Particle-hole symmetry and the composite Fermi liquid

The half-filled Landau level is widely believed to be described by the Halperin-Lee-Read theory of the composite Fermi liquid (CFL). In this paper, we develop a theory for the particle-hole conjugate of the CFL, the anti-CFL, which we argue to be a distinct phase of matter as compared with the CFL. The anti-CFL provides a possible explanation of a recent experiment [D. Kamburov et al., Phys. Rev. Lett. 113, 196801 (2014)] demonstrating that the density of composite fermions in GaAs quantum wells corresponds to the electron density when the filling fraction $\nu <\frac{1}{2}$ and to the hole density when $\nu >\frac{1}{2}$. We introduce a local field theory for the CFL and anti-CFL in the presence of a boundary, which we use to study CFL-insulator-CFL junctions, and the interface between the anti-CFL and CFL. We show that the CFL–anti-CFL interface allows partially fused boundary phases in which “composite electrons” can directly tunnel into “composite holes,” providing a nontrivial example of transmutation between topologically distinct quasiparticles. We discuss several observable consequences of the anti-CFL, including a predicted resistivity jump at a first-order transition between uniform CFL and anti-CFL phases. We also present a theory of a continuous quantum phase transition between the CFL and anti-CFL. We conclude that particle-hole symmetry requires a modified view of the half-filled Landau level, in the presence of strong electron-electron interactions and weak disorder, as a critical point between the CFL and the anti-CFL.