Two-scale localization of wave functions in disordered wires in a weak magnetic field

Using the supersymmetry technique combined with the transfer matrix method we calculate different physical quantities characterizing localization in disordered wires. In particular, we analyze the density-density correlation function and study effects of an external magnetic field $H$ on tails of wave functions. At zero and very strong magnetic fields, we obtain explicit expressions, valid at arbitrary distances, for all moments and for the entire distribution function of the density-density correlations. A two-scale decay is shown to be a typical feature of infinitely long wires at weak magnetic fields: The far tail of the wave functions decays twice as slow as their main body. Extending Mott's physical picture for the localized states we present a qualitative description of the crossover in the magnetic field. Our arguments can be used for any dimensionality indicating that the concept of the two-scale localization in a weak magnetic field is a general feature of localization. The effect is very sensitive to a level smearing and cannot be seen in the transmittance of a system with metallic leads. This means that the Borland conjecture may not be used for a numerical check of the two-scale localization.