Dimensional Crossover of Localisation and Delocalisation in a Quantum Hall Bar

The 2-- to 1--dimensional crossover of the localisation length of electrons confined to a disordered quantum wire of finite width $L_y$ is studied in a model of electrons moving in the potential of uncorrelated impurities. An analytical formula for the localisation length is derived, describing the dimensional crossover as function of width $L_y$, conductance $g$ and perpendicular magnetic field $B$ . On the basis of these results, the scaling analysis of the quantum Hall effect in high Landau levels, and the delocalisation transition in a quantum Hall wire are reconsidered.Comment: 12 pages, 7 figure

Super universality of the quantum Hall effect and the "large NN picture" of the ϑ\vartheta angle

It is shown that the "massless chiral edge excitations" are an integral and universal aspect of the low energy dynamics of the $\vartheta$ vacuum that has historically gone unnoticed. Within the $SU(M+N)/S(U(M) \times U(N))$ non-linear sigma model we introduce an effective theory of "edge excitations" that fundamentally explains the quantum Hall effect. In sharp contrast to the common beliefs in the field our results indicate that this macroscopic quantization phenomenon is, in fact, a {\em super universal} strong coupling feature of the $\vartheta$ angle with the replica limit $M=N=0$ only playing a role of secondary importance. To demonstrate super universality we revisit the large $N$ expansion of the $CP^{N-1}$ model. We obtain, for the first time, explicit scaling results for the quantum Hall effect including quantum criticality of the quantum Hall plateau transition. Consequently a scaling diagram is obtained describing the cross-over between the weak coupling "instanton phase" and the strong coupling "quantum Hall phase" of the large $N$ theory. Our results are in accordance with the "instanton picture" of the $\vartheta$ angle but fundamentally invalidate all the ideas, expectations and conjectures that are based on the historical "large $N$ picture."Comment: 40 pages, 9 figure