  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, $m(k)=$ $m+uk^{2}$. However, in the Chern insulator based on Dirac
fermions, only the sign of the mass $m$ is relevant. Recently, it is reported
that the quantum metric is observable by means of the optical conductivity,
which is significantly affected by the parabolic coefficient. We analytically
obtain the quantum metric and the optical conductivity in the Dirac Hamiltonian
in arbitrary dimensions, where the Dirac mass has parabolic dispersion. We also
study the quantum metric and the optical conductivity in the Qi-Wu-Zhang model
and the Haldane model. We find that the optical conductivity has a significant
dependence on $u$ in the Qi-Wu-Zhang model, while it is not the case in the
Haldane model. We also analytically study the quantum metric and the optical
conductivity in the Su-Schrieffer-Heeger model.
