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.

Popular Summary

One of the most fascinating phases of matter in modern quantum physics is the “composite Fermi-liquid” state in a half-filled Landau level—a metallic state formed by electrons in a strong magnetic field and confined to two dimensions. This state can be described in terms of an emergent particle called a “composite fermion,” loosely understood as an electron attached to two quanta of magnetic flux. Composite fermions, in contrast to bare electrons, can move in straight lines despite the strong magnetic field, giving rise to the metallic behavior observed experimentally. In certain situations, it is possible to formulate a composite fermion theory in terms of holes—quasiparticles that represent the absence of electrons—but it creates a picture that appears to be at odds with the theory formulated in terms of electrons. A fundamental question is whether these different pictures actually represent distinct phases of matter. Our analysis suggests that there is a unique phase for which multiple theoretical descriptions are equivalent.

It has been widely believed, for two decades, that the two composite fermion theories—formulated in terms of electrons and holes, respectively—lead to qualitatively different physical predictions, implying that the different pictures describe different phases. However, by carefully evaluating certain key physically observable quantities, we show that the two pictures produce results that are surprisingly identical. This is true even when the symmetry relating these two pictures (known as particle-hole symmetry) is absent at the microscopic level; we call this an “emergent particle-hole symmetry.”

We expect that our results will motivate future research that will reveal deeper theoretical structures of the composite Fermi-liquid states and connect the theories to experimental observations.

Figure 1

Magnetoroton spectrum at fractions $\nu =20/41$ and $\nu =21/41$. Plots show the reduced frequency $\tilde{\omega }\equiv \omega /|\mathrm{\Delta }{\omega }^{*}|$ versus the reduced wave vector $\widehat{q}\equiv q{l}_B/|2\nu -1|$. The curve labeled “$\mathrm{RPA}+\text{Coulomb}$” shows the magnetoroton spectrum computed in the HLR approach, including the correction due to the Coulomb interaction. The curves labeled “RPA” and “Semiclassical” show the locations of the poles in the electron conductivity tensor $\widehat{\sigma }(\mathbf{q},\omega )$, which does not include the interaction effect, computed in the random phase approximation and semiclassical approximation, respectively. Curves for $\nu =20/41$ and $\nu =21/41$ could not be distinguished in these plots. The expanded figure in the lower panel includes, for comparison, a naive approximation, which identifies the magnetoroton spectrum with the zeros of the determinant of the composite-fermion conductivity ${\widehat{\sigma }}^{\mathrm{cf}}(\mathbf{q},\omega )$. Although the naive approximation coincides with the RPA and semiclassical approximations to leading order in the deviation from $\nu =1/2$, it deviates from them at second order and is not symmetric about $\nu =1/2$ at this order.