Composite fermi liquids and non-commutative field theory

The interplay between topology, strong correlations, and kinetic energy presents a new challenge for the theory of quantum matter. In this talk I will describe some recent progress on understanding a simple class of problems where these effects can all be analytically handled. I will first present results on a microscopic lowest Landau theory of the composite fermi liquid state of bosons at filling 1.  Building on work from the 1990s I will derive an effective field theory for this system that takes the form of a non-commutative field theory. I will show that an approximate mapping of this theory to a commutative field theory yield the familiar Halperin-Lee-Read action but with parameters correctly described by the interaction strength. I will describe the effect of a finite bandwidth  introduced to the Landau Level and describe the evolution between the composite Fermi liquid and a boson superfluid. Time permitting, I will describe some generalizations that will include the evolution between a Quantum Anomalous Hall state and a Landau Fermi liquid that may be experimentally accessible in moire graphene systems.