Coherent transport in non-Abelian quantum graphs

We study quantum charge transport in two-dimensional networks in the presence of a magnetic field and spin-orbit interaction. The interplay of the corresponding Abelian and non-Abelian gauge fields leads to an intricate behavior of the conductance, which has different periodicities in the diffusive and ballistic regimes. We classify all configurations of magnetic and spin-orbit fields where a logarithmically divergent weak-(anti)localization correction appears in the diffusive regime. The conductivity of topologically distinct configurations is the same in the diffusive regime but different in the ballistic regime. The proposed setup provides a feasible realization of quantum graphs with non-Abelian gauge fields.