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Bulk-Edge Correspondence for Two-Dimensional Topological Insulators

Abstract

Topological insulators can be characterized alternatively in terms of bulk or edge properties. We prove the equivalence between the two descriptions for two-dimensional solids in the single-particle picture. We give a new formulation of the \({\mathbb{Z}_{2}}\)-invariant, which allows for a bulk index not relying on a (two-dimensional) Brillouin zone. When available though, that index is shown to agree with known formulations. The method also applies to integer quantum Hall systems. We discuss a further variant of the correspondence, based on scattering theory.

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Correspondence to Marcello Porta.

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Communicated by M. Aizenman

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Graf, G.M., Porta, M. Bulk-Edge Correspondence for Two-Dimensional Topological Insulators. Commun. Math. Phys. 324, 851–895 (2013). https://doi.org/10.1007/s00220-013-1819-6

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  • DOI: https://doi.org/10.1007/s00220-013-1819-6

Keywords

  • Vector Bundle
  • Edge State
  • Topological Insulator
  • Quantum Hall Effect
  • Fermi Line