Journal of Statistical Physics

, Volume 158, Issue 5, pp 1147–1180 | Cite as

Exact and Asymptotic Features of the Edge Density Profile for the One Component Plasma in Two Dimensions

Article

Abstract

There is a well known analogy between the Laughlin trial wave function for the fractional quantum Hall effect, and the Boltzmann factor for the two-dimensional one-component plasma. The latter requires continuation beyond the finite geometry used in its derivation. We consider both disk and cylinder geometry, and focus attention on the exact and asymptotic features of the edge density. At the special coupling \(\Gamma := q^2/k_BT=2\) the system is exactly solvable. In particular the \(k\)-point correlation can be written as a \(k \times k\) determinant, allowing the edge density to be computed to first order in \(\Gamma - 2\). A double layer structure is found, which in turn implies an overshoot of the density as the edge of the leading support is approached from the interior. Asymptotic analysis shows that the deviation from the leading order (step function) value is different for the interior and exterior directions. For general \(\Gamma \), a Gaussian fluctuation formula is used to study the large deviation form of the density for \(N\) large but finite. This asymptotic form involves thermodynamic quantities which we independently study, and moreover an appropriate scaling gives the asymptotic decay of the limiting edge density outside of the plasma.

Keywords

Two dimensional Coulomb system Fractional quantum Hall effect Edge density profile 

