# Original reference for gapped chiral fermion k·p model?

In eq. 1 of this reference, the authors present the following Hamiltonian for the 2-band gapped chiral fermion model that can describe many systems including various semiconductors, N-layer graphene, etc: $${H}({k}) = \begin{pmatrix} \Delta & \alpha(|{k}|)e^{i w \phi_k}\\ \alpha(|{k}|)e^{-i w \phi_k} & -\Delta \end{pmatrix},$$ where the energy gap is $$2\Delta$$, $$\phi_{{k}}=\tan^{-1}({k_y}/{k_x})$$ and $$\alpha(|{k}|)=\alpha|{k}|^\gamma$$.

Does anyone know where this Hamiltonian was first used in the literature (specifically in condensed matter)? All the several works I have seen use it (or some version of it) take its form for granted. For instance, I looked at citations [5-10], [12-17] of the referenced work. Additionally I looked at some Google Scholar results (including this thesis). But could not find it. Perhaps because it probably was not named 'gapped chiral fermion' in its original form?

• +1 but please note the edit I made to your title and keep it in mind for future questions. Regarding the sentence "All the several works I have seen use it (or some version of it) take its form for granted" could you share links to these works in your question? I started doing some research to try to track down the earliest references, but it would help if I could see the examples you've already considered. Dec 25, 2022 at 19:28
• @NikeDattani thanks! I just added a quick summary of most references I looked at. I can be more thorough once I am done traveling. Dec 26, 2022 at 3:52
• Perhaps it's equation E.2 in the thesis (Appendix E in kilthub.cmu.edu/articles/thesis/…), which cites "S. Dresselhaus, G. Dresselhaus, and A. Jorio, Group theory: application to the physics of condensed matter". Dec 26, 2022 at 14:07
• You mean that's possibly the earliest presentation of this model? Dec 26, 2022 at 14:09
• Good question. It's unlikely the earliest. Dec 29, 2022 at 10:05