A Constructive Description of Ground States and Gibbs Measures for Ising Model with Two-Step Interactions on Cayley Tree

2006 ◽  
Vol 122 (2) ◽  
pp. 217-235 ◽  
Author(s):  
U. A. Rozikov
Keyword(s):  
Ising Model ◽  
Cayley Tree ◽  
Gibbs Measures ◽  
Ground States ◽  
Constructive Description

Ground states for the ising model with an external field on the Cayley tree

2018 ◽  
Vol 2018 (3) ◽  
pp. 147-155
Author(s):  
M.M. Rakhmatullaev ◽  
M.A. Rasulova
Keyword(s):  
Ising Model ◽  
External Field ◽  
Cayley Tree ◽  
Ground States

scholarly journals GIBBS MEASURES FOR SOS MODELS ON A CAYLEY TREE

2006 ◽  
Vol 09 (03) ◽  
pp. 471-488 ◽  
Author(s):  
U. A. ROZIKOV ◽  
Y. M. SUHOV
Keyword(s):  
Ising Model ◽  
Normal Subgroup ◽  
Finite Index ◽  
Cayley Tree ◽  
Nearest Neighbor ◽  
Gibbs Measures ◽  
Natural Generalization ◽  
Mirror Reflection ◽  
Sos Model ◽  
Translation Invariant

We consider a nearest-neighbor solid-on-solid (SOS) model, with several spin values 0, 1,…, m, m ≥ 2, and zero external field, on a Cayley tree of order k (with k + 1 neighbors). The SOS model can be treated as a natural generalization of the Ising model (obtained for m = 1). We mainly assume that m = 2 (three spin values) and study translation-invariant (TI) and "splitting" (S) Gibbs measures (GMs). (Splitting GMs have a particular Markov-type property specific for a tree.) Furthermore, we focus on symmetric TISGMs, with respect to a "mirror" reflection of the spins. [For the Ising model (where m = 1), such measures are reduced to the "disordered" phase obtained for free boundary conditions, see Refs. 9, 10.] For m = 2, in the antiferromagnetic (AFM) case, a symmetric TISGM (and even a general TISGM) is unique for all temperatures. In the ferromagnetic (FM) case, for m = 2, the number of symmetric TISGMs and (and the number of general TISGMs) varies with the temperature: this gives an interesting example of phase transition. Here we identify a critical inverse temperature, [Formula: see text] such that [Formula: see text], there exists a unique symmetric TISGM μ* and [Formula: see text] there are exactly three symmetric TISGMs: [Formula: see text] (a "bottom" symmetric TISGM), [Formula: see text] (a "middle" symmetric TISGM) and [Formula: see text] (a "top" symmetric TISGM). For [Formula: see text] we also construct a continuum of distinct, symmertric SGMs which are non-TI. Our second result gives complete description of the set of periodic Gibbs measures for the SOS model on a Cayley tree. A complete description of periodic GMs means a characterisation of such measures with respect to any given normal subgroup of finite index in the representation group of the tree. We show that (i) for an FM SOS model, for any normal subgroup of finite index, each periodic SGM is in fact TI. Further, (ii) for an AFM SOS model, for any normal subgroup of finite index, each periodic SGM is either TI or has period two (i.e. is a chess-board SGM).

Periodic and weakly periodic ground states corresponding to subgroups of index three for the ising model on cayley tree

2021 ◽  
Vol 88 (2) ◽  
pp. 247-257
Author(s):  
M.M. Rahmatullaev ◽  
D.O. Egamov ◽  
F.H. Haydarov
Keyword(s):  
Ising Model ◽  
Cayley Tree ◽  
Ground States

Uniqueness of Gibbs Measures for an Ising Model with Continuous Spin Values on a Cayley Tree

2020 ◽  
Vol 86 (3) ◽  
pp. 293-302
Author(s):  
Farhod Halimjonovich Haydarov ◽  
Shamshod A. Akhtamaliyev ◽  
Madalixon A. Nazirov ◽  
Behzod Boyxonovich Qarshiyev
Keyword(s):  
Ising Model ◽  
Cayley Tree ◽  
Gibbs Measures ◽  
Continuous Spin

scholarly journals Про конструктивні описи мір Гіббса для моделі Поттса на дереві Келі

2021 ◽  
Vol 73 (7) ◽  
pp. 938-950
Author(s):  
M. Rahmatullaev ◽  
F. К. Rafikov ◽  
Sh. Kh. Azamov
Keyword(s):  
Ising Model ◽  
Cayley Tree ◽  
Gibbs Measures ◽  
Stat Phys

УДК 517.9 Розглядається модель Поттса на деревi Келi. Доведено iснування мiр Гiббса, побудованих аналогiчним методом iз [H. Akin, U. A. Rozikov, S. Temir, <em>A new set of limiting Gibbs measures for the Ising model on a Cayley tree</em>, J. Stat. Phys., <strong>142</strong>, № 2, 314 – 321 (2011)] i -трансляцiйно-iнварiантних мiр Гiббса для моделi Поттса на деревi Келi. Обчислено вiльнi енергiї цих мiр Гiббса.

A New Set of Limiting Gibbs Measures for the Ising Model on a Cayley Tree

2010 ◽  
Vol 142 (2) ◽  
pp. 314-321 ◽  
Author(s):  
H. Akin ◽  
U. A. Rozikov ◽  
S. Temir
Keyword(s):  
Ising Model ◽  
Cayley Tree ◽  
Gibbs Measures
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