Geometrical Description of the Fractional Quantum Hall Effect

TL;DR

This paper introduces a geometric framework for understanding the fractional quantum Hall effect, highlighting the role of a spatial metric and its quantum fluctuations in describing the system's collective behavior.

Contribution

It proposes a novel geometric description of fractional quantum Hall states using a unimodular spatial metric and topologically-quantized guiding-center spin.

Findings

Identifies the local shape of correlations as a spatial metric

Links charge fluctuations to Gaussian curvature

Describes quantum fluctuations via guiding-center spin

Abstract

The fundamental collective degree of freedom of fractional quantum Hall states is identified as a unimodular two-dimensional spatial metric that characterizes the local shape of the correlations of the incompressible fluid. Its quantum fluctuations are controlled by a topologically-quantized "guiding-center spin". Charge fluctuations are proportional to its Gaussian curvature.