Critical Transport in Weakly Disordered Semiconductors and Semimetals

Motivated by Weyl semimetals and weakly doped semiconductors, we study transport in a weakly disordered semiconductor with a power-law quasiparticle dispersion ξₖ∝kα. We show, that in 2α dimensions short-correlated disorder experiences logarithmic renormalization from all energies in the band. We study the case of a general dimension d using a renormalization group, controlled by an ϵ=2α-d expansion. Above the critical dimensions, conduction exhibits a localization-delocalization phase transition or a sharp crossover (depending on the symmetries of the Hamiltonian) as a function of disorder strength. We utilize this analysis to compute the low-temperature conductivity in Weyl semimetals and weakly doped semiconductors near and below the critical disorder point.