The Hall conductivity is given by $$\sigma_{xy}^{2D}=\frac{e^2}{\hbar} \int \frac{d\vec{k}}{(2\pi)^d} f(\epsilon(\vec{k}))\Omega_{k_xk_y} \tag{1} $$ in which $f$ is the Fermi distribution function and $\Omega$ the Berry curvature. If we write the Berry curvature in terms of the Berry vector poten...