Noncommutative field theory approach to the fractional quantum Hall effect with the filling factor one–half

Sendic Estrada–Jiménez and Hugo García–Compeán

Departamento de Física Centro de Investigación y de Estudios Avanzados del IPN Apdo. Postal 14–740, 07000, México D.F., México

Recibido el 18 de julio de 2005
Aceptado el 14 de marzo de 2005

Abstract

The fractional quantum Hall effect is studied in the context of the noncommutative quantum field theory in (2+1) dimensions. For the filling factor v = 1/2, the noncommutative effective field theory incorporates a Chern–Simons gauge field (in the temporal gauge) coupled to the matter in the presence of a suitable quenched external magnetic field. After providing the Feynman rules for this system, the noncommutative corrections to the self–energy of quasiparticles are computed, showing that it is zero at Hartree–Fock approximation. Finally, in this approach it is proved that the density ρ satisfies a noncommutative deformation of the w_∞–algebra.

Keywords: Noncommutative field theory; fractional quantum Hall effect; Chern–Simons theory.

Resumen

El efecto Hall cuántico fraccionario se estudia en el contexto de la teoría cuántica de campos no conmutativa en (2+1) dimensiones. Para el factor de llenado v = 1/2, la teoría de campos efectiva incorpora un campo de norma de Chern–Simons (en la norma temporal) acoplado a la materia en la presencia de un campo magnético externo apropiadamente cancelado. Después de dar las reglas de Feynman para este sistema, las correcciones no conmutativas de la autoenegía de las cuasipartículas son calculadas y se muestra que son cero en la aproximación de Hartree–Fock. Finalmente, en este enfoque se prueba que la densidad ρ satisface una deformación no conmutativa del álgebra w_∞.

Descriptores: Teoría de campos no conmutativa; efecto Hall cuántico fraccionario; teoría de Chern–Simons.

PACS: 11.10.Nx;73.43.Lp; 11.10.Gh

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Acknowledgments

This work was supported in part by a CONACyT México grant No. 33951E. The work of S.E.–J. is supported by a CONACyT graduate fellowship. S. Estrada–Jiménez whishes to thank Prof. Yong–Shi Wu for useful discussions and suggestions concerning the subject of this paper and for his kind hospitality at the University of Utah, where part of this work was outlined.

References

1. M.R. Douglas and N. A. Nekrasov, Rev. Mod. Phys. 73 (2002) 977.        [ Links ]

2. R.J. Szabo, Phys. Rep. 378 (2003) 207.        [ Links ]

3. L. Susskind, "The Quantum Hall Fluid and Noncommutative Chern–Simons Theory", hep–th/0101029.        [ Links ]

4. E. Fradkin, V. Jejjala and R.G. Leigh, Nucl. Phys. B 642 (2002) 483.        [ Links ]

5. J.L.F. Barbon and A. Paredes, Int. J. Mod. Phys. A 17 (2002) 3589.        [ Links ]

6. J. Gomis, K. Landsteiner and E. Lopez, Phys. Rev D 62 (2000) 105006.        [ Links ]

7. S. Hellerman and M. V. Raamsdonk, JHEP 0110 (2001) 039.        [ Links ]

8. A. Jellal, Mod. Phys. Lett. A 18 (2003) 1473.        [ Links ]

9.  J. Polchinski, "Effective Field Theory and the Fermi Surface", in Recent Directions in Particle Theory: From Superstrings and Black Holes to the Standard Model (Tasi 92) Eds. J. Harvey and J. Polchinski, (River Edge, N.J., World Scientific, 1993) p. 827, hep–th/9210046.        [ Links ]

10. N. Macris and S. Ouvry, J. Phys. A 35 (2002) 4477.        [ Links ]

11. B.I. Halperin, P.A. Lee and N. Read, Phys. Rev. B 47 (1993) 7312.        [ Links ]

12. A. Stern and B.I. Halperin, Phys. Rev. B 52 (1995) 5890.        [ Links ]

13. Y. Yu, Z.–B. Su, and X. Dai, Phys. Rev. B 57 (1998) 9897.        [ Links ]

14. F. Wilczek (ed), Fractional Statistics and Anyon Superconductivity, (World Scientific, Singapore, 1990)        [ Links ]

15. R. Shankar and G. Murthy, Phys. Rev. Lett. 79 (1997) 4437.        [ Links ]

16. S. M. Girvin and T. Jach, Phys. Rev. A 29 (1984) 5617, afterwords in Physics in Noncommutative World Eds.         [ Links ] Miao Li and Yong–Shi Wu. (Rinton Press, Princeton NJ, 2002).        [ Links ]

17. N. Read, Phys. Rev B 58 (1998) 16262.        [ Links ]