Understanding Topological Insulators in Real Space.

Angel Martín Pendás,Carlos Cardenas,Francisco Muñoz,Julia Contreras-García

doi:10.3390/molecules26102965

Abstract

A real space understanding of the Su–Schrieffer–Heeger model of polyacetylene is introduced thanks to delocalization indices defined within the quantum theory of atoms in molecules. This approach enables to go beyond the analysis of electron localization usually enabled by topological insulator indices—such as IPR—enabling to differentiate between trivial and topological insulator phases. The approach is based on analyzing the electron delocalization between second neighbors, thus highlighting the relevance of the sublattices induced by chiral symmetry. Moreover, the second neighbor delocalization index, , also enables to identify the presence of chirality and when it is broken by doping or by eliminating atom pairs (as in the case of odd number of atoms chains). Hints to identify bulk behavior thanks to are also provided. Overall, we present a very simple, orbital invariant visualization tool that should help the analysis of chirality (independently of the crystallinity of the system) as well as spreading the concepts of topological behavior thanks to its relationship with well-known chemical concepts.

Highlights

Topological insulators (TIs) constitute one of the big discoveries of solid state physics in the last decades [1,2,3,4,5]
We show that the bond-alternating polyacetylene model provides a good basis for understanding the meaning of edge states, topological protection and other basic concepts in the theory on TIs—all in terms of standard chemical concepts such as Lewis resonance structures and spatiallocalization [41]
A number of the structural features of the wavefunction of a chemical system may be unveiled by using chemical bonding descriptors

Summary

Introduction

Topological insulators (TIs) constitute one of the big discoveries of solid state physics in the last decades [1,2,3,4,5]. This enables identifying the disappearance of the TI organization through DIs and to assign a simple bond order chemical meaning to their changes. The second solution of the SSH model can be understood as the simplest case of topologically protected edge states This leads to the bond-alternating polyacetylene having an extra symmetry known as “chiral symmetry”. Since (de)localization indices refer to the full multielectronic wavefunction, these indices can be used when edge states are difficult to identify or isolate

Localization and Delocalization in Real Space

Bond Alternation and Resonance

The Number of Centers

The On-Site Energy

Conclusions