Periodic and weakly periodic ground states corresponding to the subgroups of index three for the Ising model on the Cayley tree of order three

Dilshod O.  Egamov

doi:10.37863/umzh.v75i6.7108

Dilshod O. Egamov V. I. Romanovskiy Institute of Mathematics, Tashkent, Uzbekistan

DOI: https://doi.org/10.37863/umzh.v75i6.7108

Keywords: Cayley tree, configuration, Ising model, ground state.

Abstract

UDC 517.9

We determine periodic and weakly periodic ground states with subgroups of index three for the Ising model on the Cayley tree of order three.

References

U. A. Rozikov, A constructive description of ground states and Gibbs measures for Ising model with two-step interactions on Cayley tree, J. Stat. Phys., 122, 217–235 (2006). DOI: https://doi.org/10.1007/s10955-005-8029-3

R. J. Baxter, Exactly solved models in statistical mechanics, Acad. Press, London, New York (1982).

P. M. Bleher, J. Ruiz, V. A. Zagrebnov, On the purity of the limiting Gibbs state for the Ising model on the Bethe lattice, J. Stat. Phys., 79, № 1, 473–482 (1995). DOI: https://doi.org/10.1007/BF02179399

P. M. Bleher, N. N. Ganikhodjaev, On pure phases of the Ising model on the Bethe lattice, Theory Probab. and Appl., 35, 216–227 (1990). DOI: https://doi.org/10.1137/1135031

P. M. Bleher, J. Ruiz, R. H. Schonmann, S. Shlosman, V. A. Zagrebnov, Rigidity of the critical phases on a Cayley tree, Moscow Math. J., 3, (2001). DOI: https://doi.org/10.17323/1609-4514-2001-1-3-345-363

N. N. Ganikhodjaev, U. A. Rozikov, A description of periodic extremal Gibbs measures of some lattice models on the Cayley tree, Theor. and Math. Phys., 111, № 1, 109–117 (1997). DOI: https://doi.org/10.1007/BF02634202

F. M. Mukhamedov, U. A. Rozikov, On Gibbs measures of models with competing ternary and binary interactions and corresponding von Neumann algebras, J. Stat. Phys., 114, 825–848 (2004). DOI: https://doi.org/10.1023/B:JOSS.0000012509.10642.83

F. H. Haydarov, New normal subgroups for the group representation of the Cayley tree, Lobachevskii J. Math., 39, № 2, 213–217 (2018). DOI: https://doi.org/10.1134/S1995080218020142

F. H. Haydarov, R. A. Ilyasova, On periodic Gibbs measures of Ising model corresponding to new subgroups of the group representation of the Cayley tree, Theor. and Math. Phys., 210, № 2, 261–274 (2022). DOI: https://doi.org/10.1134/S0040577922020076

Kh. A. Nazarov, U. A. Rozikov, Periodic Gibbs measures for the Ising model with competing interactions, Theor. and Math. Phys., 135, 881–888 (2003).

C. Preston, Gibbs states on countable sets, Cambridge Univ. Press, London (1974). DOI: https://doi.org/10.1017/CBO9780511897122

U. A. Rozikov, Partition structures of the group representation of the Cayley tree into cosets by finite-index normal subgroups and their applications to the description of periodic Gibbs distributions, Theor. and Math. Phys., 112, 929–933 (1997). DOI: https://doi.org/10.1007/BF02634109

Ya. G. Sinai, Theory of phase transitions: rigorous results, Pergamon, Oxford (1982).

S. Zachary, Countable state space Markov random fields and Markov chains on trees, Ann. Probab., 11, № 4, 894–903 (1983). DOI: https://doi.org/10.1214/aop/1176993439

U. A. Rozikov, F. H. Haydarov, Invariance property on group representations of the Cayley tree and its applications; arXiv:1910.13733.

U. A. Rozikov, Gibbs measures on a Cayley tree, World Sci. Publ., Singapore (2013). DOI: https://doi.org/10.1142/8841

R. A. Minlos, Introduction to mathematical statistical physics, Amer. Math. Soc., Providence (2000). DOI: https://doi.org/10.1090/ulect/019

U. A. Rozikov, M. M. Rakhmatullaev, Weakly periodic ground states and Gibbs measures for the Ising model with competing interactions on the Cayley tree, Theor. and Math. Phys., 160, № 3, 507–516 (2009). DOI: https://doi.org/10.1007/s11232-009-0116-1

U. A. Rozikov, F. H. Haydarov, Normal subgroups of finite index for the group represantation of the Cayley tree, TWMS J. Pure. and Appl. Math., 5, 234–240 (2014).

M. M. Rakhmatullaev, Description of weak periodic ground states of Ising model with competing interactions on Cayley tree, Appl. Math. Inf. Sci., 4, № 2, 237–241 (2010).

M. M. Rakhmatullaev, Weakly periodic Gibbs measures and ground states for the Potts model with competing interactions on the Cayley tree, Theor. and Math. Phys., 176, № 3, 477–493 (2013).

M. M. Rakhmatullaev, M. A. Rasulova, Existence of weakly periodic ground states for the Potts model with competing interactions on the Cayley tree, Dokl. Akad. Nauk Resp. Uzbekistan, 3, № 10 (2013). DOI: https://doi.org/10.1007/s11232-013-0103-4

M. M. Rahmatullaev, M. A. Rasulova, Periodic and weakly periodic ground states for the Potts model with competing interactions on the Cayley tree, Sib. Adv. Math., 26, № 3, 215–229 (2016). DOI: https://doi.org/10.3103/S1055134416030056

M. M. Rahmatullaev, D. O. Egamov, F. H. Haydarov, Periodic and weakly periodic ground states corresponding to subgroups of index three for the Ising model on Cayley tree, Rep. Math. Phys., 88, № 2 (2021). DOI: https://doi.org/10.1016/S0034-4877(21)00072-0