Deconstructing Kubo formula usage: Exact conductance of a mesoscopic system from weak to strong disorder limit

In both research and textbook literature one often finds two “different” Kubo formulas for the zero-temperature conductance of a noninteracting Fermi system. They contain a trace of the product of velocity operators and single-particle (retarded and advanced) Green operators: $\mathrm{Tr}\left(v_x{G}^r{v}_x{G}^a\right)$ or $\mathrm{Tr}\left(v_x\mathrm{Im}Ĝv_x\mathrm{Im}Ĝ\right).\mathrm{}$ The study investigates the relationship between these expressions, as well as the requirements of current conservation, through exact evaluation of such quantum-mechanical traces for a nanoscale (containing 1000 atoms) mesoscopic disordered conductor. The traces are computed in the semiclassical regime (where disorder is weak) and, more importantly, in the nonperturbative transport regime (including the region around localization-delocalization transition) where the concept of a mean free path ceases to exist. Since quantum interference effects for such strong disorder are not amenable to diagrammatic or nonlinear $\sigma$-model techniques, the evolution of different Green function terms with disorder strength provides insight into the development of an Anderson localized phase.