I am looking to study Berry phase-like phenomena in an un-gapped material model. 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 and seeing how bands 'invert'.
The closest I got was some effective Hamiltonians for bilayer graphene in https://arxiv.org/pdf/1205.6953.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! Thank you!