  Correlated Dirac semimetal phases emerge in lightly doped (Tb- or La-doped)
Mott insulator Sr$_2$IrO$_4$, where a d-wave symmetry-breaking order underlying
a pseudogap plays a crucial role in determining the nature of Dirac degeneracy,
i.e., whether it is a Dirac line node or Dirac point node. Here, using a
realistic five-orbital tight-binding model with a Hubbard U and a semiclassical
Boltzmann transport theory, we systematically study the low-energy
electrodynamic properties of the Dirac semimetal phases in the paramagnetic
lightly doped Sr$_2$IrO$_4$. We investigate the effects of the d-wave
electronic order and electron doping concentration on the electronic band
structures and optical properties of various Dirac semimetal phases. We
calculate the intraband optical conductivity and obtain electrodynamic
parameters of dc conductivity, scattering rate, and Drude weight for three
Dirac semimetal phases: two are Dirac point-node states observed in the 3%
Tb-doped and 5% La-doped Sr$_2$IrO$_4$, and the other is a Dirac line-node
state. Our results show that the temperature dependence of the electrodynamic
parameters is strong in the Tb-doped system while weak in the La-doped and
Dirac line-node systems, which are consistent with available experimental data.
Moreover, using the low-energy effective theory, we also compare the
temperature-dependent screening effect in the Tb- and La-doped systems using
graphene as a reference. Our paper provides valuable insight for understanding
the transport and optical properties of correlated Dirac semimetal phases in
the doped Sr$_2$IrO$_4$.
