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.
Highlights
JHEP01(2016)021 the Coulomb energy is much smaller than the cyclotron energy
We prove three sum rules relating these spectral densities with the shift S, the q4 coefficient of the static structure factor S4, and the high-frequency shear modulus of the ground state μ∞, which is precisely defined
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
Summary
One of the authors has proposed the use of nonrelativistic general coordinate invariance as a way to constrain the dynamics of quantum Hall systems [10] (for related work, see refs. [11, 12]). One of the authors has proposed the use of nonrelativistic general coordinate invariance as a way to constrain the dynamics of quantum Hall systems [10] In this paper we will relax the latter condition, allowing for energies comparable to the gap. In this regime, terms with arbitrary number of time derivatives must be taken into account in the effective action. The effective Lagrangian describing the response of a gapped quantum Hall state to external electromagnetic (A0, Ai) and gravitational perturbations (hij), in the massless limit, is: L ν 4π εμνλaμ∂ν aλ ρvμ(∂μφ. In ref. [10] it was found that these two parameters control some quantities, most notably the q2 part of the Hall conductivity at zero frequency (q being the wavenumber of the perturbation)
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
