# Example of a standard/archetypal/simple 4-band un-gapped condensed matter model with analytic results?

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!

• I gave you +1 a while ago, but what's wrong with the answer you got from Anyon (I noticed you changed the question, and at that time had not upvoted the answer). Is the BHZ model not gapped? If it's not gapped, you can't blame the answerer because the question did not ask for a gapped model. If the BHZ model is not gapped, then changing the question after the answer was written, makes the answer invalid, which is unfair. After 48 hours you can put a bounty up asking for new answers that give gapped models, or perhaps ask a new question about gapped models and leave this one for ungapped. – Nike Dattani Apr 8 at 2:02
• @NikeDattani thank you. I am not blaming the answer (I upvoted it and appreciate the answerer). But to my understanding, the degeneracy makes study of Berry phase phenomena in that model unclear numerically (I stated these in the original question). This is because the Abelian Berry curvature is not well-defined. However, I realized that this should have been clearer, which is why I edited the question to specify "gapped". You make a fair point. So I will change this question and ask again. – TribalChief Apr 8 at 2:23
• @TribalChief Since you mentioned band inversion I figured you'd be interested in TIs where degeneracies are of interest (at least at some points). These can be tricky numerically, for sure. Are you more interested in a fully gapped model with e.g. Chern-ful bands? Maybe one can construct a nicely behaved model with the number of bands you're after. – Anyon Apr 8 at 2:29
• @Anyon, thanks for the insight. But yes, I am looking for a fully gapped model that whose Chern numbers are usually calculated. Is the three-band model you had in mind by chance for a 3-band optical Haldane model; or Di Xiao's 3-band effective Hamiltonian for monolayer TMDs? If it's not either, I would appreciate it. However, a 4-band model would be most ideal. That said, I just posted the question again here: mattermodeling.stackexchange.com/questions/4678/… – TribalChief Apr 8 at 2:39
• @TribalChief I was thinking about the 3-band kagome model discussed in this review, but I realized it included less analytical insights than I remembered. – Anyon Apr 8 at 2:45