najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2024-15-5-576-585najo-91Research ArticleMATHEMATICSМАТЕМАТИКАTranslation-invariant Gibbs measures for the mixed spin-1/2 and spin-1 Ising model with an external field on a Cayley treeТрансляционно-инвариантные меры Гиббса для смешанной модели Изинга со спинами 1/2 и 1 с внешним полем на дереве Кэлиhttps://orcid.org/0000-0003-2987-7714РахматуллаевМузаффар М.RahmatullaevMuzaffar M.

Muzaffar M. Rahmatullaev – V. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences,

4-b, University str, 100174; 100000, Tashkent;  316, Uychi str, Namangan

mrahmatullaev@rambler.ruhttps://orcid.org/0000-0002-0133-0883ЭгамовДилшод О.EgamovDilshod O.

Dilshod O. Egamov – V. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences

4-b, University str,100174, Tashkent; 316, Uychi str, Namangan

dilshodbekegamov87@gmail.comV. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences; New Uzbekistan University; Namangan state UniversityUzbekistanV. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences; Namangan state UniversityUzbekistan202403062025155576585Copyright © Rahmatullaev M.M., Egamov D.O., 20252025Рахматуллаев М.М., Эгамов Д.О.Rahmatullaev M.M., Egamov D.O.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://nanojournal.ifmo.ru/jour/article/view/91

Phase transitions of the mixed spin-1/2 and spin-1 Ising model under the presence of an external field on the general order Cayley tree are investigated within the framework of the tree-indexed Markov chains. We find the conditions that ensure the existence of at least three translation-invariant Gibbs measures for the model on the Cayley tree of order k. We are able to solve the model exactly on the binary tree (k = 2) under the specific external field. The main attention is paid to the systematic study of the structure of the set of the Gibbs measures. We find the extremality and non-extremality regions of the disordered phase of the model on the binary tree.

Фазовые переходы смешанной модели Изинга со спинами 1/2 и 1 при наличии внешнего поля на дереве Кэли произвольной степени исследуются в рамках древовидно-индексированных цепей Маркова. Получены условия, гарантирующие существование не менее трех трансляционно-инвариантных мер Гиббса для модели на дереве Кэли порядка k. Для модели на бинарном дереве (k=2) при определенном значении внешнего поля найдено точное решение. Основное внимание уделено анализу структуры множества мер Гиббса. Определены области экстремальности и неэкстремальности неупорядоченной фазы модели на бинарном дереве.

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The authors declare that there are no conflicts of interest present.