I will discuss the half-filled Landau level problem in two different filling fractions.
At ν=1/2, the Halperin-Lee-Read (HLR) composite fermi liquid state was thought to be incompatible with particle-hole symmetry of the half-filled Landau level. We found, however, that when properly evaluated, the HLR theory gives results for long-wavelength and low-energy physical properties, including the Hall conductance at ν=1/2 and the magnetoroton spectra at nearby fillings, that are particle-hole symmetric. In fact, the HLR theory predicts an emergent particle-hole symmetry near half filling, even when the symmetry is microscopically absent.
At ν=5/2, the topologically-ordered ground state is predicted, through numerical studies, to be either the Moore-Read Pfaffian or its particle hole conjugate, the anti-Pfaffian state. However, recent experiments, very surprisingly, appear to favor a quantized thermal hall conductivity with quantum number K = 5/2, rather than the value K = 7/2 or K = 3/2 expected for the Pfaffian or anti-Pfaffian state, respectively. I will discuss a mechanism for disorder to induce a set of new phases in the ν=5/2 quantum Hall system. These include a “thermal metal” phase with quantized electric Hall response, and a somewhat new topological order called PH-Pfaffian, which has a quantized thermal Hall conductivity K=5/2, in agreement with the recent measurement.