Lattice BF Theory, Dumbbells, and Composite Fermions

TL;DR

This paper develops a lattice formulation of $U(1)$ BF theory applicable to all abelian groups, providing a rigorous framework for composite fermion theory in the fractional quantum Hall effect and exploring higher form gauge models.

Contribution

It introduces a lattice method for $U(1)$ BF theory applicable to any abelian group, enabling rigorous analysis of composite fermions and fractional quantum Hall states.

Findings

Derived Jain's fractions for FQHE

Provided a rigorous lattice framework for composite fermions

Generalized to higher form gauge theories

Abstract

We formulate $U(1)$ $bda$ Chern-Simons theory, which is also called BF theory, on a lattice, adapting a method proposed by Kantor and Susskind for the groups $\mathbb{R}$ and $\mathbb{Z}_N$. Our method applies to any finite or infinite abelian group. We study the discrete symmetries and use the model to provide a rigorous treatment of the composite fermion theory of the fractional quantum Hall effect (FQHE), with no ambiguities relating to intersecting Wilson/'t Hooft lines. We derive Jain's fractions, and one can also calculate corrections to the mean field solution within this framework. We also generalize the formalism to higher form gauge models in arbitrary dimension, and suggest a possible non-Abelian extension.