Diffusive density response of electrons in anisotropic multiband systems

We explicitly calculate the density-density response function with conserving vertex corrections for anisotropic multiband systems in the presence of impurities including long-range disorder. The direction dependence of the vertex corrections is correctly considered to obtain the diffusion constant which is given by a combination of componentwise transport relaxation times and velocities on the Fermi surface. We also investigate the diffusive density response of various anisotropic systems, propose some empirical rules for the corresponding diffusion constant, and demonstrate that it is crucial to consider the component dependence of the transport relaxation times to correctly interpret the transport properties of anisotropic systems, especially various topological materials with a different power-law dispersion in each direction.

Figure 1

Feynman diagrams for (a) the disorder-averaged Green's function within the Born approximation, (b) the density-density response function without vertex corrections, (c) the density-density response function with vertex corrections, and (d) the ladder approximation for the charge vertex.

Figure 2

Anisotropy ${\mathcal{D}}_{yy}/{\mathcal{D}}_{xx}$ of the diffusion constant normalized by ${(v_{\text{F}}^{\left(y\right)}/v_{\text{F}}^{\left(x\right)})}^2$ as a function of the screening factor $Q$ for (a), (d) anisotropic 2DEG, (b), (e) anisotropic graphene, and (c), (f) fBP at the semi-Dirac transition point, obtained from (a)–(c) Eq. (13) considering the component dependence of the transport relaxation time and from (d)–(f) Eq. (19) neglecting the component dependence of the transport relaxation time as in isotropic systems. Here, $Q\equiv q_{\text{TF}}/k_{\text{F}}$ and $A\equiv k_{\text{F}}^{\left(x\right)}/k_{\text{F}}^{\left(y\right)}$. The values for the short-range disorder are represented by the dashed lines with the corresponding colors or by the black dashed lines if they coincide.