Composite-fermion pairing at half-filled and quarter-filled lowest Landau level

The Halperin-Lee-Read Fermi sea of composite fermions at half-filled lowest Landau level is the realization of a fascinating metallic phase that is a strongly correlated "non-Fermi liquid" from the electrons' perspective. Remarkably, experiments have found that, as the width of the quantum well is increased, this state makes a transition into a fractional quantum Hall state, the origin of which has remained an important puzzle since its discovery more than three decades ago. We perform detailed and accurate quantitative calculations using a systematic variational framework for the pairing of composite fermions that closely mimics the Bardeen-CooperSchrieffer theory of superconductivity. Our calculations show that, (i) as the quantum-well width is increased, the single-component composite-fermion Fermi sea occupying the lowest symmetric subband of the quantum well undergoes an instability into a single-component p-wave paired state of composite fermions; (ii) the theoretical phase diagram in the quantum-well width-electron-density plane is in excellent agreement with experiments; (iii) a sufficient amount of asymmetry in the charge distribution of the quantum well destroys the fractional quantum Hall effect, as observed experimentally; and (iv) the two-component 331 state is energetically less favorable than the single-component paired state. Evidence for fractional quantum Hall effect has been seen in wide quantum wells also at quarter-filled lowest Landau level; here our calculations indicate an f-wave paired state of composite fermions. We further investigate bosons in the lowest Landau level at filling factor equal to one and show that a p-wave pairing instability of composite fermions, which are bosons carrying a single vortex, occurs for the short range as well as the Coulomb interaction, in agreement with exact diagonalization studies. The general consistency of the composite-fermion Bardeen-Cooper-Schrieffer approach with experiments lends support to the notion of composite-fermion pairing as the primary mechanism of fractional quantum Hall effects at even-denominator filling factors. Various experimental implications are mentioned.