Theory of the half-filled Landau level

TLDR

A two‑dimensional electron system at half filling can be mapped to composite fermions coupled to a Chern‑Simons gauge field with zero average magnetic field, implying a well‑defined Fermi surface in the absence of gauge fluctuations and disorder. The study aims to predict the energy gap of the fractional quantum Hall state at ν=p/(2p+1) and to explore its consequences for specific heat, other physical properties, and the phase diagram at ν=1/2 and related even‑denominator fillings. The authors analyze the half‑filled Landau level by treating composite fermions coupled to a Chern‑Simons gauge field, incorporating gauge‑field fluctuations, impurity scattering, and electron‑electron interactions to compute effective mass corrections, resistivity, and acoustic‑wave anomalies. They find that gauge‑field fluctuations produce interaction‑dependent infrared divergences in the effective mass—no divergence for long‑range, power‑law for short‑range, logarithmic for Coulomb—yet the Fermi surface remains robust, yielding a finite low‑temperature resistivity and an acoustic‑wave anomaly in agreement with experiments.

Abstract

A two-dimensional electron system in an external magnetic field, with Landau-level filling factor \ensuremath{\nu}=1/2, can be transformed to a mathematically equivalent system of fermions interacting with a Chern-Simons gauge field such that the average effective magnetic field acting on the fermions is zero. If one ignores fluctuations in the gauge field, this implies that for a system with no impurity scattering, there should be a well-defined Fermi surface for the fermions. When gauge fluctuations are taken into account, we find that there can be infrared divergent corrections to the quasiparticle propagator, which we interpret as a divergence in the effective mass ${\mathit{m}}^{\mathrm{*}}$, whose form depends on the nature of the assumed electron-electron interaction v(r). For long-range interactions that fall off slower than 1/r at large separation r, we find no infrared divergences; for short-range repulsive interactions, we find power-law divergences; while for Coulomb interactions, we find logarithmic corrections to ${\mathit{m}}^{\mathrm{*}}$. Nevertheless, we argue that many features of the Fermi surface are likely to exist in all these cases. In the presence of a weak impurity-scattering potential, we predict a finite resistivity ${\mathrm{\ensuremath{\rho}}}_{\mathit{x}\mathit{x}}$ at low temperatures, whose value we can estimate. We compute an anomaly in surface acoustic wave propagation that agrees qualitatively with recent experiments. We also make predictions for the size of the energy gap in the fractional quantized Hall state at \ensuremath{\nu}=p/(2p+1), where p is an integer. Finally, we discuss the implications of our picture for the electronic specific heat and various other physical properties at \ensuremath{\nu}=1/2, we discuss the generalization to other filling fractions with even denominators, and we discuss the overall phase diagram that results from combining our picture with previous theories that apply to the regime where impurity scattering is dominant.