Non-commutative field theory and composite Fermi Liquids in some quantum Hall systems

TL;DR

This paper develops a non-commutative field theory framework for composite Fermi liquids in quantum Hall systems, clarifying the HLR theory's validity at the microscopic level for bosons at filling factor 1.

Contribution

It introduces a non-commutative field theory description of composite Fermi liquids, connecting it to the traditional HLR theory with accurately determined microscopic parameters.

Findings

Formulation of a non-commutative field theory for bosonic composite Fermi liquids.

Derivation of an effective commutative field theory resembling HLR with correct parameters.

Discussion of extensions to other composite Fermi liquids and related states.

Abstract

Composite Fermi liquid metals arise at certain special filling fractions in the quantum Hall regime and play an important role as parent states 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 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 {\em non-commutative} field theory. This theory has a Fermi surface of composite fermions coupled to a $U(1)$ gauge field with no Chern-Simons term but with the feature that all fields are defined in a non-commutative spacetime. 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.