Group Theoretical Methods in Physics pp 12-16 | Cite as

# Landau theory, symmetry breaking and the chain criterion

Atomic, Molecular, Solid-State, and Statistical Physics

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## Abstract

A study of absolute minima of bounded below, real, G-invariant polynomials on R^{m} is initiated. The group Im(G), acting on R^{m}, 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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## References

- 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.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.In the Landau theory this linear form is usually referred to as Landau's density function.Google Scholar
- 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
- 5.M.V. Jarić, Phys. Rev. (To be published).Google Scholar
- 6.M.V. Jarić, Ph.D. Thesis (CUNY, New York, 1977).Google Scholar
- 7.M.V. Jarić, and J.L. Birman, Phys. Rev. B16, 2564 (1977).ADSGoogle Scholar
- 8.M.V. Jarić, Phys. Rev. B (To be published).Google Scholar
- 9.Independently L. Michel proved the same, Rev. Mod. Physics (To be published).Google Scholar
- 10.I will only consider finite, orthogonal and irreducible groups Im(G).Google Scholar
- 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.A similar theorem was proved by L. Michel, preprint Ref. TH.2716-CERN.Google Scholar

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© Springer-Verlag 1980