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For an isotropic single-band system, it is well known that the semiclassicalBoltzmann transport theory within the relaxation time approximation and theKubo formula with the vertex corrections provide the same result with the$(1-\cos\theta)$ factor in the inverse transport relaxation time. Inanisotropic multiband systems, the semiclassical Boltzmann transport equationis generalized to coupled integral equations relating transport relaxationtimes at different angles in different bands. Using the Kubo formula, we studythe vertex corrections to the dc conductivity in anisotropic multiband systemsand derive the relation satisfied by the transport relaxation time for bothelastic and inelastic scatterings, verifying that the result is consistent withthe semiclassical approach.

Vertex corrections to the dc conductivity in anisotropic multiband systems