Is the Composite Fermion a Dirac Particle?

TL;DR

This paper proposes that composite fermions at half-filled Landau levels are massless Dirac particles with particle-hole symmetry, offering a new theoretical framework that unifies various quantum Hall states and their paired phases.

Contribution

It introduces a particle-hole symmetric Dirac fermion theory for the half-filled Landau level, connecting it to known quantum Hall states and their pairing mechanisms.

Findings

Composite fermions are described as massless Dirac particles with a Berry phase of π.

Particle-hole conjugate states map to the same half-integer filling factor in the Dirac picture.

Paired states correspond to d-wave and s-wave BCS states of the Dirac fermion.

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 $\nu=\frac12$ 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 $n+\frac12$ 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.