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

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 $C{P}^{N-1}$ 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."

40 pages, 9 figures

References in corpus (5)

• The instanton vacuum of generalized $C{P}^{N-1}$ models
• Theta renormalization, electron-electron interactions and super universality in the quantum Hall regime
• Coulomb Blockade and Super Universality of the Theta-Angle
• General topological features and instanton vacuum in quantum Hall and spin liquids
• Exact Haldane mapping for all $S$ and super universality in spin chains

Cited by in corpus (5)

• Resilience of Majorana Fermions in the face of Disorder
• Theta-vacuum and large N limit in CP^{N-1} sigma models
• Edge states and conformal boundary conditions in super spin chains and super sigma models
• Topological phase diagram of the disordered 2XY model in presence of generalized Dzyaloshinskii-Moriya Interaction
• Super universality of dimerised $SU(N+M)$ spin chains