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Rozikov, U. A.; Rakhmatullaev, M. M.; Khakimov, R. M.

Periodic Gibbs measures for the Potts model in translation-invariant and periodic external fields on the Cayley tree. (English. Russian original) Zbl 1489.82019

Theor. Math. Phys. 210, No. 1, 135-153 (2022); translation from Teor. Mat. Fiz. 210, No. 1, 156-176 (2022).
In the present paper, the authors deal with a Potts model in which spin variables have values in the range \(\Phi=\{1, 2,\dots, q\}, q\geq 2\) on a Cayley tree of arbitrary order. The authors investigate translation-invariant and periodic Gibbs measures for the Potts model having translation invariant and periodic external fields on the Cayley tree of arbitrary order. The existence and non-uniqueness of translation-invariant Gibbs measures (TISGMs) with a translational non-invariant external field are proved. Based on the given parameter values, the authors determine the number of Gibbs measures corresponding to the Potts model. For the Potts model in a translation non-invariant external field, the authors prove the nonexistence of translation invariant Gibbs measures. Let \(H\) be a normal divisor of finite index, and \(G^{(2)}_k\) be a subgroup consisting of words of even length in \(G_k\). For the Potts model with an external field, the authors prove that all \(H\)-periodic Gibbs measures are either \(G^{(2)}_k\)-periodic or translation invariant. For the \(q\)-state Potts model having a translation-invariant external field on the Cayley tree, the authors examine all \(G^{(2)}_k\)-periodic Gibbs measures under the given conditions. Lastly, for the Potts model with a periodic external field under given conditions, the authors prove that there is at least one \(G^{(2)}_k\)-periodic (translation-non-invariant) Gibbs measure.
Reviewer: Hasan Akin (Trieste)

Cited in 3 Documents

MSC:

82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
05C05 Trees

Keywords:

Cayley tree; Gibbs measure; Potts model; periodic external field; translation-invariant external field
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References:

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