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

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.