Quantum geometric contribution to the diffusion constant in metals and semimetals

We discuss the quantum geometric contribution to the diffusion constant and the DC conductivity in metals and semimetals with linear Dirac dispersion. We demonstrate that, for systems with perfectly linear dispersion, there exists a clear and rigorous separation of the quantum geometric from the ordinary band velocity contributions to the diffusion constant, which turns out to be directly related to the separation of a rank two tensor into transverse and longitudinal parts. We also demonstrate that, within the self-consistent Born approximation and for Gaussian-distributed disorder, the diffusion constant of three-dimensional Dirac fermions at charge neutrality is entirely quantum geometric in origin, which is not the case for two-dimensional Dirac fermions. This is the result of an accidental perfect cancellation of the band velocity contribution in three dimensions.