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The \(p\)-adic Ising model in an external field on a Cayley tree: periodic Gibbs measures

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Abstract

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.

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Authors and Affiliations

  1. Department of Mathematical Sciences, College of Science, United Arab Emirates University, Abu Dhabi, United Arab Emirates

    F. M. Mukhamedov

  2. Akfa University, Tashkent, Uzbekistan

    F. M. Mukhamedov

  3. Romanovsky Institute of Mathematics, Academy of Sciences of Uzbekistan, Tashkent, Uzbekistan

    F. M. Mukhamedov & M. M. Rahmatullaev

  4. Namangan State University, Namangan, Uzbekistan

    M. M. Rahmatullaev, A. M. Tukhtabaev & R. Mamadjonov

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  1. F. M. Mukhamedov
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  2. M. M. Rahmatullaev
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  3. A. M. Tukhtabaev
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  4. R. Mamadjonov
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Correspondence to F. M. Mukhamedov.

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Prepared from an English manuscript submitted by the author; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, 2023, Vol. 216, pp. 383–400 https://doi.org/10.4213/tmf10509.

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Mukhamedov, F.M., Rahmatullaev, M.M., Tukhtabaev, A.M. et al. The \(p\)-adic Ising model in an external field on a Cayley tree: periodic Gibbs measures. Theor Math Phys 216, 1238–1253 (2023). https://doi.org/10.1134/S0040577923080123

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  • DOI: https://doi.org/10.1134/S0040577923080123

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