Quantum geometric contribution to the diffusion constant

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 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.

💡 Research Summary

The paper investigates how quantum geometry influences the diffusion constant and the DC conductivity in metals and semimetals that possess a perfectly linear Dirac dispersion. Starting from a generic d‑dimensional (d ≥ 2) mass‑less Dirac Hamiltonian H = v_F σ·k, the authors introduce Gaussian scalar disorder with correlator ⟨V(r)V(r′)⟩ = γ²δ(r−r′) and treat disorder within the self‑consistent Born approximation (SCBA). In the heavily doped regime (ε_F τ ≫ 1) the SCBA yields a simple momentum‑independent scattering rate τ⁻¹ = πγ²g_d, where g_d is the density of states at the Fermi energy. Summing ladder diagrams for the particle‑hole propagator gives the diffusion propagator D(q,Ω) = 1/