Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences


We present analytic formulas for the ground state energy of two-dimensional (2D) anyon gas in strong magnetic field (Landau level filling factor νL ≤ 1). The formulas are obtained by applying harmonic potential regularization for the vanishing confinement to harmonically confined Coulomb anyon gas. In a case of absence of the Coulomb interaction our analytic result provides an exact solution. It contains a contribution of the anyon gauge field characterized by the anyon parameters ν and νL. In a case of presence of the Coulomb interaction we introduce a function depending on the parameters ν, νL and the density parameter rs. The function is determined by fitting the Fano-Ortolani interpolation equation in the fractional quantum Hall effect regime for spin-polarized electrons and by consistence requirement with known results for the ground state energy of the 2D Coulomb Bose gas in strong magnetic field. We show that our formulas are valid not only for fermions (ν=1) but quite generally for anyons (0 ≤ ν ≤ 1).

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1. Prange R.E. and Girvin S.M. The Quantum Hall Effect, Springer-Verlag, Berlin, (1987).

2. Kato Y.K., Myers R.C., Gossard A.C., and Awschalom D.D. Observation of the Spin Hall Effect in Semiconductors. Science, Vol. 306, 1910–1913 (2004).

3. Wunderlich J., Kaestner B., Sinova J. and Jungwirth T. Experimental Observation of the Spin-Hall Effect in a Two-Dimensional Spin-Orbit. Coupled Semiconductor System. Phys. Rev. Lett., Vol. 94, 047204 (2005).

4. Abdullaev B., Choi M. and Park C.-H. Examination of Current-Induced Magnetic Field in the Slab Geometry: Possible Origin of Spin Hall Effect. J. Korean Phys. Soc., Vol. 49, S638–S641 (2006).

5. Abdullaev B. Implicit Anyon or Single Particle Mechanism of HTCS and Pseudogap Regime. In: A.V. Ling (Ed.), Trends in Boson Research, Nova Science Publishers, N.Y., pp 139-161, (2006); Abdullaev B. and Park C.-H. Bosonization of 2D Fermions due to Spin and Statistical Magnetic Field Coupling and Possible Nature of Superconductivity and Pseudogap Phases Below E g. J. Korean Phys. Soc., Vol. 49, S642-S646 (2006).

6. Laughlin R.B. Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally Charged Excitations. Phys. Rev. Lett. Vol. 50, 1395 (1983).

7. Levesque D., Weis J.J. and MacDonald A.H. Crystallization of the incompressible quantum-fluid state of a two-dimensional electron gas in a strong magnetic field. Phys. Rev. B., Vol. 30, 1056 (1984).

8. Fano G. and Ortolani F. Interpolation formula for the energy of a two-dimensional electron gas in the lowest Landau level. Phys. Rev. B., Vol. 37, 8179 (1988).

9. Ortiz G., Ceperley D.M. and Martin R.M. New stochastic method for systems with broken time-reversal symmetry: 2D fermions in a magnetic field. Phys. Rev. Lett., Vol. 71, 2777 (1993).

10. Price R., Platzman P.M. and He S. Fractional quantum Hall liquid, Wigner solid phase boundary at finite density and magnetic field. Phys. Rev. Lett., Vol. 70, 339 (1993).

11. Wilczek F. Magnetic Flux, Angular Momentum, and Statistics. Phys. Rev. Lett., Vol. 48, 1144–1147 (1982).

12. Leinaas J.M. and Myrheim J. On the Theory of Identical Particles. Nuovo Cimento B., Vol. 37, 1–23 (1977).

13. Halperin B. Statistics of Quasiparticles and the Hierarchy of Fractional Quantized Hall States. Phys. Rev. Lett., Vol. 52, 1583 (1984); Erratum, Phys. Rev. Lett., Vol. 52, 2390 (1984).

14. Forte S. Quantum mechanics and field theory with fractional spin and statistics. Rev. Mod. Phys., Vol. 64, 193 (1992); Quantum Hall Effect, Ed. Stone M., World Scientific, Singapore, (1992).

15. Johnson M.D. and Canright G.S. Anyons in a magnetic field. Phys. Rev. B., Vol. 41, 6870 (1990); Vercin A., Phys. Lett. B., Vol. 260, 120 (1991).

16. Grundberg J., Hansson T.H., Karlhede A. and Westerberg E. Landau levels for anyons. Phys. Rev. B., Vol. 44, 8373 (1991).

17. Roy B., Roy P. and Varshni Y.P. Two anyons in a constant magnetic field: a nonperturbative approach. Mod. Phys. Lett. B., Vol. 8, 159 (1994).

18. Perez R. and Gonsales A. Few-anyon systems in a parabolic dot. Phys. Rev. B., Vol. 58, 7412 (1998).

19. Khare A., Fractional Statistics and Quantum Theory, World Scientific, Singapore, (1997).

20. Hansson T.H., Leinaas J.M. and Myrheim J. Dimensional reduction in anyon systems. Nucl. Phys. B., Vol. 384, 559–580 (1992).

21. Calogero F. Solution of a Three-Body Problem in One Dimension. J. Math. Phys., Vol. 10, 2191–2197 (1969).

22. Haldane F.D. Fractional statistics in arbitrary dimensions: A generalization of the Pauli principle Phys. Rev. Lett., Vol. 67, 937 (1991).

23. Ouvry S. On the relation between the anyon and Calogero models. Phys. Lett. B., Vol. 510, 335–340 (2001).

24. Abdullaev B., Rössler U. and Musakhanov M. An analytic approach to the ground state energy of charged anyon gases. Phys. Rev. B., Vol. 76, 075403(1–7) (2007).

25. Abdullaev B., Ortiz G., Rössler U., Musakhanov M. and Nakamura A. Approximate ground state of a confined Coulomb anyon gas in an external magnetic field. Phys. Rev. B., Vol. 68, 165105(1–9) (2003).

26. Wensauer A. and Rössler U. Exchange-correlation energy densities for two-dimensional systems from quantum dot ground states. Phys. Rev. B., Vol. 69, 155301 (2004).

27. Yoshioka D. Ground state of the two-dimensional charged particles in a strong magnetic field and the fractional quantum Hall effect. Phys. Rev. B., Vol. 29, 6833 (1984).

28. Lerda A. Anyons. Springer-Verlag, Berlin, (1992).

29. Chitra R. and Sen D. Ground state of many anyons in a harmonic potential. Phys. Rev. B., Vol. 46, 10923 (1992).

30. Wu Y.-S. Multiparticle Quantum Mechanics Obeying Fractional Statistics. Phys. Rev. Lett., Vol. 53, 111–115 (1984); Erratum, Phys. Rev. Lett., Vol. 53, 1028 (1984); Laughlin R.B. Superconducting Ground State of Noninteracting Particles Obeying Fractional Statistics. Phys. Rev. Lett., Vol. 60, 2677 (1988).

31. Fetter A.L., Hanna C.B. and Laughlin R.B. Random-phase approximation in the fractional-statistics gas. Phys. Rev. B., Vol. 39, 9679 (1989).

32. Galicki V.M., Karnakov B.M. and Kogan V.I. Problems in Quantum Mechanics, Nauka, Moscow, (1981) (in russian).

33. Dasnieres de Veigy A. and Ouvry S. Equation of state of an anyon gas in a strong magnetic field. Phys. Rev. Lett., Vol. 72, 600 (1994).

34. Sen D. and Chitra R. Anyons as perturbed bosons. Phys. Rev. B., Vol. 45, 881 (1991).



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