References

  1. 1.
    Alastuey, A., Jancovici, B.: On the two-dimensional one-component Coulomb plasma. J. Phys. 42, 1–12 (1981)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Ameur, Y., Hedenmalm, H., Makarov, N.: Fluctuations of eigenvalues of random matrices. Duke Math. J. 159, 31–81 (2011)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Bernevig, B.A., Haldane, F.D.M.: Model fractional quantum Hall states and Jack polynomials. Phys. Rev. Lett. 100, 246802 (2008)CrossRefADSGoogle Scholar
  4. 4.
    Bernevig, B.A., Regnault, N.: The anatomy of Abelian and non-Abelian fractional quantum Hall states. Phys. Rev. Lett. 103, 206801 (2009)CrossRefADSGoogle Scholar
  5. 5.
    Can, T., Forrester, P.J., Téllez, G., Wiegmann, P.: Singular behaviour at the edge of Laughlin states. Phys. Rev. B 89, 235137 (2014)CrossRefADSGoogle Scholar
  6. 6.
    Choquard, Ph, Forrester, P.J., Smith, E.R.: The two-dimensional one-component plasma at \(\Gamma = 2\): the semiperiodic strip. J. Stat. Phys. 33, 13–22 (1983)CrossRefADSMATHMathSciNetGoogle Scholar
  7. 7.
    Ciftja, O., Wexler, C.: Monte Carlo simulation method for Laughlin-like states in a disk geometry. Phys. Rev. B 67, 075304 (2003)CrossRefADSGoogle Scholar
  8. 8.
    Datta, N., Morf, R., Ferrari, R.: Edge of the Laughlin droplet. Phys. Rev. B 53, 10906–10915 (1996)CrossRefADSGoogle Scholar
  9. 9.
    Forrester, P.J.: Finite size corrections to the free energy of Coulomb systems with a periodic boundary condition. J. Stat. Phys. 63, 491–504 (1991)CrossRefADSMathSciNetGoogle Scholar
  10. 10.
    Forrester, P.J.: Fluctuation formula for complex random matrices. J. Phys. A 32, L159–L163 (1999)CrossRefADSMATHMathSciNetGoogle Scholar
  11. 11.
    Forrester, P.J.: Log-Gases and Random Matrices. Princeton University Press, Princeton (2010)CrossRefMATHGoogle Scholar
  12. 12.
    Forrester, P.J.: Spectral density asymptotics for Gaussian and Laguerre \(\beta \)-ensembles in the exponentially small region. J. Phys. A 45, 075206 (2012)CrossRefADSMathSciNetGoogle Scholar
  13. 13.
    Forrester, P.J.: Large deviation eigenvalue density for the soft edge Laguerre and Jacobi \(\beta \)-ensembles. J. Phys. A 45, 145201 (2012)CrossRefADSMathSciNetGoogle Scholar
  14. 14.
    Forrester, P.J., Mays, A.: A method to calculate correlation functions for \(\beta = 1\) random matrices of odd size. J. Stat. Phys. 134, 443–462 (2009)CrossRefADSMATHMathSciNetGoogle Scholar
  15. 15.
    Ginibre, J.: Statistical ensembles of complex, quaternion, and real matrices. J. Math. Phys. 6, 440–440 (1965)CrossRefADSMATHMathSciNetGoogle Scholar
  16. 16.
    Jancovici, B.: Exact results for the two-dimensional one-component plasma. Phys. Rev. Lett. 46, 386–388 (1981)CrossRefADSMathSciNetGoogle Scholar
  17. 17.
    Jancovici, B., Manificat, G., Pisani, C.: Coulomb systems seen as critical systems: finite-size effects in two dimensions. J. Stat. Phys. 76, 307–330 (1994)CrossRefADSMATHGoogle Scholar
  18. 18.
    Laughlin, R.B.: Anomalous quantum Hall effect: an incompressible quantum fluid with fractionally charge excitations. Phys. Rev. Lett. 50, 1395–1398 (1983)CrossRefADSGoogle Scholar
  19. 19.
    Morf, R., Halperin, B.I.: Monte Carlo evaluation of trial wave functions for the fractional quantized Hall effect: disk geometry. Phys. Rev. B 33, 2221–2246 (1986)CrossRefADSGoogle Scholar
  20. 20.
    Rider, B., Virág, B.: The noise in the circular law and the Gaussian free field, IMRN 2007 (2007), rnm006Google Scholar
  21. 21.
    Saff, E.B., Totik, V.: Logarithmic Potentials with External Fields. Springer, Berlin (1997)CrossRefMATHGoogle Scholar
  22. 22.
    Šamaj, L., Wagner, J., Kalinay, P.: Translation symmetry breaking in the one-component plasma on the cylinder. J. Stat. Phys. 117, 159–178 (2004)CrossRefADSMATHGoogle Scholar
  23. 23.
    Sari, R.R., Merlini, D.: On the \(\nu \)-dimensional one-component classical plasma: the thermodynamic limit revisited. J. Stat. Phys. 76, 91–100 (1976)CrossRefADSMathSciNetGoogle Scholar
  24. 24.
    Shakirov, S.: Exact solution for mean energy of 2d Dyson gas at \(\beta = 1\). Phys. Lett. A 375, 984–989 (2011)CrossRefADSMATHMathSciNetGoogle Scholar
  25. 25.
    Téllez, G.: Exactly solvable models in statistical mechanics of Coulomb systems. Rev. Acad. Colomb. Cienc. 37, 61–74 (2013)Google Scholar
  26. 26.
    Téllez, G., Forrester, P.J.: Finite size study of the 2dOCP at \(\Gamma =4\) and \(\Gamma =6\). J. Stat. Phys. 97, 489–521 (1999)CrossRefADSMATHGoogle Scholar
  27. 27.
    Téllez, G., Forrester, P.J.: Expanded Vandermonde powers and sum rules for the two-dimensional one-component plasma. J. Stat. Phys. 148, 824–855 (2012)CrossRefADSMATHMathSciNetGoogle Scholar
  28. 28.
    Thouless, D.J.: Theory of the quantised Hall effect. Surf. Sci. 142, 147–154 (1984)CrossRefADSGoogle Scholar
  29. 29.
    Wiegmann, P.: Nonlinear hydrodynamics and fractionally quantized solitons at the fractional quantum Hall edge. Phys. Rev. Lett. 108, 206810 (2012)CrossRefADSGoogle Scholar
  30. 30.
    Zabrodin, A., Wiegmann, P.: Large-\(N\) expansion for the 2D Dyson gas. J. Phys. A 39, 8933 (2006)CrossRefADSMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • T. Can
    • 1
    • 2
  • P. J. Forrester
    • 3
  • G. Téllez
    • 4
  • P. Wiegmann
    • 1
  1. 1.Department of PhysicsUniversity of ChicagoChicagoUSA
  2. 2.Simons Center for Geometry and PhysicsStony Brook UniversityStony BrookUSA
  3. 3.Department of Mathematics and StatisticsUniversity of MelbourneParkvilleAustralia
  4. 4.Departamento de FísicaUniversidad de Los AndesBogotáColombia

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