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

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.