Geodesic
The~$p$-adic Ising model in an~external field on a~Cayley tree: periodic Gibbs measures
Teoretičeskaâ i matematičeskaâ fizika, Tome 216 (2023) no. 2, pp. 383-400

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We consider the generalized Gibbs measures corresponding to the $p$-adic Ising model in an external field on the Cayley tree of order two. It is established that if $p\equiv 1\,(\operatorname{mod}\, 4)$, then there exist three translation-invariant and two $G_2^{(2)}$-periodic non-translation-invariant $p$-adic generalized Gibbs measures. It becomes clear that if $p\equiv 3\,(\operatorname{mod}\, 4)$, $p\neq3$, then one can find only one translation-invariant $p$-adic generalized Gibbs measure. Moreover, the considered model also exhibits chaotic behavior if $|\eta-1|_p|\theta-1|_p$ and $p\equiv 1\,(\operatorname{mod}\, 4)$. It turns out that even without $|\eta-1|_p|\theta-1|_p$, one could establish the existence of $2$-periodic renormalization-group solutions when $p\equiv 1\,(\operatorname{mod}\, 4)$. This allows us to show the existence of a phase transition.
Keywords: $p$-adic numbers, Ising model, $p$-adic generalized Gibbs measure, translation invariance, periodicity
Mots-clés : phase transition. 
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     author = {F. M. Mukhamedov and M. M. Rahmatullaev and A. M. Tukhtabaev and R. Mamadjonov},
     title = {The~$p$-adic {Ising} model in an~external field on {a~Cayley} tree: periodic {Gibbs} measures},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {383--400},
     publisher = {mathdoc},
     volume = {216},
     number = {2},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2023_216_2_a11/}
}
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F. M. Mukhamedov; M. M. Rahmatullaev; A. M. Tukhtabaev; R. Mamadjonov. The~$p$-adic Ising model in an~external field on a~Cayley tree: periodic Gibbs measures. Teoretičeskaâ i matematičeskaâ fizika, Tome 216 (2023) no. 2, pp. 383-400. http://geodesic.mathdoc.fr/item/TMF_2023_216_2_a11/