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Chern and Fu–Kane–Mele Invariants as Topological Obstructions

  • Domenico MonacoEmail author
Chapter
Part of the Springer INdAM Series book series (SINDAMS, volume 18)

Abstract

The use of topological invariants to describe geometric phases of quantum matter has become an essential tool in modern solid state physics. The first instance of this paradigmatic trend can be traced to the study of the quantum Hall effect, in which the Chern number underlies the quantization of the transverse Hall conductivity. More recently, in the framework of time-reversal symmetric topological insulators and quantum spin Hall systems, a new topological classification has been proposed by Fu, Kane and Mele, where the label takes value in \(\mathcal{Z}_{2}\).

We illustrate how both the Chern number \(c \in \mathbb{Z}\) and the Fu–Kane–Mele invariant \(\delta \in \mathbb{Z}_{2}\) of 2-dimensional topological insulators can be characterized as topological obstructions. Indeed, c quantifies the obstruction to the existence of a frame of Bloch states for the crystal which is both continuous and periodic with respect to the crystal momentum. Instead, δ measures the possibility to impose a further time-reversal symmetry constraint on the Bloch frame.

Keywords

Chern numbers Fu–Kane–Mele invariants Obstruction theory Quantum hall effect Quantum spin hall effect Topological insulators 

Notes

Acknowledgements

The author is indebted with D. Fiorenza and G. Panati for valuable discussions, as well as with G. dell’​Antonio and A. Michelangeli for the invitation to the INdAM meeting “Contemporary Trends in the Mathematics of Quantum Mechanics”. Financial support from INdAM and from the German Science Foundation (DFG) within the GRK 1838 “Spectral theory and dynamics of quantum systems” is gratefully acknowledged.

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Eberhard Karls Universität TübingenTübingenGermany

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