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

TL;DR

This paper demonstrates that the traditional fermion-Chern-Simons theory, when correctly evaluated, aligns with the Dirac fermion approach in describing particle-hole symmetry at half-filled Landau levels, revealing an emergent symmetry.

Contribution

It shows that the Halperin, Lee, Read (HLR) theory inherently predicts particle-hole symmetry near half filling, matching the Dirac fermion description.

Findings

HLR theory results match Dirac fermion predictions for Hall conductance.

Long-wavelength properties in HLR theory exhibit particle-hole symmetry.

Emergent particle-hole symmetry exists in HLR theory even at finite cyclotron energy.

Abstract

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, so one can ignore mixing between Landau levels. This symmetry is explicit in the description of a half-filled Landau level recently introduced by D. T. Son, using Dirac fermions, but it was thought to be absent in the older fermion-Chern- Simons approach, developed by Halperin, Lee, and Read and subsequent authors. We show here, however, that when properly evaluated, the Halperin, Lee, Read (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.