# Composite fermions in Fock space: Operator algebra, recursion relations, and order parameters

## TL;DR

This paper introduces a second-quantized algebraic framework for composite fermion states, providing recursion relations, operator algebra, and order parameters that deepen understanding of fractional quantum Hall states.

## Contribution

It develops a purely second-quantized approach to composite fermion wave functions, generalizing Read's recursion and introducing flux attachment operators and edge excitation algebra.

## Key findings

- Recursion relations for all unprojected Jain CF wave functions
- Second-quantized representation of flux attachment operator
- Explicit formulas for non-local order parameters

## Abstract

We develop recursion relations, in particle number, for all (unprojected) Jain composite fermion (CF) wave functions. These recursions generalize a similar recursion originally written down by Read for Laughlin states, in mixed first-second quantized notation. In contrast, our approach is purely second-quantized, giving rise to an algebraic, `pure guiding center' definition of CF states that de-emphasizes first quantized many-body wave functions. Key to the construction is a second-quantized representation of the flux attachment operator that maps any given fermion state to its CF counterpart. An algebra of generators of edge excitations is identified. In particular, in those cases where a well-studied parent Hamiltonian exists, its properties can be entirely understood in the present framework, and the identification of edge state generators can be understood as an instance of `microscopic bosonization'. The intimate connection of Read's original recursion with `non-local order parameters' generalizes to the present situation, and we are able to give explicit second quantized formulas for non-local order parameters associated with CF states.