Noncommutative field theory and composite Fermi liquids in some quantum Hall systems

Composite Fermi liquid metals arise at certain special filling fractions in the quantum Hall regime and play an important role as parents of gapped states with quantized Hall response. They have been successfully described by the Halperin-Lee-Read (HLR) theory of a Fermi surface of composite fermions coupled to a U(1) gauge field with a Chern-Simons term. However, the validity of the HLR description when the microscopic system is restricted to a single Landau level has not been clear. Here for the specific case of bosons at filling nu = 1, we build on earlier work from the 1990s to formulate a low-energy description that takes the form of a noncommutative field theory. This theory has a Fermi surface of composite fermions coupled to a U (1) gauge field with no ChernSimons term but with the feature that all fields are defined in a noncommutative space-time. An approximate mapping of the long-wavelength, small-amplitude gauge fluctuations yields a commutative effective-field theory which, remarkably, takes the HLR form but with microscopic parameters correctly determined by the interaction strength. Extensions to some other composite Fermi liquids, and to other related states of matter are discussed.