Holomorphicity, Vortex Attachment, Gauge Invariance and the Fractional Quantum Hall Effect

TL;DR

This paper introduces a gauge invariant reformulation of nonrelativistic fermions in magnetic fields, enabling exact derivation of Laughlin and Jain wave functions through vortex attachment within a mean-field framework, applicable to spherical geometries.

Contribution

It develops a novel gauge invariant approach that replaces gauge symmetry with holomorphic symmetry, allowing exact vortex attachment and wave function construction in mean-field theory.

Findings

Exact Laughlin and Jain wave functions derived within MFT

Holomorphic invariance enables analytical vortex attachment methods

Framework extended to spherical geometries with exact solutions

Abstract

A gauge invariant reformulation of nonrelativistic fermions in background magnetic fields is used to obtain the Laughlin and Jain wave functions as exact results in Mean Field Theory (MFT). The gauge invariant framework trades the U(1) gauge symmetry for an emergent holomorphic symmetry and fluxes for vortices. The novel holomorphic invariance is used to develop an analytical method for attaching vortices to particles. Vortex attachment methods introduced in this paper are subsequently employed to construct the Read operator within a second quantized framework and obtain the Laughlin and Jain wave functions as exact results entirely within a mean-field approximation. The gauge invariant framework and vortex attachment techniques are generalized to the case of spherical geometry and spherical counterparts of Laughlin and Jain wave functions are also obtained exactly within MFT.