μ= 12 Landau level: Half-empty versus half-full

We show here that an extension of the Hamiltonian theory developed by us over the years furnishes a composite fermion (CF) description of the ν=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 quasiparticle with a certain dipole moment related to its momentum. We also a provide an extension away from ν=1/2, which has the features from ν=1/2 and implements the PH transformation on the LLL as an antiunitary operator T with T2=-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 antisemion of charge ±e/2. Along the way, we also found a representation with all the features listed above except that now T2=+1. We suspect it corresponds to an emergent charge-conjugation symmetry of the μ=1 boson problem analyzed by Read.