Fermion liquids as quantum Hall liquids in non-commutative phase space

The discovery of many strongly correlated metallic phases have inspired novel routes to generalize or go beyond the celebrated Landau Fermi liquid theory. To this end, from universal consideration of symmetries and anomalies, Else, Thorngren and Senthil (ETS) have introduced a class of theories called ersatz Fermi liquids which possess a Fermi surface and satisfy a generalized Luttinger's theorem. In this work, we imagine all such fermion liquids obeying the Luttinger theorem as incompressible quantum Hall liquids in higher-dimensional noncommutative phase space. The non-commutativity naturally introduces additional terms that correspond to the correct topological Chern-Simons action postulated by ETS in one, two and three dimensions. Additionally, our approach also reproduces all the non-topological responses that characterize the semi-classical equations of motion. Finally, our derivation of the Chern-Simons term also verifies a longstanding conjecture in non-commutative field theory.