Particle-Hole Symmetry in the Fermion-Chern-Simons and Dirac Descriptions of a Half-Filled Landau Level

It is well known that there is a particle-hole symmetry for spin-polarized electrons with two-body interactions in a partially filled Landau level, which becomes exact in the limit where the cyclotron energy is large compared to the interaction strength; thus, one can ignore mixing between Landau levels. This symmetry is explicit in the description of a half-filled Landau level recently introduced by Son, using Dirac fermions, but it was thought to be absent in the older fermion-Chern-Simons approach, developed by Halperin, Lee, and Read (HLR) and subsequent authors. We show here, however, that when properly evaluated, the HLR theory gives results for long-wavelength low-energy physical properties-including the Hall conductance in the presence of impurities and the positions of minima in the magnetoroton spectra for fractional quantized Hall states close to half-filling-that are identical to predictions of the Dirac formulation. In fact, the HLR theory predicts an emergent particle-hole symmetry near half-filling, even when the cyclotron energy is finite.