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.
Highlights
The theory of the fractional quantum Hall (FQH) effect [1,2] is based on the paradigm of the composite fermion [3,4,5], which provides a unified explanation of a large amount of observed phenomena, among which the most early ones are the Jain sequences—series of quantum Hall plateaux at filling factors ν near 1=2, 1=4, etc
Our task is to understand the ground state of the system in the finite magnetic field B 1⁄4 Fxy and its excitations. This problem is nonperturbative even at e2 ≪ 1 since it maps to a FQH problem
We have argued that the problem of finding the ground state of a Dirac fermion in a magnetic field, in the weak coupling regime e2 ≪ 1, is equivalent to finding the ground state of a nonrelativistic fermion in the LLL limit m → 0
Summary
The theory of the fractional quantum Hall (FQH) effect [1,2] is based on the paradigm of the composite fermion [3,4,5], which provides a unified explanation of a large amount of observed phenomena, among which the most early ones are the Jain sequences—series of quantum Hall plateaux at filling factors ν near 1=2, 1=4, etc. It is not obvious that the flux attachment procedure can be carried out for Dirac fermions; the usual workaround is to work in the limit of zero Landau-level mixing where the projected Hamiltonian is identical to the nonrelativistic one [25] This method explicitly breaks particle-hole symmetry and, does not work at finite. We propose an explicitly particle-hole symmetric effective theory describing the low-energy dynamics of the Fermi liquid state of a half-filled. The resolution to the puzzle is not the noncommutativity of the m → 0 limit and the clean limit; rather, it is in an ingredient of the Fermi liquid theory missing in all treatments of the CFs so far: the global Berry phase Beside these differences, there are many similarities between the picture proposed here and the standard CF picture.
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