Gibbs measures for the~HC Blume--Capel model with countably many states on a~Cayley tree
Teoretičeskaâ i matematičeskaâ fizika, Tome 211 (2022) no. 3, pp. 491-501
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We study the Blume–Capel model with a countable set $\mathbb Z$ of spin values and a force $J\in \mathbb R$ of interaction between the nearest neighbors on a Cayley tree of order $k\geq 2$. The following results are obtained. Let $\theta=e^{-J/T}$, $T>0$, be the temperature. For $\theta\geq1$, there exist no translation invariant Gibbs measures or $2$-periodic Gibbs measures. For $0\theta1$, we prove the uniqueness of a translation-invariant Gibbs measure. Let $\Theta=\sum_i\theta^{(k+1)i^2}$ and $\Theta_\mathrm{cr}(k)=k^k/(k-1)^{k+1}$. If $0\Theta\leq\Theta_\mathrm{cr}$, then there exists exactly one $2$-periodic Gibbs measure that is translation invariant. For $\Theta>\Theta_\mathrm{cr}$, there exist exactly three $2$-periodic Gibbs measures, one of which is a translation-invariant Gibbs measure.
Keywords:
Cayley tree, HC Blume–Capel model, Gibbs measure.
@article{TMF_2022_211_3_a6,
author = {N. N. Ganikhodzhaev and U. A. Rozikov and N. M. Khatamov},
title = {Gibbs measures for {the~HC} {Blume--Capel} model with countably many states on {a~Cayley} tree},
journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
pages = {491--501},
publisher = {mathdoc},
volume = {211},
number = {3},
year = {2022},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TMF_2022_211_3_a6/}
}
TY - JOUR AU - N. N. Ganikhodzhaev AU - U. A. Rozikov AU - N. M. Khatamov TI - Gibbs measures for the~HC Blume--Capel model with countably many states on a~Cayley tree JO - Teoretičeskaâ i matematičeskaâ fizika PY - 2022 SP - 491 EP - 501 VL - 211 IS - 3 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/TMF_2022_211_3_a6/ LA - ru ID - TMF_2022_211_3_a6 ER -
%0 Journal Article %A N. N. Ganikhodzhaev %A U. A. Rozikov %A N. M. Khatamov %T Gibbs measures for the~HC Blume--Capel model with countably many states on a~Cayley tree %J Teoretičeskaâ i matematičeskaâ fizika %D 2022 %P 491-501 %V 211 %N 3 %I mathdoc %U http://geodesic.mathdoc.fr/item/TMF_2022_211_3_a6/ %G ru %F TMF_2022_211_3_a6
N. N. Ganikhodzhaev; U. A. Rozikov; N. M. Khatamov. Gibbs measures for the~HC Blume--Capel model with countably many states on a~Cayley tree. Teoretičeskaâ i matematičeskaâ fizika, Tome 211 (2022) no. 3, pp. 491-501. http://geodesic.mathdoc.fr/item/TMF_2022_211_3_a6/