  With the use of simple models, we investigated the optical conductivity of a
nodal-line semimetal (NLSM) whose crossing of the conduction and valence bands
near the origin ($O$ point) in the ($k_x,k_y$) plane of a small cubic region
can be adjusted by a parameter $\alpha$. The Hamiltonian of the NLSM is based
on the ${\bf k}\cdot {\bf p}$ model for the low-lying energy bands. When
$\alpha=0$, these bands touch each other along a continuous closed loop but the
opening of a band gap corresponding to finite values of $\alpha$ and the
varying of the carrier concentration can be adjusted. This provides a tunable
semiconductor gap, around the $O$ point and the valence and conduction bands
can meet at a pair of points within the small cubic region in ${\bf k }$ space.
The optical conductivity of such a NLSM is calculated using the Kubo formula
with emphasis on the optical spectral weight redistribution, deduced from
appropriate Green's functions, brought about by changes in gap and chemical
potential due to modifying $\alpha$. We derived closed-form semi-analytic
expressions for the longitudinal components of the optical conductivity for
these model systems of NLSM and compare results for chosen $\alpha$ and
chemical potential. We also present results for the heat capacity when the
system is in thermal equilibrium for various chosen $\alpha$ and chemical
potential.
