Super universality of the quantum Hall effect and the "large $N$ picture" of the $\vartheta$ angle

TL;DR

This paper reveals the super universality of the quantum Hall effect as a fundamental feature of the $ heta$ vacuum, introducing an effective edge excitation theory and revisiting the large N expansion to demonstrate quantum criticality and phase crossover.

Contribution

It introduces a universal edge excitation framework within the $ heta$ vacuum and revises the large N analysis to show quantum Hall criticality and phase transition scaling.

Findings

Edge excitations are integral to the $ heta$ vacuum's low energy dynamics.

Quantum Hall effect exhibits super universality across coupling regimes.

Explicit scaling results and a phase crossover diagram are obtained for the large N theory.

Abstract

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."