Graphene, Lattice Field Theory and Symmetries

Abstract

Borrowing ideas from tight binding model, we propose a board class of lattice field models that are classified by non simply laced Lie algebras. In the case of AN − 1 ≃ su(N) series, we show that the couplings between the quantum states living at the first nearest neighbor sites of the lattice \documentclass[12pt]{minimal}\begin{document}$\mathcal {L}_{su\left( N\right) }$\end{document}LsuN are governed by the complex fundamental representations \documentclass[12pt]{minimal}\begin{document}${{\mathbf {\underline N}}}$\end{document}N̲ and \documentclass[12pt]{minimal}\begin{document}$\overline{{\mathbf {\bm\rm N}}}$\end{document}N¯ of su(N) and the second nearest neighbor interactions are described by its adjoint \documentclass[12pt]{minimal}\begin{document}${{ \underline{\mathbf {\bm\rm N}}}}$\end{document}N̲\documentclass[12pt]{minimal}\begin{document}$\otimes \overline{{\mathbf {\bm\rm N}}}$\end{document}⊗N¯. The lattice models associated with the leading su(2), su(3), and su(4) cases are explicitly studied and their fermionic field realizations are given. It is also shown that the su(2) and su(3) models describe the electronic properties of the acetylene chain and the graphene, respectively. It is established as well that the energy dispersion of the first nearest neighbor couplings is completely determined by the AN roots \documentclass[12pt]{minimal}\begin{document}$\mathbf {\bm\rm \alpha }$\end{document}α through the typical dependence \documentclass[12pt]{minimal}\begin{document}$N/2+\sum _{{\rm roots}}\cos \left( \mathbf {\bm\rm k}.\alpha \right)$\end{document}N/2+∑ roots cosk.α with \documentclass[12pt]{minimal}\begin{document}$\mathbf {\bm\rm k}$\end{document}k the wave vector. Other features such as the SO(2N) extension and other applications are also discussed.