Abstract
This chapter further supports the case for the Ising spin as the Drosophila of statistical mechanics, that is the system that can be used to model virtually every interesting thermodynamic phenomenon and to test every theoretical method. Exact solutions are extremely rare in many-body physics. Here, we consider three of them: (1) the one-dimensional Ising model, (2) the one-dimensional Ising model in a transverse field, the simplest quantum spin system, and (3) the two-dimensional Ising model in zero field. The results are paradigms for a host of more complex systems and situations that cannot be solved exactly but which can be understood qualitatively and even quantitatively on the basis of simulations and asymptotic methods such as series expansions and the renormalization group.
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References
H.W.J. Blöte, J.L. Cardy, M.P. Nightingale, Conformal invariance, the central charge, and universal finite-size amplitudes at criticality. Phys. Rev. Lett. 56, 742 (1986)
R.P. Feynman, Statistical Mechanics; a set of lectures (1972) (Notes taken by R. Kikuchi and H.A. Fiveson, W.A. Benjamin)
K. Huang, Statistical Mechanics, 2nd edn. (Wiley, 1987)
E. Ising, Contribtions to the theory of ferromagnetism. Ph.D. thesis, and Z. Phys. 31, 253 (1925)
M. Kac, Toeplitz matrices, transition kernels and a related problem in probability theory. Duke Math. J. 21, 501 (1954)
W. Lenz, Contributions to the understanding of magnetic properties in solid bodies (in German). Physikalische Zeitschrift 21, 613 (1920)
E.W. Montroll, R.B. Potts, J.C. Ward, Correlations and spontaneous magnetization of the two-dimensional ising model. J. Math. Phys. 4, 308 (1963)
L. Onsager, Crystal statistics. I. A two-dimensional model with an order-disorder transition. Phys. Rev. 65, 117 (1944)
M. Plischke, B. Bergersen, Equilibrium Statistical Physics, 2nd edn. (World Scientific, 1994)
T.D. Schultz, D.C. Mattis, E.H. Lieb, Two-dimensional ising model as a soluble problem of many fermions. Rev. Mod. Phys. 36, 856 (1964)
G. Szego, Orthogonal Polynomials, American Mathematical Society, Vol. XXIII (Colloquium Publications, 1939)
C.N. Yang, The spontaneous magnetization of a two-dimensional ising model, Phys. Rev. 85, 808 (1952). (For the rectangular lattice see C.H. Chang, The spontaneous magnetization of a two-dimensional rectangular ising model. Phys. Rev. 88, 1422, 1952)
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Berlinsky, A.J., Harris, A.B. (2019). The Ising Model: Exact Solutions. In: Statistical Mechanics. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-28187-8_17
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DOI: https://doi.org/10.1007/978-3-030-28187-8_17
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