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?

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    $\begingroup$ +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. $\endgroup$ Commented Dec 25, 2022 at 19:28
  • $\begingroup$ @NikeDattani thanks! I just added a quick summary of most references I looked at. I can be more thorough once I am done traveling. $\endgroup$ Commented Dec 26, 2022 at 3:52
  • $\begingroup$ 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". $\endgroup$ Commented Dec 26, 2022 at 14:07
  • $\begingroup$ You mean that's possibly the earliest presentation of this model? $\endgroup$ Commented Dec 26, 2022 at 14:09
  • $\begingroup$ Good question. It's unlikely the earliest. $\endgroup$ Commented Dec 29, 2022 at 10:05

1 Answer 1


I am not sure whether the following are right, but these are what I ended up using:

  • F. D. M. Haldane, Model for a Quantum Hall Effect without Landau Levels: Condensed-Matter Realization of the “Parity Anomaly”, Phys. Rev. Lett. 61, 2015 (1988).
  • X. Zhang, W.-Y. Shan, and D. Xiao, Optical Selection Rule of Excitons in Gapped Chiral Fermion Systems, Phys. Rev. Lett. 120, 077401 (2018).
  • X. Zhang, Topological effects in two-dimensional systems, Ph.D. thesis, Carnegie Mellon University, 2019.
  • M. S. Dresselhaus, G. Dresselhaus, and A. Jorio, Group Theory: Application to the Physics of Condensed Matter (Springer Science & Business Media, 2007).
  • $\begingroup$ +1 but the last one is the book by Dresselhaus which was basically used as the course notes for her course at MIT right? Where in her course notes (I have the PDF) is this Hamiltonian discussed? $\endgroup$ Commented Dec 11, 2023 at 20:22

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