Electron localization

At low temperatures, the resistivity of a metal is dominated by the elastic scattering of electrons by impurities in the system. If we treat the electrons as classical particles, we would expect their trajectories to resemble random walks after many collisions, i.e., their motion is diffusive when observed over length scales much greater than the mean free path. This diffusion becomes slower with increasing disorder, and can be measured directly as a decrease in the electrical conductance.

When the scattering is so frequent that the distance travelled by the electron between collisions is comparable to its wavelength, quantum interference becomes important. Quantum interference between different scattering paths has a drastic effect on electronic motion: the electron wavefunctions are localized inside the sample so that the system becomes an insulator. This mechanism (Anderson localization) is quite different from that of a band insulator for which the absence of conduction is due to the lack of any electronic states at the Fermi level.

Localization by a disordered potential has been studied extensively. For a review, see B. Kramer and A. MacKinnon, Rep. Prog. Phys. 56, 1469 (1993). I have been interested in the problem of scattering by a random magnetic field. A random field with zero average breaks time-reversal symmetry without introducing a finite Hall conductance. Conventional wisdom suggests that electronic motion in this random field is localized in two dimensions. With improving fabrication techniques, a phase-coherent 2D electron gas with random magnetic fields can soon be realized in semiconductor structures. This also has connections with the composite-fermion theory of the half-filled Landau level in fractional quantum Hall systems. 

We mapped this problem in the limit of smooth disorder to a multi-channel Chalker-Coddington network model which was originally developed for a particle in a strong uniform magnetic field. We find that the localization length is finite in this model, but it is large compared to the correlation length of the random field. This study was restricted to systems with small conductances at short distances. To study the limit of large conductance, we have also mapped the network model to a set of coupled SU(N) spin chains. Perturbative renormalization group calculations on this model show that the conductance scales downwards with increasing system size, consistent with conventional localization theory. 

Surprisingly, the network-model approach reveals that conclusions about the random-field problem have implications for quantum Hall systems where the disorder broadening of the Landau band exceeds the Zeeman splitting. In particular, the centre of such a spin-degenerate Landau band should be localized, and the system should have a pair of delocalization transitions away from the band centre, in the same universality class as the spin-split case. 
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