American Journal of Modern Physics

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Self-Similar Transformations of Lattice-Ising Models at Critical Temperatures

Received: 18 July 2014    Accepted: 29 July 2014    Published: 10 August 2014
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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.

DOI 10.11648/j.ajmp.20140304.16
Published in American Journal of Modern Physics (Volume 3, Issue 4, July 2014)
Page(s) 184-194
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Ising, Renormalization, Fractal

References
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  • APA Style

    You-Gang Feng. (2014). Self-Similar Transformations of Lattice-Ising Models at Critical Temperatures. American Journal of Modern Physics, 3(4), 184-194. https://doi.org/10.11648/j.ajmp.20140304.16

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    ACS Style

    You-Gang Feng. Self-Similar Transformations of Lattice-Ising Models at Critical Temperatures. Am. J. Mod. Phys. 2014, 3(4), 184-194. doi: 10.11648/j.ajmp.20140304.16

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    AMA Style

    You-Gang Feng. Self-Similar Transformations of Lattice-Ising Models at Critical Temperatures. Am J Mod Phys. 2014;3(4):184-194. doi: 10.11648/j.ajmp.20140304.16

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  • @article{10.11648/j.ajmp.20140304.16,
      author = {You-Gang Feng},
      title = {Self-Similar Transformations of Lattice-Ising Models at Critical Temperatures},
      journal = {American Journal of Modern Physics},
      volume = {3},
      number = {4},
      pages = {184-194},
      doi = {10.11648/j.ajmp.20140304.16},
      url = {https://doi.org/10.11648/j.ajmp.20140304.16},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmp.20140304.16},
      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.},
     year = {2014}
    }
    

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  • TY  - JOUR
    T1  - Self-Similar Transformations of Lattice-Ising Models at Critical Temperatures
    AU  - You-Gang Feng
    Y1  - 2014/08/10
    PY  - 2014
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    T2  - American Journal of Modern Physics
    JF  - American Journal of Modern Physics
    JO  - American Journal of Modern Physics
    SP  - 184
    EP  - 194
    PB  - Science Publishing Group
    SN  - 2326-8891
    UR  - https://doi.org/10.11648/j.ajmp.20140304.16
    AB  - 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.
    VL  - 3
    IS  - 4
    ER  - 

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Author Information
  • College of Science, Guizhou University, Huaxi, Guiyang, 550025 China

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