I am looking to study Berry phase-like phenomena in a gapped 4-band material model. In particular, I want to numerically and analytically calculate the Abelian Berry curvature integral of each band over some region of k-space. It should be easy to calculate the Chern number of this system.
However, I am having trouble finding a widely-used 4-band model with analytic expressions for wavefunctions and energies. I am looking to compare results from numerical methods vs analytical ones; and would ideally already have these models in terms of k-space variables in some reasonable basis. Ultimately, I want to toy with this model by changing its diagonal values to see how bands 'invert' after a gap-closing, and how their Berry curvatures change accordingly. Ideally, it would be easy to see edge states in a domain-wall-like setup, due to varying of an x-dependent diagonal/potential factor.
I first tried the BHZ model for the Z2 invariant (as recommended by the answer to this poorly-asked previous attempt of this question here). However, the band degeneracies make it hard to study what I want to study: the Abelian Berry curvature is not well-defined; and when I tune the diagonal with some constant to force it to "gap", the band crossings are too complicated. For example, the gap can close at several points besides the Gamma point.
The closest I got was some effective Hamiltonians for bilayer graphene in this PDF (such as equation 76). However, solving for wavefunctions took a while using Mathematica, and led to messy-looking algebraic expressions. Isn't there a simpler model out there, of theoretical value? Or just a model that justifies using not-so-simple expressions? I would find any appropriate 4-band condensed matter models very useful! Even suggestions for a simple/absurd toy model with potential nontrivial topology are welcome. Thank you!
