Hall conductivity of strained $Z_2$ crystals

TL;DR

This paper investigates the topological Hall conductivity in $Z_2$ crystals like graphene under inhomogeneous perturbations such as mechanical stress and magnetic fields, using lattice Weyl-Wigner formalism.

Contribution

It introduces a method to analyze the topological Hall response of $Z_2$ crystals under non-uniform conditions, linking topological invariants to level counting.

Findings

Topological Hall conductivity is established for strained $Z_2$ crystals.

The lattice Weyl-Wigner formalism effectively describes inhomogeneous perturbations.

Relation between topological invariants and energy level counting is clarified.

Abstract

We establish topological nature of Hall conductivity of graphene and other $Z_2$ crystals in 2D and 3D in the presence of inhomogeneous perturbations. To this end the lattice Weyl-Wigner formalism is employed. The non-uniform mechanical stress is considered, along with spatially varying magnetic field. The relation of the obtained topological invariant to level counting is clarified.