Two-Scale Localization in Disordered Wires in a Magnetic Field

We study localization properties of wave functions in a magnetic field. At arbitrarily weak magnetic field, the far tail is shown to decay with the length $L_{\mathrm{cu}}{=2L}_{\mathrm{co}}$, where $L_{\mathrm{co}}$ and $L_{\mathrm{cu}}$ are the localization lengths in zero magnetic field and in a strong magnetic field, respectively. Increasing the magnetic field broadens the region of the decay with the length $L_{\mathrm{cu}}$, leading finally to the decay with $L_{\mathrm{cu}}$ at all distances. Thus, the crossover between the orthogonal and unitary ensembles is characterized by two localization lengths. This peculiar behavior results in two different temperature regimes in the hopping conductivity with the boundary between them depending on the magnetic field.