2004/09/03 by U. A. Rozikov, Rozikov, U. A., Yu. M. Suhov +1
Materials Science · Mathematics · Physics and Astronomy · #60K35 #82B05 #82B20 #FOS: Mathematics #FOS: Physical sciences #Magnetism in coordination complexes #Mathematical Physics (math-ph) #Probability (math.PR) #Quantum many-body systems #Theoretical and Computational Physics #math-ph #math.MP #math.PR #msc:60K35 #msc:82B05 #msc:82B20
paper · pdf · doi:10.48550/arxiv.math/0409047
16 pages
arxiv created 2004/09/03 · openalex publication_date 2004/09/03 · arxiv updated 2011/02/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider a nearest-neighbor SOS model, spin values 0,1,..., m, m≥ 2, on a Cayley tree of order k . We mainly assume that m=2 and study translation-invariant (TI) and `splitting' (S) Gibbs measures (GMs). For m=2, in the anti-ferromagnetic (AFM) case, a symmetric TISGM is unique for all temperatures. In the ferromagnetic (FM) case, for m=2, the number of symmetric TISGMs varies with the temperature: here we identify a critical inverse temperature, β1_\rmcr (=T_\rmcr^\rmSTISG) ∈ (0,∞) such that ∀ 0≤ β≤β1_\rmcr, there exists a unique symmetric TISGM μ^* and ∀ β>β1_\rmcr there are exactly three symmetric TISGMs : μ^*+, μ^*\rm m and μ^*- For β>β1_\rmcr 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. 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).