Magnetic Properties of Some Itinerant-Electron Systems at T > 0

  • Elliott H. Lieb
  • Michael Aizenman


The Lieb-Mattis theorem on the absence of one-dimensional ferromagnetism is extended here from ground states to T> 0 by proving, inter alia, that M(ß,h), the magnetization of a quantum system in a field h> 0, is always less than the pure paramagnetic value M o(ß,h)=lanh(ßh), with ß=1/kT. Our proof rests on a new formulation in terms of path integrals that holds in any dimension; another of its applications is that the Nagaoka-Thouless theorem on the Hubbard model also extends to T > 0 in the sense that M(ß,h ) exceeds M 0(ß,h ).


Hubbard Model World Line Physical Review Letter Wiener Measure Spin Assignment 
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  1. 1.
    E. H. Lieb and D. C. Mattis, Phys. Rev. 125, 164–172 (1962).zbMATHCrossRefGoogle Scholar
  2. 2.
    C. Herring, Phys. Rev. B 11, 2056 (1975).CrossRefGoogle Scholar
  3. 3.
    Y. Nagaoka, Phys. Rev. 147, 392 (1967).CrossRefGoogle Scholar
  4. 4.
    D. Thouless, Proc. Phys. Soc. London 86, 893 (1965).MathSciNetCrossRefGoogle Scholar
  5. 5.
    E. H. Lieb, in Phase Transitions, Proceedings of the Fourteenth Chemistry Solvay Conference (Wiley, New York, 1971), pp. 45–63.Google Scholar
  6. 6.
    The restrictions on V are fairly laxGoogle Scholar
  7. 8.
    C. M. Fortuin and P. W. Kasteleyn, Physica (Utrecht) 57, 536 (1972).MathSciNetCrossRefGoogle Scholar
  8. 9.
    At T=0 all the powers of P occur with equal weights in Eq. (13). By Eq. (9), Zj(ß,N/2-l)/Z j(ß,/N/2) →0 as ß→ ∞.Google Scholar
  9. 10.
    B. Simon, Functional Integration and Quantum Physics (Academic, New York, 1979).zbMATHGoogle Scholar
  10. 11.
    M. Donsker, in Four Papers on Probability, Memoirs of the American Mathematical Society Vol. 6 (American Mathematical Society, Providence, RI, 1951).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Elliott H. Lieb
    • 2
  • Michael Aizenman
    • 1
  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  2. 2.Department of Physics and MathematicsPrinceton UniversityPrincetonUSA

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