Understanding Topological Insulators in Real Space
Abstract
1. Introduction
2. Theoretical Background
2.1. The SSH Model
2.2. Localization and Delocalization in Real Space
3. Real Space Characteristics of the SSH Model
3.1. Bond Alternation and Resonance
3.2. Short vs. Long Chains: Simulating Crystalline Environments
3.3. Revealing the Breaking of Chiral Symmetry
3.3.1. The Number of Centers
3.3.2. The On-Site Energy
4. Conclusions
- The chiral symmetry is present if bond orders with second-neighbors, , are zero in the bulk (i.e., equivalent to the absence of delocalization in meta carbons in benzene).
- A topological transition in the bulk can be detected by a change in the pattern of maxima and minima bond orders with first-neighbors, , which reflects the two resonance forms.
- The topological phase and its protected edge states can be detected by a non-zero second-neighbor bond order at both edges of the molecule. These bond orders decay exponentially to zero far from the edges.
- The visualization of these edge states enables the quantification of edge decoupling, i.e., of the appearance of bulk properties in linear chains.
- In the case of impurities, due to the on-site .
- In the case of odd number of atoms, only one edge state appears upon changing from to , showing that both cases are equivalent, so that there is no phase transition.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Sample Availability
References
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Martín Pendás, A.; Muñoz, F.; Cardenas, C.; Contreras-García, J. Understanding Topological Insulators in Real Space. Molecules 2021, 26, 2965. https://doi.org/10.3390/molecules26102965
Martín Pendás A, Muñoz F, Cardenas C, Contreras-García J. Understanding Topological Insulators in Real Space. Molecules. 2021; 26(10):2965. https://doi.org/10.3390/molecules26102965
Chicago/Turabian StyleMartín Pendás, Angel, Francisco Muñoz, Carlos Cardenas, and Julia Contreras-García. 2021. "Understanding Topological Insulators in Real Space" Molecules 26, no. 10: 2965. https://doi.org/10.3390/molecules26102965
APA StyleMartín Pendás, A., Muñoz, F., Cardenas, C., & Contreras-García, J. (2021). Understanding Topological Insulators in Real Space. Molecules, 26(10), 2965. https://doi.org/10.3390/molecules26102965