In this write-up, I present further informal supplementary results from some preliminary investigations on my published work: Quantized interband topological index in two-dimensional systems PhysRevB.108.L081111 (Physical Review version, arXiv version).
In particular, using the interband index, I present: An algorithm for detecting band reorderings; A potential abstract visual representation of the associated geometric and Hilbert space; Several correlations in 2-band models, between the index, Berry curvature, winding number, orientation of the normal to the Bloch sphere (a.k.a. pseudospin surface), and rotation number; Phase diagrams for loops that do not enclose a single valley (i.e., no valleys and two valleys); and Further clarification on the tangent vector dependence of the interband winding element. At the end, I list some other potentially interesting directions one could pursue on this topic.
This document was compiled hastily using various outdated write-ups from several years ago, and may not contain the most accurate, rigorous or technically correct information. These results have not been peer-reviewed. So, please use them at your own discretion and remember to cite this write-up appropriately.
