Description of weakly periodic Gibbs measures for the isingmodel on a cayley tree

Abstract

We introduce the concept of a weakly periodic Gibbs measure. For the Ising model, we describe a set of such measures corresponding to normal subgroups of indices two and four in the group representation of a Cayley tree. In particular, we prove that for a Cayley tree of order four, there exist critical values Tc < Tcr of the temperature T > 0 such that there exist five weakly periodic Gibbs measures for 0 < T < Tc or T > Tcr, three weakly periodic Gibbs measures for T = Tc, and one weakly periodic Gibbs measure for Tc < T ≤ Tcr.

Weakly periodic ground states and Gibbs measures for the Ising model with competing interactions on the Cayley tree

“…In Sec. 4, we present new weakly periodic Gibbs measures for the Ising model, significantly improving the result in [3]. In Sec.…”

“…(15) are therefore obtained from the solutions of (14) by replacing θ with −θ. Theorem 2 in [3] can therefore be supplemented with new solutions, and we obtain the following theorem. 1.…”

“…Then G (4) H 2 , H 3 , H 4 } be the quotient group. We note that I(G (4) k ) = {(1, 3), (3,1), (1,4), (4,1), (2,3), (3,2), (2,4), (4, 2)}, and any G (4) k -weakly periodic configuration therefore has the form…”

“…The problem of describing weakly periodic Gibbs measures in several special cases was reduced in [3] to the problem of describing solutions of the equation…”

“…At low temperatures, a periodic ground state corresponds to a periodic Gibbs measure [1], [2]. The notion of a weakly periodic Gibbs measure was introduced in [3], and the set of such measures for the Ising model was described there. The problem of describing weakly periodic ground states therefore arises.…”