# Algebraic approach to fractional quantum Hall effect

## Abstract

We construct an algebraic description for the ground state and for the static response of the quantum Hall plateaus with a filling factor ν = N/(2N + 1) in the large N limit. Here, by analyzing the algebra of the fluctuations of the shape of the Fermi surface of the composite fermions, we find the explicit form of the projected static structure (SSF) factor at large N and fixed z = (2N + 1)qℓ _{B} ~ 1, where q is the wave number at which the system is probed, and ℓ _{B} is the magnetic length. When z < 3.8, we obtain the exact universal formula for the projected SSF. The formula does not depend on the particular form of the Hamiltonian.

- Authors:

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

- Publication Date:

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

- Sponsoring Org.:
- USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25)

- OSTI Identifier:
- 1594024

- Alternate Identifier(s):
- OSTI ID: 1488626

- Grant/Contract Number:
- SC0009924; FG02-13ER41958

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Physical Review B

- Additional Journal Information:
- Journal Volume: 98; Journal Issue: 24; Journal ID: ISSN 2469-9950

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; Fractional quantum Hall effect; Bosonization; Fermi liquid theory

### Citation Formats

```
Nguyen, Dung Xuan, and Son, Dam Thanh. Algebraic approach to fractional quantum Hall effect. United States: N. p., 2018.
Web. doi:10.1103/PhysRevB.98.241110.
```

```
Nguyen, Dung Xuan, & Son, Dam Thanh. Algebraic approach to fractional quantum Hall effect. United States. doi:10.1103/PhysRevB.98.241110.
```

```
Nguyen, Dung Xuan, and Son, Dam Thanh. Sat .
"Algebraic approach to fractional quantum Hall effect". United States. doi:10.1103/PhysRevB.98.241110. https://www.osti.gov/servlets/purl/1594024.
```

```
@article{osti_1594024,
```

title = {Algebraic approach to fractional quantum Hall effect},

author = {Nguyen, Dung Xuan and Son, Dam Thanh},

abstractNote = {We construct an algebraic description for the ground state and for the static response of the quantum Hall plateaus with a filling factor ν = N/(2N + 1) in the large N limit. Here, by analyzing the algebra of the fluctuations of the shape of the Fermi surface of the composite fermions, we find the explicit form of the projected static structure (SSF) factor at large N and fixed z = (2N + 1)qℓB ~ 1, where q is the wave number at which the system is probed, and ℓB is the magnetic length. When z < 3.8, we obtain the exact universal formula for the projected SSF. The formula does not depend on the particular form of the Hamiltonian.},

doi = {10.1103/PhysRevB.98.241110},

journal = {Physical Review B},

number = 24,

volume = 98,

place = {United States},

year = {2018},

month = {12}

}

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Works referenced in this record:

##
Two-Dimensional Magnetotransport in the Extreme Quantum Limit

journal, May 1982

- Tsui, D. C.; Stormer, H. L.; Gossard, A. C.
- Physical Review Letters, Vol. 48, Issue 22

##
Is the Composite Fermion a Dirac Particle?

journal, September 2015

- Son, Dam Thanh
- Physical Review X, Vol. 5, Issue 3

##
Hall Viscosity and Electromagnetic Response

journal, February 2012

- Hoyos, Carlos; Son, Dam Thanh
- Physical Review Letters, Vol. 108, Issue 6

##
Particle-hole symmetry and composite fermions in fractional quantum Hall states

journal, May 2018

- Nguyen, Dung Xuan; Golkar, Siavash; Roberts, Matthew M.
- Physical Review B, Vol. 97, Issue 19

##
Non-Abelian adiabatic statistics and Hall viscosity in quantum Hall states and ${p}_{x}+i{p}_{y}$ paired superfluids

journal, January 2009

- Read, N.
- Physical Review B, Vol. 79, Issue 4

##
Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally Charged Excitations

journal, May 1983

- Laughlin, R. B.
- Physical Review Letters, Vol. 50, Issue 18

##
Bimetric Theory of Fractional Quantum Hall States

journal, November 2017

- Gromov, Andrey; Son, Dam Thanh
- Physical Review X, Vol. 7, Issue 4

##
Microscopic theory of the fractional quantum Hall effect

journal, April 1992

- Jain, J. K.
- Advances in Physics, Vol. 41, Issue 2

##
Framing Anomaly in the Effective Theory of the Fractional Quantum Hall Effect

journal, January 2015

- Gromov, Andrey; Cho, Gil Young; You, Yizhi
- Physical Review Letters, Vol. 114, Issue 1

