Gibbs measures for the HC Blume–Capel model with countably many states on a Cayley tree

• Published in: Theoretical and mathematical physics Vol. 211; no. 3; pp. 856 - 865
• Main Authors: Ganikhodzhaev, N. N., Rozikov, U. A., Khatamov, N. M.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 2022; Springer Nature B.V
• Subjects: 14/34; 639/766/189; 639/766/530; 639/766/747; Applications of Mathematics; Invariants; Mathematical and Computational Physics; Physics; Physics and Astronomy; Theoretical; Gibbs measure; Cayley tree; HC Blume–Capel model
• ISSN: 0040-5779; 1573-9333
• DOI: 10.1134/S0040577922060071

We study the Blume–Capel model with a countable set
of spin values and a force
of interaction between the nearest neighbors on a Cayley tree of order
. The following results are obtained. Let
,
, be the temperature. For
, there exist no translation invariant Gibbs measures or
-periodic Gibbs measures. For
, we prove the uniqueness of a translation-invariant Gibbs measure. Let
and
. If
, then there exists exactly one
-periodic Gibbs measure that is translation invariant. For
, there exist exactly three
-periodic Gibbs measures, one of which is a translation-invariant Gibbs measure.