Signatures of Lifshitz transition in the optical conductivity of two-dimensional tilted Dirac materials

Lifshitz transition is a kind of topological phase transition in which the Fermi surface is reconstructed. It can occur in the two-dimensional (2D) tilted Dirac materials when the energy bands change between the type-I phase ($01$) through the type-III phase ($t=1$), where different tilts are parametrized by the values of $t$. In order to characterize the Lifshitz transition therein, we theoretically investigate the longitudinal optical conductivities (LOCs) in type-I, type-II, and type-III Dirac materials within linear response theory. In the undoped case, the LOCs are constants either independent of the tilt parameter in both type-I and type-III phases or determined by the tilt parameter in the type-II phase. In the doped case, the LOCs are anisotropic and possess two critical frequencies determined by $\omega=\omega_1(t)$ and $\omega=\omega_2(t)$, which are also confirmed by the joint density of state. The tilt parameter and chemical potential can be extracted from optical experiments by measuring the positions of these two critical boundaries and their separation $\Delta\omega(t)=\omega_2(t)-\omega_1(t)$. With increasing the tilting, the separation becomes larger in the type-I phase whereas smaller in the type-II phase. The LOCs in the regime of large photon energy are exactly the same as that in the undoped case. The type of 2D tilted Dirac bands can be determined by the asymptotic background values, critical boundaries and their separation in the LOCs. These can therefore be taken as signatures of Lifshitz transition therein. The results of this work are expected to be qualitatively valid for a large number of 2D tilted Dirac materials, such as 8-\emph{Pmmn} borophene monolayer, $\alpha$-SnS$_2$, TaCoTe$_2$, TaIrTe$_4$, and $1T^\prime$ transition metal dichalcogenides, due to the underlying intrinsic similarities of 2D tilted Dirac bands.