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Topological insulators and geometry of vector bundles

by A. S. Sergeev

Submission summary

As Contributors: Alexander Sergeev
Arxiv Link: (pdf)
Date submitted: 2020-11-11 11:24
Submitted by: Sergeev, Alexander
Submitted to: SciPost Physics Lecture Notes
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Mathematical Physics
Approach: Theoretical


For a long time, band theory of solids has been focused on the energy spectrum, or Hamiltonian eigenvalues. Recently, it was realized that the collection of eigenvectors also contains important physical information. The local geometry of eigenspaces determines the electric polarization, while their global twisting gives rise to the metallic surface states in topological insulators. These phenomena are central topics of the present notes. The shape of eigenspaces is also responsible for many intriguing physical analogies, which have their roots in the theory of vector bundles. We give an informal introduction to the geometry and topology of vector bundles and describe various physical models from this mathematical perspective.

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Submission 2011.05004v1 on 11 November 2020

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