Chiral Graviton Theory of Fractional Quantum Hall States

Abstract: Recent polarized Raman scattering experiments indicate that fractional quantum Hall systems host a chiral spin-2 neutral collective mode, the long-wavelength limit of the magnetoroton, which behaves as a condensed-matter graviton. We present a nonlinear, gauge-invariant effective theory by gauging area-preserving diffeomorphisms (APDs) with a unimodular spatial metric as the gauge field. A Stueckelberg construction introduces an APD-invariant local potential that aligns the dynamical metric with a reference geometry, opening a tunable gap while preserving gauge redundancy. Together with a geometric Maxwell kinetic sector and the Wen-Zee and gravitational Chern-Simons terms, the theory yields a gapped chiral spin-2 excitation consistent with universal long-wavelength constraints. The tunable gap emerges naturally from symmetry and provides a route to an isotropic-nematic quantum critical point where the spin-2 mode softens. We further establish a linear dictionary to quadrupolar deformations in composite Fermi liquid bosonization, and outline applications to fractional Chern insulators as well as higher-dimensional generalizations. Finally, the approach can be extended to non-Abelian fractional quantum Hall states, capturing both spin-2 and spin-3/2 neutral modes.