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It is well known that there is a particle-hole symmetry for spin-polarizedelectrons with two-body interactions in a partially filled Landau level, whichbecomes exact in the limit where the cyclotron energy is large compared to theinteraction strength, so one can ignore mixing between Landau levels. Thissymmetry is explicit in the description of a half-filled Landau level recentlyintroduced by D. T. Son, using Dirac fermions, but it was thought to be absentin the older fermion-Chern- Simons approach, developed by Halperin, Lee, andRead and subsequent authors. We show here, however, that when properlyevaluated, the Halperin, Lee, Read (HLR) theory gives results forlong-wavelength low-energy physical properties, including the Hall conductancein the presence of impurities and the positions of minima in the magnetorotonspectra for fractional quantized Hall states close to half-filling, that areidentical to predictions of the Dirac formulation. In fact, the HLR theorypredicts an emergent particle-hole symmetry near half filling, even when thecyclotron energy is finite.

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