Faster method to calculate total Chern number

[sp94]

Currently we have Wilson loop methods which are very general and can be used to calculate

• Total Chern number of a set of bands
• Some symmetry-protected Chern numbers
• Multiband/non-Abelian Berry curvature

However, when calculating the total Chern number

1. you must manually count the winding of the Wilson loop spectrum to find the Chern number
2. using the Wilson loop winding method is slower than other 'plaquette' based methods

We should add a direct Chern number calculation based on plaquettes, as in eg
a. Fukui et al https://doi.org/10.1143/JPSJ.74.1674
b. Zhao et al https://doi.org/10.1364/OE.380077

[kabume]

Some topological invariant of high order topological insulator, e.g. quadrupole moment $Q_{xy}$, may also be considered, which can be computed by Wilson loop.

Some paper about high order TI:

1. He et al https://doi.org/10.1038/s41467-020-16916-z
2. Xie et al https://doi.org/10.1103/PhysRevB.98.205147

[sp94]

Good point - we should add better support for nested Wilson loops!

When I have time I will add an example of calculating nested Wilson loops for the Xie et al paper you linked