Density-curvature response and gravitational anomaly

TL;DR

The work derives Ward-identity constraints from local Galilean invariance for two-dimensional gapped systems in a background magnetic field, linking electromagnetic and elastic responses to gravitational data. By decomposing background fields and constructing a quadratic effective action, it connects Hall conductivity, Hall viscosity, and orbital spin to geometric terms (Wen–Zee, gravitational Chern–Simons) and introduces a bulk density-curvature response coefficient $b$ tied to the chiral central charge $c$. A central result is the relation $b = \nu \bar{s}(1-\bar{s}) + \frac{c}{12}$, generalizable to Abelian ($K$-matrix) states, which, together with $K$-matrix expressions for the thermal Hall effect, links bulk bulk responses to edge physics. The framework reproduces known limits (Kohn’s theorem, Wen–Zee shift) and yields predictions for a broad class of quantum Hall states, with potential extensions beyond strictly Galilean-invariant systems.

Abstract

We study constraints imposed by the Galilean invariance on linear electromagnetic and elastic responses of two-dimensional gapped systems in background magnetic field. Exact relations between response functions following from the Ward identities are derived. In addition to viscosity-conductivity relations known in literature we find new relations between the density-curvature response and the chiral central charge.