Precise Wigner-Weyl calculus for the honeycomb lattice

In this paper we propose the precise Wigner-Weyl calculus for the lattice models defined on the honeycomb lattice. We construct two symbols of operators: the $\mathscr{B}$-symbol, which is similar to the symbol introduced by F. Buot, and the $W$ (or, Weyl) symbol. The latter possesses the set of useful properties. These identities allow us to use it in physical applications. In particular, we derive topological expression for the Hall conductivity through the Wigner transformed Green function. This expression may be used for the description of quantum Hall effect in the systems with artificial honeycomb lattice, when magnetic flux through the lattice cell is of the order of elementary quantum of magnetic flux.