Linear dependencies between Composite Fermion states

TL;DR

This paper develops a systematic method to identify linear dependencies among composite fermion states, simplifying calculations in quantum Hall systems by revealing redundancies before projection.

Contribution

It introduces a new approach to uncover all linear dependencies among CF states at the Slater determinant level, reducing computational complexity in quantum Hall studies.

Findings

Identified all linear dependencies among bosonic CF states in the lowest sub-band.

Extended the method to higher sub-bands and generic states, applicable to both one- and two-species systems.

Provided analytically exact identities for wave functions in disk geometry.

Abstract

It has been observed that the composite fermion (CF) approach tends to overcount the number of linearly independent candidate states for fixed sets of quantum numbers [number of particles, total angular momentum, and (pseudo)spin if applicable]. That is, CF Slater determinants that are orthogonal before projection, may lead to wave functions that are identical, or possess linear dependencies, after projection. This has been pointed out both in the context of rotating bosons in the lowest Landau level, and for excited bands of the (fermionic) fractional quantum Hall effect. We present a systematic approach that enables us to reveal all linear dependencies between bosonic compact states in the lowest CF "cyclotron energy" sub-band, and almost all dependencies in higher sub-bands, at the level of the CF Slater determinants, i.e. before projection, which implies a major computational simplification. Our approach is introduced for so-called simple states of two-species rotating bosons, and then generalised to generic compact bosonic states, both one- and two-species. Some perspectives also apply to fermionic systems. The identities and linear dependencies we find, are analytically exact for "brute force" projection in the disk geometry.