## Spectral sum rules and magneto-roton as emergent graviton in fractional quantum Hall effect

## Abstract

Here, 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 $$\bar{p}$$ _{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 _{∞}, 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.

- Authors:

- Univ. of Chicago, Chicago, IL (United States)

- Publication Date:

- Research Org.:
- Univ. of Chicago, IL (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1327326

- Grant/Contract Number:
- FG02-90ER40560

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Journal of High Energy Physics (Online)

- Additional Journal Information:
- Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2016; Journal Issue: 1; Journal ID: ISSN 1029-8479

- Publisher:
- Springer Berlin

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; field theories in lower dimensions; sum rules; effective field theories

### Citation Formats

```
Golkar, Siavash, Nguyen, Dung X., and Son, Dam T. Spectral sum rules and magneto-roton as emergent graviton in fractional quantum Hall effect. United States: N. p., 2016.
Web. doi:10.1007/JHEP01(2016)021.
```

```
Golkar, Siavash, Nguyen, Dung X., & Son, Dam T. Spectral sum rules and magneto-roton as emergent graviton in fractional quantum Hall effect. United States. doi:10.1007/JHEP01(2016)021.
```

```
Golkar, Siavash, Nguyen, Dung X., and Son, Dam T. Tue .
"Spectral sum rules and magneto-roton as emergent graviton in fractional quantum Hall effect". United States. doi:10.1007/JHEP01(2016)021. https://www.osti.gov/servlets/purl/1327326.
```

```
@article{osti_1327326,
```

title = {Spectral sum rules and magneto-roton as emergent graviton in fractional quantum Hall effect},

author = {Golkar, Siavash and Nguyen, Dung X. and Son, Dam T.},

abstractNote = {Here, 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, ρT(ω) and $\bar{p}$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 q4 coefficient of the static structure factor S4, and the high-frequency shear modulus of the ground state μ∞, which is precisely defined. We confirm an inequality, first suggested by Haldane, that S4 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.},

doi = {10.1007/JHEP01(2016)021},

journal = {Journal of High Energy Physics (Online)},

number = 1,

volume = 2016,

place = {United States},

year = {2016},

month = {1}

}

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