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