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We consider a two-dimensional electron gas with long range disorder. Assumingthat 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 disorderitself, we derive a supermatrix $\sigma $-model. As an intermediate step, weprovide a microscopic derivation of the ballistic $\sigma$-model, and find thatcertain corrections to its usual form may become important. We then integrateout degrees of freedom corresponding to short length scales to derive alow-energy supermatrix $\sigma $-model. We find an extra term in the freeenergy that couples to the correlator of local currents. Use of a properultraviolet regularisation procedure that preserves gauge invariance indicatesthat 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 thescaling behaviour of the delta-correlated random magnetic field model from thatof the conventional unitary ensemble. We then generalize the discussion toinclude models of even longer-ranged disorder, plus short-range disorder. Whenthe disorder is sufficiently long-ranged that the local currents becomedelta-correlated, a new term appears in the free energy that does give rise tologarithmic corrections to the conductivity. A renormalisation group analysisof the free energy yields a scaling form for the diffusion coefficient whichcontains both a positive correction, that represents classical superdiffusion,and a negative correction, which is the usual weak localization correction. Thefact that both corrections are of the same order and opposite sign leads to theinteresting possibility of a quantum phase transition at weak disorder in twodimensions, tuned by the relative disorder strengths.Comment: 40 pages, 1 figur

Influence of long-range disorder on electron motion in two dimensions