Abstract
Composite fermions (CFs) of the fractional quantum Hall effect (FQHE) are described as spherical products of electron and vortex spinors, built from underlying ladder operators aligned so that the spinor angular momenta, and , are maximal. We identify the CF's quantum numbers as the angular momentum in , its magnetic projection , the electron number , and magnetic spin, . Translationally invariant FQHE states are formed by fully filling subshells with their respective CFs, in order of ascending for fixed and , beginning with the lowest allowed value, . CF subshells are contained entirely within the first Landau level (FLL). Alternatively, we provide an equivalent hierarchical wave function in which the underlying objects are vortices with , correlated pairwise via . We show that CFs can be written as a valence operator carrying the angular momentum quantum numbers acting on a scalar half-filled intrinsic state. This scalar state serves as the vacuum for the valence electron () and vortex () ladder operators that create FQHE states. With respect to this vacuum, FQHE states can be grouped into -spin multiplets mirror symmetric around , in which is held constant. states have a net electron particle or hole number. Particle-hole conjugation with respect to this vacuum is identified as the mirror symmetry relating FQHE states of the same but distinct fillings and , e.g., . Alternatively, mirror symmetric -spin multiplets can be constructed in which the magnetic field strength is held fixed: the valence states are electron particle-vortex hole excitations relative to the half-filled vacuum () and their mirror conjugates (). Multiplet members are linked by the -spin raising/lowering operators, . Particle-hole (PH) symmetry—relating the -particle FQHE state of filling to the -particle state of filling , e.g., —is shown to be equivalent to electron-vortex exchange, and . The -particle states and are connected by this mirror symmetry. In this construction, CFs of the state occupy an extra zero-mode subshell that is annihilated by . We link this structure, familiar from supersymmetric quantum mechanics, to the CF Pauli Hamiltonian, which we show is isospectral, quadratic in the -spin raising and lowering operators , and fourfold degenerate in , and . On linearization, it takes a Dirac form similar to that found in the integer quantum Hall effect (IQHE).
2 More- Received 12 February 2018
- Revised 21 July 2018
DOI:https://doi.org/10.1103/PhysRevB.98.115140
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