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Ordering Energy Levels of Interacting Spin Systems

  • Elliott Lieb
  • Daniel Mattis

Abstract

The total spin S is a good quantum number in problems of interacting spins. We have shown that for rather general antiferromagnetic or ferrimagnetie Hamiltonians, which need not exhibit translational invariance, the lowest energy eigenvalue for each value of S [denoted E(S) ] is ordered in a natural way. In antiferromagnetism, E(S + 1) > E(S) for S0. In ferrimagnetism, E(S + 1) > E(S) for S ≥ S, and in addition the ground state belongs to S ≤ S. S is defined as follows: Let the maximum spin of the A sublattice be S A and of the B sublattice S B; then S = S AS B. Antiferromagnetism is treated as the special case of S = 0. We also briefly discuss the structure of the lowest eigenfunctions in an external magnetic field.

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Note

  1. 1.
    W. Marshall, Proc. Roy. Soc. (London) A232, 48 (1955).Google Scholar
  2. 2.
    E. Lieb, T. Schultz, D. Mattis, Ann. Phys. 16, 407 (1961), particularly Appendix B.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    E. Lieb and D. Mattis, Phys. Rev. 125, 164 (1962).zbMATHCrossRefGoogle 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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