Band Representations and Topological Quantum Chemistry

Jennifer Cano, Barry Bradlyn

Research output: Contribution to journalReview articlepeer-review


In this article, we provide a pedagogical review of the theory of topological quantum chemistry and topological crystalline insulators. We begin with an overview of the properties of crystal symmetry groups in position and momentum space. Next, we introduce the concept of a band representation, which quantifies the symmetry of topologically trivial band structures. By combining band representations with symmetry constraints on the connectivity of bands in momentum space, we show how topologically nontrivial bands can be cataloged and classified. We present several examples of new topological phases discovered using this paradigm and conclude with an outlook toward future developments.

Original languageEnglish (US)
Pages (from-to)225-246
Number of pages22
JournalAnnual Review of Condensed Matter Physics
StatePublished - Mar 10 2021


  • Topological insulators
  • band theory of solids
  • crystal symmetry

ASJC Scopus subject areas

  • Materials Science(all)
  • Condensed Matter Physics


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