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Theory of Ferromagnetism and the Ordering of Electronic Energy Levels

  • Elliott Lieb
  • Daniel Mattis

Abstract

Consider a system of N electrons in one dimension subject to an arbitrary symmetric potential, V(x 1…,XN), and let E(S) be the lowest energy belonging to the total spin value S. We have proved the following theorem: E(S) <E(S’) if S>Sr. Hence, the ground state is unmagnetized. The theorem also holds in two or three dimensions (although it is possible to have degeneracies) provided V(x 1 ,y 1 ,Z 1 ; …; x n ,yn, Z N ) is separately symmetric in the x i y i and Z i. The potential need not be separable, however. Our theorem has strong implications in the theory of ferromagnetism because it is generally assumed that for certain repulsive potentials, the ground state is magnetized. If such be the case, it is a very delicate matter, for a plausible theory must not be so general as to give ferromagnetism in one dimension, nor in three dimensions with a separately symmetric potential

Keywords

Linear Chain Kronecker Product Symmetry Class Pauli Principle Symmetric Potential 
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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Note

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    The authors believe this theorem is due to E. P. Wigner. 2 h2/2m=l.Google Scholar
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    The authors believe this theorem is due to E. P. Wigner. 2 h2/2m=l.Google Scholar
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    See, for example, D. E. Rutherford, Substilutional Analysis (Edinburgh University Press, Edinburgh, 1948). Young’s scheme proceeds by symmetrization, instead of antisymmetrization as used here.Google Scholar
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    We have departed slightly from the notation in Eq. (9). Formerly the bar was regarded as possibly movable leftwards.Google Scholar
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    R. E. Peierls, Quantum Theory of Solids (Oxford University Press, New York, 1955).zbMATHGoogle Scholar
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    David I. Paul, Phys. Rev. 118, 92 (1960), and Phys. Rev. 120, 463 (1960).MathSciNetzbMATHCrossRefGoogle Scholar
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    W. J. Carr, Jr., Phys. Rev. 122, 1437 (1961).CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Elliott Lieb
    • 1
  • Daniel Mattis
    • 1
  1. 1.Thomas J. Watson Research CenterInternational Business Machines CorporationNew YorkUSA

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