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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
M.C. Gutzwiller, Phys. Rev. Lett. 10 (1963) 159-162; Phys. Rev. 134 (1964) A923-941, 137 (1965) A1726 - 1735.
J. Hubbard, Proc. Roy. Soc. (London), Ser. A 276 (1963) 238-257, 277 (1964) 237 - 259.
J. Kanamori, Prog. Theor. Phys. 30 (1963) 275 - 289.
T. Kennedy and E.H. Lieb, An itinerant electron model with crystalline or magnetic long range order, Physica 138A (1986) 320.
R.L. Dobrushin, Theory Probab. Appl. 13 (1968) 197-224.
L. Gross, Commun. Math. Phys. 68 (1979) 9 - 27.
H. Föllmer, J. Funct. Anal. 46 (1982) 387 - 395.
B. Simon, Commun. Math. Phys. 68 (1979) 183 - 185.
J.M. Combes and L. Thomas, Commun. Math. Phys. 34 (1973) 251 - 270.
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 ).
R.E. Peierls, Quantum Theory of Solids ( Clarendon, Oxford 1955 ), p. 108.
W.P. Su, J.R. Schrieffer and A.J. Heeger, Phys. Rev. Lett. 42 (1979) 1698 - 1701.
S.A. Brazovskii, N.E. Dzyaloshinskii and I.M. Krichever, Sov. Phys. JETP 56 (1982) 212 - 225.
H. Fröhlich, Proc. Roy. Soc. A 223 (1954) 296 - 305.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Lieb, E.H. (2004). A Model for Crystallization: A Variation on the Hubbard Model. In: Nachtergaele, B., Solovej, J.P., Yngvason, J. (eds) Condensed Matter Physics and Exactly Soluble Models. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-06390-3_8
Download citation
DOI: https://doi.org/10.1007/978-3-662-06390-3_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-06093-9
Online ISBN: 978-3-662-06390-3
eBook Packages: Springer Book Archive