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

F. M. Mukhamedov, M. M. Rahmatullaev, A. M. Tukhtabaev, R. Mamadjonov

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)1238-1253
Number of pages16
JournalTheoretical and Mathematical Physics(Russian Federation)
Issue number2
Publication statusPublished - Aug 2023


  • Ising model
  • p -adic generalized Gibbs measure
  • p -adic numbers
  • periodicity
  • phase transition
  • translation invariance

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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