Spectral Sum Rules and Magneto-Roton as Emergent Graviton in Fractional Quantum Hall Effect

TL;DR

This paper establishes spectral sum rules linking graviton absorption probabilities to fundamental quantum Hall parameters, revealing the magneto-roton as an emergent graviton-like excitation and connecting it to the Laughlin state and internal metric oscillations.

Contribution

It introduces spectral densities and sum rules that relate graviton absorption to quantum Hall parameters, identifying the magneto-roton as an emergent graviton and linking it to internal metric oscillations.

Findings

Sum rules relate spectral densities to shift, structure factor, and shear modulus.

Laughlin wavefunction saturates the bound, indicating polarization preference.

Magneto-roton arises from mixing internal metric oscillations and hydrodynamics.

Abstract

We consider gapped fractional quantum Hall states on the lowest Landau level when the Coulomb energy is much smaller than the cyclotron energy. We introduce two spectral densities, \rho_T(\omega) and \bar \rho_T(\omega), which are proportional to the probabilities of absorption of circularly polarized gravitons by the quantum Hall system. We prove three sum rules relating these spectral densities with the shift S, the q^4 coefficient of the static structure factor S_4, and the high-frequency shear modulus of the ground state \mu_\infty, which is precisely defined. We confirm an inequality, first suggested by Haldane, that S_4 is bounded from below by |S-1|/8. The Laughlin wavefunction saturates this bound, which we argue to imply that systems with ground state wavefunctions close to Laughlin's absorb gravitons of predominantly one circular polarization. We consider a nonlinear model where the sum rules are saturated by a single magneto-roton mode. In this model, the magneto-roton arises from the mixing between oscillations of an internal metric and the hydrodynamic motion. Implications for experiments are briefly discussed.