Topological insulators and geometry of vector bundles

TL;DR

This paper explores the geometric and topological properties of eigenspaces in band theory, highlighting their role in phenomena like electric polarization and surface states in topological insulators, through the lens of vector bundle theory.

Contribution

It provides an informal introduction to the geometry and topology of vector bundles and connects these mathematical concepts to physical models in topological insulators.

Findings

Eigenspace geometry influences electric polarization.

Global twisting of eigenspaces causes metallic surface states.

Vector bundle theory explains physical phenomena in topological insulators.

Abstract

For a long time, band theory of solids has 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.