Analytic approach to quantum metric and optical conductivity in Dirac models with parabolic mass in arbitrary dimensions

The imaginary part of the quantum geometric tensor is the Berry curvature, while the real part is the quantum metric. Dirac fermions derived from a tight-binding model naturally contains a mass term $m(k)$ with parabolic dispersion, $m(k)=\phantom{\rule{4pt}{0ex}}m+u{k}^{2}$. However, in the Chern insulator based on Dirac fermions, only the sign of the mass $m$ is relevant. Recently, it was reported that the quantum metric is observable by means of the optical conductivity, which is significantly affected by the parabolic coefficient $u$. We analytically obtain the quantum metric and the optical conductivity in the Dirac Hamiltonian in arbitrary dimensions, where the Dirac mass has parabolic dispersion. The optical conductivity at the band-edge frequency significantly depends on the dimensions. We also undertake an analytical study of the quantum metric and the optical conductivity in the Su-Schrieffer-Heeger model, the Qi-Wu-Zhang model, and the Haldane model.