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

Abstract

It is shown that the "massless chiral edge excitations" are an integral and universal aspect of the low energy dynamics of the ϑ vacuum that has historically gone unnoticed.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 super universal strong coupling feature of the ϑ 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 ϑ angle but fundamentally invalidate all the ideas, expectations and conjectures that are based on the historical "large N picture."

Super universality of dimerised $SU(N+M)$ spin chains

“…We mention, in particular, the "large N " picture of the ϑ vacuum concept [19][20][21][22][23] as opposed to the distinctly different "instanton" picture [24][25][26]. However, these different views have in fact turned out to be complementary [4,15]. This new development is consistent with the original critique by A. Jevicki [27] and is diametrically opposite to what historically has been dubbed "the arena of bloody controversies" [28].…”

“…However, it is also well known that this regime is numerically accessible, namely based on the density matrix renormalisation group (DMRG) approach to quantum spin chains [12,13]. The main purpose of the present paper, therefore, is to use the DMRG as an unequivocal test of the distinctly different strong coupling ideas that over the years have emerged in the study of both SU (N + M ) quantum spin chains [9,14] and the closely related grassmannian U (N + M )/U (N ) × U (M ) sigma model [4,15].…”

Edge states and conformal boundary conditions in super spin chains and super sigma models

“…We note that the role of edge states in sigma models has recently been revisited in[30], although the relation with our and other's work is not clear to us.…”

Resilience of Majorana fermions in the face of disorder

“…It would be desirable to incorporate the above observations to construct an effective order parameter theory to address the interplay between the p-wave superconductivity and disorder 25 . The analogy with plateau-to-plateau transitions in quantum Hall effect and possible effective theories with theta term 26 or possible RG interpretation 27 is worth exploration.…”