Abstract
We propose a particle-hole symmetric theory of the Fermi-liquid ground state of a half-filled Landau level. This theory should be applicable for a Dirac fermion in the magnetic field at charge neutrality, as well as for the quantum Hall ground state of nonrelativistic fermions in the limit of negligible inter-Landau-level mixing. We argue that when particle-hole symmetry is exact, the composite fermion is a massless Dirac fermion, characterized by a Berry phase of around the Fermi circle. We write down a tentative effective field theory of such a fermion and discuss the discrete symmetries, in particular, . The Dirac composite fermions interact through a gauge, but non-Chern-Simons, interaction. The particle-hole conjugate pair of Jain-sequence states at filling factors and , which in the conventional composite fermion picture corresponds to integer quantum Hall states with different filling factors, and , is now mapped to the same half-integer filling factor of the Dirac composite fermion. The Pfaffian and anti-Pfaffian states are interpreted as -wave Bardeen-Cooper-Schrieffer paired states of the Dirac fermion with orbital angular momentum of opposite signs, while -wave pairing would give rise to a particle-hole symmetric non-Abelian gapped phase. When particle-hole symmetry is not exact, the Dirac fermion has a -breaking mass. The conventional fermionic Chern-Simons theory is shown to emerge in the nonrelativistic limit of the massive theory.
- Received 19 February 2015
DOI:https://doi.org/10.1103/PhysRevX.5.031027
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Published by the American Physical Society
Popular Summary
The fractional quantum Hall effect—in which a voltage is produced given a very strong magnetic field and a current—is the most striking example of emergent quantum phenomena in condensed-matter physics. The quasiparticle in the fractional quantum Hall effect is the so-called composite fermion, whose existence was established experimentally in the 1990s. The most common way of thinking about the composite fermion is to view it as an electron with two units of flux quanta bound to it. This picture is, however, not symmetric under the so-called particle-hole symmetry, which is an exact symmetry in the limit of a very high magnetic field. Here, we develop an effective field theory to propose that the composite fermion is a truly emergent particle, characterized by different quantum numbers than the original electrons (or holes).
Under particle-hole symmetry, the composite fermion transforms into itself, not into its hole (i.e., it is its own particle-hole conjugate). Moreover, it is a Dirac particle, characterized by a Berry phase of around the Fermi surface. Our proposed theory predicts a new gapped phase, called PH-Pfaffian, which is a particle-hole symmetric version of the Pfaffian state. We additionally discuss the physical implications of our theory, and we propose experimental signatures that could distinguish our new theory from old ones.
We anticipate that our theoretical results will be verified in physical systems and numerical simulations.