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

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 chain, 1D trans-polyacetylene and 2D graphene are found to have similar conjugated π-bond systems, and the Dirac cone is correlated with 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 to be consistent with the band picture. The real-space valence-bond analysis of topological materials with a conjugated π-bond system suggests that these topological materials must be classified with concepts borrowed from group theory and topology, so that a supersymmetry may absorb the fluctuating broken symmetry. Restoration of a thermodynamic system with higher entropy (i.e., the lower Gibbs free energy) is more appropriate to describe such topological materials instead of the traditional material classification with the lowest enthalpy for the presumed rigid crystal structure.