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

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!

• How about Kane's four-band model? fen.bilkent.edu.tr/~bulutay/573/notes/ders_4.pdf
– Jack
Apr 10 at 6:24
• @Jack thanks for the suggestion. I checked it out. I am slightly confused here because it seems as if Kane's model is usually used in its 6-band or 8-band versions. Did you mean the 4-band model in slide 11? If so, I'm not sure how to tune the HH band with free-e mass, and make the system fully gapped. I am also not too sure about values for P. I understand slide 18 provides some equation I can try using to get approximate values, but if there is something more obvious please let me know. Apr 13 at 15:44
• I also found this thesis with some data tables, however, I think their notation is different: web.wpi.edu/Pubs/ETD/Available/etd-120905-095359/unrestricted/… Apr 13 at 15:44
• I think the top users of our condensed-matter tag and model-hamiltonians tag have taken a thorough look at this question and haven't been able to come up with anything yet. Have you tried at the larger but less-specialized Physics.SE? Apr 20 at 22:05
• @NikeDattani thank you, I will take a look. May 2 at 3:44