Effects of spatial dimensionality and band tilting on the longitudinal optical conductivities in Dirac bands

We report a unified theory based on linear response, for analyzing the longitudinal optical conductivity (LOC) of materials with tilted Dirac cones. Depending on the tilt parameter $t$, the Dirac electrons have four phases: untilted, type-I, type-II, and type-III; the Dirac dispersion can be isotropic or anisotropic; the spatial dimension of the material can be one-, two-, or three-dimensions (1D, 2D and 3D). The interband LOCs and intraband LOCs in $d$ dimension (with $d\ge2$) are found to scale as $\sigma_{0}\omega^{d-2}$ and $\sigma_{0}\mu^{d-1}\delta(\omega)$, respectively, where $\omega$ is the frequency and $\mu$ the chemical potential. The interband LOC vanishes in 1D due to lack of extra spatial dimension. In contrast, the interband LOCs in 2D and 3D are nonvanishing and share many similar properties. A universal and robust fixed point of interband LOCs appears at $\omega=2\mu$ no matter $d=2$ or $d=3$, which can be intuitively understood by the geometric structures of Fermi surface and energy resonance contour. The intraband LOCs and the carrier density for 2D and 3D tilted Dirac bands are both closely related to the geometric structure of Fermi surface and the cutoff of integration. The angular dependence of LOCs is found to characterize both spatial dimensionality and band tilting and the constant asymptotic background values of LOC reflect features of Dirac bands. The LOCs in the anisotropic tilted Dirac cone can be connected to its isotropic counterpart by a ratio that consists of Fermi velocities for both 2D and 3D. Most of the findings are universal for tilted Dirac materials and hence valid for a great many Dirac materials in the spatial dimensions of physical interest.