Graphene, Lattice QFT and Symmetries

TL;DR

This paper introduces a class of lattice quantum field theory models classified by ADE Lie algebras, analyzing their couplings, symmetries, and applications to materials like graphene and acetylene.

Contribution

It proposes a new framework for lattice QFT models based on ADE Lie algebras, explicitly studying su(N) cases and their physical realizations.

Findings

su(2) and su(3) models describe acetylene and graphene properties

Energy dispersion relates to ADE root systems and cosine functions

Explicit fermionic realizations for su(2), su(3), and su(4) models

Abstract

Borrowing ideas from tight binding model, we propose a board class of Lattice QFT models that are classified by the ADE Lie algebras. In the case of su(N) series, we show that the couplings between the quantum states living at the first nearest neighbor sites of the lattice $\mathcal{L}_{su(N)}$ are governed by the complex fundamental representations \underline{${{\mathbf{N}}}$} and $\bar{{\mathbf{N}}}$ of $su(N)$; and the second nearest neighbor interactions are described by its adjoint $\underline{\mathbf{N}} \otimes \bar{\mathbf{N}}$. 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 respectively the electronic properties of the acetylene chain and the graphene. It is established as well that the energy dispersion of the first nearest neighbor couplings is completely determined by the $A_{N}$ roots $ \mathbf{\alpha}$ through the typical dependence $N/2+\sum_{roots}\cos(\mathbf{k}.\alpha) $ with $\mathbf{k}$ the wave vector. Other features such as DE extension and other applications are also discussed. Keywords: Tight Binding Model, Graphene, Lattice QFT, ADE Symmetries.