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

Abstract

We introduce the notion of a weakly periodic configuration. For the Ising model with competing interactions, we describe the set of all weakly periodic ground states corresponding to normal divisors of indices 2 and 4 of the group representation of the Cayley tree. In addition, we study new Gibbs measures for the Ising model.

Ising model on Cayley trees: a new class of Gibbs measures and their comparison with known ones

“…-measures corresponding to solutions on I 3 are weakly periodic, which coincide with the measures given in Theorem 1 for a 1 = k − |A|, a 2 = 0, a 3 = 0, a 4 = |A|, Fig. 5) Moreover, the system (3.8) was solved only in cases |A| = 1 and |A| = k (see [13], [14] [8]), Higuchi's non-translation-invariant measures (see [9]), Alternating Gibbs measures (see [6]) and weakly periodic measures for subgroups of index 4 (see Chapter 2 of [12] with more than 4 distinct values. Thus these measures are different from the measures mentioned in Theorem 1.…”

Weakly periodic Gibbs measures for two and three state HC models on a Cayley tree

“…In [8], [9] the notion of the periodic Gibbs measure was extended to a more general notion called weakly periodic Gibbs measure, where the authors have constructed such measures for the Ising model on the Cayley tree.…”

Weakly periodic Gibbs measures of the Ising model with an external field on the Cayley tree

“…Under some conditions on the parameters of some invariant sets, weakly periodic (nonperiodic) Gibbs measures for the Ising model on the Cayley tree were found in [8] and [9]. But weakly periodic Gibbs measures for Ising models with external fields have not yet been studied.…”