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

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.
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.


  • [Snapshot]
  • [Snapshot]
  • [Snapshot]
  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


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
[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.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2017 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+