2016, page 14). In topological phases of matter, the material’s behavior derives from the connectedness of the band structure rather than the material’s symmetries, which explain most states of matter.

When a wave—for example, an electron wavefunction—travels around a topologically nontrivial path, it gains a phase after completing a closed loop rather than returning to its initial state. Although the results of the band structure’s topology are complicated to understand in detail, an essential feature is the emergence of dynamic excitations...

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