  Within linear response theory, the absorptive part of highly anisotropic
optical conductivities are analytically calculated for distinct tilts in
two-dimensional (2D) tilted semi-Dirac bands (SDBs). The transverse optical
conductivities always vanish. The interband longitudinal optical conductivities
(LOCs) in 2D tilted SDBs differ qualitatively in the power-law scaling of
$\omega$ as
$\mathrm{Re}\sigma_{\perp}^{\mathrm{IB}}(\omega)\propto\sigma_0\sqrt{\omega}$
and
$\mathrm{Re}\sigma_{\parallel}^{\mathrm{IB}}(\omega)\propto\sigma_0/\sqrt{\omega}$.
By contrast, the intraband LOCs in 2D tilted SDBs depend on $\mu$ in the
power-law scaling as
$\mathrm{Re}\sigma_{\perp}^{\mathrm{D}}(\omega)\propto\sigma_0\mu \sqrt{\mu}$
and
$\mathrm{Re}\sigma_{\parallel}^{\mathrm{D}}(\omega)\propto\sigma_0\mu/\sqrt{\mu}$.
The tilt-dependent behaviors of LOCs could qualitatively characterize distinct
impact of band tilting in 2D tilted SDBs. In particular, for arbitrary tilt $t$
satisfying $0<t\le 2$, the interband LOCs always possess a robust fixed point
at $\omega=2\mu$. The power-law scalings and tilt-dependent behaviors further
dictate significant differences in the asymptotic background values and angular
dependence of LOCs. Our theoretical predictions should be valid for a broad
class of 2D tilted SDB materials, and can also be used to fingerprint 2D tilted
SDB from 2D untilted SDB as well as tilted Dirac bands.
