Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Siavash Golkar, Dung X. Nguyen, Dam T. Son

TL;DR

This work derives three exact sum rules linking the stress-tensor spectral densities of gapped fractional quantum Hall states to the topological shift, the quartic structure-factor coefficient, and the high-frequency shear modulus, thereby connecting dynamic gravitational response to static and elastic properties. It proves a Haldane-type bound on $S_4$ and demonstrates Laughlin states saturating it, while introducing a gravitational model in which a single magneto-roton mode saturates the sum rules via an internal metric degree of freedom, yielding an explicit dispersion. The analysis suggests experimental probes, including polarized Raman scattering, to detect the spin-2 magneto-roton at $q=0$ and its momentum-dependent polarization. Overall, the paper ties together topological data, high-frequency elasticity, and gravitational responses in fractional quantum Hall fluids and presents a concrete phenomenology for the magneto-roton as an emergent graviton.

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, ρ_T(ω) and \bar ρ_T(ω), 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 μ_\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.

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

TL;DR

This work derives three exact sum rules linking the stress-tensor spectral densities of gapped fractional quantum Hall states to the topological shift, the quartic structure-factor coefficient, and the high-frequency shear modulus, thereby connecting dynamic gravitational response to static and elastic properties. It proves a Haldane-type bound on and demonstrates Laughlin states saturating it, while introducing a gravitational model in which a single magneto-roton mode saturates the sum rules via an internal metric degree of freedom, yielding an explicit dispersion. The analysis suggests experimental probes, including polarized Raman scattering, to detect the spin-2 magneto-roton at and its momentum-dependent polarization. Overall, the paper ties together topological data, high-frequency elasticity, and gravitational responses in fractional quantum Hall fluids and presents a concrete phenomenology for the magneto-roton as an emergent graviton.

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, ρ_T(ω) and \bar ρ_T(ω), 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 μ_\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.

Paper Structure

This paper contains 10 sections, 42 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The effective action
  3. Review of the Newton-Cartan formalism
  4. Physics at the Coulomb energy scale
  5. Gravitational response, spectral representations and shift sum rule
  6. Static structure factor and high-frequency shear modulus
  7. Inequalities following from the sum rules
  8. A gravitational model of the magneto-roton
  9. Dispersion relation for the magneto-roton
  10. Conclusion

Figures (2)

  • Figure 1: Dispersion relation of the collective mode with $\nu=1/3$ and $\lambda/ (l_B\Delta)=0.3$. Minimum dispersion appears at $k_{min} \approx 1.69 \,l^{-1}_B$.
  • Figure 2: Polarization function $f(k)$, with $\nu=1/3$ , $\lambda/ (l_B\Delta)=0.3$. Linear polarization appears at $k\approx 1.49 l^{-1}_B$.