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
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Author
You-Gang Feng, College of Science, Guizhou University, Huaxi, Guiyang, 550025 China
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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
References
[1]
HA Kramers and GH Wannier, Phys. Rev. 60, 247, 1941.
[2]
L. Onsager, Phys. Rev. 65, 117, 1944.
[3]
You-Gang Feng, EJTP 7, 12, 2005
[4]
H. Arisue, T. Fujwara, and K. Tabata, Nucl. Phys. B (Proc. Suppl) 129&130, 774, 2004.
[5]
Nan-zhi Zhou, Da-fang Zheng and You-yan Lin, Phys. Rev. A 42, 3259, 1990.
[6]
Gyan Bhanot, Michael Creutz, and Jan Lacki, Phys. Rev. Lett. 69, 1841, 1992.
[7]
Zhi-Dong Zhang, Phil. Mag. B 87, 5307, 2007.
[8]
F. Y. Wu, B. M. McCoy, M. E. Fisher, and L. Chayes, Phil. Mag. B 88, 3093, 2008.
[9]
J Als-Nielsen, J. and R. J. Birgeneau, American Journal of Physics 45, 554, 1977.
[10]
Fisher M. E., and Essam J. W., J. Math. Phys. 2, 609, 1961.
[11]
B. Widom, J. Chem. Phys. 43, 3898, 1965.
[12]
Kadanoff L. P., Physics 2, 263, 1966.
[13]
Wilson K. G., Phys.Rev. B 4, 3174, 1971. Wilson K. G. and Kogut J., Phys.Rep. 12, 75, 1974
[14]
Wilson K. G., Rev. Mod. Phys. 55, 595, 1983.
[15]
Zhi-Dong Zhang, Phil. Mag. B 88, 3097, 2008.
[16]
Pascal Monceau, Michel Perreau and Frèdèric Hèbert, Phys. Rev. B 58, 6386, 1998.
[17]
Wolfhard Janke and Adriuan M. J. Schakel, Phys. Rev. E 71, 036703, 2005.
[18]
Armstrong, M. A., Basic Topology (Springer, New York, 1983) p.119-120, 59- 60.
[19]
Chen, S. S., Chen, W. H. and Lam, K. S., Lectures on differential geometry (World scientific, Beijing,1999) p.43-50.
[20]
Falconer, K. J., Fractal Geometry Second edition (: John Wiley, Chichester, 2003) p41-43, xxiv.
[21]
T. H. Berlin and M. Kac, Phys.Rev. 86, 821, 1952; H. E. Stanley, Phys.Rev. 176, 718, 1968; M. Kac and C. J. Thompson, Physica Norvegica 5, 163, 1971.
[22]
Landau, L. D. and Lifshitz, E. M., Quantum mechanics Third edition (Pergamon, Oxford, 1977) p.231.
[23]
Jun Kigami, Analysis of fractals (Cambridge University, Cambridge, 2001) p.9.
[24]
Kouichi Okonishi and Tomotoshi Nishino, Prog. Theor. Phys. 103, 54, 2000.
[25]
H. W. J. Blöte, Erik Luijten, and J. R. Heringa J. Phys. A 28, 6289, 1995.
[26]
A. L. Talapov and H. W. J. Blöte, J. Phys. A 29, 5727, 1996.
[27]
Tullio Regge and Riccardo Zecchina J.Phys. A 33, 741, 2000.
[28]
Pathria, R. K., Statistical mechanics Second edition (Elsevier Pte., Singapore, 2001) p 383
[29]
R. Kubo, Rep. Prog. Phys. 29, 255, 1996.
[30]
You-Gang Feng arxiv 1111.2233, You-Gang Feng arxiv 1204.1807
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