Super Universality of the Quantum Hall Effect and the “Large N Picture” of the ϑ Angle

It is shown that the "massless chiral edge excitations" are an integral and univer- sal aspect of the low energy dynamics of the ϑ vacuum that has historically gone unnoticed. Within the SU(M + N )/S(U (M) × U( N )) non-linear sigma model we introduce an effec- tive 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 super universal strong coupling feature of the ϑ angle with the replica limit M = N = 0 only playing a role of secondary importance. To demon- strate 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 crit- icality 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 cou- pling "quantum Hall phase" of the large N theory. Our results are in accordance with the "instanton picture" of the ϑ angle but fundamentally invalidate all the ideas, expectations and conjectures that are based on the historical "large N picture."