Alternatives for calculating topological invariants in topological materials My questing is regarding the different alternatives for calculating topological invariants in topological materials protected by symmetry, specially time-reversal invariant topological insulators, topological crystalline insulators and higher-order topological insulators.
I have seen a few different methods but I am a little bit confused about how many of them are there and which ones of them are actually equivalent. The methods that come to my mind are calculation of the Berry phase, Wilson loop, elementary band representations and symmetry indicators.
Could someone help me to better understand the difference between each of them and let me know if there are other ways of calculating/defining the topological invariants in these materials?
Thanks a lot,
Warlley
 A: This is too big a question to answer fully.  Let me start by explaining why in a  single symmetry class we need so many invariants. 
Consider just Chern insulators, and ignore interactions.  We want to work with crystals and quasicrystals and amorphous systems, for both infinite and finite area systems.  We want to work with various boundaries and boundary conditions.  We want to be able to understand disordered systems.  Analytic (hand) calculations are important, but we want to be able to use computers too, and need invariants that are conducive to fast numerical algorithms.  Defects and adjacent areas of different Chern numbers cannot be ignored, so we need local invariants.
While no one formula will do all of the above, at least we have the fact that K-theory is a very flexible subject.  It is not just about vector bundles, as there is also the K-theory of C*-algebras.  We can create many, many index formulas that do parts of what we want.  Explaining the relations between these can be tricky, as some of the arguments are dozens of pages of C*-algebra homotopy calculations that have no intuitive physical explanation, at present.
Here are three invariants for Chern insulators.  One is the standard one involving Berry curvature, that can be computed by hand but is not even defined except fot a disorder-free perfect cyrstalline system of infinite area.   The second is the Bott index, that only is defined for finite systems with periodic boundary conditions and generally requires a computer program to evaluate.  The third is the local index formula of Kitaev.  This was defined in infinite area, but can be made to work in finite systems.
It has been stated that the Bott index is the Chern number (topocondmat.org/w8_general/invariants.html) but this is not a fully accurate statement.  What is true is that we can prove that if we take large enough finite patches of a clean infinite system, then the Bott index of the finite system with periodic boundary equals the Chern number of the infinite system.  It is easy to find small systems where the Bott index comes out wrong.  I think that at present we don’t know how big a system is needed so that we can guarantee the numbers are equal.  In a physics paper, one tends to just check the 10-by-10, 20-by-20 and 30-by-30 cases and back up ones claims with something like modeling a two-terminal conductance test.
A local index, like the one by Kitaev, allows us to examine finite systems that are a mashup of Hamiltoninans of different Chern numbers.  Neither the Bott index or the Berry curvature can even be defined here.  The sort of equivalence one wants to prove (I don’t think this has been done) is that if one takes a patch of an infinite area crystalline system, imposes open boundary conditions on it, then as long as one computes the local index far from the edge one gets a real number that is very close to the Chern number.
