Clifford algebra apprach to graphene flatland

Abstract

Quantum properties of a monolayer graphene are studied in terms of the Clifford algebras. The simplest of algebras suitable for this purpose appears to be Cl3,0 algebra, the basis vectors of which represent 3D rather than 2D Euclidean space. If electron spin is included, higher dimensional algebras should be used. It is shown that Cl3,1 algebra which describes Minkowski’s spacetime is suitable for this purpose. It is shown that in both algebras, Cl3,0 and Cl3,1, the conduction and valence band spinors are rotors of 3D Euclidean space. Properties of electron spin and Berry phase when exchange and spin-orbit interactions are taken into account are illustrated within Clifford algebra formalism.