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
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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 
@article{TMF_2019_199_3_a9,
     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},
     pages = {497--510},
     year = {2019},
     volume = {199},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2019_199_3_a9/}
}
TY  - JOUR
AU  - I. M. Ratner
TI  - Eigenvalues of the transfer matrix of the three-dimensional Ising model in the particular case $n=m=2$
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2019
SP  - 497
EP  - 510
VL  - 199
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/TMF_2019_199_3_a9/
LA  - ru
ID  - TMF_2019_199_3_a9
ER  - 
%0 Journal Article
%A I. M. Ratner
%T Eigenvalues of the transfer matrix of the three-dimensional Ising model in the particular case $n=m=2$
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2019
%P 497-510
%V 199
%N 3
%U http://geodesic.mathdoc.fr/item/TMF_2019_199_3_a9/
%G ru
%F TMF_2019_199_3_a9
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/

[1] E. Ising, “Beitrag zur Theorie des Ferromagnetismus”, Z. Phys., 31:1 (1925), 253–258 | DOI

[2] L. Onsager, “Crystal statistics. I. A two-dimensional model with an order-disorder transition”, Phys. Rev., 65:3–4 (1944), 117–149 | DOI | MR

[3] B. Kaufman, “Crystal statistics. II. Partition function evaluated by spinor analysis”, Phys. Rev., 76:8 (1949), 1231–1243 | DOI

[4] Yu. B. Rumer, “Termodinamika ploskoi dipolnoi reshetki”, UFN, 53:2 (1954), 245–284 | DOI

[5] M. Kac, J. C. Ward, “A combinatorial solution of the two-dimensional Ising model”, Phys. Rev., 88:6 (1952), 1332–1337 | DOI

[6] S. Sherman, “Combinatorial aspects of the Ising model for ferromagnetism. I. A conjecture of Feynman on paths and graphs”, J. Math. Phys., 1:3 (1960), 202–217 | DOI | MR

[7] C. A. Hurst, H. S. Green, “New solution of the Ising problem for a rectangular lattice”, J. Chem. Phys., 33:4 (1960), 1059–1062 | DOI | MR

[8] P. W. Kasteleyn, “Dimer statistic and phase transitions”, J. Math. Phys., 4:2 (1963), 287–293 | DOI | MR

[9] N. V. Vdovichenko, “Vychislenie statisticheskoi summy ploskoi dipolnoi reshetki”, ZhETF, 47:2 (1964), 715–731 | MR

[10] Yu. M. Zinovev, “Formula Onzagera”, Tr. MIAN, 228 (2000), 298–319 | MR | Zbl

[11] Yu. M. Zinovev, “Model Izinga i $L$-funktsiya”, TMF, 126:1 (2001), 84–101 | DOI | DOI | MR | Zbl

[12] Yu. M. Zinovev, “Spontannaya namagnichennost v dvumernoi modeli Izinga”, TMF, 136:3 (2003), 444–462 | DOI | DOI | MR | Zbl

[13] I. M. Ratner, “Translyatsionnaya simmetriya kristalla i ee opisanie s pomoschyu mnogomernykh matrits”, Issledovaniya, sintez, tekhnologiya krupnotonnazhnykh lyuminoforov, Sb. trudov VNII lyuminoforov, vyp. 36, Stavropol, 1989, 88–99

[14] K. Khuang, Statisticheskaya mekhanika, Mir, M., 1966 | MR

[15] N. P. Sokolov, Prostranstvennye matritsy i ikh prilozheniya, Fizmatlit, M., 1960

[16] M. A. Yurischev, “Nizhnie i verkhnie granitsy dlya kriticheskoi temperatury v trekhmernoi anizotropnoi modeli Izinga”, ZhETF, 125:6 (2004), 1349–1366 | DOI

[17] I. M. Ratner, “Matematicheskoe modelirovanie trekhmernoi reshetki Izinga”, Infokommunikatsionnye tekhnologii v nauke, proizvodstve i obrazovanii, SevKavGTU, Stavropol, 2004, 514–521