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Topological phases of amorphous matter

Topological phases of matter are often understood and predicted with the help of crystal symmetries, although they don't rely on them to exist. In this chapter we review how topological phases have been recently shown to emerge in amorphous systems. We summarize the properties of topological states and discuss how disposing of translational invariance has motivated the surge of new tools to characterize topological states in amorphous systems, both theoretically and experimentally. The ubiquity of amorphous systems combined with the robustness of topology has the potential to bring new fundamental understanding in our classification of phases of matter, and inspire new technological developments.