DC conductivity of tilted Dirac Fermions across the Lifshitz Transition: short- versus long-range impurities

We theoretically investigate the DC conductivity of two-dimensional tilted Dirac systems subject to short- and long-range impurity scattering. Using the Kubo formalism, we systematically study transport across the subcritical (Type I), critical, and overcritical (Type II) tilt regimes. In the subcritical phase, short-range impurities yield a frequency-independent conductivity that decreases monotonically with tilt. Conversely, long-range Coulomb scattering results in a strongly energy-dependent conductivity governed by a tilt-independent scattering rate. At the Lifshitz transition ($t = 1$), the transport signatures of these impurities diverge fundamentally: the van Hove singularity in the density of states induces a localized conductivity dip for short-range disorder, but a pronounced macroscopic peak for Coulomb impurities. In the overcritical regime, an ultraviolet momentum cutoff is required to regularize the open Fermi surface, leading to distinct behaviors for each impurity type. Notably, the conductivity perpendicular to the tilt direction ($\sigma_{xx}$) exhibits a cutoff-dependent, non-monotonic peak near $t = \sqrt{2}$ for short-range defects, while it decays monotonically with increasing tilt for long-range scattering. For both potentials, the conductivity along the tilt axis ($\sigma_{yy}$) increases without bound, revealing extreme transport anisotropy. For long-range impurities, the energy dependence of the conductivity becomes nearly quadratic and linear for Type I and II, respectively. Furthermore, vertex corrections vanish identically at the Lifshitz transition for both impurity types. Finally, we provide a unified geometric framework for these phenomena, establishing the tilt parameter as a powerful knob for engineering macroscopic transport in Dirac materials.