The $\nu={1\over2}$ Landau level: Half-full or half-empty?

TL;DR

This paper develops a particle-hole symmetric composite fermion theory for the half-filled Landau level, incorporating Dirac fermion features and extending the framework to include symmetries and interpretations inspired by recent theoretical advances.

Contribution

It extends the Hamiltonian theory of composite fermions to incorporate particle-hole symmetry and Dirac fermion characteristics at half-filling, connecting to recent theoretical insights.

Findings

A PH-symmetric CF description with magnetic translation algebra.

Extension away from ν=1/2 with anti-unitary PH transformation.

Representation with T^2=+1 suggesting charge-conjugation symmetry.

Abstract

We show here that an extension of the Hamiltonian theory developed by us over the years furnishes a composite fermion (CF) description of the $\nu =\frac{1}{2}$ state that is particle-hole (PH) symmetric, has a charge density that obeys the magnetic translation algebra of the lowest Landau level (LLL), and exhibits cherished ideas from highly successful wave functions, such as a neutral quasi-particle with a certain dipole moment related to its momentum. We also a provide an extension away from $\nu=\frac{1}{2}$ which has the features from $\nu=\frac{1}{2}$ and implements the the PH transformation on the LLL as an anti-unitary operator ${\cal T}$ with ${\cal T}^2=-1$. This extension of our past work was inspired by Son, who showed that the CF may be viewed as a Dirac fermion on which the particle-hole transformation of LLL electrons is realized as time-reversal, and Wang and Senthil who provided a very attractive interpretation of the CF as the bound state of a semion and anti-semion of charge $\pm {e\over 2}$. Along the way we also found a representation with all the features listed above except that now ${\cal T}^2=+1$. We suspect it corresponds to an emergent charge-conjugation symmetry of the $\nu =1$ boson problem analyzed by Read.