##
Particle-vortex duality of two-dimensional Dirac fermion from electric-magnetic duality of three-dimensional topological insulators

journal, June 2016

- Metlitski, Max A.; Vishwanath, Ashvin
- Physical Review B, Vol. 93, Issue 24

##
Dual Dirac Liquid on the Surface of the Electron Topological Insulator

journal, November 2015

- Wang, Chong; Senthil, T.
- Physical Review X, Vol. 5, Issue 4

##
Geometry of quantum Hall states: Gravitational anomaly and transport coefficients

journal, November 2015

- Can, Tankut; Laskin, Michael; Wiegmann, Paul B.
- Annals of Physics, Vol. 362

##
Effective-Field-Theory Model for the Fractional Quantum Hall Effect

journal, January 1989

- Zhang, S. C.; Hansson, T. H.; Kivelson, S.
- Physical Review Letters, Vol. 62, Issue 1

##
Anomalous dynamical scaling from nematic and U(1) gauge field fluctuations in two-dimensional metals

journal, July 2015

- Holder, Tobias; Metzner, Walter
- Physical Review B, Vol. 92, Issue 4

##
Particle-Hole Duality in the Lowest Landau Level

journal, May 2017

- Nguyen, Dung Xuan; Can, Tankut; Gromov, Andrey
- Physical Review Letters, Vol. 118, Issue 20

##
Fractional quantum Hall systems near nematicity: Bimetric theory, composite fermions, and Dirac brackets

journal, May 2018

- Nguyen, Dung Xuan; Gromov, Andrey; Son, Dam Thanh
- Physical Review B, Vol. 97, Issue 19

##
Magneto-roton theory of collective excitations in the fractional quantum Hall effect

journal, February 1986

- Girvin, S. M.; MacDonald, A. H.; Platzman, P. M.
- Physical Review B, Vol. 33, Issue 4

##
Theory of the half-filled Landau level

journal, March 1993

- Halperin, B. I.; Lee, Patrick A.; Read, Nicholas
- Physical Review B, Vol. 47, Issue 12

##
Fermi wave vector for the partially spin-polarized composite-fermion Fermi sea

journal, December 2017

- Balram, Ajit C.; Jain, J. K.
- Physical Review B, Vol. 96, Issue 23

##
Fractional Quantum Hall Effect in a Curved Space: Gravitational Anomaly and Electromagnetic Response

journal, July 2014

- Can, T.; Laskin, M.; Wiegmann, P.
- Physical Review Letters, Vol. 113, Issue 4

##
Geometrical Description of the Fractional Quantum Hall Effect

journal, September 2011

- Haldane, F. D. M.
- Physical Review Letters, Vol. 107, Issue 11

##
Higher-Spin Theory of the Magnetorotons

journal, November 2016

- Golkar, Siavash; Nguyen, Dung Xuan; Roberts, Matthew M.
- Physical Review Letters, Vol. 117, Issue 21

##
Fractional quantum Hall effect and Chern-Simons gauge theories

journal, September 1991

- Lopez, Ana; Fradkin, Eduardo
- Physical Review B, Vol. 44, Issue 10

##
Shift and Spin Vector: New Topological Quantum Numbers for the Hall Fluids

journal, November 1992

- Wen, X. G.; Zee, A.
- Physical Review Letters, Vol. 69, Issue 20

##
Conformal invariance of chiral edge theories

journal, June 2009

- Read, N.
- Physical Review B, Vol. 79, Issue 24

##
Composite-fermion approach for the fractional quantum Hall effect

journal, July 1989

- Jain, J. K.
- Physical Review Letters, Vol. 63, Issue 2

##
Erratum: Framing Anomaly in the Effective Theory of the Fractional Quantum Hall Effect [Phys. Rev. Lett. **114** , 016805 (2015)]

journal, April 2015

- Gromov, Andrey; Cho, Gil Young; You, Yizhi
- Physical Review Letters, Vol. 114, Issue 14

##
Quantum phase transitions of metals in two spatial dimensions. I. Ising-nematic order

journal, August 2010

- Metlitski, Max A.; Sachdev, Subir
- Physical Review B, Vol. 82, Issue 7