Composite fermions and the first-Landau-level fine structure of the fractional quantum Hall effect

Abstract

A set of scalar operators are employed to generate explicit representations of both hierarchy states (e.g., the series of fillings 1/3, 2/5, 3/7, ... ) and their conjugates (fillings 1, 2/3, 3/5, ...) as non-interacting quasi-electrons filling fine-structure sub-shells within the FLL. This yields, for planar and spherical geometries, a quasi-electron representation of the incompressible FLL state of filling p/(2p +1) in a magnetic field of strength B that is algebraically identical to the IQHE state of filling p in a magnetic field of strength B/(2p+1). The construction provides a precise definition of the quasi-electron/composite fermion that differs in some respects from common descriptions: they are eigenstates of L,Lz; they and the FLL subshells they occupy carry a third index I that is associated with breaking of scalar pairs; they absorb in their internal wave functions one, not two, units of magnetic flux; and they share a common, simple structure as vector products of a spinor creating an electron and one creating magnetic flux. We argue that these properties are a consequence of the breaking of the degeneracy of noninteracting electrons within the FLL by the scale-invariant Coulomb potential. We discuss the sense in which the wave function construction supports basic ideas of both composite fermion and hierarchical descriptions of the FQHE. We describe symmetries of the quasi-electrons at half filling, where a deep Fermi sea of quasi-electrons forms, and the quasi-electrons take on Majorana and pseudo-Dirac characters. Finally, we show that the wave functions can be viewed as fermionic excitations of the bosonic half-filled shell, producing at half filling an operator that differs from but plays the same role as the Pfaffian.