Four Dimensional Graphene

TL;DR

This paper explores a 4D lattice QCD model inspired by graphene, utilizing SU(5) symmetry and hyperdiamond lattices to refine the Dirac operator and analyze zero modes, connecting algebraic structures with lattice geometry.

Contribution

It introduces a refined 4D lattice QCD model based on SU(5) symmetry and hyperdiamond lattices, providing explicit solutions for Dirac operator zeros and linking to previous models and quaternionic approaches.

Findings

Zeros of the Dirac operator are located at H_4* lattice sites.

The continuum limit of the Dirac operator is explicitly derived.

Connections between the model, Creutz's quaternionic approach, and previous BBTW model are established.

Abstract

Mimicking pristine 2D graphene, we revisit the BBTW model for 4D lattice QCD given in ref.[5] by using the hidden SU(5) symmetry of the 4D hyperdiamond lattice H_4. We first study the link between the H_4 and SU(5); then we refine the BBTW 4D lattice action by using the weight vectors \lambda_1, \lambda_2, \lambda_3, \lambda_4, \lambda_5 of the 5-dimensional representation of SU(5) satisfying {\Sigma}_i\lambda_i=0. After that we study explicitly the solutions of the zeros of the Dirac operator D in terms of the SU(5) simple roots \alpha_1, \alpha_2, \alpha_3, \alpha_4 generating H_4; and its fundamental weights \omega_1, \omega_2, \omega_3, \omega_4 which generate the reciprocal lattice H_4^\ast. It is shown, amongst others, that these zeros live at the sites of H_4^\ast; and the continuous limit D is given by ((id\surd5)/2) \gamma^\muk_\mu with d, \gamma^\mu and k_\mu standing respectively for the lattice parameter of H_4, the usual 4 Dirac matrices and the 4D wave vector. Other features such as differences with BBTW model as well as the link between the Dirac operator following from our construction and the one suggested by Creutz using quaternions, are also given. Keywords: Graphene, Lattice QCD, 4D hyperdiamond, BBTW model, SU(5) Symmetry.