Geodesic
The Minimal $LM(3,4)$ Lattice Model and the Two-Dimensional Ising Model with Cylindrical Boundary Conditions
Teoretičeskaâ i matematičeskaâ fizika, Tome 126 (2001) no. 1, pp. 63-83 
A. A. Belavin; R. A. Usmanov. The Minimal $LM(3,4)$ Lattice Model and the Two-Dimensional Ising Model with Cylindrical Boundary Conditions. Teoretičeskaâ i matematičeskaâ fizika, Tome 126 (2001) no. 1, pp. 63-83. http://geodesic.mathdoc.fr/item/TMF_2001_126_1_a1/
@article{TMF_2001_126_1_a1,
     author = {A. A. Belavin and R. A. Usmanov},
     title = {The {Minimal} $LM(3,4)$ {Lattice} {Model} and the {Two-Dimensional} {Ising} {Model} with {Cylindrical} {Boundary} {Conditions}},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {63--83},
     year = {2001},
     volume = {126},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2001_126_1_a1/}
}
TY  - JOUR
AU  - A. A. Belavin
AU  - R. A. Usmanov
TI  - The Minimal $LM(3,4)$ Lattice Model and the Two-Dimensional Ising Model with Cylindrical Boundary Conditions
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2001
SP  - 63
EP  - 83
VL  - 126
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TMF_2001_126_1_a1/
LA  - ru
ID  - TMF_2001_126_1_a1
ER  - 
%0 Journal Article
%A A. A. Belavin
%A R. A. Usmanov
%T The Minimal $LM(3,4)$ Lattice Model and the Two-Dimensional Ising Model with Cylindrical Boundary Conditions
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2001
%P 63-83
%V 126
%N 1
%U http://geodesic.mathdoc.fr/item/TMF_2001_126_1_a1/
%G ru
%F TMF_2001_126_1_a1

Voir la notice de l'article provenant de la source Math-Net.Ru

We consider an integrable $XXZ$ model with some special open boundary conditions and one-dimensional Ising quantum chains with four different boundary conditions. We show that each of the Ising chains coincides with the minimal $LM(3,4)$ lattice model resulting from the quantum group reduction of the $XXZ$ model and the number of nodes in the former model is determined by the type of boundary conditions. The relation between the two-dimensional Ising model with four different types of boundary conditions and the $LM(3,4)$ model is established.

[1] R. E. Behrend, P. A. Pearce, D. L. O'Brien, Interaction-round-a-face models with fixed boundary conditions: The ABF fusion hierarchy, Preprint University of Melbourne E-print hep-th/9507118 | MR

[2] D. L. O'Brien, P. A. Pearce, S. Ole Warnaar, Physica A, 228 (1996), 63 | DOI | MR

[3] F. Alcaraz, M. Barber, M. Batchelor, Phys. Rev. Lett., 58 (1987), 771 | DOI | MR

[4] C. Hamer, J. Phys. A, 14 (1981), 2981 | DOI | MR

[5] F. Alcaraz, M. Baake, U. Grimm, V. Rittenberg, J. Phys. A, 22 (1989), L5 | DOI | MR | Zbl

[6] V. Pasqier, H. Saleur, Nucl. Phys. B, 330 (1990), 523 | DOI | MR

[7] A. Belavin, Yu. Stroganov, Minimal models of integrable lattice theory and truncated functional equations, E-print hep-th/9908050 | MR

[8] E. K. Sklyanin, J. Phys. A, 21 (1988), 2375 | DOI | MR | Zbl

[9] P. P. Kulish, E. K. Sklyanin, J. Phys. A, 24 (1991), L435 | DOI | MR | Zbl

[10] F. C. Alcaraz, A. A. Belavin, R. A. Usmanov, Correspondence between the $XXZ$ model in roots of unity and the one-dimensional quantum Ising chain with different boundary conditions, E-print hep-th/0007151 | MR

[11] E. Lusztig, Contemp. Math., 82, 1989, 59 | DOI | MR | Zbl

[12] P. P. Martin, Potts Models and Related Problems in Statistical Mechanics, World Scientific, Singapore, 1991 | MR