Abstract
The 16th-order transfer matrix of the three-dimensional Ising model in the particular case n = m = 2 (n × m is number of spins in a layer) is specified by the interaction parameters of three basis vectors. The matrix eigenvectors are divided into two classes, even and odd. Using the symmetry of the eigenvectors, we find their corresponding eigenvalues in general form. Eight of the sixteen eigenvalues related to odd eigenvectors are found from quadratic equations. Four eigenvalues related to even eigenvectors are found from a fourth-degree equation with symmetric coefficients. Each of the remaining four eigenvalues is equal to unity.
Similar content being viewed by others
References
E. Ising, “Beitrag zur Theorie des Ferromagnetismus,” Z. Phys., 31, 253–258 (1925).
L. Onsager, “Crystal statistics: I. A two-dimensional model with an order-disorder transition,” Phys. Rev., 65, 117–149 (1944).
B. Kaufman, “Crystal statistics: II. Partition function evaluated by spinor analysis,” Phys. Rev., 76, 1231–1243 (1949).
Yu. B. Rumer, “Thermodynamics of a two-dimensional dipole lattice [in Russian],” Usp. Fiz. Nauk, 53, 245–284 (1954).
M. Kac and J. C. Ward, “A combinatorial solution of the two-dimensional Ising model,” Phys. Rev., 88, 1332–1337 (1952).
S. Sherman, “Combinatorial aspects of the Ising model for ferromagnetism: I. A conjecture of Feynman on paths and graphs,” J. Math. Phys., 1, 202–217 (1960).
C. A. Hurst and H. S. Green, “New solution of the Ising problem for a rectangular lattice,” J. Chem. Phys., 33, 1059–1062 (1960).
P. W. Kasteleyn, “Dimer statistic and phase transitions,” J. Math. Phys., 4, 287–293 (1963).
N. V. Vdovichenko, “A calculation of the partition function for a plane dipole lattice,” Soviet Phys. JETP, 20, 477–479 (1965).
Yu. M. Zinoviev, “The Onsager formula,” Proc. Steklov Inst. Math., 228, 286–306 (2000).
Yu. M. Zinoviev, “Ising model and L-function,” Theor. Math. Phys., 126, 66–80 (2001).
Yu. M. Zinoviev, “Spontaneous magnetization in the two-dimensional Ising model,” Theor. Math. Phys., 136, 1280–1296 (2003).
I. M. Ratner, “Translation symmetry of a crystal and its description using multidimensional matrices [in Russian],” in: Research, Synthesis, Technology of Bulk Luminophores (Collected works, All-Russian Scientific Research Institute for Luminophores, No. 36), Stavropol (1989), pp. 88–99.
K. Huang, Statistical Mechanics, Wiley, New York (1963).
N. P. Sokolov, Spatial Matrices and Their Applications [in Russian], Fizmatlit, Moscow (1960).
M. A. Yurishchev, “Lower and upper bounds on the critical temperature for anisotropic three-dimensional Ising model,” JETP, 98, 1183–1197 (2004).
I. M. Ratner, “Mathematical modeling of the three-dimensional Ising lattice [in Russian],” in: Info-Communication Technology in Science, Industry, and Education, North-Caucasus Technical Univ., Stavropol (2004), pp. 514–521.
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 199, No. 3, pp. 497–510, June, 2019.
Rights and permissions
About this article
Cite this article
Ratner, I.M. Eigenvalues of the Transfer Matrix of the Three-Dimensional Ising Model in the Particular Case n = m = 2. Theor Math Phys 199, 909–921 (2019). https://doi.org/10.1134/S0040577919060102
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0040577919060102