# Fundamental insights to topological quantum materials: A real-space view of 13 cases by supersymmetry of valence bonds approach

## TL;DR

This paper offers a real-space valence-bond perspective on 13 topological materials across 1D to 3D, linking bonding electron characteristics with their topological electronic properties.

## Contribution

It introduces a valence-bond model analysis for various topological materials, connecting real-space bonding with reciprocal-space band features.

## Key findings

- Graphene and trans-polyacetylene share similar conjugated bond systems.
- Strain and symmetry influence topological insulator properties.
- Valence-bond models align with band theory for Dirac and Weyl semimetals.

## Abstract

We present a real-space view of one-dimensional (1D) to three-dimensional (3D) topological materials with 13 representative samples selected from each class, including 1D trans-polyacetylene, two-dimensional (2D) graphene, and 3D topological insulators, Dirac semimetals, Weyl semimetals, and nodal-line semimetals. This review is not intended to present a complete up-to-date list of publications on topological materials, nor to provide a progress report on the theoretical concepts and experimental advances, but rather to focus on an analysis based on the valence-bond model to help the readers gain a more balanced view of the real-space bonding electron characteristics at the molecular level versus the reciprocal-space band picture of topological materials. Starting from a brief review of low-dimensional magnetism with `toy models' for a 1D Heisenberg antiferromagnetic (HAF) chain, 1D trans-polyacetylene and 2D graphene are found to have similar conjugated (\pi)-bond systems, and the Dirac cone is correlated to their unconventional 1D and 2D conduction mechanisms. Strain-driven and symmetry-protected topological insulators are introduced from the perspective of material preparation and valence-electron sharing in the valence-bond model analysis. The valence-bond models for the newly developed Dirac semimetals, Weyl semimetals, and nodal line semimetals are examined with more emphasis on the bond length and electron sharing, which is found consistent with the band picture.