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.[more]
The right side shows the three terms of the extended SSH Hamiltonian model: for electron hopping between two sites inside each unit cell of the model chain, for electronic hopping between different sites in two nearest neighbor unit cells and , which is an optional additional term that describes electronic hopping between the same types of sites in nearest neighbor cells. When , the extended model reduces to the simple SSH model.
The electronic energy dispersion relation in the BZ is obtained from the diagonalization of the Hamiltonian matrix .
The winding plot of the Hamiltonian vector as the quantum number sweeps through the BZ allows discrimination between trivial and topological insulating phases; the winding number gives the actual number of times that the Hamiltonian goes around the origin of the BZ, depending on the parameters , , . When the Hamiltonian winding plot does not enclose the origin of the BZ, the winding number and the insulator is considered trivial. This occurs when intercell hopping dominates . When the Hamiltonian winding performs a single complete loop around the BZ origin, the insulating phase is considered topological and the winding number . This occurs when intracell hopping dominates . Higher winding numbers give winding plots with double loops, triple loops and so on. The higher winding plots can be obtained by switching from the simple to the extended SSH model. The highest possible winding number for the Hamiltonian defined in this Demonstration is , which is obtained by adding a nonzero term to the matrix. When the system is in a metallic phase ( and ), the winding circle crosses the BZ origin: in this case the winding number is undefined.[less]
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
 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.
 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.