Two-scale localization in disordered wires in a magnetic field

Calculating the density-density correlation function for disordered wires, we study localization properties of wave functions in a magnetic field. The supersymmetry technique combined with the transfer matrix method is used. It is demonstrated that at arbitrarily weak magnetic field the far tail of the wave functions decays with the length $L_{{\rm cu}}=2L_{{\rm co}}$, where $L_{{\rm co}}$ and $L_{{\rm cu}}$ are the localization lengths in the absence of a magnetic field and in a strong magnetic field, respectively. At shorter distances, the decay of the wave functions is characterized by the length $L_{{\rm co}}$. Increasing the magnetic field broadens the region of the decay with the length $L_{{\rm cu}}$, leading finally to the decay with $L_{{\rm cu}}$ at all distances. In other words, the crossover between the orthogonal and unitary ensembles in disordered wires is characterized by two localization lengths. This peculiar behavior must result in two different temperature regimes in the hopping conductivity with the boundary between them depending on the magnetic field.