Precise Wigner-Weyl calculus for the honeycomb lattice

TL;DR

This paper develops a precise Wigner-Weyl calculus tailored for honeycomb lattice models, introducing new operator symbols and deriving a topological expression for Hall conductivity applicable to quantum Hall systems with artificial honeycomb lattices.

Contribution

It introduces a new Wigner-Weyl calculus for honeycomb lattice models, including two operator symbols, and derives a topological Hall conductivity formula for systems with artificial lattices.

Findings

Defined the $ ext{B}$-symbol and Weyl symbol for honeycomb lattice operators.

Derived a topological expression for Hall conductivity using the Wigner-transformed Green function.

Applicable to quantum Hall effect in artificial honeycomb lattice systems.

Abstract

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.