Noncommutative Geometry, Quantum Hall Effect and Berry Phase

TL;DR

This paper explores the fractional quantum Hall effect using noncommutative geometry, linking Berry phase, symplectic structure deformation, and noncommutative field theory to explain the filling factor.

Contribution

It demonstrates the equivalence of fractional quantum Hall effect on noncommutative manifolds with noncommutative field theory, connecting Berry phase and chiral anomaly.

Findings

Fractional quantum Hall effect modeled on noncommutative manifold.

Filling factor related to Berry phase and symplectic area deformation.

Equivalence established between noncommutative geometry approach and noncommutative field theory.

Abstract

Taking resort to Haldane's spherical geometry we can visualize fractional quantum Hall effect on the noncommutative manifold $M_4 \times Z_N$ with $N>2$ and odd. The discrete space leads to the deformation of symplectic structure of the continuous manifold such that the symplectic area is given by $\triangle p.\triangle q=2\pi m \hbar$ with $m$ an odd integer which is related to the Berry phase and the filling factor is given by $\frac{1}{m}$. We here argue that this is equivalent to the noncommutative field theory as prescribed by Susskind and Polychronakos which is characterized by area preserving diffeomorphism. The filling factor $\frac{1}{m}$ is determined from the change in chiral anomaly and hence the Berry phase as envisaged by the star product.