Elastic deformations and wigner-weyl formalism in graphene

We discuss the tight-binding models of solid state physics with the Z₂ sublattice symmetry in the presence of elastic deformations in an important particular case-the tight binding model of graphene. In order to describe the dynamics of electronic quasiparticles, the Wigner-Weyl formalism is explored. It allows the calculation of the two-point Green's function in the presence of two slowly varying external electromagnetic fields and the inhomogeneous modification of the hopping parameters that result from elastic deformations. The developed formalism allows us to consider the influence of elastic deformations and the variations of magnetic field on the quantum Hall effect.