Landau theory, symmetry breaking and the chain criterion

  • Marko V. Jarić
Atomic, Molecular, Solid-State, and Statistical Physics
Part of the Lecture Notes in Physics book series (LNP, volume 135)


A study of absolute minima of bounded below, real, G-invariant polynomials on Rm is initiated. The group Im(G), acting on Rm, is assumed orthogonal, irreducible and finite. Aforementioned polynomials are used for the free energy in a theory of phase transitions. Symmetry of an absolute minimum of such a polynomial is the broken symmetry. Several theorems on possible broken symmetries are proven.


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  1. 1.
    L.D. Landau, Phys. Z. Sowj. Un. 11, 26 and 545 (1937). See also L.D. Landau and E.M. Lifshitz, Statisticheskaya Fizika (Nauka, Moskva 1976).zbMATHGoogle Scholar
  2. 2.
    The theory was most successful in applications to structural phase transitions; see G. Ya. Lyubarskii, The Application of Group Theory in Physics (Pergamon, Oxford, 1960).Google Scholar
  3. 3.
    In the Landau theory this linear form is usually referred to as Landau's density function.Google Scholar
  4. 4.
    Note that the relevant group is the image of G defined by the representation. Thus, when we say G we will actually refer to Im(G).Google Scholar
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    M.V. Jarić, Phys. Rev. (To be published).Google Scholar
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    M.V. Jarić, Ph.D. Thesis (CUNY, New York, 1977).Google Scholar
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    M.V. Jarić, and J.L. Birman, Phys. Rev. B16, 2564 (1977).ADSGoogle Scholar
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    M.V. Jarić, Phys. Rev. B (To be published).Google Scholar
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    Independently L. Michel proved the same, Rev. Mod. Physics (To be published).Google Scholar
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    I will only consider finite, orthogonal and irreducible groups Im(G).Google Scholar
  11. 11.
    Due to the theorem Eq. 4, even a local minimum (associated with a quasi-stable phase) can be obtained in the same fashion.Google Scholar
  12. 12.
    A similar theorem was proved by L. Michel, preprint Ref. TH.2716-CERN.Google Scholar

Copyright information

© Springer-Verlag 1980

Authors and Affiliations

  • Marko V. Jarić
    • 1
  1. 1.Department of PhysicsUniversity of CaliforniaBerkeleyUSA

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