Abstract
Most of the theoretical wisdom on the phase transition of the ferromagnetic type and the related symmetry breaking is based on the two-dimensional Ising model, which also played the role of a laboratory for ideas and strategies and it is now regarded as a corner stone in the foundations of statistical mechanics. Anyone interested in critical phenomena and in the functional integral approach to quantum field theory should have a look at the model. Even if a discussion of the two-dimensional Ising model would be very appropriate for our purposes, we refer the reader to the very good accounts which can be found in literature116. We restrict our discussion to the one-dimensional version of the model, which is almost trivial, but nevertheless provides an interesting simple example for testing the constructive strategies of symmetry breaking discussed above.
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References
For the history of the model, see S.G. Brush, Rev. Mod. Phys. 39, 883 (1967). The model is now part of the basic knowledge in statistical mechanics and the theory of phase transitions; for textbook accounts, see e.g. K. Huang, Statistical Mechanics, Wiley 1987, Chap. 14, 15; G. Gallavotti, Statistical Mechanics: A Short Treatise, Springer 1999, Sect. 6; B. Simon, The Statistical Mechanics of Lattice Gases, Vol. I, Princeton Univ. Press 1993, Sect. II.6. An extensive treatment, which also emphasizes the links with quantum field theory and general theoretical physics problems, is in B.M. McCoy and T.T. Wu, The Two Dimensional Ising Model, Harvard Univ. Press 1973.
For the basic elements of statistical mechanics, see e.g. K. Huang, Statistical Mechanics, Wiley 1987; a brief account is given in the following section.
See T.D. Schulz, D.C. Mattis and E.H. Lieb, Rev. Mod. Phys. 36, 856 (1964) and references therein; E. Lieb, in Boulder Lectures in Theoretical Physics, Vol.XI D, K.T. Mahantappa and W.E. Brittin eds., Gordon and Breach 1969, p.329; J.B. Kogut, Rev. Mod. Phys. 51, 659 (1979).
For the proof of this result, and its relevance in the functional integral approach to quantum theories, see J. Glimm and A. Jaffe, Quantum Physics. A Functional Integral Point of View, 2nd ed., Springer 1987, p. 51.
This approximation is at the basis of the Curie-Weiss theory of magnetic phase transitions, also called molecular field approximation; see e.g. H.E. Stanley, Introduction to Phase Transitions and Critical Phenomena, Oxford Univ. Press 1974, Chap.6; C.J. Thompson, Mathematical Statistical Mechanics, Princeton Univ. Press 1972, Sect. 4.5.
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Strocchi, F. (2008). Symmetry Breaking in the Ising Model. In: Symmetry Breaking. Lecture Notes in Physics, vol 732. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73593-9_21
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