Fermion liquids as quantum Hall liquids in phase space: A unified approach for anomalies and responses

TL;DR

This paper presents a unified framework viewing fermion liquids satisfying Luttinger's theorem as quantum Hall liquids in phase space, deriving their effective field theories and responses using noncommutative geometry and the Seiberg-Witten map.

Contribution

It introduces a novel approach to derive effective theories for fermion liquids as quantum Hall states in phase space, connecting anomalies, topological responses, and noncommutative geometry.

Findings

Derives Chern-Simons actions in multiple dimensions consistent with previous conjectures.

Reproduces key response terms including semiclassical equations of motion.

Verifies a longstanding conjecture in noncommutative field theory.

Abstract

The discovery of many strongly correlated metallic phases has inspired different 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 view all such fermion liquids obeying the Luttinger theorem as incompressible quantum Hall liquids in higher-dimensional phase space and use it as the starting point to derive their effective low-energy field theory. The noncommutativity of phase space motivates us to use the Seiberg-Witten map to derive the field theory in an ordinary (commutative) space and naturally leads to 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 terms that characterize important contributions to the response, including the semiclassical equations of motion. Finally, our derivations of Chern-Simons terms from the Seiberg-Witten map also verify a longstanding conjecture in noncommutative field theory.