Fundamentals of the Physics of Solids / 06-Consequences of Symmetries

6.3 Symmetry Breaking and Its Consequences

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however the longitudinal mode does not become soft. Goldstone bosons are absent in superconductors, too. However, in liquid helium-4, where interactions are short-ranged, the excitation spectrum is gapless. The bosonic excitations that acquire a finite energy due to long-range forces are called Higgs bosons15 in solid-state physics, too.

Goldstone’s theorem is concerned with the spectrum of excitations for systems in which a continuous symmetry of the Hamiltonian is broken. It does not tell anything about when this happens. A partial answer to this question is given by Coleman’s theorem.16 It states that quantum fluctuations will restore the continuous symmetry in the ground state of one-dimensional systems; continuous symmetry cannot be broken in the ground state, except when the order parameter associated with symmetry breaking is conserved. It was already mentioned in Chapter 2 that thermal fluctuations hinder any ordering in one-dimensional systems at finite temperatures unless the forces are long-ranged (Landau–Peierls instability). Coleman’s theorem states that even the ground state cannot be ordered, since quantum fluctuations play a similar role there. The instability against breaking a continuous symmetry in two-dimensional systems at finite temperatures will be discussed in Chapters 12 and 15.

Further Reading

1.J. P. Elliott and P. G. Dawber, Symmetry in Physics, Vol. 1., Principles and Simple Applications, Macmillan Education Ltd., (1989).

2.R. A. Evarestov and V. P. Smirnov, Site Symmetry in Crystals, Theory and Applications, Second edition, Springer Series in Solid-State Sciences, Vol. 108, Springer-Verlag, Berlin (1997).

3.V. Heine, Group Theory in Quantum Mechanics, Pergamon Press, New York (1960).

4.L. D. Landau and E. M. Lifshitz, Statistical Physics (Course of Theoretical Physics Volume 5), Third Edition, Butterworth-Heinemann, (1984).

5.W. Ludwig and C. Falter, Symmetries in Physics, Group Theory Applied to Physical Problems, Second Extended Edition, Springer-Series in SolidState Sciences, Springer-Verlag, New York (1996).

6.G. Ya. Lyubarskii, The Application of Group Theory in Physics, Pergamon Press, Oxford (1960).

7.M. Tinkham, Group Theory and Quantum Mechanics, McGraw–Hill, New York (1964).

15P. W. Higgs, 1964.

16S. Coleman, 1973.