Is the Composite Fermion a Dirac Particle?

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 $\nu =\frac{1}{2}$ 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 $\pi$ around the Fermi circle. We write down a tentative effective field theory of such a fermion and discuss the discrete symmetries, in particular, $\mathcal{C}\mathcal{P}$. 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 $\frac{n}{2n+1}$ and $\frac{n+1}{2n+1}$, which in the conventional composite fermion picture corresponds to integer quantum Hall states with different filling factors, $n$ and $n+1$, is now mapped to the same half-integer filling factor $n+\frac{1}{2}$ of the Dirac composite fermion. The Pfaffian and anti-Pfaffian states are interpreted as $d$-wave Bardeen-Cooper-Schrieffer paired states of the Dirac fermion with orbital angular momentum of opposite signs, while $s$-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 $\mathcal{C}\mathcal{P}$-breaking mass. The conventional fermionic Chern-Simons theory is shown to emerge in the nonrelativistic limit of the massive theory.

13 pages; v2: added discussion of experimental signatures, Kohn's theorem; v3: typo fixed, published version

References in corpus (8)

• Topological origin of subgap conductance in insulating bilayer graphene
• Two-dimensional Dirac fermions in a topological insulator: transport in the quantum limit
• Particle-hole symmetry and the Pfaffian state
• Particle-Hole Symmetry and the $\nu =5/2$ Quantum Hall State
• Tunable Fractional Quantum Hall Phases in Bilayer Graphene
• Composite Dirac liquids: parent states for symmetric surface topological order
• Composite Dirac fermions in graphene
• Effective Field Theory of Relativistic Quantum Hall Systems

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