Abstract
We discuss the tight-binding models of solid state physics with the Z 2 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.
Highlights
There has been a revival of interest in Wigner–Weyl formalism in both condensed matter and high energy physics
In the present paper we proceeded with the development of Wigner–Weyl formalism for tight-binding models of solid state physics
The developed technique was applied to the class of inhomogeneous models that include, in particular, the tight-binding model of graphene in the presence of both inhomogeneous magnetic field and nontrivial elastic deformations
Summary
There has been a revival of interest in Wigner–Weyl formalism in both condensed matter and high energy physics. In [57] it was shown that in the lattice models (i.e., in the tight-binding models) of solid state physics with essential inhomogeneity (caused by the varying external magnetic field), the Hall conductance integrated over the whole space is given by a topological invariant in phase space This quantity is expressed through the Wigner transformation of the one-fermion Green’s functions. We prove that for the noninteracting 2D condensed matter model with slowly varying electromagnetic fields and elastic deformations, the Hall conductivity integrated over the whole area of the sample is given by the topological invariant in phase space composed of the Wigner transformed one-particle Green’s function. The Weyl symbols of operators are denoted by the sub-index W: ( Q)W ≡ QW
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