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TMF, 2019, Volume 199, Number 3, Pages 497–510 (Mi tmf9599)  

Eigenvalues of the transfer matrix of the three-dimensional Ising model in the particular case $n=m=2$

I. M. Ratner

Institute of Information Technologies and Telecommunications, North-Caucasus Federal University, Stavropol, Russia

Abstract: The $16$th-order transfer matrix of the three-dimensional Ising model in the particular case $n=m=2$ $(n\times m$ 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.

Keywords: partition function, three-dimensional Ising model, spin, transfer matrix, eigenvalue, eigenvector, symmetry.

DOI: https://doi.org/10.4213/tmf9599

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English version:
Theoretical and Mathematical Physics, 2019, 199:3, 909–921

Bibliographic databases:

Received: 22.06.2018
Revised: 22.06.2018

Citation: I. M. Ratner, “Eigenvalues of the transfer matrix of the three-dimensional Ising model in the particular case $n=m=2$”, TMF, 199:3 (2019), 497–510; Theoret. and Math. Phys., 199:3 (2019), 909–921

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  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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