Wigner Crystallization in the Quantum 1D Jellium at All Densities

Abstract

The jellium is a model, introduced by Wigner (Phys Rev 46(11):1002, 1934), for a gas of electrons moving in a uniform neutralizing background of positive charge. Wigner suggested that the repulsion between electrons might lead to a broken translational symmetry. For classical one-dimensional systems this fact was proven by Kunz (Ann Phys 85(2):303–335, 1974), while in the quantum setting, Brascamp and Lieb (Functional integration and its applications. Clarendon Press, Oxford, 1975) proved translation symmetry breaking at low densities. Here, we prove translation symmetry breaking for the quantum one-dimensional jellium at all densities.

This is a preview of subscription content, access via your institution.

References

  1. AGL01

    Aizenman M., Goldstein S., Lebowitz J.L.: Bounded fluctuations and translation symmetry breaking in one-dimensional particle systems. J. Stat. Phys. 103(3), 601–618 (2001)

    MATH  MathSciNet  Article  Google Scholar 

  2. AJJ10

    Aizenman M., Jansen S., Jung P.: Symmetry breaking in quasi-1D Coulomb systems. Ann. Henri Poincaré 11(8), 1–33 (2010)

    MathSciNet  Article  Google Scholar 

  3. AM80

    Aizenman M., Martin P.A.: Structure of Gibbs states of one dimensional Coulomb systems. Commun. Math. Phys. 78(1), 99–116 (1980)

    ADS  MathSciNet  Article  Google Scholar 

  4. Bax63

    Baxter, R.J.: Statistical mechanics of a one-dimensional Coulomb system with a uniform charge background. In: Le Couteur, K.J. (ed.) Mathematical Proceedings of the Cambridge Philosophical Society, vol. 59, pp. 779–787. Cambridge University Press, Cambridge (1963)

  5. BL75

    Brascamp, H.J., Lieb, E.H.: Some inequalities for Gaussian measures and the long-range order of the one-dimensional plasma. In: Arthurs, A.M. (ed.) Functional Integration and Its Applications. Clarendon Press, Oxford (1975)

  6. BR97

    Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics 2: Equilibrium States. Models in Quantum Statistical Mechanics. 2nd edn. Springer, Berlin (1997)

  7. DB08

    Deshpande V.V., Bockrath M.: The one-dimensional Wigner crystal in carbon nanotubes. Nat. Phys. 4(4), 314–318 (2008)

    Article  Google Scholar 

  8. DVJ03

    Daley D.J., Vere-Jones D.: An Introduction to the Theory of Point Processes, vol. 1. Elementary Theory and Methods. 2nd edn. Springer, Berlin (2003)

    Google Scholar 

  9. Gin65

    Ginibre J.: Reduced density matrices of quantum gases. I. Limit of infinite volume. J. Math. Phys. 6, 238 (1965)

    ADS  MATH  MathSciNet  Article  Google Scholar 

  10. KM59

    Karlin S., McGregor J.: Coincidence probabilities. Pac. J. Math. 9(4), 1141–1164 (1959)

    MATH  MathSciNet  Article  Google Scholar 

  11. KR48

    Krein M.G., Rutman M.A.: Linear operators leaving invariant a cone in a Banach space. Uspekhi Mat. Nauk. 3(1), 3–95 (1948)

    MATH  MathSciNet  Google Scholar 

  12. Kun74

    Kunz H.: The one-dimensional classical electron gas. Ann. Phys. 85(2), 303–335 (1974)

    ADS  MathSciNet  Article  Google Scholar 

  13. LN76

    Lieb E.H., Narnhofer H.: The thermodynamic limit for jellium. J. Stat. Phys. 14(5), 465–465 (1976)

    ADS  MathSciNet  Article  Google Scholar 

  14. RS80

    Reed M.C., Simon B.: Methods of Modern Mathematical Physics, vol. 2: Fourier Analysis, Self-adjointness. Gulf Professional Publishing, Houston (1980)

    Google Scholar 

  15. Sch93

    Schulz H.J.: Wigner crystal in one dimension. Phys. Rev. Lett. 71(12), 1864–1867 (1993)

    ADS  Article  Google Scholar 

  16. Sim79

    Simon B.: Functional Integration and Quantum Physics, vol. 86. Academic Press, New York (1979)

    Google Scholar 

  17. Wig34

    Wigner E.: On the interaction of electrons in metals. Phys. Rev. 46(11), 1002 (1934)

    ADS  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to P. Jung.

Additional information

Communicated by H. Spohn

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Jansen, S., Jung, P. Wigner Crystallization in the Quantum 1D Jellium at All Densities. Commun. Math. Phys. 331, 1133–1154 (2014). https://doi.org/10.1007/s00220-014-2032-y

Download citation

Keywords

  • Point Process
  • Open Loop
  • Thermodynamic Limit
  • Weyl Chamber
  • Brownian Bridge