The $\theta$ vacuum reveals itself as the fundamental theory of the quantum Hall effect. I. Confronting controversies

TL;DR

This paper reexamines the $ heta$ dependence in a non-linear sigma model, revealing that the quantum Hall effect is a fundamental topological feature independent of the replica limit, challenging previous large N expansion assumptions.

Contribution

It demonstrates that the quantum Hall effect is a universal topological property of the model, not limited to the replica limit, contradicting earlier large N expansion theories.

Findings

Quantum Hall effect exists for all non-negative m, n values.

The effect is independent of the replica limit m=n=0.

Contradicts large N expansion expectations.

Abstract

The $\theta$ dependence of the Grassmannian $U(m+n)/U(m)\times U(n)$ non-linear $\sigma$ model is reexamined. This general theory provides an important laboratory for studying the quantum Hall effect, in the special limit $m=n=0$ (replica limit). We discover however that the quantum Hall effect is in fact independent of this limit and exists as a generic topological feature of the theory for all non-negative values of $m$ and $n$. The results are in concflict with many of the historical ideas and expectations on the basis of the large $N$ expansion of the $CP^{N-1}$ or $SU(N)/U(N-1)$ model