Abstract
This work is concerned with the critical properties of Ising models on self similar fractal lattices [1]. Fractal lattices are not translationally invariant, in contrast to the usual integer dimensional lattices. A fractal’s simplest characteristic is its fractal dimensionality, D, which is ordinarily not an integer. (The ε-expansion arguments are predicated upon the existence of lattices that combine non integer dimensionality with translational invariance, but no such lattice has been actually exhibited). Numerous real physical systems, e.g. polymers and critical percolation clusters, are usefully viewed as self similar fractals [1,2].
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References
B. B. Mandelbrot, Fractals: Form, Chance and Dimension (W. H. Freeman, San Francisco 1977) and forthcoming sequel.
H. E. Stanley, R. J. Birgenau, P. J. Reynolds, and J. F. Nicoll, J. Phys. C9, L553 (1976)
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H. E. Stanley, R. J. Birgenau, P. J. Reynolds, and J. F. Nicoll B. B. Mandelbrot D. Stauffer, S. Kirkpatrick, Les Houches Summer School on III-Condensed Matter 1978, ed. by R. Balian et al (North Holland, 1979), p. 323.
Y. Gefen, B. B. Mandelbrot and A. Aharony, to be published.
See e.g. the papers by R. B. Griffiths and by C. J. Thompson in Phase Transitions and Critical Phenomena, edited by C. Domb and M. S. Green (Academic, New York, 1976) vol. 6, p. 357.
See also P. Suranyi, Phys. Rev. Lett. 38, 1436 (1977), who used a similar approach to obtain approximate results near TC for the triangular lattice.
L. P. Kadanoff, Ann. Phys. (N.Y.) 100, 359 (1976).
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© 1981 Springer-Verlag Berlin Heidelberg
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Gefen, Y., Aharony, A., Mandelbrot, B.B. (1981). Critical Phenomena and Fractals with Dimensionality Near 1. In: Bernasconi, J., Schneider, T. (eds) Physics in One Dimension. Springer Series in Solid-State Sciences, vol 23. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-81592-8_41
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