Level broadening for localized electrons: Change of the magnetoconductance sign

The distribution of the hopping probabilities in the strong localization regime is calculated via the independent-directed-path formalism, allowing for level broadening of the intermediate site energies of the hopping process. The distribution belongs to the Gaussian unitary ensemble even in the absence of a magnetic field, whereas in the absence of level broadening it belongs to the orthogonal ensemble. In the presence of the field it leads to an orbital magnetoconductance quadratic in the field, changing from positive to negative as the temperature is lowered and the number of paths increases.