# Topological Winding Number in 1D Su-Schrieffer-Heeger Model

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This Demonstration shows the electronic energy dispersion relation and the winding of the Hamiltonian in the Brillouin zone (BZ) of the extended one-dimensional (1D) Su–Schrieffer–Heeger (SSH) tight-binding model. The SSH model is often used as a parametric toy model for explaining the appearance of topological insulating phases in low-dimensional condensed matter systems such as polyacetylene chains. It is also often used as a pedagogical introduction to the more advanced topic of topological insulator 2D systems.

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Contributed by: Jessica Alfonsi (September 2016)

Padova, Italy

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

Snapshot 1: topological insulating phase: the circle in the winding plot encloses the origin of the BZ; winding number

Snapshot 2: trivial insulating phase: the circle in the winding plot does not include the origin of the BZ; winding number

Snapshot 3: metallic phase: the circle in the winding plot crosses the origin of the BZ; winding number is undefined

Snapshot 4: trivial insulating phase and full dimerization limit due to dominating intracell hopping amplitude (winding plot reduces to a point off the origin)

Snapshot 5: topological insulating phase and full dimerization limit due to dominating intercell hopping amplitude

Snapshot 6: topological insulating phase with next-nearest neighbor hopping added to the SSH Hamiltonian, winding number

References

[1] J. K. Asbóth, L. Oroszlány and A. Pályi, *A Short Course on Topological Insulators*, Cham, Switzerland: Springer International Publishing, 2016. doi:10.1007/978-3-319-25607-8. Pre-print available at arxiv.org/abs/1509.02295.

[2] L. Li, C. Yang and S. Chen, "Winding Numbers of Phase Transition Points for One-Dimensional Topological Systems," *Europhysics Letters, *112(1), 2015 10004. iopscience.iop.org/0295-5075/112/1/10004.

## Permanent Citation