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A Model for Crystallization: A Variation on the Hubbard Model

  • Elliott H. Lieb
Chapter

Abstract

A quantum mechanical lattice model of fermionic electrons interacting with infinitely massive nuclei is considered. (It can be viewed as a modified Hubbard model in which the spin-up electrons are not allowed to hop.) The electron—nucleus potential is “on-site” only. Neither this potential alone nor the kinetic energy alone can produce long range order. Thus, if long range order exists in this model, it must come from an exchange mechanism. N, the electron plus nucleus number, is taken to be less than or equal to the number of lattice sites. We prove the following: (i) For all dimensions, d, the ground state has long range order; in fact it is a perfect crystal with spacing \(sqrt 2\) times the lattice spacing. A gap in the ground state energy always exists at the half-filled band point (N = number of lattices sites). (ii) For small, positive temperature, T, the ordering persists when d ≥ 2. If T is large there is no long range order and there is exponential clustering of all correlation functions.

Keywords

Long Range Ground State Energy Range Order Hubbard Model Translation Invariance 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1).
    M.C. Gutzwiller, Phys. Rev. Lett. 10 (1963) 159-162; Phys. Rev. 134 (1964) A923-941, 137 (1965) A1726 - 1735.Google Scholar
  2. 2).
    J. Hubbard, Proc. Roy. Soc. (London), Ser. A 276 (1963) 238-257, 277 (1964) 237 - 259.Google Scholar
  3. 3).
    J. Kanamori, Prog. Theor. Phys. 30 (1963) 275 - 289.MATHGoogle Scholar
  4. 4).
    T. Kennedy and E.H. Lieb, An itinerant electron model with crystalline or magnetic long range order, Physica 138A (1986) 320.MathSciNetCrossRefGoogle Scholar
  5. R.L. Dobrushin, Theory Probab. Appl. 13 (1968) 197-224.Google Scholar
  6. 6).
    L. Gross, Commun. Math. Phys. 68 (1979) 9 - 27.MATHGoogle Scholar
  7. 7).
    H. Föllmer, J. Funct. Anal. 46 (1982) 387 - 395.MathSciNetMATHCrossRefGoogle Scholar
  8. 8).
    B. Simon, Commun. Math. Phys. 68 (1979) 183 - 185.MATHGoogle Scholar
  9. 9).
    J.M. Combes and L. Thomas, Commun. Math. Phys. 34 (1973) 251 - 270.MathSciNetADSMATHCrossRefGoogle Scholar
  10. 10).
    Int. Conf. Mathematical Aspects of Statistical Mechanics and Field Theory, Groningen, The Netherlands, 1985, N.M. Hugenholtz and M. Winnink, eds., Springer Lecture Notes in Physics, Vol. 257 ( Springer, New York, 1986 ).Google Scholar
  11. 11).
    R.E. Peierls, Quantum Theory of Solids ( Clarendon, Oxford 1955 ), p. 108.MATHGoogle Scholar
  12. 12).
    W.P. Su, J.R. Schrieffer and A.J. Heeger, Phys. Rev. Lett. 42 (1979) 1698 - 1701.ADSCrossRefGoogle Scholar
  13. 13).
    S.A. Brazovskii, N.E. Dzyaloshinskii and I.M. Krichever, Sov. Phys. JETP 56 (1982) 212 - 225.MathSciNetGoogle Scholar
  14. 14).
    H. Fröhlich, Proc. Roy. Soc. A 223 (1954) 296 - 305.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Elliott H. Lieb
    • 1
  1. 1.Departments of Mathematics and PhysicsPrinceton UniversityPrincetonUSA

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