Electromagnetic response of composite Dirac fermions in the half-filled Landau level

Johannes Hofmann

doi:10.1103/physrevb.104.115401

Abstract

An effective field theory of composite Dirac fermions was proposed by Son [Phys. Rev. X 5, 031027 (2015)] as a theory of the half-filled Landau level with explicit particle-hole symmetry. We compute the electromagnetic response of this Son-Dirac theory on the level of the random phase approximation (RPA), where we pay particular attention to the effect of an additional composite-fermion dipole term that is needed to restore Galilean invariance. We find that once this dipole correction is taken into account, spurious interband transitions and collective modes that are present in the response of the unmodified theory either cancel or are strongly suppressed. We demonstrate that this gives rise to a consistent theory of the half-filled Landau level valid at all frequencies, at least to leading order in the momentum. In addition, the dipole contribution modifies the Fermi-liquid response at small frequency and momentum, which is a prediction of the Son-Dirac theory within the RPA that distinguishes it from a separate description of the half-filled Landau level by Halperin, Lee, and Read within the RPA.

Highlights

The fractional quantum Hall effect (FQHE) in the lowest Landau level (LLL) is a prototypical example of a stronginteraction phenomenon, where, due to the quenching of the kinetic electron energy in a magnetic field, only a single scale remains [1]
In the special case of the half-filled Landau level, mean-field theory predicts that the AharonovBohm flux attached to each electron precisely cancels the external magnetic field [8], and the composite fermions form a Fermi liquid, a field-theoretical analysis of which including random phase approximation (RPA) excitations was first given by Halperin, Lee, and Read
We have discussed the response of the SonDirac theory of the half-filled Landau level using the random phase approximation

Summary

Introduction

The fractional quantum Hall effect (FQHE) in the lowest Landau level (LLL) is a prototypical example of a stronginteraction phenomenon, where, due to the quenching of the kinetic electron energy in a magnetic field, only a single (interaction) scale remains [1]. The field-theoretical description of composite fermions is based on a Chern-Simons theory that is obtained from the Hamiltonian of interacting electrons in a magnetic field by a formally exact singular gauge transformation, which attaches a number of flux quanta to each electron [3,4]. The advantage of this formulation is that standard many-body approximations—such as a mean-field approximation for the ground state and a random phase approximation (RPA) for the fluctuations [5,6,7]—provide an accurate description of the FQHE.

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