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Chern character and Fermi point

Kyouhei Horie

Abstract

This paper expresses the Chern character for topological K-theory based on the formulation of the family of Fredholm operators, by using the points at which the Fredholm operator becomes singular (Fermi points). In particular, we explain that the odd Chern character can be thought of as a generalization of the spectral flow. As applications, we give elementary proofs of the evenness of the edge index and the bulk-edge correspondence for four-dimensional topological insulators with time-reversal symmetry of class AI.

Chern character and Fermi point

Abstract

This paper expresses the Chern character for topological K-theory based on the formulation of the family of Fredholm operators, by using the points at which the Fredholm operator becomes singular (Fermi points). In particular, we explain that the odd Chern character can be thought of as a generalization of the spectral flow. As applications, we give elementary proofs of the evenness of the edge index and the bulk-edge correspondence for four-dimensional topological insulators with time-reversal symmetry of class AI.
Paper Structure (24 sections, 45 theorems, 153 equations, 1 table)

This paper contains 24 sections, 45 theorems, 153 equations, 1 table.

Table of Contents

  1. Introduction
  2. Topological $K$-theory and spectral flow
  3. Main result
  4. Applications to Topological Insulators
  5. Outline
  6. Clifford algebra and its representation
  7. Formulation of topological $K$-theory via Fredholm operators
  8. Definition of the $K$-group
  9. Description of the generator of $K^n(D^n,\partial D^n)$
  10. Chern character and Push-forward
  11. Expression of Chern character by using Fermi points
  12. Approximation of family of Fredholm operators
  13. Sign coordinate
  14. Proof of the main theorem
  15. Specific calculation example
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $k$ be a non-negative integer. Let $X$ be a compact, oriented, $2k$-dimensional differentiable manifold, and $\hat{A}\colon X\to \mathcal{F}_{0}(\mathcal{\hat{H}})$ a continuous map. Then, the following holds.

Theorems & Definitions (113)

  • Theorem 1.1: The Main Theorem for Even Degrees
  • Theorem 1.2: The Main Theorem for Odd Degrees
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Proposition 2.6
  • proof
  • Definition 2.7
  • ...and 103 more