# An Introduction to Topological Insulators

**Authors:** Michel Fruchart, David Carpentier  
**arXiv:** [1310.0255](https://arxiv.org/abs/1310.0255)  
**Date:** 2013-10-01  
**Summary page:** https://www.summarizepaper.com/en/paper/1310.0255v2/

## Key points

- The paper explores the concept of topological insulators in electronic bands in crystals
- Insulators have a distinct energy gap separating valence and conduction bands
- Valence bands can exhibit non-trivial or twisted topological properties, leading to classification as a topological insulator
- Topological order serves as an obstruction to defining Bloch wave functions across the entire Brillouin Zone using a single phase convention
- Various historical models showcasing topological order in two dimensions are examined, highlighting different expressions of the corresponding topological index
- The paper provides a comprehensive overview of topological insulators and their fundamental properties, offering insights into their implications for condensed matter physics

## Summary

The paper "An Introduction to Topological Insulators" by Michel Fruchart and David Carpentier explores the concept of topological insulators. The authors delve into the electronic bands in crystals and their associated energies, which are characterized by Bloch wave functions indexed by momenta in the first Brillouin Zone. In insulators, a distinct energy gap separates valence bands from conduction bands around the chemical potential. This ensemble of valence bands can exhibit non-trivial or twisted topological properties, leading to the classification of the insulator as a topological insulator. The paper introduces the notion of topological order in insulators and highlights how it serves as an obstruction to defining Bloch wave functions across the entire Brillouin Zone using a single phase convention. Various historical models showcasing topological order in two dimensions are examined, shedding light on different expressions of the corresponding topological index. Overall, this pedagogical review provides a comprehensive overview of topological insulators and offers insights into their fundamental properties and implications for condensed matter physics. The authors' analysis contributes to our understanding of these intriguing materials and their potential applications in future research and technology development.