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TMF, 2009, Volume 160, Number 3, Pages 507–516 (Mi tmf6412)  

This article is cited in 18 scientific papers (total in 18 papers)

Weakly periodic ground states and Gibbs measures for the Ising model with competing interactions on the Cayley tree

U. A. Rozikova, M. M. Rakhmatullaevb

a Institute for Mathematics and Information Technologies, Academy of Sciences of the Republic of Uzbekistan, Tashkent, Uzbekistan
b Namangan State University, Namangan, Uzbekistan

Abstract: We introduce the notion of a weakly periodic configuration. For the Ising model with competing interactions, we describe the set of all weakly periodic ground states corresponding to normal divisors of indices $2$ and $4$ of the group representation of the Cayley tree. In addition, we study new Gibbs measures for the Ising model.

Keywords: Cayley tree, Gibbs measure, Ising model, weakly periodic ground state

DOI: https://doi.org/10.4213/tmf6412

Full text: PDF file (413 kB)
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English version:
Theoretical and Mathematical Physics, 2009, 160:3, 1292–1300

Bibliographic databases:

Received: 06.08.2008

Citation: U. A. Rozikov, M. M. Rakhmatullaev, “Weakly periodic ground states and Gibbs measures for the Ising model with competing interactions on the Cayley tree”, TMF, 160:3 (2009), 507–516; Theoret. and Math. Phys., 160:3 (2009), 1292–1300

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Akin H., Rozikov U.A., Temir S., “A New Set of Limiting Gibbs Measures for the Ising Model on a Cayley Tree”, J Stat Phys, 142:2 (2011), 314–321  crossref  mathscinet  zmath  adsnasa  isi  scopus
    2. U. A. Rozikov, R. M. Khakimov, “The uniqueness condition for a weakly periodic Gibbs measure for the hard-core model”, Theoret. and Math. Phys., 173:1 (2012), 1377–1386  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    3. R. M. Khakimov, “Uniqueness of Weakly Periodic Gibbs Measure for HC-Models”, Math. Notes, 94:5 (2013), 834–838  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    4. M. M. Rakhmatullaev, “Weakly periodic Gibbs measures and ground states for the Potts model with competing interactions on the Cayley tree”, Theoret. and Math. Phys., 176:3 (2013), 1236–1251  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    5. Gandolfo D. Rakhmatullaev M.M. Rozikov U.A. Ruiz J., “On Free Energies of the Ising Model on the Cayley Tree”, J. Stat. Phys., 150:6 (2013), 1201–1217  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    6. Rozikov U.A., “Gibbs Measures on Cayley Trees: Results and Open Problems”, Rev. Math. Phys., 25:1 (2013), 1330001  crossref  mathscinet  isi  elib  scopus  scopus
    7. M. M. Rakhmatullaev, “The existence of weakly periodic Gibbs measures for the Potts model on a Cayley tree”, Theoret. and Math. Phys., 180:3 (2014), 1019–1029  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    8. N. M. Khatamov, G. T. Madgoziev, “Gibbs measures for a generalized Potts model with the interaction radius two on a Cayley tree”, Theoret. and Math. Phys., 183:3 (2015), 836–845  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    9. M. M. Rakhmatullaev, “Weakly periodic Gibbs measures of the Ising model with an external field on the Cayley tree”, Theoret. and Math. Phys., 183:3 (2015), 822–828  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    10. M. M. Rakhmatullaev, “New weakly periodic Gibbs measures of Ising model on Cayley tree”, Russian Math. (Iz. VUZ), 59:11 (2015), 45–53  mathnet  crossref
    11. M. M. Rahmatullaev, “A weakly periodic Gibbs measure for the ferromagnetic Potts model on a Cayley tree”, Siberian Math. J., 56:5 (2015), 929–935  mathnet  crossref  crossref  isi  elib
    12. M. A. Rasulova, M. M. Rahmatullaev, “Periodic and weakly periodic ground states for the Potts model with competing interactions on the Cayley tree”, Siberian Adv. Math., 26:3 (2016), 215–229  mathnet  crossref  crossref  mathscinet  elib
    13. M. M. Rahmatullaev, “On weakly periodic Gibbs measures of the Potts model with a special external field on a Cayley tree”, Zhurn. matem. fiz., anal., geom., 12:4 (2016), 302–314  mathnet  crossref  mathscinet
    14. Mukhamedov F. Barhoumi A. Souissi A., “Phase Transitions for Quantum Markov Chains Associated with Ising Type Models on a Cayley Tree”, J. Stat. Phys., 163:3 (2016), 544–567  crossref  mathscinet  zmath  isi  elib  scopus
    15. Rahmatullaev M., “On new weakly periodic Gibbs measures of the Ising model on the Cayley tree of order ? 5”, Algebra, Analysis and Quantum Probability, Journal of Physics Conference Series, 697, ed. Ayupov S. Chilin V. Ganikhodjaev N. Mukhamedov F. Rakhimov I., IOP Publishing Ltd, 2016, 012020  crossref  isi  scopus
    16. Ganikhodjaev N. Rahmatullaev M. Rodzhan Mohd Hirzie Bin Mohd, “Weakly Periodic Gibbs Measures of the Ising Model on the Cayley Tree of Order Five and Six”, Math. Phys. Anal. Geom., 21:1 (2017), 2  crossref  mathscinet  isi  scopus  scopus
    17. Rahmatullaev M., “Ising Model on Trees: (K(0)) - Non Translation-Invariant Gibbs Measures”, 37Th International Conference on Quantum Probability and Related Topics (Qp37), Journal of Physics Conference Series, 819, ed. Accardi L. Mukhamedov F. Hee P., IOP Publishing Ltd, 2017, UNSP 012019  crossref  mathscinet  isi  scopus  scopus
    18. Mukhamedov F. Pah Ch.H. Rahimatullaev M. Jamil H., “Periodic and Weakly Periodic Ground States For the Lambda-Model on Cayley Tree”, 4Th International Conference on Mathematical Applications in Engineering 2017 (Icmae'17), Journal of Physics Conference Series, 949, ed. Rakhimov A. Ural B. Daoud J. Saburov K. Chowdhury M., IOP Publishing Ltd, 2018, UNSP 012021  crossref  isi  scopus  scopus
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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