Effect of Long Range Disorder on Electron Motion in Two Dimensions

We consider a two-dimensional electron gas with long range disorder. Assuming that time reversal symmetry is broken either by an external magnetic field or, as in the case of a delta-correlated random magnetic field, by the disorder itself, we derive a supermatrix $σ$-model. As an intermediate step, we provide a microscopic derivation of the ballistic $σ$-model, and find that certain corrections to its usual form may become important. We then integrate out degrees of freedom corresponding to short length scales to derive a low-energy supermatrix $σ$-model. We find an extra term in the free energy that couples to the correlator of local currents. Use of a proper ultraviolet regularisation procedure that preserves gauge invariance indicates that the contribution of the extra term seems finally to become irrelevant. Within the scope of our analyis, we therefore do not find any deviation of the scaling behaviour of the delta-correlated random magnetic field model from that of the conventional unitary ensemble. We then generalize the discussion to include models of even longer-ranged disorder, plus short-range disorder. When the disorder is sufficiently long-ranged that the local currents become delta-correlated, a new term appears in the free energy that does give rise to logarithmic corrections to the conductivity. A renormalisation group analysis of the free energy yields a scaling form for the diffusion coefficient which contains both a positive correction, that represents classical superdiffusion, and a negative correction, which is the usual weak localization correction. The fact that both corrections are of the same order and opposite sign leads to the interesting possibility of a quantum phase transition at weak disorder in two dimensions, tuned by the relative disorder strengths.