Dirac's spectrum from Newton laws in graphene

Abstract

In the present work, we give a phenomenological theory of the monolayer graphene where two worlds quantum and classical meet together and complete each other in the most natural way. It appears that the graphene is the unique material where this complementarity could be explained in an effective way due to its exceptional band structure properties. We introduce the electron mass-vortex representation and we define surface tension excitation states in the monolayer graphene. By abstracting from the usual band energy dispersion we calculate the band mass of the electrons at the Dirac point by introducing the mathematical mass-dispersion relation. As a result, we obtain the Dirac energy dispersion in monolayer graphene from the classical Newton law. Within the semiclassical theory, we show the presence of the surface spin tension vectorial field which, possibly, closely relates the surface tension and spin tension states on the helical surface. We calculate the surface tension related with the electron band mass-vortex formation at the Dirac's point and we predict accurately the surface tension value related to the excitonic binding at the Dirac point as being formed from the electron and hole band mass-vortices. Moreover, we give the solution to a long-standing problem in the spin group theory and we construct an example which shows, phenomenologically, that the manifolds on $\rm S^{(6)}$ are not integrable. The principal reason for this is attributed to the irreducibility of the spinorial group $\rm Spin(6)^{\rm R}$ at the Dirac's point, due to the band mass formation via gravitational field.