Abstract
It is shown how correlations in spin systems can be described in terms of geometrical clusters. For the q-state Potts model exact relations are given between connectivity and thermal properties. A cluster dynamics developed by Swendsen and Wang is described. At criticality these clusters are shown to have a fractal structure made of links and blobs as in percolation. In two dimensions the value of the fractal dimension of the links or red bonds and the fractal dimension of the hull of the incipient infinite cluster can be found exactly for all values of 0 ≤ q ≤ 4. Extension of these geometrical concepts to Ising spin glasses is also made. The clusters now represent fluctuations which interfere with each other and give rise to net correlations which are shorter than the typical cluster radius. In fact it is found numerically that the mean cluster size diverges well above the spin glass transition.
Lecture notes prepared by Jysoo Lee and Sona Prakash.
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Coniglio, A. (1990). Correlations in Thermal & Geometrical Systems. In: Stanley, H.E., Ostrowsky, N. (eds) Correlations and Connectivity. NATO ASI Series, vol 188. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-2157-3_2
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DOI: https://doi.org/10.1007/978-94-009-2157-3_2
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