Newton-Cartan Geometry and the Quantum Hall Effect

TL;DR

This work constructs a nonrelativistic, diffeomorphism-invariant EFT for quantum Hall states by adopting Newton-Cartan geometry as the natural formalism. The approach yields a massless limit that remains regular (notably at $g=2$) and separates universal transport properties from nonuniversal details via an $S_0$ functional. A key result is a diffeomorphism-covariant action that combines a Chern-Simons term with a spin-connection-coupled composite boson, encoding the shift and Hall viscosity, and predicting a universal $q^2$ correction to the Hall conductivity. The framework clarifies the geometric origin of universal QH responses and sets the stage for extensions to edge dynamics and holographic models, while highlighting the role of the internal metric and higher-derivative terms in capturing beyond-hydrodynamic behavior.

Abstract

We construct an effective field theory for quantum Hall states, guided by the requirements of nonrelativistic general coordinate invariance and regularity of the zero mass limit. We propose Newton-Cartan geometry as the most natural formalism to construct such a theory. Universal predictions of the theory are discussed.