Numerical Evaluation of Topological Invariants

Band structure of a Chern PTI

Berry curvature of a valley PTI

Wilson loops of a spin PTI

The explosion of research on topological physics has brought with it a slew of interesting mathematical quantities that define the behavior of systems, often without prior knowledge beyond numerical simulation. Despite this, there has been a great lack of any helpful tools for calculating these, and most descriptions in the literature for numerically evaluating them merely provide brief accounts of the math, along with examples, but rarely give implementations.

To help remedy this, as well as to introduce many of the fascinating features of photonic topological insulators have to the broader electromagnetics community, my collaborators and I have written a tutorial paper (available in IEEE Antennas and Propagation Magazine) with a corresponding GitHub repository that provides a comprehensive toolkit for the interested researcher. The paper covers the three primary types of photonic topological insulator (which we refer to as the Chern, valley, and spin types), and the code base gives all the tools used to generate the figures and relevant topological quantities, including

  • Berry curvature

  • Chern/valley Chern/spin Chern numbers

  • Wilson loops

The tutorial makes use of the Berry curvature, but we also provide the code for generating Wilson loops, which are another powerful numerical tool for understanding topological behavior, especially for spin-based designs. All simulations were performed using Ansys HFSS, and a suite of Python scripts is also provided to facilitate easy exporting of data (no small feat in HFSS!), but any software can be used so long as the final data is shaped into the proper form. Further details on the code base can be found on the official Applied Electromagnetics GitHub page, where it is hosted and maintained.

Associated Publications

  1. D. Bisharat, R. J. Davis, Y. Zhou, P. Bandaru, and D. F. Sievenpiper, “Photonic Topological Insulators: A Beginner’s Introduction [Electromagnetic Perspectives],” IEEE Antennas and Propagation Magazine, vol. 63, no. 3, pp. 112–124, Jun. 2021, doi: 10.1109/MAP.2021.3069276. (arxiv)