IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v9y2021i7p776-d529246.html
   My bibliography  Save this article

A Method of Riemann–Hilbert Problem for Zhang’s Conjecture 1 in a Ferromagnetic 3D Ising Model: Trivialization of Topological Structure

Author

Listed:
  • Osamu Suzuki

    (Department of Computer and System Analysis, College of Humanities and Sciences, Nihon University, Sakurajosui 3-25-40, Setagaya-ku, Tokyo 156-8550, Japan)

  • Zhidong Zhang

    (Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy of Sciences, 72 Wenhua Road, Shenyang 110016, China)

Abstract

A method of the Riemann–Hilbert problem is applied for Zhang’s conjecture 1 proposed in Philo. Mag. 87 (2007) 5309 for a ferromagnetic three-dimensional (3D) Ising model in the zero external field and the solution to the Zhang’s conjecture 1 is constructed by use of the monoidal transform. At first, the knot structure of the ferromagnetic 3D Ising model in the zero external field is determined and the non-local behavior of the ferromagnetic 3D Ising model can be described by the non-trivial knot structure. A representation from the knot space to the Clifford algebra of exponential type is constructed, and the partition function of the ferromagnetic 3D Ising model in the zero external field can be obtained by this representation (Theorem I). After a realization of the knots on a Riemann surface of hyperelliptic type, the monodromy representation is realized from the representation. The Riemann–Hilbert problem is formulated and the solution is obtained (Theorem II). Finally, the monoidal transformation is introduced for the solution and the trivialization of the representation is constructed (Theorem III). By this, we can obtain the desired solution to the Zhang’s conjecture 1 (Main Theorem). The present work not only proves the Zhang’s conjecture 1, but also shows that the 3D Ising model is a good platform for studying in deep the mathematical structure of a physical many-body interacting spin system and the connections between algebra, topology, and geometry.

Suggested Citation

  • Osamu Suzuki & Zhidong Zhang, 2021. "A Method of Riemann–Hilbert Problem for Zhang’s Conjecture 1 in a Ferromagnetic 3D Ising Model: Trivialization of Topological Structure," Mathematics, MDPI, vol. 9(7), pages 1-28, April.
  • Handle: RePEc:gam:jmathe:v:9:y:2021:i:7:p:776-:d:529246
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/9/7/776/pdf
    Download Restriction: no

    File URL: https://www.mdpi.com/2227-7390/9/7/776/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Wang, Xue-Sheng & Zhang, Fan & Si, Nan & Meng, Jing & Zhang, Yan-Li & Jiang, Wei, 2019. "Unique magnetic and thermodynamic properties of a zigzag graphene nanoribbon," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 527(C).
    2. Li, Qi & Li, Run-dong & Wang, Wei & Geng, Rui-ze & Huang, Han & Zheng, Shu-juan, 2020. "Magnetic and thermodynamic characteristics of a rectangle Ising nanoribbon," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 555(C).
    3. Lu, Zhao-Ming & Si, Nan & Wang, Ya-Ning & Zhang, Fan & Meng, Jing & Miao, Hai-Ling & Jiang, Wei, 2019. "Unique magnetism in different sizes of center decorated tetragonal nanoparticles with the anisotropy," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 523(C), pages 438-456.
    4. Yang, Min & Wang, Wei & Li, Bo-chen & Wu, Hao-jia & Yang, Shao-qing & Yang, Jun, 2020. "Magnetic properties of an Ising ladder-like graphene nanoribbon by using Monte Carlo method," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 539(C).
    5. Siyuan Ma & Hongzhong Zhang & Hongshu Chen, 2021. "Opinion Expression Dynamics in Social Media Chat Groups: An Integrated Quasi-Experimental and Agent-Based Model Approach," Complexity, Hindawi, vol. 2021, pages 1-14, January.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Si, Nan & Su, Xin & Meng, Jing & Miao, Hai-Ling & Zhang, Yan-Li & Jiang, Wei, 2020. "Magnetic properties of decorated 2D kagome-like lattice," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 560(C).
    2. Si, Nan & Guan, Yin-Yan & Gao, Wei-Chun & Guo, An-Bang & Zhang, Yan-Li & Jiang, Wei, 2022. "Ferrimagnetism and reentrant behavior in a coronene-like superlattice with double-layer," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 589(C).
    3. Benjamin Cabrera & Björn Ross & Daniel Röchert & Felix Brünker & Stefan Stieglitz, 2021. "The influence of community structure on opinion expression: an agent-based model," Journal of Business Economics, Springer, vol. 91(9), pages 1331-1355, November.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:9:y:2021:i:7:p:776-:d:529246. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.