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Self-Similar Transformations of Lattice-Ising Models at Critical Temperatures
American Journal of Modern Physics
Volume 3, Issue 4, July 2014, Pages: 184-194
Received: Jul. 18, 2014; Accepted: Jul. 29, 2014; Published: Aug. 10, 2014
Volume 3, Issue 4, July 2014, Pages: 184-194
Received: Jul. 18, 2014; Accepted: Jul. 29, 2014; Published: Aug. 10, 2014
Views 3409 Downloads 161
Author
You-Gang Feng, College of Science, Guizhou University, Huaxi, Guiyang, 550025 China
Abstract
We classify geometric blocks that serve as spin carriers into simple blocks and compound blocks by their topologic connectivity, define their fractal dimensions and describe the relevant transformations. By the hierarchical property of transformations and a block-spin scaling law we obtain a relation between the block spin and its carrier’s fractal dimension. By mapping we set up a block-spin Gaussian model and get a formula connecting the critical point and the minimal fractal dimension of the carrier, which guarantees the uniqueness of a fixed point corresponding to the critical point, changing the complicated calculation of critical point into the simple one of the minimal fractal dimension. The numerical results of critical points with high accuracy for five conventional lattice-Ising models prove our method very effective and may be suitable to all lattice-Ising models. The origin of fluctuations in structure at critical temperature is discussed. Our method not only explains the problems met in the renormalization-group theory, but also provides a useful tool for deep investigation of the critical behaviour.
Keywords
Ising, Renormalization, Fractal
To cite this article
You-Gang Feng,
Self-Similar Transformations of Lattice-Ising Models at Critical Temperatures, American Journal of Modern Physics.
Vol. 3, No. 4,
2014, pp. 184-194.
doi: 10.11648/j.ajmp.20140304.16
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