Exact Shape of the Lowest Landau Level in a Double-Layer System and a Superlattice with Uncorrelated Disorder.

We extend Wegner’s exact solution for the 2D density of states at the lowest Landau level with a short–range disorder to the cases of a double–layer system and a superlattice. For the double–layer system, an analytical expression for the density of states, illustrating the interplay between the tunnel splitting of Landau levels and the disorder–induced broadening, is obtained. For the superlattice, we derive an integral equation, the eigenvalue of which determines the exact density of states. By solving this equation numerically, we trace the disappearance of the miniband with increasing disorder. PACS numbers: 73.20.Dx, 73.40.Hm Typeset using REVTEX 1 The shape of the Landau levels (LL) in a 2D system in the presence of a disorder was the subject of intensive study during the last two decades.1–19 The complexity of the problem arises from the fact that in the absence of the disorder the energy spectrum is discrete. As a result, the self–energy of an electron appears to be real in any finite order of the perturbation theory. Therefore, obtaining a finite width of the LL requires summation of the entire diagram expansion. It was demonstrated that such a summation is possible when the number of the LL is large. The simplifications, arising in this limit, are different in the case of a short–range and a smooth disorder. In the former case only a subsequence of diagrams without self–intersections contributes to the self–energy, or, in other words, the self–consistent Born approximation becomes asymptotically exact. The shape of the LL in this case is close to semielliptical. For a smooth disorder, with correlation radius larger than the magnetic length, all diagrams are of the same order of magnitude, but in this case magnetic phases, caused by self–intersections of impurity lines, become small. The origin of these phases lies in an uncertainty in the position of the center of the Larmour orbit. Having the phases dropped, the entire perturbation series can be summed up with the help of the Ward identity, resulting in the Gaussian shape of the LL. For low LL numbers and short–range disorder, the magnetic phases in diagrams are of the order of unity. A small parameter appears in the problem only if the energy ε (measured from the lowest LL) is much larger than the LL width Γ, making possible a calculation of the density of states (DOS) in the tails of LL. Such calculations were carried out in the framework of the instanton approach and the tails were shown to be Gaussian. In the domain ε ∼ Γ the problem has no small parameter and no simplifications are possible. However, for the lowest LL, the exact DOS was found by Wegner for an arbitrary ratio ε/Γ. Wegner has shown that the diagrammatic expansion of the disorder–averaged Green function, G(ε), can be mapped onto that of the zero–dimensional complex φ–model with the partition function Z (1) 0 given by a simple integral Z (1) 0 (ε,Γ) = ∫ dφdφ exp [ iεφφ− Γ 2 4 (φφ) ]