Geodesic
Eigenvalues of the~transfer matrix of the~three-dimensional Ising model in the~particular case $n=m=2$
Teoretičeskaâ i matematičeskaâ fizika, Tome 199 (2019) no. 3, pp. 497-510

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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, eigenvalue, eigenvector, symmetry.
Mots-clés : transfer matrix 
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     author = {I. M. Ratner},
     title = {Eigenvalues of the~transfer matrix of the~three-dimensional {Ising} model in the~particular case $n=m=2$},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
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     publisher = {mathdoc},
     volume = {199},
     number = {3},
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     url = {http://geodesic.mathdoc.fr/item/TMF_2019_199_3_a9/}
}
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I. M. Ratner. Eigenvalues of the~transfer matrix of the~three-dimensional Ising model in the~particular case $n=m=2$. Teoretičeskaâ i matematičeskaâ fizika, Tome 199 (2019) no. 3, pp. 497-510. http://geodesic.mathdoc.fr/item/TMF_2019_199_3_a9/