Analytical results on quantum interference and magnetoconductance for   strongly localized electrons in a magnetic field: Exact summation of   forward-scattering paths
  We study quantum interference effects on the transition strength for strongly localized electrons hopping on 2D square and 3D cubic lattices in the presence of a magnetic field B. These effects arise from the interference between phase factors associated with different electron paths connecting two distinct sites. For electrons confined on a square lattice, with and without disorder, we obtain closed-form expressions for the tunneling probability, which determines the conductivity, between two arbitrary sites by exactly summing the corresponding phase factors of all forward-scattering paths connecting them. An analytic field-dependent expression, valid in any dimension, for the magnetoconductance (MC) is derived. A positive MC is clearly observed when turning on the magnetic field. In 2D, when the strength of B reaches a certain value, which is inversely proportional to twice the hopping length, the MC is increased by a factor of two compared to that at zero field. We also investigate transport on the much less-studied and experimentally important 3D cubic lattice case, where it is shown how the interference patterns and the small-field behavior of the MC vary according to the orientation of B. The effect on the low-flux MC due to the randomness of the angles between the hopping direction and the orientation of B is also examined analytically. 
Yeong-Lieh Lin, Franco Nori